Pulsed light source and system using the same

Abstract

To provide a pulse light source capable of generating convex or concave pulses (positive bright, positive dark, negative bright, and negative dark) with a positive or negative amplitude in any shape directly from a mode-locked laser and a system using the same.SOLUTION: The pulse light source has a frequency-modulated mode-locked laser including an optical phase modulator and an optical filter in a laser resonator. Amplitude and phase characteristics are tunable by setting the amplitude and phase characteristics of the optical filter according to the shape of a desired optical pulse and the amount of continuous wave offset. The pulse light source is configured to generate dark and bright pulses having the desired optical pulse shape having the continuous wave offset amount.SELECTED DRAWING: Figure 2

Description

[Technical Field]

[0001] The present invention relates to a pulsed light source capable of realizing bright pulses, in which the amplitude of the light pulse is convex, and dark pulses, in which the amplitude is concave, in both positive and negative phases and in various shapes, and to a system using the same. [Background technology]

[0002] Mode-locked lasers have a wide range of applications as optical pulse light sources used in optical communication, optical measurement, optical signal processing, and the like. The pulse shape generated from mode-locked lasers is generally known to be a Gaussian pulse (see, for example, Non-Patent Documents 1 and 2). In addition, mode-locked lasers can generate sech pulses by using a nonlinear optical effect called the soliton effect within the resonator (see, for example, Non-Patent Document 3).

[0003] On the other hand, as a method for directly generating pulses other than Gaussian and sech pulses from a mode-locked laser, the inventors have proposed an optical function generator capable of generating pulse trains with desired time waveforms, even if they are not Gaussian or sech pulses, by inserting an optical filter, whose amplitude and phase characteristics are appropriately designed according to the desired optical pulse shape, into the resonator of the mode-locked laser (see, for example, Non-Patent Documents 4, 5, 6, and 7). Using this method, it has been demonstrated that Nyquist pulses with single exponential functions, double exponential functions, triangular, parabolic, rectangular, and even arbitrary roll-off ratios can be generated from an amplitude-modulated (AM) or frequency-modulated (FM) mode-locked laser.

[0004] The waveforms of optical pulses realized so far with these mode-locked lasers have all been positive pulses, as shown in Figure 1(a), where the amplitude is convex at the value of the positive electric field and the tails at both ends are zero. On the other hand, there are also dark pulses, which have a concave shape. A dark pulse is a pulse in which a part of continuous wave (CW) light is cut off in a certain shape, as shown in Figure 1(c). A dark soliton is known as an example of this. A dark soliton is defined as having an amplitude of tanh(t / T) and an intensity of tanh 2 (t / T) = 1 - sech 2 This is an optical pulse given by (t / T). Dark solitons are nonlinear pulses that propagate stably when the normal dispersion and the Kerr effect in an optical fiber are balanced (see, for example, Non-Patent Documents 8 and 9), and are useful not only as pulses for optical transmission (see, for example, Non-Patent Document 10) but also as optical frequency combs (see, for example, Non-Patent Document 11). [Prior art documents] [Non-patent literature]

[0005] [Non-Patent Document 1] D. Kuizenga and A. Siegman, “FM and AM mode locking of the homogeneous laser - Part I: Theory”, IEEE J. Quantum Electron., November 1970, vol. 6, no. 11, pp. 694-708 [Non-Patent Document 2] HA Haus, “A theory of forced mode locking”, IEEE J. Quantum Electron., July 1975, vol. QE-11, no. 7, pp. 323-330 [Non-Patent Document 3] M. Nakazawa and E. Yoshida, "A 40-GHz 850-fs regeneratively FM mode-locked polarization-maintaining erbium fiber ring laser," IEEE Photonics Technology Letters, Dec. 2000, vol. 12, no. 12, pp. 1613-1

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[0006] However, while dark solitons, as shown in Figure 1(c), can be obtained in optical fibers when normal dispersion and the Kerr effect balance each other, the conditions for obtaining them are limited, and above all, they are difficult to obtain with mode-locked lasers, which has been a challenge to their practical application. On the other hand, in contrast to dark pulses, bright pulses, as shown in Figure 1(b), can be considered, in which a CW offset is superimposed on a convex optical pulse.

[0007] Conventional optical pulses are positive pulses (see Figure 1(a)) whose amplitude is defined by a positive value, but negative pulses (see Figure 1(d)) also exist, where the amplitude is defined by the value of the negative electric field. Because negative pulses can be generated in this way, it becomes possible to switch light on and off with a high extinction ratio through interference with positive pulses, leading to expectations for a wide range of applications in optical signal processing.

[0008] Therefore, if the aforementioned bright pulses and dark pulses can be easily generated, it could greatly expand the potential for a wide range of applications in optical signal processing, similar to the relationship between positive and negative pulses.

[0009] As is clear from the above, by combining positive / negative CW and positive / negative pulses, four types of pulses can be defined: (1) a positive bright pulse (see Figure 1(b)), (2) a positive dark pulse (see Figure 1(c)), (3) a negative bright pulse (see Figure 1(e)), and (4) a negative dark pulse (see Figure 1(f)). (1) is defined as a combination of a positive pulse + positive CW offset, (2) as a negative pulse + positive CW offset, (3) as a positive pulse + negative CW offset, and (4) as a combination of a negative pulse + negative CW offset.

[0010] The present invention aims to solve these problems and provides a pulse light source and a system using the same that can directly generate convex or concave pulses (positive bright, positive dark, negative bright, negative dark) with positive or negative amplitude in any shape from a mode-locked laser. [Means for solving the problem]

[0011] To achieve this objective, the pulse light source according to the present invention has a frequency-modulated mode-locked laser comprising an optical phase modulator (frequency modulator) and an optical filter in a laser resonator, and is configured to generate dark pulses and bright pulses having the desired optical pulse shape and continuous wave offset amount by setting the amplitude and phase characteristics of the optical filter according to the desired optical pulse shape and continuous wave offset amount.

[0012] Furthermore, in the pulse light source according to the present invention, the optical phase modulator has a repetition angular frequency Ω m Driven by a sine wave, the amplitude and phase characteristics of the optical filter are determined by a sinc function S(ω) = sin(ωτ / 2) / ω(τ= 2π / Ω) that gives the modulation degree of the optical phase modulator, the spectrum A(ω) of the pulse to be output, and the continuous wave offset amount. m ), as well as A(ω) and S(ω) as the repetition angular frequency Ω mThe function A(ω - nΩ) or A(ω + nΩ) shifted positively or negatively by an integer multiple of m ), A(ω + nΩ m ), S(ω - nΩ m ), S(ω + nΩ m ), (n: integer) is preferably used. Further, in the optical filter, a band limitation may be provided in its amplitude and phase characteristics according to the spectral width of the optical pulse to be output.

[0013] Also, in the pulse light source according to the present invention, the optical filter may be configured to generate four types of pulses: positive bright pulses, positive dark pulses, negative bright pulses, and negative dark pulses according to the combination by taking the sign of the sinc function S(ω) that gives the spectral A(ω) and the continuous wave offset amount as positive or negative.

[0014] A system using the pulse light source according to the present invention is characterized by using the pulse light source according to the present invention.

Advantages of the Invention

[0015] According to the present invention, it is possible to provide a pulse light source capable of directly generating convex or concave pulses (positive bright, positive dark, negative bright, negative dark) having positive or negative amplitudes in an arbitrary shape from a mode - locked laser, and a system using the same. The pulse light source according to the present invention can generate positive / negative and bright / dark pulse trains with various waveforms at a high optical signal - to - noise ratio (OSNR; Optical Signal to Noise Ratio) according to the amplitude and phase characteristics of an optical filter inserted into the resonator of a mode - locked laser. Therefore, since arbitrary positive / negative and bright / dark pulses can be generated with a simple configuration using a mode - locked laser, it is possible to provide a high - performance and high - quality pulse light source for optical measurement and optical signal processing.

