Optical Function Generator

Abstract

To provide an optical function generator that can generate directly from an FM-mode-locked laser high output, high quality and chirp-free Fourier-limited optical pulses having arbitrary time waveforms.SOLUTION: A FM mode-locked laser includes an optical phase modulator, an optical amplifier, and an optical filter in a laser cavity. The optical filter has variable amplitude and phase characteristics and is configured to generate an optical pulse having an arbitrary shape by setting amplitude and phase characteristics in accordance with the shape of the optical pulse to be output.SELECTED DRAWING: Figure 1

Description

[Technical Field]

[0001] The present invention relates to an optical function generator. [Background technology]

[0002] Ultrashort optical pulses are widely used in applications such as ultrafast optical communications, optical measurement, and optical signal processing. Mode-locked lasers are widely used as light sources for such ultrashort optical pulses. There are two types of mode-locking: passive mode-locking and forced (active) mode-locking. Passive mode-locked lasers can easily generate ultrashort pulses in the femtosecond range by inserting a saturable absorber into the resonator or by inducing nonlinear polarization rotation. However, their repetition rates are generally slow, ranging from tens to hundreds of MHz. On the other hand, actively (forced) mode-locked lasers can generate pulses synchronized with an externally supplied modulation signal by inserting an optical modulator into the resonator. Although the pulse width is on the order of picoseconds to subpicoseconds, the repetition rate can be increased to tens of GHz.

[0003] In forced mode-locking, when an amplitude modulator is used, it is called AM (Amplitude Modulation) mode-locking, and when a phase modulator is used, it is called FM (Frequency Modulation) mode-locking. FM mode-locking generates pulses by generating a linear chirp using a phase modulator and removing the high-frequency components of the spectrum corresponding to the pulse's base using an optical filter. Here, chirp refers to changing the frequency modulation linearly with time. In AM mode-locking, the modulation index achievable with an intensity modulator is limited to 1, while in FM mode-locking, modulation indexes exceeding 1 can easily be achieved with a phase modulator, making FM mode-locking particularly useful for generating short pulses.

[0004] It is known that the pulse shape generated from an FM mode-locked laser is generally a Gaussian pulse (see, for example, Non-Patent Documents 1 and 2). Furthermore, by using a nonlinear optical effect called the soliton effect within the resonator, it is possible to generate a sech pulse (see, for example, Non-Patent Document 3). The pulse output from an FM mode-locked laser generally has a chirped frequency. Therefore, to obtain a chirp-free Fourier-limited pulse, it is necessary to insert a dispersion compensation element outside the laser to compensate for the frequency chirp.

[0005] On the other hand, to generate pulses other than Gaussian and Sech, a technique for shaping the waveform outside the laser is used. Let u(t) be the waveform of the pulse directly output from the laser, and a(t) be the waveform to be generated. If the frequency spectra of u(t) and a(t) are U(ω) and A(ω), respectively, then by inserting an optical filter with a transfer function given by F(ω) = A(ω) / U(ω) outside the laser, an optical pulse with a waveform given by a(t) can be obtained at the filter output. In this way, an optical filter with a freely designable transfer function can be easily realized using an LCoS (Liquid Crystal on Silicon) element (see, for example, Non-Patent Document 4).

[0006] Regarding AM mode locking, the inventors have investigated the change in the spectral shape A(ω) of the optical pulse a(t) due to amplitude modulation (repeated positive and negative changes from A(ω) to the angular frequency Ω m A(ω+Ω) shifted by 1 minute m ) and A(ω-Ω mThey proposed that optical pulses with desired waveforms can be generated by designing the transfer function of an optical filter taking into account the generation of a sinusoidal wave. However, they pointed out that it is difficult to generate pulses with constant amplitude, such as square waves, using an AM mode-locked laser because the intensity modulator used in AM mode-locking modulates the amplitude of the pulse. Similarly, they pointed out that pulses whose intensity (the square of the electric field) is triangular or parabolic also contain components whose amplitude rises vertically, and that such abrupt amplitude change is difficult to achieve with a sinusoidal wave intensity modulator, making it difficult to achieve with an AM mode-locked laser (see, for example, Non-Patent Document 5). [Prior art documents] [Non-patent literature]

[0007] [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. 1613-1615 [Non-patent document 4] G. Baxter, S. Frisken, D. Abakoumov, H. Zhou, I. Clarke, A. Bartos, and S. Poole, “Highly programmable wavelength selective switch based on liquid crystal on silicon elements switching”, in OFC 2006, OTuF2 [Non-patent document 5] M. Nakazawa and T. Hirooka, “Theory of AM Mode-Locking of a Laser as an Arbitrary Optical Function Generator”, IEEE J. Quantum Electron., December 2021, vol. 57, no. 6, 1300320 Summary of the Invention [Problem to be solved by the invention]

[0008] However, the waveform shaping method described in Non-Patent Document 4 reduces much of the power of the original optical signal due to the shaping of the spectral shape, in addition to the insertion loss of the optical filter itself. Therefore, it is necessary to compensate for these optical losses using an optical amplifier after shaping, which results in the problem of spontaneous emission noise being superimposed on the pulse and degrading the optical signal-to-noise ratio (OSNR). In other words, conventional waveform shaping methods have the problem of making it difficult to generate high-quality optical pulses with high output and high OSNR.

[0009] Furthermore, even in the Gaussian pulses and sech pulses that can be directly generated by conventional FM mode-locked lasers as described in Non-Patent Documents 1 to 3, the frequency generally chirps immediately after laser output. Therefore, in order to obtain Fourier-limited pulses, there is a problem in that a dispersion compensation optical element must be inserted outside the laser to compensate for the frequency chirp.

[0010] The present invention is intended to solve these problems, and aims to provide an optical function generator that can generate high-power, high-quality, chirp-free Fourier-limited optical pulses with arbitrary temporal waveforms directly from an FM mode-locked laser. [Means for solving the problem]

[0011] In order to achieve this object, the optical function generator of the present invention has an FM mode-locked laser equipped with an optical phase modulator (frequency modulator), an optical amplifier, and an optical filter within a laser resonator, and the optical filter has variable amplitude and phase characteristics, and is characterized in that it generates optical pulses of any shape by setting the amplitude and phase characteristics according to the shape of the optical pulse to be output.

[0012] In the optical function generator according to the present invention, the optical phase modulator has a repetition rate of Ω m and the optical filter is provided so as to be drivable by a sine wave of Ω. The amplitude and phase characteristics of the optical filter are determined by the modulation degree of the optical phase modulator and the spectrum A(ω) of the optical pulse to be output, and by converting A(ω) into Ω. m A function A(ω-nΩ) shifted positively or negatively by an integer multiple of m ), A(ω+nΩ m ) (n: integer). Furthermore, the optical filter may be configured to provide a band limit to the amplitude and phase characteristics in accordance with the spectral width of the optical pulse to be output, thereby obtaining a TL (Transform-Limited) pulse.

[0013] In the optical function generator according to the present invention, the optical filter calculates the amplitude and phase characteristics by a Gaussian function A(ω) and a frequency band that repeats A(ω) at an angular frequency Ω. m A function A(ω-nΩ) shifted positively or negatively by an integer multiple of m ), A(ω+nΩ mThe optical filter may be configured to generate a Gaussian pulse by applying a frequency Ω(ω) to the amplitude and phase characteristics of the optical filter. m A function A(ω-nΩ) shifted positively or negatively by an integer multiple of m ), A(ω+nΩ m The optical filter may be configured to generate a sech pulse by applying a Lorentz function A(ω) and a repeating function A(ω) at an angular frequency Ω. m A function A(ω-nΩ) shifted positively or negatively by an integer multiple of m ), A(ω+nΩ m ) to generate a pulse having a bi-exponential shape.

[0014] The optical filter also calculates the amplitude and phase characteristics by a squared sinc function A(ω) and a function A(ω) that is repeated at an angular frequency Ω m A function A(ω-nΩ) shifted positively or negatively by an integer multiple of m ), A(ω+nΩ m ) to generate a pulse having a triangular electric field amplitude. 2k A(ω) is given by the series of (ω) (k: integer), and A(ω) is repeated at angular frequency Ω m A function A(ω-nΩ) shifted positively or negatively by an integer multiple of m ), A(ω+nΩ m ) to generate a pulse whose intensity (square of the electric field) has a triangular shape. 3 and cos ω / ω 2 A(ω) is given by the sum of A(ω) and A(ω) is repeated at angular frequency Ω m A function A(ω-nΩ) shifted positively or negatively by an integer multiple of m ), A(ω+nΩ m ) to generate a pulse whose electric field amplitude has a parabolic shape.

[0015] The optical filter also calculates the amplitude and phase characteristics by repeating A(ω) given by the function J1(ω) / ω (J1(ω): first-order Bessel function of the first kind) and A(ω) at an angular frequency Ω m A function A(ω-nΩ) shifted positively or negatively by an integer multiple of m ), A(ω+nΩ m ) to generate a pulse whose intensity (square of the electric field) has a parabolic shape. Also, the optical filter may be configured to calculate the amplitude and phase characteristics by applying a sinc function A(ω) and a sinc function A(ω) to the optical filter by repeating A(ω) at an angular frequency Ω m A function A(ω-nΩ) shifted positively or negatively by an integer multiple of m ), A(ω+nΩ m ) to generate a pulse having a rectangular shape. [Effects of the Invention]

[0016] The optical function generator according to the present invention can generate a pulse train with any temporal waveform and a high OSNR, depending on the amplitude and phase characteristics of the optical filter inserted in the resonator of the mode-locked laser. Therefore, according to the present invention, it is possible to provide an optical function generator that can generate high-power, high-quality, chirp-free Fourier-limited optical pulses with any temporal waveform directly from an FM mode-locked laser. [Brief explanation of the drawings]