Brief Description of the Drawings

[0016] [Figure 1]These waveform diagrams illustrate the definitions of (a) positive pulse, (b) positive bright pulse, (c) positive dark pulse, (d) negative pulse, (e) negative bright pulse, and (f) negative dark pulse. [Figure 2] This is a schematic diagram showing the configuration of a pulse light source according to an embodiment of the present invention. [Figure 3] This graph shows the shapes (black: absolute value, gray: phase) of the optical filters used to generate (a) a positive bright Gaussian pulse, (b) a positive dark Gaussian pulse, (c) a negative bright Gaussian pulse, and (d) a negative dark Gaussian pulse in the first embodiment of the present invention. [Figure 4] This graph shows the waveforms (black: electric field amplitude, gray: phase) of steady-state pulses (a) positive bright Gaussian pulse, (b) positive dark Gaussian pulse, (c) negative bright Gaussian pulse, and (d) negative dark Gaussian pulse, obtained by computer analysis using the optical filter shown in Figure 3. [Figure 5] This graph shows the shape of the optical filter (black: absolute value, gray: phase) for generating (a) a positive bright sech pulse, (b) a positive dark sech pulse, (c) a negative bright sech pulse, and (d) a negative dark sech pulse in a second embodiment of the present invention. [Figure 6] This graph shows the waveforms (black: electric field amplitude, gray: phase) of steady-state pulses (a) positive bright sech pulse, (b) positive dark sech pulse, (c) negative bright sech pulse, and (d) negative dark sech pulse, obtained by computer analysis using the optical filter shown in Figure 5. [Figure 7] This graph shows the shape of the optical filter (black: absolute value, gray: phase) for generating (a) positive bright exponential pulses, (b) positive dark exponential pulses, (c) negative bright exponential pulses, and (d) negative dark exponential pulses in a third embodiment of the present invention. [Figure 8]This graph shows the waveforms (black: electric field amplitude, gray: phase) of steady-state pulses (a) positive bright double exponential pulse, (b) positive dark double exponential pulse, (c) negative bright double exponential pulse, and (d) negative dark double exponential pulse, obtained by computer analysis using the optical filter shown in Figure 7. [Figure 9] This graph shows the shape of the optical filter (black: absolute value, gray: phase) for generating (a) a positive bright triangular pulse, (b) a positive dark triangular pulse, (c) a negative bright triangular pulse, and (d) a negative dark triangular pulse, as defined by the electric field amplitude in the fourth embodiment of the present invention. [Figure 10] This graph shows the waveforms (black: electric field amplitude, gray: phase) of steady-state pulses (a) positive bright triangular pulse, (b) positive dark triangular pulse, (c) negative bright triangular pulse, and (d) negative dark triangular pulse, defined by the electric field amplitude obtained by computer analysis using the optical filter shown in Figure 9. [Figure 11] This graph shows the shape of the optical filter (black: absolute value, gray: phase) for generating (a) a positive bright triangular pulse, (b) a positive dark triangular pulse, (c) a negative bright triangular pulse, and (d) a negative dark triangular pulse, as defined by the electric field strength in the fifth embodiment of the present invention. [Figure 12] This graph shows the waveforms (black: field amplitude, gray: phase) of steady-state pulses (a) positive bright triangular pulse, (b) positive dark triangular pulse, (c) negative bright triangular pulse, and (d) negative dark triangular pulse, defined by the field strength obtained by computer analysis using the optical filter shown in Figure 11. [Figure 13] This graph shows the shapes (black: absolute value, gray: phase) of optical filters for generating (a) a positive bright parabolic pulse, (b) a positive dark parabolic pulse, (c) a negative bright parabolic pulse, and (d) a negative dark parabolic pulse, as defined by the electric field amplitude in the sixth embodiment of the present invention. [Figure 14]This graph shows the waveforms (black: electric field amplitude, gray: phase) of steady-state pulses (a) positive bright parabolic pulse, (b) positive dark parabolic pulse, (c) negative bright parabolic pulse, and (d) negative dark parabolic pulse, defined by the electric field amplitude obtained by computer analysis using the optical filter shown in Figure 13. [Figure 15] This graph shows the shapes of the optical filters (black: absolute value, gray: phase) for generating (a) a positive bright parabolic pulse, (b) a positive dark parabolic pulse, (c) a negative bright parabolic pulse, and (d) a negative dark parabolic pulse, as defined by the electric field strength in the seventh embodiment of the present invention. [Figure 16] This graph shows the waveforms (black: field amplitude, gray: phase) of steady-state pulses (a) positive bright parabolic pulse, (b) positive dark parabolic pulse, (c) negative bright parabolic pulse, and (d) negative dark parabolic pulse, defined by the field strength obtained by computer analysis using the optical filter shown in Figure 15. [Figure 17] This graph shows the shape of the optical filter (black: absolute value, gray: phase) for generating (a) a positive bright rectangular pulse, (b) a positive dark rectangular pulse, (c) a negative bright rectangular pulse, and (d) a negative dark rectangular pulse in the eighth embodiment of the present invention. [Figure 18] This graph shows the waveforms (black: electric field amplitude, gray: phase) of steady-state pulses (a) positive bright rectangular pulse, (b) positive dark rectangular pulse, (c) negative bright rectangular pulse, and (d) negative dark rectangular pulse, obtained by computer analysis using the optical filter shown in Figure 17. [Figure 19] This graph shows the shape of the optical filter (black: absolute value, gray: phase) for generating (a) a positive bright Nyquist pulse, (b) a positive dark Nyquist pulse, (c) a negative bright Nyquist pulse, and (d) a negative dark Nyquist pulse in the ninth embodiment of the present invention. [Figure 20]This graph shows the waveforms (black: electric field amplitude, gray: phase) of steady-state pulses (a) positive bright Nyquist pulse, (b) positive dark Nyquist pulse, (c) negative bright Nyquist pulse, and (d) negative dark Nyquist pulse, obtained by computer analysis using the optical filter shown in Figure 19. [Figure 21] This graph shows the shape ((a) absolute value, (b) phase) of the optical filter for dark soliton generation in the tenth embodiment of the present invention. [Figure 22] This graph shows the waveform ((a) electric field strength, (b) phase) of a steady-state pulse of a dark soliton, obtained by computer analysis using the optical filter shown in Figure 21. [Figure 23] This is a schematic diagram showing the configuration of the optical fiber ring resonator used in the experiment for the pulse light source according to an embodiment of the present invention. [Figure 24] These are (a) the transmittance characteristics and (b) the phase characteristics of the optical filter FFpd(ω) used to generate a dark Gaussian pulse implemented with an LCoS element in the first embodiment of the present invention. [Figure 25] Figure 24 shows the waveform of a dark Gaussian pulse obtained using the optical filter ((a) intensity waveform, (b) waveform converted to amplitude by taking the square root of (a)), and (c) optical spectrum. [Figure 26] These are (a) the transmittance characteristics and (b) the phase characteristics of the optical filter FFpb(ω) used to generate a Bright Gauss pulse implemented with an LCoS element in the first embodiment of the present invention. [Figure 27] Figure 26 shows the waveform of a Bright Gauss pulse obtained using the optical filter ((a) intensity waveform, (b) waveform converted to amplitude by taking the square root of (a)), and (c) optical spectrum. [Figure 28] The (a) transmittance characteristics and (b) phase characteristics of the optical filter FFpd(ω) used to generate dark sech pulses implemented with an LCoS element in a second embodiment of the present invention. [Figure 29] Figure 28 shows the waveform of the dark pulse obtained using the optical filter ((a) intensity waveform, (b) waveform converted to amplitude by taking the square root of (a)), and (c) optical spectrum. [Figure 30] The (a) transmittance characteristics and (b) phase characteristics of the optical filter FFpb(ω) used to generate a bright sech pulse implemented with an LCoS element in a second embodiment of the present invention. [Figure 31] Figure 30 shows the waveform of a bright spot pulse obtained using the optical filter ((a) intensity waveform, (b) waveform converted to amplitude by taking the square root of (a)), and (c) optical spectrum. [Figure 32] The third embodiment of the present invention shows (a) the transmittance characteristics and (b) the phase characteristics of the optical filter FFpd(ω) used to generate dark exponential pulses implemented with an LCoS element. [Figure 33] Figure 32 shows the waveforms of the dark exponential pulses obtained using the optical filter ((a) intensity waveform, (b) waveform converted to amplitude by taking the square root of (a)), and (c) optical spectrum. [Figure 34] The third embodiment of the present invention shows (a) the transmittance characteristics and (b) the phase characteristics of the optical filter FFpb(ω) used to generate bright double exponential pulses implemented with an LCoS element. [Figure 35] Figure 34 shows the waveforms of the Bright exponential pulses obtained using the optical filter ((a) intensity waveform, (b) waveform converted to amplitude by taking the square root of (a)), and (c) optical spectrum. [Figure 36] The fourth embodiment of the present invention shows (a) the transmittance characteristics and (b) the phase characteristics of an optical filter FFpd(ω) implemented with an LCoS element and used to generate a dark triangle pulse defined by the electric field amplitude. [Figure 37] Figure 36 shows the waveform of a dark triangle pulse defined by the electric field amplitude, obtained using the optical filter ((a) intensity waveform, (b) waveform converted to amplitude by taking the square root of (a)), and (c) optical spectrum. [Figure 38] The fourth embodiment of the present invention shows (a) the transmittance characteristics and (b) the phase characteristics of an optical filter FFpb(ω) implemented with an LCoS element and used to generate a bright triangular pulse defined by the electric field amplitude. [Figure 39]Figure 38 shows the waveform of a Bright triangular pulse defined by the electric field amplitude, obtained using the optical filter ((a) intensity waveform, (b) waveform converted to amplitude by taking the square root of (a)), and (c) optical spectrum. [Figure 40] The (a) transmittance characteristics and (b) phase characteristics of the optical filter FFpd(ω), which is implemented with an LCoS element and used to generate a dark triangle pulse defined by electric field strength, according to the fifth embodiment of the present invention. [Figure 41] Figure 40 shows the waveform of a dark triangle pulse defined by electric field strength, obtained using the optical filter ((a) intensity waveform, (b) waveform converted to amplitude by taking the square root of (a)), and (c) optical spectrum. [Figure 42] The (a) transmittance characteristics and (b) phase characteristics of the optical filter FFpb(ω), which is implemented with an LCoS element and used to generate a bright triangular pulse defined by electric field strength, according to the fifth embodiment of the present invention. [Figure 43] Figure 42 shows the waveform of a Bright triangular pulse defined by electric field strength, obtained using the optical filter ((a) intensity waveform, (b) waveform converted to amplitude by taking the square root of (a)), and (c) optical spectrum. [Figure 44] The (a) transmittance characteristics and (b) phase characteristics of the optical filter FFpd(ω), which is implemented with an LCoS element and used to generate a dark parabolic pulse defined by the electric field amplitude, according to the sixth embodiment of the present invention. [Figure 45] Figure 44 shows the waveform of a dark parabolic pulse defined by the electric field amplitude, obtained using the optical filter ((a) intensity waveform, (b) waveform converted to amplitude by taking the square root of (a)), and (c) optical spectrum. [Figure 46] The (a) transmittance characteristics and (b) phase characteristics of the optical filter FFpb(ω), which is implemented with an LCoS element and used to generate a bright parabolic pulse defined by the electric field amplitude, according to the sixth embodiment of the present invention. [Figure 47] Figure 46 shows the waveform of a bright parabolic pulse defined by the electric field amplitude, obtained using the optical filter ((a) intensity waveform, (b) waveform converted to amplitude by taking the square root of (a)), and (c) optical spectrum. [Figure 48]These are (a) the transmittance characteristics and (b) the phase characteristics of an optical filter FFpd(ω) implemented with an LCoS element in a seventh embodiment of the present invention, used for generating dark parabolic pulses defined by electric field strength. [Figure 49] Figure 48 shows the waveform of a dark parabolic pulse defined by electric field strength, obtained using the optical filter ((a) intensity waveform, (b) waveform converted to amplitude by taking the square root of (a)), and (c) optical spectrum. [Figure 50] These are (a) the transmittance characteristics and (b) the phase characteristics of an optical filter FFpb(ω) implemented with an LCoS element in a seventh embodiment of the present invention, used for generating bright parabolic pulses defined by electric field strength. [Figure 51] Figure 50 shows the waveform of a bright parabolic pulse defined by electric field strength, obtained using the optical filter ((a) intensity waveform, (b) waveform converted to amplitude by taking the square root of (a)), and (c) optical spectrum. [Figure 52] These are (a) the transmittance characteristics and (b) the phase characteristics of the optical filter FFpd(ω) used to generate dark rectangular pulses implemented with an LCoS element in the eighth embodiment of the present invention. [Figure 53] Figure 52 shows the waveform of the dark rectangular pulse obtained using the optical filter ((a) intensity waveform, (b) waveform converted to amplitude by taking the square root of (a)), and (c) optical spectrum. [Figure 54] These are (a) the transmittance characteristics and (b) the phase characteristics of the optical filter FFpb(ω) used to generate a bright rectangular pulse implemented with an LCoS element in the eighth embodiment of the present invention. [Figure 55] Figure 54 shows the waveform of a bright rectangular pulse obtained using the optical filter ((a) intensity waveform, (b) waveform converted to amplitude by taking the square root of (a)), and (c) optical spectrum. [Figure 56] These are (a) the transmittance characteristics and (b) the phase characteristics of the optical filter FFpd(ω) used to generate a dark Nyquist pulse implemented with an LCoS element in the ninth embodiment of the present invention. [Figure 57]Figure 56 shows the waveform of the dark Nyquist pulse obtained using the optical filter ((a) intensity waveform, (b) waveform converted to amplitude by taking the square root of (a)), and (c) optical spectrum. [Figure 58] These are (a) the transmittance characteristics and (b) the phase characteristics of the optical filter FFpb(ω) used to generate a bright Nyquist pulse implemented with an LCoS element in the ninth embodiment of the present invention. [Figure 59] Figure 58 shows the waveform of the bright Nyquist pulse obtained using the optical filter ((a) intensity waveform, (b) waveform converted to amplitude by taking the square root of (a)), and (c) optical spectrum. [Figure 60] These are (a) the transmittance characteristics and (b) the phase characteristics of the optical filter Ftanh(ω) used for generating dark solitons implemented with an LCoS element in the tenth embodiment of the present invention. [Figure 61] Figure 60 shows the waveform of the dark soliton obtained using the optical filter ((a) intensity waveform, (b) waveform converted to amplitude by taking the square root of (a)), and (c) optical spectrum. [Modes for carrying out the invention]

[0017] Embodiments of the present invention will be described below with reference to the drawings. Figure 2 shows a schematic diagram of the configuration of a pulse light source in an embodiment of the present invention. An optical fiber 1 constitutes a laser ring resonator, and the resonator includes an optical amplifier 2 used as a gain medium, an optical phase modulator 3 used as a mode rocker, and an optical filter 4 whose amplitude and phase characteristics can be arbitrarily set. A general optical amplifier is used for the optical amplifier 2. Preferably, an erbium-doped fiber amplifier (EDFA), a semiconductor optical amplifier (SOA), a solid-state laser element, etc., can be used. For the optical phase modulator 3, a modulator is used that modulates the phase of the output light by changing the refractive index of an electro-optic crystal with an electrical signal. Preferably, a phase modulator using an LN(LiNbO3) crystal, etc., can be used. The optical phase modulator 3 is controlled by an external frequency f m(Angular frequency Ω m = 2πf m ), modulation degree M PM Sine wave M PM cos(Ω m It is driven by t). A programmable optical filter composed of LCoS or the like can be used for the optical filter 4. As will be described later, assuming the generation of pulse waveforms with asymmetrical shapes, it is desirable that the optical filter 4 can control not only the amplitude (transmission) characteristics but also the phase characteristics.

[0018] Using this pulsed light source, the amplitude is a p To generate the optical pulse given by (t) (the positive pulse in Figure 1(a)), the transfer function F of the optical filter 4 is Fp (ω) can be designed as shown in equation (1) (see, for example, Non-Patent Document 5). Here, A p (ω) is a p The spectrum of (t), where n is an integer from -∞ to ∞, J n (x) is the nth-order Bessel function of the first kind. The transfer function in equation (1) is the phase-modulated exp(iM PM cosΩ m t) and optical filter F Fp Light pulse a given by (ω) p This is obtained from equation (2), which is the master equation for (t) (where F represents the Fourier transform). Here, G is the gain and L is the loss of the resonator, and GL = 1 is set.

[0019]

number

[0020] In equation (2), the amplitude is a n (t) = -a p (t), the spectrum is A n (ω) = - A p Let's consider the master equation for the light pulse (negative pulse) given by (ω). n The transfer function of the optical filter 4 to generate (t) is F Fn Let (ω), then a nThe master equation for (t) is given by equation (3). Since equation (3) can also be written as equation (4), from equations (2) and (4), F Fn (ω) is F Fn (ω)=-F Fp It is given by (ω).

[0021]

number

[0022] Next, the positive bright pulse a shown in Figure 1(b) pb Let's discuss the generation of (t). a pb (t) is a p (t) is defined as a pulse with a positive CW offset superimposed on it. This CW offset is defined as one period (time width τ = 2π / Ω). m ) each a p If we consider it to be superimposed on (t), then a pb The waveform of (t) is a square wave R with pulse width τ. τ Using (t), it is given by equation (5). Here, γ represents the amplitude of the square wave and the height of the offset. a pb Spectrum A of (t) pb (ω) is given by equation (6) from the Fourier transform of equation (5). Here, the first term on the right-hand side is A. p (ω) is a p The Fourier transform of (t), the sinc function 2sin(ωτ / 2) / ω in the second term of the right-hand side is a square wave R. τ This is the Fourier transform of (t). Therefore, a pb Transfer function F of optical filter 4 for generating (t) Fpb (ω) is given by substituting equation (6) into equation (1) and obtaining equation (7).

[0023]

number

[0024] Similarly, for the positive dark pulse shown in Figure 1(c), its amplitude a pd (t) and spectrum A pd(ω) is given by equations (8) and (9), respectively. Therefore, a pd Transfer function F of optical filter 4 for generating (t) Fpb (ω) is given by substituting equation (9) into equation (1) and obtaining equation (10).

[0025]

number

[0026] Similarly, for the negative bright pulse and negative dark pulse shown in Figure 1(e) and Figure 1(f), respectively, their amplitude a nb (t), a nd (t) and spectrum A nb (ω), A nd (ω) is given by equations (11), (12), (13), and (14), respectively. Therefore, a nb (t) and a nd Transfer function F of optical filter 4 for generating (t) Fpb (ω) is given by substituting equations (13) and (14) into equation (1), resulting in equations (15) and (16).

[0027]

number

[0028] In other words, the spectrum A(ω) of the pulse you want to output is the transfer function F of the optical filter 4. Fpb Let (ω) be the sinc function that gives the W offset S(ω) = sin(ωτ / 2) / ω(τ= 2π / Ω m ), as well as A(ω) and S(ω) repeating at angular frequency Ω m A function A(ω-nΩ) that is shifted positively or negatively by an integer multiple of the value. m ), A(ω+nΩ m ), S(ω-nΩ m ), S(ω+nΩ m It is given using (n: integer).

[0029] [First Embodiment] In the first embodiment of the present invention, positive or negative bright or dark Gaussian pulses can be generated. Waveforms a of positive and negative Gaussian pulses p (t), a n (t) is given by equation (17). From the Fourier transform of equation (17), the spectra of positive and negative Gaussian pulses A p (ω), A n (ω) is given by equation (18). Substituting equation (18) into equations (7), (10), (15), and (16), we can find the shapes of the optical filters 4 that generate positive bright Gaussian pulses, positive dark Gaussian pulses, negative bright Gaussian pulses, and negative dark Gaussian pulses, respectively.

[0030]

number

[0031] For example, the transfer function F of the optical filter 4 for generating a positive Bright Gauss pulse. Fpb (ω) is given by equation (19). Also, the transfer function F of the optical filter 4 for generating a positive dark Gaussian pulse is given. Fpd (ω) is given by equation (20). By inserting these optical filters into the resonator, positive bright or dark Gaussian pulses can be generated. Optical filter F for generating negative bright and dark pulses Fnb (ω), F Fnd (ω) is F in equations (19) and (20). Fpb (ω), F Fpd Using (ω), F Fnb (ω) = - F Fpd (ω), F Fnd (ω) = - F Fpb It is given by (ω).

[0032]

number

[0033] In this embodiment, the transfer function F Fpb (ω) of the optical filter 4 used for generating a positive bright Gaussian pulse is shown in FIG. 3(a). Here, the pulse width T = 6 ps, the modulation angular frequency Ω m = 2π×10 GHz, the offset amplitude γ = 1, and the modulation index M PM = 2. The black dots in the figure represent the frequencies at which the longitudinal modes (10 GHz intervals) of the optical spectrum exist. F Fpb (ω) is given as a complex function. The black line in FIG. 3(a) represents its absolute value |F Fpb (ω)|, and the gray line represents the phase arg F Fpb (ω). F Fpb (ω) may be band-limited to such an extent that the waveform is not impaired. In FIG. 3(a), a band limit of 200 GHz (± 100 GHz) is given. The result of obtaining the steady-state solution of the laser by computer analysis using this filter is shown in FIG. 4(a). The black line in FIG. 4(a) represents the waveform a pb (t) of the steady-state pulse, and the gray line represents its phase arg a pb (t). It can be seen that the phase of the steady-state pulse is almost constant, and a bright Gaussian pulse without chirp can be output.

[0034] Next, in this embodiment, an example of the shape of the transfer function F Fpd (ω) of the optical filter 4 used for generating a positive dark Gaussian pulse is shown in FIG. 3(b). Here, the pulse width T = 6 ps, the modulation angular frequency Ω m = 2π×10 GHz, the offset amplitude γ = 1, and the modulation index M PM = 1. Here, F Fpd (ω) is given a band limit of 200 GHz (± 100 GHz). The result of obtaining the steady-state solution of the laser by computer analysis using this filter is shown in FIG. 4(b). The black line in FIG. 4(b) represents the waveform a pd (t) of the steady-state pulse, and the gray line represents its phase arg a pd (t). It can be seen that the phase of the steady-state pulse is almost constant, and a dark Gaussian pulse without chirp can be output.

[0035] Furthermore, in this embodiment, the transfer function F Fnb (Ω) of the optical filter 4 used for generating the negative bright pulse and the dark pulse, and F Fnd (Ω) are respectively shown in FIGS. 3(c) and (d). The results of calculating the steady-state solution of the laser by computer analysis using this filter are shown in FIGS. 4(c) and (d). The steady-state pulse has a phase difference of π compared to a pd (t) in FIG. 4(b) and a pb (t) in FIG. 4(a), indicating that a negative bright-dark Gaussian pulse can be output.

[0036] [Second Embodiment] In the second embodiment of the present invention, positive or negative bright or dark sech pulses can be generated. The waveforms a p (t) and a n (t) of the positive and negative sech pulses are given by Equation (21). From the Fourier transform of Equation (21), the spectra A p (ω) and A n (ω) of the positive and negative sech pulses are given by Equation (22). Substituting Equation (22) into Equations (7), (10), (15), and (16), the shapes of the optical filter 4 for generating positive bright sech pulses, positive dark sech pulses, negative bright sech pulses, and negative dark sech pulses can be obtained respectively.

[0037] [Equation]

[0038] For example, the transfer function F Fpb (ω) of the optical filter 4 for generating a positive bright sech pulse is given by Equation (23). Also, the transfer function F Fpd (ω) of the optical filter 4 for generating a positive dark sech pulse is given by Equation (24). By inserting these optical filters into the resonator, positive bright or dark sech pulses can be generated. The optical filter F Fnb(ω), F Fnd (ω) is F in equations (23) and (24). Fpb (ω), F Fpd Using (ω), F Fnb (ω) = - F Fpd (ω), F Fnd (ω) = - F Fpb It is given by (ω).

[0039]

number

[0040] In this embodiment, the transfer function F of the optical filter 4 used to generate positive bright pulses. Fpb An example of the shape of (ω) is shown in Figure 5(a). Here, the pulse width T = 6 ps and the modulation angular frequency Ω. m = 2π × 10 GHz, offset amplitude γ = 1, modulation index M PM We set F = 2. Fpb (ω) is given a bandwidth limit of 200 GHz (±100 GHz). The black dots in the figure represent the frequencies at which the vertical modes (10 GHz intervals) of the optical spectrum exist. Fpb (ω) is given by a complex function, and the black line in Figure 5(a) is its absolute value |F Fpb (ω)|, the gray line represents the phase arg F Fpb (ω) is shown. The steady-state solution of the laser obtained by computer analysis using this filter is shown in Figure 6(a). The black line in Figure 6(a) is the waveform a of the steady pulse. pb (t), the gray line is its phase arg a pb (t) is shown. The phase of the steady pulse is almost constant, and it can be seen that a bright sech pulse without chirp can be output.

[0041] Next, in this embodiment, the transfer function F of the optical filter 4 used to generate positive dark pulses. Fpd An example of the shape of (ω) is shown in Figure 5(b). Here, the pulse width T = 6 ps and the modulation angular frequency Ω. m = 2π × 10 GHz, offset amplitude γ = 1, modulation index MPM We set = 1. Here, F Fpd A bandwidth limit of 200 GHz (±100 GHz) is applied to (ω). The steady-state solution of the laser obtained by computer analysis using this filter is shown in Figure 6(b). The black line in Figure 6(b) represents the waveform a of the steady-state pulse. pd (t), the gray line is its phase arg a pd (t) is shown. The phase of the steady pulse is almost constant, and it can be seen that a dark sech pulse without chirp can be output.

[0042] Furthermore, in this embodiment, the transfer function F of the optical filter 4 used to generate negative bright sech pulses and dark sech pulses. Fnb (ω), F Fnd Examples of the shape of (ω) are shown in Figures 5(c) and (d), respectively. The results of obtaining the steady-state solution of the laser using this filter by computer analysis are shown in Figures 6(c) and (d). The steady-state pulse is shown in Figure 6(b) a pd (t), a in Figure 6(a) pb Compared to (t), it has a phase difference of π, indicating that negative bright-dark pulses can be output.