[0017] [Figure 1] 1 is a block diagram showing the configuration of an optical function generator according to an embodiment of the present invention. [Figure 2] 1 is a block diagram for explaining the operating principle of an optical function generator according to an embodiment of the present invention. [Figure 3](a) Absolute value and (b) phase of an optical filter for generating Gaussian pulses in the optical function generator of the first embodiment of the present invention, and (c) absolute value and (d) phase of a filter that imparts a band limit of 200 GHz (±100 GHz). [Figure 4] (a) Absolute value, (b) phase, and (c) spectrum of a stationary pulse (Gaussian pulse) obtained by computer analysis using the filters shown in Figures 3(c) and (d). [Figure 5] (a) Absolute value and (b) phase of the optical filter for generating SECH pulses in the optical function generator of the second embodiment of the present invention, and (c) absolute value and (d) phase of the filter that provides a band limit of 200 GHz (±100 GHz). [Figure 6] Using the filters shown in Figures 5(c) and (d), (a) the absolute value, (b) the phase, and (c) the spectrum of a stationary pulse (sech pulse) obtained by computer analysis. [Figure 7] (a) Absolute value and (b) phase of the optical filter for generating biexponential pulses in the optical function generator according to the third embodiment of the present invention, and the filter that provides a band limit of 440 GHz (±220 GHz) [Figure 8] (a) Absolute value, (b) phase, and (c) spectrum of a stationary pulse (biexponential pulse) obtained by computer analysis using the filters shown in Figures 7(c) and (d). [Figure 9] (a) Absolute value and (b) phase of an optical filter for generating a triangular pulse defined by the electric field amplitude in an optical function generator according to the fourth embodiment of the present invention, and (c) absolute value and (d) phase of a filter that imparts a band limit of 240 GHz (±120 GHz). [Figure 10] Using the filters shown in Figures 9(c) and (d), (a) the absolute value, (b) the phase, and (c) its spectrum of a steady pulse (a triangular pulse defined by the electric field amplitude) obtained by computer analysis. [Figure 11](a) Absolute value and (b) phase of an optical filter for generating a triangular pulse defined by electric field strength in an optical function generator according to the fifth embodiment of the present invention, and (c) absolute value and (d) phase of a filter with a band limit of 640 GHz (±320 GHz). [Figure 12] Using the filters shown in Figures 11(c) and (d), (a) the absolute value, (b) the phase, and (c) its spectrum of a steady pulse (a triangular pulse defined by the electric field strength) obtained by computer analysis. [Figure 13] (a) Absolute value and (b) phase of an optical filter for generating a parabolic pulse defined by the electric field amplitude in the optical function generator of the sixth embodiment of the present invention, and (c) absolute value and (d) phase of a filter that imparts a band limit of 640 GHz (±320 GHz). [Figure 14] Using the filters shown in Figures 13(c) and (d), (a) the absolute value, (b) the phase, and (c) its spectrum of a steady pulse (parabolic pulse defined by the electric field amplitude) obtained by computer analysis. [Figure 15] (a) Absolute value and (b) phase of an optical filter for generating a parabolic pulse defined by electric field strength in an optical function generator according to the seventh embodiment of the present invention, and (c) absolute value and (d) phase of a filter that imparts a band limit of 640 GHz (±320 GHz). [Figure 16] Using the filters shown in Figures 15(c) and (d), (a) the absolute value, (b) the phase, and (c) its spectrum of a steady pulse (parabolic pulse defined by electric field strength) obtained by computer analysis. [Figure 17] (a) Absolute value and (b) phase of an optical filter for generating rectangular pulses in an optical function generator according to the eighth embodiment of the present invention, and (c) absolute value and (d) phase of a filter that provides a band limit of 640 GHz (±320 GHz). [Figure 18](a) Absolute value, (b) phase, and (c) spectrum of a stationary pulse (rectangular pulse) obtained by computer analysis using the filters shown in Figures 17(c) and (d). [Figure 19] FIG. 1 is a block diagram showing the configuration of a 1.56 μm wavelength band harmonic FM mode-locked erbium fiber laser used in an experiment of an optical function generator according to an embodiment of the present invention. [Figure 20] 1A shows the amplitude characteristics and FIG. 1B shows the phase characteristics of an optical filter implemented by an LCoS element in the optical function generator according to the first embodiment of the present invention. [Figure 21] (a) Intensity waveform, (b) waveform converted to amplitude by taking the square root of (a), and (c) optical spectrum of a Gaussian pulse obtained using the optical filter shown in Figure 20. [Figure 22] 10A shows the amplitude characteristics and FIG. 10B shows the phase characteristics of an optical filter implemented by an LCoS element in the optical function generator according to the second embodiment of the present invention. [Figure 23] (a) Intensity waveform of the SECH pulse obtained using the optical filter shown in Figure 22, (b) waveform converted to amplitude by taking the square root of (a), and (c) optical spectrum. [Figure 24] 10A shows the amplitude characteristics and FIG. 10B shows the phase characteristics of an optical filter implemented with an LCoS element in the optical function generator according to the third embodiment of the present invention. [Figure 25] (a) Intensity waveform, (b) waveform converted to amplitude by taking the square root of (a), and (c) optical spectrum of a biexponential pulse obtained using the optical filter shown in Figure 24. [Figure 26] 10A shows the amplitude characteristics and FIG. 10B shows the phase characteristics of an optical filter implemented by an LCoS element in an optical function generator according to a fourth embodiment of the present invention. [Figure 27] (a) Intensity waveform of a triangular pulse defined by electric field amplitude, (b) waveform converted to amplitude by taking the square root of (a), and (c) optical spectrum, obtained using the optical filter shown in Figure 26. [Figure 28]10A shows the amplitude characteristics and FIG. 10B shows the phase characteristics of an optical filter implemented by an LCoS element in an optical function generator according to a fifth embodiment of the present invention. [Figure 29] (a) Intensity waveform of a triangular pulse defined by electric field intensity, (b) waveform converted to amplitude by taking the square root of (a), and (c) optical spectrum, obtained using the optical filter shown in Figure 28. [Figure 30] 13A shows the amplitude characteristics and (b) the phase characteristics of an optical filter implemented by an LCoS element in the optical function generator according to the sixth embodiment of the present invention. [Figure 31] (a) Intensity waveform of a parabolic pulse defined by electric field amplitude, (b) waveform converted to amplitude by taking the square root of (a), and (c) optical spectrum, obtained using the optical filter shown in Figure 30. [Figure 32] 13A shows the amplitude characteristics and (b) the phase characteristics of an optical filter implemented by an LCoS element in the optical function generator according to the seventh embodiment of the present invention. [Figure 33] (a) Intensity waveform of a parabolic pulse defined by electric field intensity, (b) waveform converted to amplitude by taking the square root of (a), and (c) optical spectrum, obtained using the optical filter shown in Figure 32. [Figure 34] 13A shows the amplitude characteristics and (b) the phase characteristics of an optical filter implemented by an LCoS element in the optical function generator according to the eighth embodiment of the present invention. [Figure 35] (a) Intensity waveform, (b) waveform converted to amplitude by taking the square root of (a), and (c) optical spectrum of a rectangular pulse obtained using the optical filter shown in Figure 34. DETAILED DESCRIPTION OF THE INVENTION

[0018] Hereinafter, an embodiment of the present invention will be described with reference to the drawings. 1 to 35 show an optical function generator according to an embodiment of the present invention. The configuration of an optical function generator according to an embodiment of the present invention is shown in Figure 1. An optical fiber 1 constitutes a laser ring resonator, and the resonator comprises an optical amplifier 2 used as a gain medium, an optical phase modulator 3 used as a mode locker, and an optical filter 4 whose amplitude and phase characteristics can be set arbitrarily. The optical amplifier 2 may be an erbium-doped fiber amplifier (EDFA), a semiconductor optical amplifier (SOA), a solid-state laser element, or the like. The optical phase modulator 3 may be a Mach-Zehnder modulator using an LN (LiNbO3) crystal, or the like, and receives an external frequency f m (Angular frequency Ω m = 2πf m ), modulation depth M PM Sine wave M PM cos(Ω m The optical filter 4 is a programmable optical filter made of LCoS or the like. Assuming the generation of asymmetric pulse waveforms, as will be described later, it is desirable that the optical filter 4 be able to control not only the amplitude (transmission) characteristics but also the phase characteristics.

[0019] The operating characteristics of this optical function generator (FM mode-locked laser) and the design method of the optical filter 4 will be explained using the block diagram in Figure 2. In Figure 1, the loss of the resonator is L, the gain is G, and the transfer function of the optical filter 4 is F. F (ω). If the electric field amplitude of the light circulating in the resonator is a(t) and its spectrum is A(ω), then in the steady state, A(ω) satisfies the following equation (1):

[0020]

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[0021] As a result, the master equation (2) is expressed as equation (4). Let GL = 1 and change equation (4) to F F (ω), we obtain equation (5). Therefore, if we want to generate a desired pulse waveform a(t) using an FM mode-locked laser, we can use its spectrum A(ω) to calculate the transfer function F of the optical filter 4. F (ω) can be designed as shown in equation (5).

[0022]

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[0023] [First embodiment] In the first embodiment of the present invention, a chirp-free Fourier-limited Gaussian pulse can be generated. The waveform a(t) of the Gaussian pulse is given by equation (6). From the Fourier transform of equation (6), the spectrum A(ω) of the Gaussian pulse is given by equation (7). The shape of the optical filter 4 for generating this Gaussian pulse is given by equation (8) by substituting equation (7) into equation (5). By inserting this optical filter 4 into a resonator, a Gaussian pulse can be generated.