[0043] [Third Embodiment] In a third embodiment of the present invention, both positive and negative bright or dark exponential pulses can be generated. Waveforms a of both positive and negative exponential pulses p (t), a n (t) is given by equation (25). From the Fourier transform of equation (25), the spectra A of both positive and negative exponential pulses are obtained. p (ω), A n (ω) is given by equation (26) (the Lorentz function, which is the Fourier transform of both exponential functions). Substituting equation (26) into equations (7), (10), (15), and (16), we can find the shapes of the optical filters 4 for generating positive bright exponential pulses, positive dark exponential pulses, negative bright exponential pulses, and negative bright exponential pulses, respectively.

[0044]

number

[0045] For example, the transfer function F of the optical filter 4 for generating positive Bright exponential pulses. Fpb (ω) is given by equation (27). Also, the transfer function F of the optical filter 4 for generating positive dark exponential pulses is given. Fpd (ω) is given by equation (28). By inserting these optical filters into the resonator, both positive bright and dark exponential pulses can be generated. Optical filter F for generating negative bright and dark pulses Fnb (ω), F Fnd (ω) is F in equations (27) and (28). Fpb (ω), F Fpd Using (ω) F Fnb (ω) = - F Fpd (ω), F Fnd (ω) = - F Fpb It is given by (ω).

[0046]

number

[0047] In this embodiment, the transfer function F of the optical filter 4 used to generate positive Bright exponential pulses. Fpb An example of the shape of (ω) is shown in Figure 7(a). Here, the pulse width T = 6.25 ps and the modulation angular frequency Ω m = 2π × 10 GHz, offset amplitude γ = 1, modulation index M PM We set F = 2. Fpb (ω) is given a bandwidth limit of 440 GHz (±220 GHz). The black dots in the figure represent the frequencies at which the longitudinal modes (10 GHz intervals) of the optical spectrum exist. F Fpb (ω) is given by a complex function, and the black line in Figure 7(a) is its absolute value |F Fpb (ω)|, the gray line represents the phase arg F Fpb(ω) is shown. The steady-state solution of the laser obtained by computer analysis using this filter is shown in Figure 8(a). The black line in Figure 8(a) is the waveform a of the steady pulse. pb (t), the gray line is its phase arg a pb (t) is shown. The phase of the steady pulse is almost constant, and it can be seen that a Bright exponential function pulse without chirp can be output.

[0048] Next, in this embodiment, the transfer function F of the optical filter 4 used to generate positive Dark exponential pulses. Fpd An example of the shape of (ω) is shown in Figure 7(b). Here, the pulse width T = 6.25 ps and the modulation angular frequency Ω m = 2π × 10 GHz, offset amplitude γ = 1, modulation index M PM We set = 1. Here, F Fpd A bandwidth limit of 440 GHz (±220 GHz) is applied to (ω). The steady-state solution of the laser obtained by computer analysis using this filter is shown in Figure 8(b). The black line in Figure 8(b) represents the waveform a of the steady-state pulse. pd (t), the gray line is its phase arg a pd (t) is shown. The phase of the steady pulse is almost constant, and it can be seen that a Dark double exponential function pulse without chirp can be output.

[0049] Furthermore, in this embodiment, the transfer function F of the optical filter 4 used to generate negative bright exponential pulses and dark exponential pulses. Fnb (ω), F Fnd Examples of the shape of (ω) are shown in Figures 7(c) and (d), respectively. The results of obtaining the steady-state solution of the laser using this filter by computer analysis are shown in Figures 8(c) and (d). The steady-state pulse is shown in Figure 8(b) a pd (t), a in Figure 8(a) pb Compared to (t), it has a phase difference of π, indicating that it can output negative bright and dark exponential function pulses.

[0050] [Fourth Embodiment] In a fourth embodiment of the present invention, it is possible to generate positive or negative bright or dark pulses whose electric field amplitude is defined by a triangle. Waveforms a of the positive and negative triangular pulses p (t), a n (t) is given by equation (29). From the Fourier transform of equation (29), the spectra of positive and negative triangular pulses A p (ω), A n (ω) is given by equation (30) (sinc function squared). Substituting equation (30) into equations (7), (10), (15), and (16), we can find the shapes of the optical filters 4 for generating positive bright triangular pulses, positive dark triangular pulses, negative bright triangular pulses, and negative bright triangular pulses, respectively.

[0051]

number

[0052] For example, the transfer function F of the optical filter 4 for generating a positive Bright triangular pulse. Fpb (ω) is given by equation (31). Also, the transfer function F of the optical filter 4 for generating positive dark triangle pulses is given. Fpd (ω) is given by equation (32). By inserting these optical filters into the resonator, positive bright or dark triangular pulses can be generated. Optical filter F for generating negative bright and dark pulses. Fnb (ω), F Fnd (ω) is F in equations (31) and (32). Fpb (ω), F Fpd Using (ω) F Fnb (ω) = - F Fpd (ω), F Fnd (ω) = - F Fpb It is given by (ω).

[0053]

number

[0054] In this embodiment, the transfer function F of the optical filter 4 used to generate positive bright triangular pulses. Fpb An example of the shape of (ω) is shown in Figure 9(a). Here, the pulse width T = 25 ps and the modulation angular frequency Ω. m = 2π × 10 GHz, offset amplitude γ = 1, modulation index M PM = 2.4. Here, F Fpd (ω) is given a bandwidth limit of 240 GHz (±120 GHz). The black dots in the figure represent the frequencies at which the longitudinal modes (10 GHz intervals) of the optical spectrum exist. F Fpb (ω) is given by a complex function, and the black line in Figure 9(a) is its absolute value |F Fpb (ω)|, the gray line represents the phase arg F Fpb (ω) is shown. The steady-state solution of the laser obtained by computer analysis using this filter is shown in Figure 10(a). The black line in Figure 10(a) is the waveform a of the steady pulse. pb (t), the gray line is its phase arg a pb (t) is shown. The phase of the steady pulse is almost constant, and it can be seen that a bright triangular pulse without chirp can be output.

[0055] Next, in this embodiment, the transfer function F of the optical filter 4 used to generate positive dark triangle pulses. Fpd An example of the shape of (ω) is shown in Figure 9(b). Here, the pulse width T = 25 ps and the modulation angular frequency Ω. m = 2π × 10 GHz, offset amplitude γ = 1, modulation index M PM We set F = 2. Fpd A bandwidth limit of 240 GHz (±120 GHz) is applied to (ω). The steady-state solution of the laser obtained by computer analysis using this filter is shown in Figure 10(b). The black line in Figure 10(b) represents the waveform a of the steady-state pulse. pd (t), the gray line is its phase arg a pd (t) is shown. The phase of the steady pulse is almost constant, and it can be seen that a dark triangle pulse without chirp can be output.

[0056] Furthermore, in this embodiment, the transfer function F of the optical filter 4 used to generate negative bright triangular pulses and dark triangular pulses. Fnb (ω), F Fnd Examples of the shape of (ω) are shown in Figures 9(c) and (d), respectively. The results of obtaining the steady-state solution of the laser using this filter by computer analysis are shown in Figures 10(c) and (d). The steady-state pulse is shown in Figure 10(b) a pd (t), a in Figure 10(a) pb It has a phase difference of π compared to (t), indicating that a negative bright-dark triangular pulse can be output.

[0057] [Fifth Embodiment] In a fifth embodiment of the present invention, instead of a triangular pulse defined by the electric field amplitude in the fourth embodiment, it is possible to generate positive or negative bright or dark pulses whose intensity (square of the electric field) is given by a triangle. Waveform (electric field amplitude) of a positive bright pulse a pb (t) is given by equation (33). In this case, |a pb (t) 2 This represents the Bright triangular pulse. From the Fourier transform of equation (33), the spectrum of this pulse A pb (ω) is given by equation (34) (the first kind of Bessel function J) 2k It is given by (ω)(a series of integers k), where φ = cos -1 (1 / (1+γ)). In particular, when γ = 1, φ = π / 3, and equation (34) is expressed as equation (35). Substituting equation (34) or (35) into equation (7), we get the shape F of the optical filter 4 for generating a positive bright pulse in which the electric field strength is given by a triangle. Fpb (ω) can be calculated.

[0058]

number

[0059] Next, the waveform (electric field amplitude) of the positive dark pulse a pd (t) is given by equation (36). In this case, |a pd (t) 2This represents the dark triangle pulse. From the Fourier transform of equation (36), the spectrum of this pulse A pd (ω) is given by equation (37). Here, φ = cos -1 (1 / (1+γ)). In particular, when γ = 1, φ = π / 2, and equation (37) is expressed as equation (38). Substituting equation (37) or (38) into equation (10), we get the shape F of the optical filter 4 for generating a positive dark pulse in which the electric field strength is given by a triangle. Fpd (ω) can be calculated.

[0060]

number

[0061] Transfer function F of optical filter 4 for generating negative bright pulses Fnb (ω) is the last ω in equation (37) or equation (38). -1 The transfer function F of the optical filter 4 for generating negative dark pulses is obtained by reversing the sign of the term from + to - and substituting it into equation (10). Fnd (ω) is the last ω in equation (34) or equation (35). -1 The answer can be found by reversing the sign of the term from + to - and substituting it into equation (7).

[0062] In this embodiment, the transfer function F of the optical filter 4 used to generate a positive bright pulse in which the electric field strength is given by a triangle. Fpb An example of the shape of (ω) is shown in Figure 11(a). Here, the pulse width T = 20 ps and the modulation angular frequency Ω. m = 2π × 10 GHz, offset amplitude γ = 1, modulation index M PM We set F = 2. Fpd (ω) is given a bandwidth limit of 240 GHz (±120 GHz). The black dots in the figure represent the frequencies at which the longitudinal modes (10 GHz intervals) of the optical spectrum exist. F Fpb (ω) is given by a complex function, and the black line in Figure 11(a) is its absolute value |F Fpb (ω)|, the gray line represents the phase arg F Fpb(ω) is shown. The steady-state solution of the laser obtained by computer analysis using this filter is shown in Figure 12(a). The black line in Figure 12(a) is the waveform a of the steady pulse. pb (t), the gray line is its intensity |a pb (t) 2 This shows that, as indicated by the gray line and the vertical axis on the right in Figure 12(a), a positive bright pulse with a triangular electric field strength is obtained.

[0063] Next, in this embodiment, the transfer function F of the optical filter 4 used to generate a positive dark pulse in which the electric field strength is given by a triangle. Fpd An example of the shape of (ω) is shown in Figure 11(b). Here, the pulse width T = 20 ps and the modulation angular frequency Ω. m = 2π × 10 GHz, offset amplitude γ = 1, modulation index M PM We set = 1. Here, F Fpd (ω) is given a bandwidth limit of 240 GHz (±120 GHz). The black dots in the figure represent the frequencies at which the longitudinal modes (10 GHz intervals) of the optical spectrum exist. F Fpd (ω) is given by a complex function, and the black line in Figure 11(b) is its absolute value |F Fpd (ω)|, the gray line represents the phase arg F Fpd (ω) is shown. The steady-state solution of the laser obtained by computer analysis using this filter is shown in Figure 12(b). The black line in Figure 12(b) is the waveform a of the steady pulse. pd (t), the gray line is its intensity |a pd (t) 2 This shows that, as indicated by the gray line and the vertical axis on the right in Figure 12(b), a positive dark pulse with a triangular electric field strength is obtained.