[0024]

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[0025] The transfer function F of the optical filter 4 used in this embodiment F An example of the shape of (ω) is shown in Figure 3. Here, the pulse width T = 6 ps (a 2 (t) full width at half maximum W p = 10 ps), modulation angular frequency Ω m = 2π×10 GHz, modulation index M PM= 1. The black dots in the figure represent the frequencies where longitudinal modes (10 GHz intervals) exist in the optical spectrum. F (ω) is given by a complex function, and its absolute value |F F (ω)|, and Fig. 3(b) shows the phase arg F F (ω). F F Although the base of (ω) gradually decays, band-limiting may be applied to the extent that the waveform is not distorted. In Figures 3(a) and (b), the shape of a filter with a band-limit of 200 GHz (±100 GHz) is shown in Figures 3(c) and (d). Figure 4 shows the results of computer analysis of a laser steady-state solution using this filter. Figure 4(a) shows the waveform of a stationary pulse (absolute amplitude |a(t)|), and Figure 4(b) shows its phase arg a(t). Figure 4(c) shows the spectrum of the stationary pulse with a solid line and the ideal spectral shape of a Gaussian pulse (A(ω) in Equation (7)) with a dashed line. In Figure 4(b), the phase of the stationary pulse is nearly constant, demonstrating that a chirp-free Gaussian pulse can be output. While chirp is usually unavoidable in Gaussian pulses obtained by FM mode-locking, this embodiment is characterized by the ability to obtain a chirp-free Fourier-limited (TL) Gaussian pulse. This is because the chirp generated by the optical phase modulator is F This is because they are cancelled out within the resonator due to the phase characteristics of (ω).

[0026] [Second embodiment] In the second embodiment of the present invention, a Sech pulse can be generated. The waveform a(t) of the Sech pulse is given by equation (9). From the Fourier transform of equation (9), the spectrum A(ω) of the Sech pulse is given by equation (10). The shape of the optical filter 4 for generating this Sech pulse is given by equation (11) by substituting equation (10) into equation (5).

[0027]

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[0028] The transfer function F of the optical filter 4 used in this embodiment F An example of the shape of (ω) is shown in Figure 5. Here, the pulse width T = 6 ps (a 2 (t) full width at half maximum W p = 10.6 ps), modulation angular frequency Ω m = 2π×10 GHz, modulation index M PM = 1. The black dots in the figure represent the frequencies where longitudinal modes (10 GHz intervals) exist in the optical spectrum. F Absolute value of (ω) |F F (ω)|, and Fig. 5(b) shows the phase arg F F (ω) is shown. |F F Because |a(t)| has a flat base, a band limit is necessary. Figures 5(c) and 5(d) show the filter shape for a band limit of 200 GHz (±100 GHz) in Figures 5(a) and 5(b). Figure 6 shows the results of a computer analysis of the steady-state solution of a laser using this filter. Figures 6(a) and 6(b) show the waveform (absolute amplitude |a(t)|) and phase arg a(t) of the steady-state pulse, and Figure 6(c) shows the spectrum and ideal spectral shape (A(ω) in Equation (10)) of the steady-state pulse. Figure 6(b) shows that the phase of the steady-state pulse is nearly constant, enabling the output of a chirp-free SECH pulse. While optical nonlinearity (soliton effect) is essential for outputting SECH pulses in a conventional mode-locked laser, this embodiment is characterized by not requiring optical nonlinearity.

[0029] [Third embodiment] In the third embodiment of the present invention, a bi-exponential pulse can be generated. The waveform a(t) of the bi-exponential pulse is given by equation (12). From the Fourier transform of equation (12), the spectrum A(ω) of the bi-exponential pulse is given by equation (13). The shape of the optical filter 4 for generating this bi-exponential pulse is given by equation (14) by substituting equation (13) into equation (5).

[0030]

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[0031] The transfer function F of the optical filter 4 used in this embodiment F An example of the shape of (ω) is shown in Figure 7. Here, the pulse width T = 6.25 ps (a 2 (t) full width at half maximum W p = 4.33 ps), modulation angular frequency Ω m = 2π×10 GHz, modulation index M PM = 1. The black dots in the figure represent the frequencies where longitudinal modes (10 GHz intervals) of the optical spectrum exist. F Absolute value of (ω) |F F (ω)|, and Fig. 7(b) shows the phase arg F F (ω) is shown. |F F Because |a(t)| has a flat, continuous base, it is necessary to set a band limit. Taking into account the broadening of the base of the spectral shape A(ω) of the stationary pulse, the filter shape for imposing a band limit of 440 GHz (±220 GHz) on Figures 7(a) and (b) is shown in Figures 7(c) and (d). The steady-state solution of the laser obtained by computer analysis using this filter is shown in Figure 8. Figures 8(a) and (b) show the waveform (absolute amplitude |a(t)|) and phase arg a(t) of the stationary pulse, and Figure 8(c) shows the spectrum of the stationary pulse and the ideal spectral shape (A(ω) in Equation (13)). Figure 8(b) shows that the phase of the stationary pulse is nearly constant, and a chirp-free exponential pulse can be output.

[0032] [Fourth embodiment] In the fourth embodiment of the present invention, an optical pulse whose electric field amplitude is given by a triangle can be generated. The waveform a(t) of the triangular pulse is given by equation (15). From the Fourier transform of equation (15), the spectrum A(ω) of the triangular pulse is given by equation (16). The shape of the optical filter 4 for generating this triangular pulse is given by equation (17) by substituting equation (16) into equation (5).

[0033]

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[0034] The transfer function F of the optical filter 4 used in this embodiment F An example of the shape of (ω) is shown in Figure 9. Here, the pulse width T = 25 ps and the modulation angular frequency Ω m = 2π×10 GHz, and the modulation index M PM In order to obtain the harmonic components required to generate a triangular wave, M PM = 2.4. The black dots in the figure represent the frequencies where longitudinal modes (10 GHz intervals) exist in the optical spectrum. F Absolute value of (ω) |F F (ω)|, and Fig. 9(b) shows the phase arg F F (ω) is shown. |F F Since |a(t)| does not attenuate and maintains a constant magnitude even as the frequency increases, a band limit is necessary. Taking into account the broadening of the base of the spectral shape A(ω) of the stationary pulse, the filter shapes for imposing a band limit of 240 GHz (±120 GHz) on the filter shown in Figures 9(a) and (b) are shown in Figures 9(c) and (d). A Nyquist filter with a roll-off factor α = 0.5 is used for band limiting. Figure 10 shows the results of computer analysis of the laser's steady-state solution using this filter. Figures 10(a) and (b) show the waveform (absolute amplitude |a(t)|) and phase arg a(t) of the stationary pulse, while Figure 10(c) shows the spectrum of the stationary pulse and the ideal spectral shape (A(ω) in Equation (16)). Figure 10(b) shows that the phase of the stationary pulse is nearly constant, and a chirp-free triangular pulse can be output.

[0035] [Fifth embodiment] In the fifth embodiment of the present invention, it is possible to generate an optical pulse whose intensity (square of the electric field) is given by a triangle, instead of the triangular pulse defined by the electric field amplitude in the fourth embodiment. The waveform (electric field amplitude) a(t) of this pulse is given by equation (18). In this case, |a(t)| 2 represents a triangle. From the Fourier transform of equation (18), the spectrum A(ω) of a(t) is given by equation (19). Here, J2n (ωT) is the 2n-th order Bessel function of the first kind, and F(n) is the function given by equation (20). The shape of the optical filter 4 for generating this pulse is given by equation (21) by substituting equation (19) into equation (5).

[0036]

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[0037] The transfer function F of the optical filter 4 used in this embodiment F An example of the shape of (ω) is shown in Figure 11. Here, the pulse width T = 25 ps and the modulation angular frequency Ω m = 2π×10 GHz, and the modulation index M PM In order to obtain the harmonic components required to generate a triangular wave, M PM = 2.0. The black dots in the figure represent the frequencies at which longitudinal modes (10 GHz intervals) of the optical spectrum exist. F Absolute value of (ω) |F F (ω)|, and Fig. 11(b) shows the phase arg F F (ω) is shown. |F F (ω)| does not attenuate and maintains a constant magnitude even as the frequency increases, so a band limit must be set. Taking into account the broadening of the base of the spectral shape A(ω) of the stationary pulse, the filter shapes given in Figures 11(a) and (b) with a band limit of 640 GHz (±320 GHz) are shown in Figures 11(c) and (d). Figure 12 shows the results of a computer analysis of the steady-state solution of the laser using this filter. Figures 12(a) and (b) show the waveform of the stationary pulse (absolute value of amplitude |a(t)| and intensity |a(t)| 2 ) and phase arg a(t), and Fig. 12(c) shows the spectrum of the stationary pulse and the ideal spectral shape (A(ω) in equation (19)). As shown on the right vertical axis in Fig. 12(a), an optical pulse with a triangular field intensity is obtained. In Fig. 12(b), the phase of the stationary pulse is almost constant, which indicates that a chirp-free pulse can be output.

[0038] [Sixth embodiment] In the sixth embodiment of the present invention, an optical pulse can be generated in which the electric field amplitude is given by a parabola. The waveform a(t) of the parabolic pulse is given by equation (22). From the Fourier transform of equation (22), the spectrum A(ω) of the parabolic pulse is given by equation (23). The shape of the optical filter 4 for generating this parabolic pulse is given by equation (24) by substituting equation (23) into equation (5).

[0039]

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[0040] The transfer function F of the optical filter 4 used in this embodiment F An example of the shape of (ω) is shown in Figure 13. Here, the pulse width T = 25.0 ps, ​​and the modulation angular frequency Ω m = 2π×10 GHz, modulation index M PM = 1. The black dots in the figure represent the frequencies where longitudinal modes (10 GHz intervals) of the optical spectrum exist. F Absolute value of (ω) |F F (ω)|, and Fig. 13(b) shows the phase arg F F (ω) is shown. |F F Since |a(t)| does not attenuate and maintains a constant magnitude even as the frequency increases, a band limit is necessary. Taking into account the broadening of the base of the spectral shape A(ω) of the stationary pulse, the filter shape with a band limit of 640 GHz (±320 GHz) is shown in Figures 13(a) and (b) in Figures 13(c) and (d). The steady-state solution of the laser obtained by computer analysis using this filter is shown in Figure 14. Figures 14(a) and (b) show the waveform (absolute amplitude |a(t)|) and phase arg a(t) of the stationary pulse, and Figure 14(c) shows the spectrum of the stationary pulse and the ideal spectral shape (A(ω) in Equation (23)). Figure 14(b) shows that the phase of the stationary pulse is nearly constant, and a chirp-free parabolic pulse can be output.