[0064] Furthermore, in this embodiment, the transfer function F of the optical filter 4 used to generate negative bright pulses and dark pulses where the electric field strength is given by a triangle. Fnb (ω), F FndExamples of the shape of (ω) are shown in Figures 11(c) and (d), respectively. The results of obtaining the steady-state solution of the laser using this filter by computer analysis are shown in Figures 12(c) and (d). The steady-state pulse is shown in Figure 12(b) a pd (t), a in Figure 12(a) pb The waveform is a phase-inverted version of (t), and as shown by the gray line and the vertical axis on the right in Figures 12(c) and (d), it can be seen that a triangular negative bright-dark pulse can be output with electric field strength.

[0065] [Sixth Embodiment] In a sixth embodiment of the present invention, it is possible to generate positive or negative bright or dark pulses whose electric field amplitude is parabolic. Waveforms a of the positive and negative parabolic pulses p (t), a n (t) is given by equation (39). From the Fourier transform of equation (39), the spectra A of the positive and negative parabolic pulses are obtained. p (ω), A n (ω) is given by equation (40) (function sin ω / ω 3 to cos ω / ω 2 It is given by the sum or difference of . Substituting equation (40) into equations (7), (10), (15), and (16), we can find the shapes of the optical filters 4 for generating positive bright parabolic pulses, positive dark parabolic pulses, negative bright parabolic pulses, and negative bright parabolic pulses, respectively.

[0066]

number

[0067] For example, the transfer function F of the optical filter 4 for generating a positive bright parabolic pulse. Fpb (ω) is given by equation (41). Also, the transfer function F of the optical filter 4 for generating positive dark parabolic pulses is given. Fpdω is given by Equation (42). By inserting these optical filters into the resonator, positive bright or dark parabolic pulses can be generated. Optical filters F for generating negative bright and dark pulses Fnb ω, F Fnd ω is F of Equations (41) and (42) Fpb ω, F Fpd Using ω for F Fnb ω = -F Fpd ω, F Fnd ω = -F Fpb is given by ω

[0068] [Number]

[0069] In this embodiment, the transfer function F of the optical filter 4 used for generating a positive bright parabolic pulse Fpb An example of the shape of ω is shown in Fig. 13(a). Here, the pulse width T = 25 ps, the modulation angular frequency Ω m = 2π × 10 GHz, the offset amplitude γ = 1, and the modulation index M PM = 2. Here, F Fpb ω has a bandwidth limitation of 640 GHz (±320 GHz). The black dots in the figure represent the frequencies where the longitudinal modes (10 GHz interval) of the optical spectrum exist. F Fpb ω is given by a complex function. The black line in Fig. 13(a) is its absolute value |F Fpb ω|, and the gray line is the phase arg F Fpb ω. The result of calculating the steady state solution of the laser using this filter by computer analysis is shown in Fig. 14(a). The black line in Fig. 14(a) is the waveform a pb (t) of the steady state pulse, and the gray line is its phase arg a pb (t). It can be seen that the phase of the steady state pulse is almost constant, and a bright parabolic pulse without chirp can be output

[0070] Next, in the present embodiment, the transfer function F of the optical filter 4 used for generating a positive dark parabola pulse Fpd (ω) is shown in Fig. 13(b) as an example. Here, the pulse width T = 25 ps, the modulation angular frequency Ω m = 2π×10 GHz, the offset amplitude γ = 1, and the modulation index M PM = 1. Here, F Fpd (ω) has a bandwidth limit of 640 GHz (±320 GHz). The result of obtaining the steady state solution of the laser by computer analysis using this filter is shown in Fig. 14(b). The black line in Fig. 14(b) is the waveform a of the steady state pulse pd (t), and the gray line shows its phase arg a pd (t). It can be seen that the phase of the steady state pulse is almost constant, and a dark parabola pulse without chirp can be output

[0071] Furthermore, in the present embodiment, the transfer functions F of the optical filter 4 used for generating a negative bright parabola pulse and a dark parabola pulse Fnb (ω), F Fnd (ω) are shown in Figs. 13(c) and (d) respectively as examples. The results of obtaining the steady state solution of the laser by computer analysis using this filter are shown in Figs. 14(c) and (d). The steady state pulse has a phase difference of π compared to a pd (t) in Fig. 14(b) and a pb (t) in Fig. 14(a), and it can be seen that a negative bright-dark parabola pulse can be output

[0072] [Embodiment 7] In the seventh embodiment of the present invention, positive or negative bright or dark pulses whose intensity (square of the electric field) is given by a parabola can be generated, instead of the parabola pulse defined by the electric field amplitude in the sixth embodiment. The waveform (electric field amplitude) a of the positive bright pulse pb (t) is given by Equation (43). At this time, |a pb (t)| 2 represents a bright parabola pulse. From the Fourier transform of Equation (43), the spectrum A of this pulse pb(ω) is given by equation (44) (function J1(ω) / ω(J1(ω): first-order Bessel function of the first kind)), where φ = sin -1 (1 / (1+γ) 1 / 2 ) In particular, when γ = 1, φ = π / 4, and equation (44) is expressed as equation (45). Substituting equation (44) or equation (45) into equation (7), we get the shape F of the optical filter 4 for generating a positive bright pulse in which the electric field strength is given by a parabola. Fpb (ω) can be calculated.

[0073]

number

[0074] Next, the waveform (electric field amplitude) of the positive dark pulse a pd (t) is given by equation (46). In this case, |a pd (t) 2 This represents the dark parabolic pulse. From the Fourier transform of equation (46), the spectrum A of this pulse is obtained. pd (ω) is given by equation (47), where φ = sin -1 (1 / (1+γ) 1 / 2 ) In particular, when γ = 1, φ = π / 2, and equation (47) is expressed as equation (48). Substituting equation (47) or equation (48) into equation (10), we get the shape F of the optical filter 4 for generating a positive dark pulse in which the electric field strength is given by a parabola. Fpd (ω) can be calculated.

[0075]

number

[0076] Transfer function F of optical filter 4 for generating negative bright pulses Fnb (ω) is the last ω in equation (47) or equation (48). -1 The transfer function F of the optical filter 4 for generating negative dark pulses is obtained by reversing the sign of the term from + to - and substituting it into equation (10). Fnd(ω) is the last ω in equation (44) or equation (45). -1 The answer can be found by reversing the sign of the term from + to - and substituting it into equation (7).

[0077] In this embodiment, the transfer function F of the optical filter 4 used to generate a positive bright pulse in which the electric field strength is provided by a parabola. Fpb An example of the shape of (ω) is shown in Figure 15(a). Here, the pulse width T = 25 ps and the modulation angular frequency Ω m = 2π × 10 GHz, offset amplitude γ = 1, modulation index M PM We set F = 2. Fpb (ω) is given a bandwidth limit of 240 GHz (±120 GHz). The black dots in the figure represent the frequencies at which the longitudinal modes (10 GHz intervals) of the optical spectrum exist. F Fpb (ω) is given by a complex function, and the black line in Figure 15(a) is its absolute value |F Fpb (ω)|, the gray line represents the phase arg F Fpb (ω) is shown. The steady-state solution of the laser obtained by computer analysis using this filter is shown in Figure 16(a). The black line in Figure 16(a) is the waveform a of the steady-state pulse. pb (t), the gray line is its intensity |a pb (t) 2 This shows that, as indicated by the gray line and the vertical axis on the right in Figure 16(a), a positive bright pulse of parabolic electric field strength is obtained.

[0078] Next, in this embodiment, the transfer function F of the optical filter 4 used to generate a positive dark pulse in which the electric field strength is given by a parabola. Fpd An example of the shape of (ω) is shown in Figure 15(b). Here, the pulse width T = 25 ps and the modulation angular frequency Ω. m = 2π × 10 GHz, offset amplitude γ = 1, modulation index M PM We set F = 2. Fpd (ω) is given a bandwidth limit of 240 GHz (±120 GHz). The black dots in the figure represent the frequencies at which the longitudinal modes (10 GHz intervals) of the optical spectrum exist. F Fpd(ω) is given by a complex function, and the black line in Figure 15(b) is its absolute value |F Fpd (ω)|, the gray line represents the phase arg F Fpd (ω) is shown. The steady-state solution of the laser obtained by computer analysis using this filter is shown in Figure 16(b). The black line in Figure 16(b) is the waveform a of the steady pulse. pd (t), the gray line is its intensity |a pd (t) 2 This shows that, as indicated by the gray line and the vertical axis on the right in Figure 16(b), a positive dark pulse of the parabolic field strength is obtained.

[0079] Furthermore, in this embodiment, the transfer function F of the optical filter 4 used to generate negative bright pulses and dark pulses where the electric field strength is parabolic Fnb (ω), F Fnd Examples of the shape of (ω) are shown in Figures 15(c) and (d), respectively. The results of obtaining the steady-state solution of the laser using this filter by computer analysis are shown in Figures 16(c) and (d). The steady-state pulse is shown in Figure 16(b) a pd (t), a in Figure 16(a) pb The waveform is a phase-inverted version of (t), and as shown by the gray line and the vertical axis on the right in Figures 16(c) and (d), it can be seen that the electric field strength can output parabolic negative bright-dark pulses.

[0080] [Eighth Embodiment] In the eighth embodiment of the present invention, positive or negative bright or dark rectangular pulses can be generated. Waveforms a of positive and negative rectangular pulses p (t), a n (t) is given by equation (49). From the Fourier transform of equation (49), the spectra of the positive and negative rectangular pulses A p (ω), A n(ω) is given by equation (50) (sin(ωT) / (ωT) given by pulse width T). Substituting equation (50) into equations (7), (10), (15), and (16), we can find the shapes of the optical filters 4 for generating positive bright rectangular pulses, positive dark rectangular pulses, negative bright rectangular pulses, and negative bright rectangular pulses, respectively.

[0081]

number

[0082] For example, the transfer function F of the optical filter 4 for generating a positive bright rectangular pulse. Fpb (ω) is given by equation (51). Also, the transfer function F of the optical filter 4 for generating a positive dark rectangle pulse is given. Fpd (ω) is given by equation (52). By inserting these optical filters into the resonator, positive bright or dark rectangular pulses can be generated. Optical filter F for generating negative bright and dark pulses. Fnb (ω), F Fnd (ω) is F in equations (51) and (52). Fpb (ω), F Fpd Using (ω) F Fnb (ω) = - F Fpd (ω), F Fnd (ω) = - F Fpb It is given by (ω).

[0083]

number

[0084] In this embodiment, the transfer function F of the optical filter 4 used to generate a positive bright rectangular pulse. Fpb An example of the shape of (ω) is shown in Figure 17(a). Here, the pulse width T = 25 ps and the modulation angular frequency Ω. m = 2π × 10 GHz, offset amplitude γ = 1, modulation index M PM We set = 1. Here, F Fpb(ω) is given a bandwidth limit of 640 GHz (±320 GHz). The black dots in the figure represent the frequencies at which the longitudinal modes (10 GHz intervals) of the optical spectrum exist. F Fpb (ω) is given by a complex function, and the black line in Figure 17(a) is its absolute value |F Fpb (ω)|, the gray line represents the phase arg F Fpb (ω) is shown. The steady-state solution of the laser obtained by computer analysis using this filter is shown in Figure 18(a). The black line in Figure 18(a) is the waveform a of the steady-state pulse. pb (t), the gray line is its phase arg a pb (t) is shown. The phase of the steady pulse is almost constant, and it can be seen that a bright rectangular pulse without chirp can be output.