[0041] [Seventh embodiment] In the seventh embodiment of the present invention, it is possible to generate an optical pulse whose intensity (square of the electric field) is given by a parabola (parabola), rather than the parabolic pulse defined by the electric field amplitude in the sixth embodiment. The waveform (electric field amplitude) a(t) of this pulse is given by equation (25). In this case, |a(t)| 2 represents a parabola. From the Fourier transform of equation (25), the spectrum A(ω) of a(t) is given by equation (26). Here, J1(ωT) is a first-order Bessel function of the first kind. The shape of the optical filter 4 used to generate this pulse is given by equation (27) by substituting equation (26) into equation (5).

[0042]

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[0043] The transfer function F of the optical filter 4 used in this embodiment F An example of the shape of (ω) is shown in Figure 15. Here, the pulse width T = 25 ps and the modulation angular frequency Ω m = 2π×10 GHz, and the modulation index is M PM = 1.0. The black dots in the figure represent the frequencies at which longitudinal modes (10 GHz intervals) of the optical spectrum exist. F Absolute value of (ω) |F F (ω)|, and Fig. 15(b) shows the phase arg F F (ω) is shown. |F F (ω)| does not attenuate and maintains a constant magnitude even as the frequency increases, so a band limit must be set. Taking into account the broadening of the base of the spectral shape A(ω) of the stationary pulse, the shape of the filter that imposes a band limit of 640 GHz (±320 GHz) on Figures 15(a) and (b) is shown in Figures 15(c) and (d). The results of a computer analysis of the steady-state solution of the laser using this filter are shown in Figure 16. Figures 16(a) and (b) show the waveform of the stationary pulse (absolute value of amplitude |a(t)| and intensity |a(t)| 2) and phase arg a(t), and Fig. 16(c) shows the spectrum of the stationary pulse and the ideal spectral shape (A(ω) in equation (26)). As shown by the right vertical axis in Fig. 16(a), an optical pulse with a parabolic field intensity is obtained, and in Fig. 16(b), the phase of the stationary pulse is almost constant, which indicates that a chirp-free pulse can be output.

[0044] [Eighth embodiment] In the eighth embodiment of the present invention, a rectangular pulse can be generated. The waveform a(t) of the parabolic pulse is given by equation (28). From the Fourier transform of equation (28), the spectrum A(ω) of the rectangular pulse is given by equation (29). The shape of the optical filter 4 for generating this rectangular pulse is given by equation (30) by substituting equation (29) into equation (5).

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[0045] The transfer function F of the optical filter 4 used in this embodiment F An example of the shape of (ω) is shown in Figure 17. Here, the pulse width T = 25.0 ps, ​​and the modulation angular frequency Ω m = 2π×10 GHz, modulation index M PM = 1. The black dots in the figure represent the frequencies where longitudinal modes (10 GHz intervals) of the optical spectrum exist. F Absolute value of (ω) |F F (ω)|, and Fig. 17(b) shows the phase arg F F (ω) is shown. |F FSince |a(t)| does not attenuate and maintains a constant magnitude even as the frequency increases, a band limit is necessary. Taking into account the broadening of the base of the spectral shape A(ω) of the stationary pulse, the filter shape with a band limit of 640 GHz (±320 GHz) is shown in Figures 17(a) and (b) in Figures 17(c) and (d). The steady-state solution of the laser obtained by computer analysis using this filter is shown in Figure 18. Figures 18(a) and (b) show the waveform (absolute amplitude |a(t)|) and phase arg a(t) of the stationary pulse, and Figure 18(c) shows the spectrum of the stationary pulse and the ideal spectral shape (A(ω) in Equation (29)). Figure 18(b) shows that the phase of the stationary pulse is nearly constant, and a chirp-free rectangular pulse can be output. Since an AM mode-locked laser modulates the pulse amplitude using an intensity modulator, it is difficult to generate pulses with a constant amplitude. Therefore, pulses with a constant amplitude like a square wave are only obtained by this invention.

[0046] A specific experimental example will be shown below. In the experiment, as shown in Fig. 19, a harmonic FM mode-locked erbium fiber laser with a repetition rate of 10 GHz, which oscillates at a wavelength of 1.56 µm, is used, and an LCoS element is inserted into the resonator (resonator length 15 m). The filter function F F The amplitude and phase characteristics of (ω) are implemented in the LCoS element using software. The frequency resolution of the LCoS element is 1 GHz.