[0085] Next, in this embodiment, the transfer function F of the optical filter 4 used to generate positive dark rectangle pulses. Fpd An example of the shape of (ω) is shown in Figure 17(b). Here, the pulse width T = 25 ps and the modulation angular frequency Ω. m = 2π × 10 GHz, offset amplitude γ = 1, modulation index M PM We set = 1. Here, F Fpd (ω) is given a bandwidth limit of 640 GHz (±320 GHz). The steady-state solution of the laser obtained by computer analysis using this filter is shown in Figure 18(b). The black line in Figure 18(b) is the waveform a of the steady-state pulse. pd (t), the gray line is its phase arg a pd (t) is shown. The phase of the steady pulse is almost constant, and it can be seen that a dark rectangular pulse without chirp can be output.

[0086] Furthermore, in this embodiment, the transfer function F of the optical filter 4 used to generate negative bright rectangular pulses and dark rectangular pulses. Fnb (ω), F Fnd Examples of the shape of (ω) are shown in Figures 17(c) and (d), respectively. The results of obtaining the steady-state solution of the laser using this filter by computer analysis are shown in Figures 18(c) and (d). The steady-state pulse is shown in Figure 18(b) a pd (t), a in Figure 18(a)pb It can be seen that it has a phase difference of π compared to (t), and a negative bright-dark rectangular pulse can be output.

[0087] [Embodiment 9] In the ninth embodiment of the present invention, positive or negative bright or dark Nyquist pulses can be generated. The waveforms a p (t), a n (t) are given by Equation (53). Here, ω N = π / T, and α (0 ≤ α ≤ 1) is a parameter called the roll-off rate. From the Fourier transform of Equation (53), the spectra A p (ω), A n (ω) are given by Equation (54). Substituting Equation (54) into Equations (7), (10), (15), and (16), the shapes of the optical filter 4 for generating positive bright Nyquist pulses, positive dark Nyquist pulses, negative bright Nyquist pulses, and negative bright Nyquist pulses can be obtained respectively.

[0088] [Number]

[0089] For example, the transfer function F Fpb (ω) of the optical filter 4 for generating positive bright Nyquist pulses is given by Equation (55). Also, the transfer function F Fpd (ω) of the optical filter 4 for generating positive dark Nyquist pulses is given by Equation (56). By inserting these optical filters into the resonator, positive bright or dark Nyquist pulses can be generated. The optical filters F Fnb (ω), F Fnd (ω) for generating negative bright and dark pulses are obtained by using F Fpb (ω), F Fpd (ω) in Equations (55) and (56) such that F Fnb (ω) = - F Fpd (ω), F Fnd (ω) = - FFpb It is given by (ω).

[0090]

number

[0091] In this embodiment, the transfer function F of the optical filter 4 used to generate a positive Bright Nyquist pulse. Fpb An example of the shape of (ω) is shown in Figure 19(a). Here, T = 12.5 ps and the modulation angular frequency Ω. m = 2π × 10 GHz, offset amplitude γ = 1, modulation index M PM We set F = 2. Fpb (ω) is band-limited by a Nyquist filter with α = 0.08. The black dots in the figure represent the frequencies at which the vertical modes (10 GHz intervals) of the optical spectrum exist. F Fpb (ω) is given by a complex function, and the black line in Figure 19(a) is its absolute value |F Fpb (ω)|, the gray line represents the phase arg F Fpb (ω) is shown. The steady-state solution of the laser obtained by computer analysis using this filter is shown in Figure 20(a). The black line in Figure 20(a) is the waveform a of the steady-state pulse. pb (t), the gray line is its phase arg a pb (t) is shown. The phase of the steady pulse is almost constant, indicating that a bright Nyquist pulse without chirp can be output.

[0092] Next, in this embodiment, the transfer function F of the optical filter 4 used to generate a positive dark Nyquist pulse. Fpd An example of the shape of (ω) is shown in Figure 19(b). Here, T = 12.5 ps and the modulation angular frequency Ω. m = 2π × 10 GHz, offset amplitude γ = 1, modulation index M PM We set = 1. Here, F Fpd(ω) is band-limited by a Nyquist filter with α = 0.08. The steady-state solution of the laser obtained by computer analysis using this filter is shown in Figure 20(b). The black line in Figure 20(b) represents the waveform a of the steady-state pulse. pd (t), the gray line is its phase arg a pd (t) is shown. The phase of the steady pulse is almost constant, and it can be seen that a dark Nyquist pulse without chirp can be output.

[0093] Furthermore, in this embodiment, the transfer function F of the optical filter 4 used to generate negative bright Nyquist pulses and dark Nyquist pulses. Fnb (ω), F Fnd Examples of the shape of (ω) are shown in Figures 19(c) and (d), respectively. The results of obtaining the steady-state solution of the laser using this filter by computer analysis are shown in Figures 20(c) and (d). The steady-state pulse is shown in Figure 20(b) a pd (t), a in Figure 20(a) pb Compared to (t), it has a phase difference of π, indicating that a negative bright-dark Nyquist pulse can be output.

[0094] [Tenth Embodiment] In the tenth embodiment of the present invention, the electric field amplitude is tanh and the electric field strength is 1 - sec 2 A dark soliton can be generated given by . The waveform d(t) of the dark soliton defined by one pulse repetition period (-τ / 2 < t < τ / 2) is given by equation (57). From the Fourier transform of equation (57), the spectrum D(ω) of the dark soliton is given by equation (58). From equation (58), the transfer function F of the optical filter 4 for generating the dark soliton is given by tanh (ω) is given by equation (59). By inserting this optical filter into the resonator, dark solitons can be generated.

[0095]

number

[0096] In this embodiment, the transfer function F of the optical filter 4 used to generate dark solitons. tanh An example of the shape of (ω) is shown in Figure 21. Here, the pulse width T = 2.5 ps and the modulation angular frequency Ω. m = 2π × 10 GHz (repetition frequency 20 GHz), modulation index M PM = 2.4. Here, F tanh (ω) is given a bandwidth limit of 200 GHz (±100 GHz). The black dots in the figure represent the frequencies at which the longitudinal modes (10 GHz intervals) of the optical spectrum exist. F tanh (ω) is given by a complex function, and Figure 21(a) shows its absolute value |F tanh (ω)| is the phase arg F in Figure 21(b). tanh (ω) is shown. The steady-state solution of the laser obtained by computer analysis using this filter is shown in Figure 22. Figure 22(a) shows the intensity of the steady pulse |d(t)| 2 Figure 22(b) shows the phase arg d(t). The phase of the steady pulse is almost constant, and it can be seen that a dark soliton without chirp can be output. It can also be seen that the phase is inverted by π at t = ± 50 ps.

[0097] The following describes a specific experimental example. The experiment uses an optical fiber ring resonator (resonator length 15 m) as shown in Figure 23. The optical fiber ring resonator consists of a high-power 1.48 μm semiconductor laser (LD; Laser Diode) 6 (a harmonic FM mode-locked erbium fiber laser oscillating at a wavelength of 1.56 μm and a repetition rate of 20 GHz), a wavelength division multiplexing (WDM; Wavelength Division Multiplexing) coupler 7 that couples the excitation light from the semiconductor laser 6 to the optical fiber resonator, a polarization-maintaining erbidium fiber 5 (fiber length 5 m; corresponding to optical amplifier 2 in Figure 2) that amplifies the excitation light coupled by the WDM coupler 7, and a PZT (lead zirconate) to vary the resonator length. The system consists of a titanate element 10, a coupler 11 that splits 20% of the light into a laser output and the remaining 80% into an optical fiber resonator, an isolator 12 that rectifies the light from the coupler 11, a phase modulator 8 (corresponding to optical phase modulator 3 in Figure 2) connected to the isolator 12, a frequency synthesizer 16 (20 GHz), a phase shifter 15, and an amplifier 14, respectively, and an LCoS element 13 (corresponding to optical filter 4 in Figure 2) that receives the light from the phase modulator 8 via a wavelength filter (etalon) 9. The light from the LCoS element 13 is coupled by a WDM coupler 7.

[0098] The filter function F shown in the 1st to 9th embodiments Fpb (ω), F Fpd (ω), F Fnb (ω), F Fnd The amplitude and phase characteristics of (ω) are implemented in the LCoS element 13 (optical filter 4) using software. The frequency resolution of the LCoS element is 1 GHz.

[0099] The following describes the optical filter shape and generated pulses for each embodiment of the present invention. Note that while the positive dark pulse and positive bright pulse described above are shown, the negative bright pulse and negative dark pulse are omitted below because their intensity waveforms are the same as those of the positive dark pulse and positive bright pulse, respectively.

[0100] Figure 24 shows the shape of the optical filter used to generate the dark Gaussian pulse in the first embodiment of the present invention. Here, T = 3 ps, M PM = 1. Figure (a) shows the transmittance characteristics, and (b) shows the phase characteristics. The gray line represents F in equation (20). Fpd (ω) (see Figure 3(b)) is a filter with a bandwidth limited to 400 GHz. The black line shows the filter shape when this is approximated by a step function every 20 GHz and implemented on an LCoS element. The LCoS element used in the experiment originally has a frequency resolution of 1 GHz, but considering fluctuations in the longitudinal mode frequency of the laser, F is set every 20 GHz, which is the longitudinal mode interval. Fpd (ω) is approximated in steps, and this is implemented.

[0101] Figure 25 shows the waveform and optical spectrum of the dark Gaussian pulse generated using this filter. Figure 25(a) shows the intensity waveform a observed with an optical sampling oscilloscope. 2 (t) and (b) are obtained by taking the square root and converting them to amplitude a(t), and (c) is the optical spectrum A(ω) observed with an optical spectrum analyzer expressed in dB (20 log|A(ω)|). The solid line is the experimental result, and the dashed line is the computer analysis result. The two are in good agreement, indicating that the dark Gaussian pulse can be generated as designed using the optical filter in Figure 24.

[0102] Next, Figure 26 shows the shape of the optical filter used to generate the Bright Gauss pulse in the first embodiment of the present invention. Here, T = 3 ps, M PM = 2. Figure (a) shows the transmittance characteristics, and (b) shows the phase characteristics. The gray line represents F in equation (19). Fpb The black line shows the filter shape when (ω) (see Figure 3(a)) is band-limited at 400 GHz and approximated by a step function every 20 GHz, then implemented on an LCoS element.

[0103] Figure 27 shows the waveform and optical spectrum of the Bright Gauss pulse generated using this filter. Figure 27(a) shows the intensity waveform a observed with an optical sampling oscilloscope. 2(t) and (b) are obtained by taking the square root and converting them to amplitude a(t), and (c) is the optical spectrum A(ω) observed with an optical spectrum analyzer expressed in dB (20 log|A(ω)|). The solid line is the experimental result, and the dashed line is the computer analysis result. The two are in good agreement, indicating that the Bright Gauss pulse can be generated as designed using the optical filter in Figure 26.

[0104] Figure 28 shows the shape of the optical filter used to generate dark sech pulses in the second embodiment of the present invention. Here, T = 3 ps, M PM = 1. Figure (a) shows the transmittance characteristics, and (b) shows the phase characteristics. The gray line represents F in equation (24). Fpd The black line shows the filter shape when (ω) (see Figure 5(b)) is band-limited at 400 GHz and approximated by a step function every 20 GHz, then implemented on an LCoS element.

[0105] Figure 29 shows the waveform and optical spectrum of the dark pulse generated using this filter. Figure 29(a) shows the intensity waveform a observed with an optical sampling oscilloscope. 2 (t) and (b) are obtained by taking the square root and converting them to amplitude a(t), and (c) is the optical spectrum A(ω) observed with an optical spectrum analyzer expressed in dB (20 log|A(ω)|). The solid line is the experimental result, and the dashed line is the computer analysis result. The two are in good agreement, indicating that the dark sech pulse can be generated as designed using the optical filter in Figure 28.

[0106] Next, Figure 30 shows the shape of the optical filter used to generate the bright sech pulse in the second embodiment of the present invention. Here, T = 3 ps, M PM = 2. Figure (a) shows the transmittance characteristics, and (b) shows the phase characteristics. The gray line represents F in equation (23). Fpb The black line shows the filter shape when (ω) (see Figure 5(a)) is band-limited at 400 GHz and approximated by a step function every 20 GHz, then implemented on an LCoS element.

[0107] Figure 31 shows the waveform and optical spectrum of the bright pulse generated using this filter. Figure 31(a) shows the intensity waveform a observed with an optical sampling oscilloscope. 2 (t) and (b) are obtained by taking the square root and converting them to amplitude a(t), and (c) is the optical spectrum A(ω) observed with an optical spectrum analyzer expressed in dB (20 log|A(ω)|). The solid line is the experimental result, and the dashed line is the computer analysis result. The two are in good agreement, indicating that the bright sech pulse can be generated as designed using the optical filter in Figure 30.

[0108] Figure 32 shows the shape of the optical filter used to generate the dark exponential pulse in the third embodiment of the present invention. Here, T = 6.25 ps, M PM = 1. Figure (a) shows the transmittance characteristics, and (b) shows the phase characteristics. The gray line represents F in equation (28). Fpd (ω) (see Figure 7(b)) is a filter with a bandwidth limited to 440 GHz, and the black line shows the filter shape when this is approximated by a step function every 20 GHz and implemented on an LCoS element.

[0109] Figure 33 shows the waveform and optical spectrum of the dark exponential pulses generated using this filter. Figure 33(a) shows the intensity waveform a observed with an optical sampling oscilloscope. 2 (t) and (b) are obtained by taking the square root and converting them to amplitude a(t), and (c) is the optical spectrum A(ω) observed with an optical spectrum analyzer expressed in dB (20 log|A(ω)|). The solid line is the experimental result, and the dashed line is the computer analysis result. The two are in good agreement, indicating that the optical filter in Figure 32 can generate both dark exponential function pulses as designed.

[0110] Next, Figure 34 shows the shape of the optical filter used to generate Bright exponential pulses in the third embodiment of the present invention. Here, T = 6.25 ps, M PM = 2. Figure (a) shows the transmittance characteristics, and (b) shows the phase characteristics. The gray line represents F in equation (27). FpbThe black line shows the filter shape when (ω) (see Figure 7(a)) is band-limited at 440 GHz and approximated by a step function every 20 GHz, then implemented on an LCoS element.

[0111] Figure 35 shows the waveforms and optical spectra of Bright exponential pulses generated using this filter. Figure 35(a) shows the intensity waveform a observed with an optical sampling oscilloscope. 2 (t) and (b) are obtained by taking the square root and converting them to amplitude a(t), and (c) is the optical spectrum A(ω) observed with an optical spectrum analyzer expressed in dB (20 log|A(ω)|). The solid line is the experimental result, and the dashed line is the computer analysis result. The two are in good agreement, indicating that the Bright exponential function pulses can be generated as designed using the optical filter in Figure 34.

[0112] Figure 36 shows the shape of the optical filter used to generate a dark triangle pulse defined by the electric field amplitude in the fourth embodiment of the present invention. Here, T = 12.5 ps, M PM = 2. Figure (a) shows the transmittance characteristics, and (b) shows the phase characteristics. The gray line represents F in equation (32). Fpd (ω) (see Figure 9(b)) is a filter with a bandwidth limited to 320 GHz, and the black line shows the filter shape when this is approximated by a step function every 20 GHz and implemented on an LCoS element.

[0113] Figure 37 shows the waveform and optical spectrum of a dark triangle pulse defined by the electric field amplitude, generated using this filter. Figure 37(a) shows the intensity waveform a observed with an optical sampling oscilloscope. 2 (t) and (b) are obtained by taking the square root and converting them to amplitude a(t), and (c) is the optical spectrum A(ω) observed with an optical spectrum analyzer expressed in dB (20 log|A(ω)|). The solid line is the experimental result, and the dashed line is the computer analysis result. The two agree well, indicating that the optical filter in Figure 36 can generate dark triangle pulses defined by the electric field amplitude as designed.

[0114] Next, Figure 38 shows the shape of the optical filter used to generate a Bright triangular pulse defined by the electric field amplitude in the fourth embodiment of the present invention. Here, T = 12.5 ps, M PM = 2.4. Figure (a) shows the transmittance characteristics, and (b) shows the phase characteristics. The gray line represents F in equation (31). Fpb The black line shows the filter shape when (ω) (see Figure 9(a)) is band-limited at 320 GHz and approximated by a step function every 20 GHz, then implemented on an LCoS element.

[0115] Figure 39 shows the waveform and optical spectrum of a Bright triangular pulse defined by the electric field amplitude, generated using this filter. Figure 39(a) shows the intensity waveform a observed with an optical sampling oscilloscope. 2 (t) and (b) are obtained by taking the square root and converting them to amplitude a(t), and (c) is the optical spectrum A(ω) observed with an optical spectrum analyzer expressed in dB (20 log|A(ω)|). The solid line is the experimental result, and the dashed line is the computer analysis result. The two agree well, indicating that the optical filter in Figure 38 can generate a Bright triangular pulse defined by the electric field amplitude as designed.

[0116] Figure 40 shows the shape of the optical filter used to generate a dark triangle pulse defined by intensity (square of the electric field) in the fifth embodiment of the present invention. Here, T = 10 ps, ​​M PM = 1. Figure (a) shows the transmittance characteristics, and (b) shows the phase characteristics. The gray line represents F Fpd The black line shows the filter shape when (ω) (see Figure 11(b)) is band-limited at 600 GHz and approximated by a step function every 20 GHz, then implemented on an LCoS element.

[0117] Figure 41 shows the waveform and optical spectrum of a dark triangle pulse, defined by its intensity (square of the electric field), generated using this filter. Figure 41(a) shows the intensity waveform a observed with an optical sampling oscilloscope. 2(t) and (b) are obtained by taking the square root and converting them to amplitude a(t), and (c) is the optical spectrum A(ω) observed with an optical spectrum analyzer expressed in dB (20 log|A(ω)|). The solid line is the experimental result, and the dashed line is the computer analysis result. The two agree well, indicating that the dark triangle pulse defined by intensity as designed can be generated using the optical filter in Figure 40.

[0118] Next, Figure 42 shows the shape of the optical filter used to generate a bright triangular pulse defined by intensity (square of the electric field) in the fifth embodiment of the present invention. Here, T = 10 ps, ​​M PM = 2. Figure (a) shows the transmittance characteristics, and (b) shows the phase characteristics. The gray line represents F Fpb The black line shows the filter shape when (ω) (see Figure 11(a)) is band-limited at 540 GHz and approximated by a step function every 20 GHz, then implemented on an LCoS element.

[0119] Figure 43 shows the waveform and optical spectrum of the intensity (field squared) Bright triangular pulse generated using this filter. Figure 43(a) shows the intensity waveform a observed with an optical sampling oscilloscope. 2 (t) and (b) are obtained by taking the square root and converting them to amplitude a(t), and (c) is the optical spectrum A(ω) observed with an optical spectrum analyzer expressed in dB (20 log|A(ω)|). The solid line is the experimental result, and the dashed line is the computer analysis result. The two are in good agreement, indicating that the bright triangular pulse defined by intensity as designed can be generated using the optical filter in Figure 42.

[0120] Figure 44 shows the shape of the optical filter used to generate a dark parabolic pulse defined by the electric field amplitude in the sixth embodiment of the present invention. Here, T = 12.5 ps, M PM = 1. Figure (a) shows the transmittance characteristics, and (b) shows the phase characteristics. The gray line represents F in equation (42). FpdThe black line shows the filter shape when (ω) (see Figure 13(b)) is band-limited at 600 GHz and approximated by a step function every 20 GHz, then implemented on an LCoS element.

[0121] Figure 45 shows the waveform and optical spectrum of a dark parabolic pulse, defined by the electric field amplitude, generated using this filter. Figure 45(a) shows the intensity waveform a observed with an optical sampling oscilloscope. 2 (t) and (b) are obtained by taking the square root and converting them to amplitude a(t), and (c) is the optical spectrum A(ω) observed with an optical spectrum analyzer expressed in dB (20 log|A(ω)|). The solid line is the experimental result, and the dashed line is the computer analysis result. The two agree well, indicating that the optical filter in Figure 44 can generate dark parabolic pulses defined by the electric field amplitude as designed.

[0122] Next, Figure 46 shows the shape of the optical filter used to generate a bright parabolic pulse defined by the electric field amplitude in the sixth embodiment of the present invention. Here, T = 12.5 ps, M PM = 2. Figure (a) shows the transmittance characteristics, and (b) shows the phase characteristics. The gray line represents F in equation (41). Fpb The black line shows the filter shape when (ω) (see Figure 13(a)) is band-limited at 600 GHz and approximated by a step function every 20 GHz, then implemented on an LCoS element.

[0123] Figure 47 shows the waveform and optical spectrum of a bright parabolic pulse, defined by the electric field amplitude, generated using this filter. Figure 47(a) shows the intensity waveform a observed with an optical sampling oscilloscope. 2 (t) and (b) are obtained by taking the square root and converting them to amplitude a(t), and (c) is the optical spectrum A(ω) observed with an optical spectrum analyzer expressed in dB (20 log|A(ω)|). The solid line is the experimental result, and the dashed line is the computer analysis result. The two are in good agreement, and it can be seen that a bright parabolic pulse defined by the electric field amplitude as designed can be generated using the optical filter in Figure 46.

[0124] Figure 48 shows the shape of the optical filter used to generate dark parabolic pulses defined by intensity (square of the electric field) in the seventh embodiment of the present invention. Here, T = 12.5 ps, M PM = 2. Figure (a) shows the transmittance characteristics, and (b) shows the phase characteristics. The gray line represents F Fpd The black line shows the filter shape when (ω) (see Figure 15(b)) is band-limited at 300 GHz and approximated by a step function every 20 GHz, then implemented on an LCoS element.

[0125] Figure 49 shows the waveform and optical spectrum of a dark parabolic pulse, defined by its intensity (square of the electric field), generated using this filter. Figure 49(a) shows the intensity waveform a observed with an optical sampling oscilloscope. 2 (t) and (b) are obtained by taking the square root and converting them to amplitude a(t), and (c) is the optical spectrum A(ω) observed with an optical spectrum analyzer expressed in dB (20 log|A(ω)|). The solid line is the experimental result, and the dashed line is the computer analysis result. The two are in good agreement, indicating that the optical filter in Figure 48 can generate dark parabolic pulses defined by intensity as designed.

[0126] Next, Figure 50 shows the shape of the optical filter used to generate a bright parabolic pulse defined by its intensity (square of the electric field) in the seventh embodiment of the present invention. Here, T = 10 ps, ​​M PM = 2. Figure (a) shows the transmittance characteristics, and (b) shows the phase characteristics. The gray line represents F Fpb The black line shows the filter shape when (ω) (see Figure 15(a)) is band-limited at 600 GHz and approximated by a step function every 20 GHz, then implemented on an LCoS element.

[0127] Figure 51 shows the waveform and optical spectrum of the intensity (field squared) bright parabolic pulse generated using this filter. Figure 51(a) shows the intensity waveform a observed with an optical sampling oscilloscope. 2(t) and (b) are obtained by taking the square root and converting them to amplitude a(t), and (c) is the optical spectrum A(ω) observed with an optical spectrum analyzer expressed in dB (20 log|A(ω)|). The solid line is the experimental result, and the dashed line is the computer analysis result. The two are in good agreement, indicating that a bright parabolic pulse defined by intensity as designed can be generated using the optical filter in Figure 50.

[0128] Figure 52 shows the shape of the optical filter used to generate dark rectangular pulses in the eighth embodiment of the present invention. Here, T = 12.5 ps, M PM = 1. Figure (a) shows the transmittance characteristics, and (b) shows the phase characteristics. The gray line represents F in equation (52). Fpd The (ω) filter (see Figure 17(b)) is band-limited at 640 GHz, and the dashed line shows the filter shape when this is approximated by a step function every 20 GHz and implemented on an LCoS element.

[0129] Figure 53 shows the waveform and optical spectrum of the dark rectangular pulse generated using this filter. Figure 53(a) shows the intensity waveform a observed with an optical sampling oscilloscope. 2 (t) and (b) are obtained by taking the square root and converting them to amplitude a(t), and (c) is the optical spectrum A(ω) observed with an optical spectrum analyzer expressed in dB (20 log|A(ω)|). The solid line is the experimental result, and the dashed line is the computer analysis result. The two agree well, indicating that the dark rectangular pulse can be generated as designed using the optical filter in Figure 52.

[0130] Next, Figure 54 shows the shape of the optical filter used to generate a bright rectangular pulse in the eighth embodiment of the present invention. Here, T = 12.5 ps, M PM = 2. Figure (a) shows the transmittance characteristics, and (b) shows the phase characteristics. The gray line represents F in equation (51). Fpb The (ω) filter (see Figure 17(a)) is band-limited at 560 GHz, and the dashed line shows the filter shape when this is approximated by a step function every 20 GHz and implemented on an LCoS element.

[0131] Figure 55 shows the waveform and optical spectrum of a bright rectangular pulse generated using this filter. Figure 55(a) shows the intensity waveform a observed with an optical sampling oscilloscope. 2 (t) and (b) are obtained by taking the square root and converting them to amplitude a(t), and (c) is the optical spectrum A(ω) observed with an optical spectrum analyzer expressed in dB (20 log|A(ω)|). The solid line is the experimental result, and the dashed line is the computer analysis result. The two are in good agreement, indicating that the bright rectangular pulse can be generated as designed using the optical filter in Figure 54.

[0132] Figure 56 shows the shape of the optical filter used to generate the dark Nyquist pulse in the ninth embodiment of the present invention. Here, the roll-off ratio is set to α = 0.5, T = 6.25 ps, M PM = 1. Figure (a) shows the transmittance characteristics, and (b) shows the phase characteristics. The gray line represents F in equation (56). Fpd The (ω) filter (see Figure 19(b)) is band-limited at 180 GHz, and the dashed line shows the filter shape when this is approximated by a step function every 20 GHz and implemented on an LCoS element.

[0133] Figure 57 shows the waveform and optical spectrum of the dark Nyquist pulse generated using this filter. Figure 57(a) shows the intensity waveform a observed with an optical sampling oscilloscope. 2 (t) and (b) are obtained by taking the square root and converting them to amplitude a(t), and (c) is the optical spectrum A(ω) observed with an optical spectrum analyzer expressed in dB (20 log|A(ω)|). The solid line is the experimental result, and the dashed line is the computer analysis result. The two are in good agreement, indicating that the dark Nyquist pulse can be generated as designed using the optical filter in Figure 56.

[0134] Next, Figure 58 shows the shape of the optical filter used to generate the bright Nyquist pulse in the ninth embodiment of the present invention. Here, the roll-off ratio is set to α = 0.5, T = 6.25 ps, M PM= 1. Figure (a) shows the amplitude characteristics, and (b) shows the phase characteristics. The gray line represents F in equation (55). Fpb The (ω) filter (see Figure 19(a)) is band-limited at 180 GHz, and the dashed line shows the filter shape when this is approximated by a step function every 20 GHz and implemented on an LCoS element.

[0135] Figure 59 shows the waveform and optical spectrum of the Bright Nyquist pulse generated using this filter. Figure 59(a) shows the intensity waveform a observed with an optical sampling oscilloscope. 2 (t) and (b) are obtained by taking the square root and converting them to amplitude a(t), and (c) is the optical spectrum A(ω) observed with an optical spectrum analyzer expressed in dB (20 log|A(ω)|). The solid line is the experimental result, and the dashed line is the computer analysis result. The two are in good agreement, indicating that the bright Nyquist pulse can be generated as designed using the optical filter in Figure 58.

[0136] Figure 60 shows the shape of the optical filter used to generate dark solitons in the tenth embodiment of the present invention. Here, T = 1.25 ps, M PM = 2.4. Figure (a) shows the transmittance characteristics, and (b) shows the phase characteristics. The gray line represents F in equation (59). tanh The (ω) filter (see Figure 21) is band-limited at 540 GHz, and the dashed line shows the filter shape when this is approximated by a step function every 20 GHz and implemented on an LCoS element.

[0137] Figure 61 shows the waveform and optical spectrum of the dark rectangular pulse generated using this filter. Figure 61(a) shows the intensity waveform a observed with an optical sampling oscilloscope. 2 (t) and (b) are obtained by taking the square root and converting them to amplitude a(t), and (c) is the optical spectrum A(ω) observed with an optical spectrum analyzer expressed in dB (20 log|A(ω)|). The solid line is the experimental result, and the dashed line is the computer analysis result. The two agree well, indicating that dark solitons can be generated as designed using the optical filter in Figure 60. [Industrial applicability]

[0138] As described in detail above, the present invention makes it possible to easily generate positive or negative bright or dark pulses with any time waveform from a mode-locked laser by appropriately designing the amplitude and phase characteristics of the optical filter inserted into the laser resonator. The positive or negative bright or dark pulse trains obtained by the present invention can be used in a wide range of applications, equipment, and systems such as ultra-high-speed optical communication, optical measurement, and optical signal processing.

[0139] Specific applications of the positive or negative bright or dark pulses of this invention include the possibility of modifying communication signals, as pulse signals can be canceled out by superimposing bright and dark pulses. Furthermore, because pulse signals can be canceled out by using bright and dark pulses, there is potential for application in complex and robust encryption technologies not found in conventional encryption techniques. In addition, the use of bright and dark pulses may open up new avenues for encoding not possible in conventional communications. [Explanation of symbols]

[0140] 1 Optical fiber 2. Optical Amplifier 3. Optical Phase Modulator 4. Light Filters 5. Polarization-maintaining erbium fiber 6 Semiconductor lasers 7. Wavelength Division Multiplexing (WDM) Coupler 8 Phase modulator 9-wavelength filter (etalon) 10 PZT elements 11 Couplers 12 Isolators 13 LCoS elements 14 Amplifier 15 Phase Shifter 16-Frequency Synthesizer

Claims

1. The laser has a frequency-modulated mode-locked laser equipped with an optical phase modulator and an optical filter within the laser resonator. By setting the amplitude and phase characteristics of the optical filter according to the desired shape of the optical pulse and the amount of continuous wave offset, the system is configured to generate dark pulses and bright pulses having the desired shape of the optical pulse with the amount of continuous wave offset, with variable amplitude and phase characteristics. The optical phase modulator is driven by a sine wave with a repetition angular frequency Ωm, The amplitude and phase characteristics of the optical filter are given by the modulation degree of the optical phase modulator, the spectrum A(ω) of the pulse to be output, the sinc function S(ω) = sin(ωτ / 2) / ω (τ = 2π / Ω m) that gives the continuous wave offset amount, and functions A(ω-nΩ m), A(ω+nΩ m), S(ω-nΩ m), S(ω+nΩ m), (n: integer) obtained by shifting A(ω) and S(ω) positively or negatively by an integer multiple of the repetition angular frequency Ω m. A distinctive feature is the pulsed light source.

2. The pulse light source according to claim 1, characterized in that the optical filter is configured to generate four types of pulses, a positive bright pulse, a positive dark pulse, a negative bright pulse, and a negative dark pulse, depending on the combination of the signs of the spectrum A(ω) and the sinc function S(ω) that gives the continuous wave offset amount, by taking the signs of the latter to be positive or negative.

3. The pulse light source according to claim 2, characterized in that the optical filter is configured to generate positive or negative bright or dark Gaussian pulses by making the spectrum A(ω) a Gaussian function.

4. The pulse light source according to claim 2, characterized in that the optical filter is configured to generate positive or negative bright or dark sech pulses by making the spectrum A(ω) a sech function.

5. The pulse light source according to claim 2, characterized in that the optical filter is configured to generate positive or negative bright or dark pulses having the shape of the two exponential functions by taking the Fourier transform of the spectrum A(ω) as the two exponential functions.

6. The pulse light source according to claim 2, characterized in that the optical filter is configured to generate positive or negative bright or dark pulses having a triangular shape in electric field amplitude by making the spectrum A(ω) the square of the sinc function.

7. The optical filter uses the first kind of Bessel function J to convert the spectrum A(ω) 2k The pulse light source according to claim 2, characterized in that it is configured to generate positive or negative bright or dark pulses with a triangular shape, where the intensity as the square of the electric field is a series of (ω) (k: integer).

8. The optical filter uses the function sin ω / ω to process the spectrum A(ω). 3 to cos ω / ω 2 The pulse light source according to claim 2, characterized in that it is configured to generate positive or negative bright or dark pulses having a parabolic shape in electric field amplitude by taking the sum or difference of the two.

9. The optical filter uses the function J to process the spectrum A(ω). 1 (ω) / ω (J 1 The pulse light source according to claim 2, characterized in that by using (ω): a first-order Bessel function of the first kind, it is configured to generate positive or negative bright or dark pulses in which the intensity as the square of the electric field has a parabolic shape.

10. The pulse light source according to claim 2, characterized in that the optical filter is configured to generate positive or negative bright or dark pulses having a rectangular shape by making the spectrum A(ω) a sin(ωT) / (ωT) given by the pulse width T.

11. The pulse light source according to claim 2, characterized in that the optical filter is configured to generate positive or negative bright or dark pulses having the shape of a Nyquist pulse by using the spectrum A(ω) as the transfer function of the Nyquist filter.

12. A frequency-modulated mode-locked laser having an optical phase modulator and an optical filter in a laser resonator, By setting the amplitude and phase characteristics of the optical filter according to the desired shape of the optical pulse and the amount of continuous wave offset, the amplitude and phase characteristics are variable, and the system is configured to generate a dark soliton having the desired shape of the optical pulse with the amount of continuous wave offset. The optical phase modulator is driven by a sine wave with a repetition angular frequency Ωm, The aforementioned optical filter takes the spectrum A(ω) of the pulse to be output as cos ω, cos ω / ω, sin ω, The amplitude and phase characteristics of the optical filter are determined by the modulation degree of the optical phase modulator, the spectrum A(ω), and the repetition angular frequency Ω of the spectrum A(ω). m The function A(ω-nΩ) is shifted to positive or negative by an integer multiple of the given value. m ), A(ω+nΩ m The fact that it is given using (n: integer) A distinctive feature is the pulsed light source.

13. A system using a pulsed light source according to any one of claims 1 to 12.

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