[0047] The shape of the optical filter used to generate a Gaussian pulse in the first embodiment of the present invention is shown in Fig. 20. (a) in the figure shows the amplitude characteristic, and (b) in the figure shows the phase characteristic. The solid line indicates F in equation (5). F The broken line shows the filter shape implemented on the LCoS element, which approximates this with a step function every 10 GHz. The LCoS element used in the experiment originally has a frequency resolution of 1 GHz, but in consideration of fluctuations in the longitudinal mode frequency of the laser, the F F(ω) is approximated as a step and implemented.

[0048] The waveform and optical spectrum of the Gaussian pulse generated using this filter are shown in Figure 21. Figure 21(a) shows the intensity waveform a observed with an optical sampling oscilloscope. 2 (t) and (b) show the square roots of these values ​​converted to amplitude a(t), and (c) shows the optical spectrum A(ω) observed with an optical spectrum analyzer, displayed in dB (20 log|A(ω)|). The solid line shows the experimental results, and the dashed line shows the computer analysis results shown in Figure 4(a) and (c). The two lines match well, demonstrating that Gaussian pulses can be generated as designed using the optical filter in Figure 20. Furthermore, the time-bandwidth product is 0.44, demonstrating that TL pulses can be obtained directly from the laser without the need for external chirp compensation as in a conventional FM mode-locked laser.

[0049] The shape of the optical filter used to generate the sech pulse in the second embodiment of the present invention is shown in Figure 22. (a) of the figure shows the amplitude characteristic, and (b) of the figure shows the phase characteristic. The solid line indicates the F F The filter (Fig. 5(c)) has a band limit of 200 GHz (ω), and the dashed line shows the filter shape implemented on the LCoS element by approximating this with a step function every 10 GHz.

[0050] The waveform and optical spectrum of the SECH pulse generated using this filter are shown in Figure 23. Figure 23 (a) shows the intensity waveform a observed with an optical sampling oscilloscope. 2 (t) and (b) show the square roots of these values ​​converted to amplitude a(t), and (c) shows the optical spectrum A(ω) observed with an optical spectrum analyzer, displayed in dB (20 log|A(ω)|). The solid line shows the experimental results, and the dashed line shows the computer analysis results shown in Figure 6(a) and (c). The two lines match well, demonstrating that sech pulses can be generated as designed using the optical filter in Figure 22. Furthermore, the time-bandwidth product is 0.32, demonstrating that TL sech pulses can be obtained directly from the laser.

[0051] The shape of the optical filter used to generate a bi-exponential pulse in the third embodiment of the present invention is shown in Fig. 24. (a) in the figure shows the amplitude characteristic, and (b) shows the phase characteristic. The solid line indicates the F F The filter (Fig. 7(c)) has a band limit of 440 GHz (ω), and the dashed line shows the filter shape implemented on an LCoS element by approximating this with a step function every 10 GHz.

[0052] The waveform and optical spectrum of the biexponential pulse generated using this filter are shown in Figure 25. Figure 25(a) shows the intensity waveform a observed with an optical sampling oscilloscope. 2 (t) and (b) show the square roots of these values ​​converted to amplitude a(t), and (c) shows the optical spectrum A(ω) observed with an optical spectrum analyzer, displayed in dB (20 log|A(ω)|). The solid line shows the experimental results, and the dashed line shows the computer analysis results shown in Figures 8(a) and (c). The two lines match well, demonstrating that the optical filter in Figure 24 can generate bi-exponential pulses as designed.

[0053] The shape of the optical filter used to generate a triangular pulse defined by the electric field amplitude in the fourth embodiment of the present invention is shown in Fig. 26. (a) of the figure shows the amplitude characteristic, and (b) shows the phase characteristic. The solid line shows the F F The filter (Fig. 9(c)) has a band limit of 240 GHz (ω), and the dashed line shows the filter shape implemented on the LCoS element by approximating this with a step function every 10 GHz.

[0054] The waveform and optical spectrum of a triangular pulse defined by the electric field amplitude generated using this filter are shown in Figure 27. Figure 27(a) shows the intensity waveform a observed with an optical sampling oscilloscope. 2(t) and (b) show the square roots of these values ​​converted to amplitude a(t), and (c) shows the optical spectrum A(ω) observed with an optical spectrum analyzer, displayed in dB (20 log|A(ω)|). The solid line shows the experimental results, and the dashed line shows the computer analysis results shown in Figures 10(a) and (c). The two lines match well, demonstrating that the optical filter in Figure 26 can generate triangular pulses defined by the electric field amplitude as designed.

[0055] The shape of the optical filter used to generate a triangular pulse defined by the intensity (square of the electric field) in the fifth embodiment of the present invention is shown in Fig. 28. (a) in the figure shows the amplitude characteristic, and (b) shows the phase characteristic. The solid line indicates F in equation (21). F The filter (ω) is band-limited at 640 GHz (Fig. 11(c)). The dashed line shows the filter shape implemented on the LCoS element by approximating this with a step function every 10 GHz.

[0056] The waveform and optical spectrum of a triangular pulse generated using this filter, defined by the intensity (square of the electric field), are shown in Figure 29. Figure 29 (a) shows the intensity waveform a observed with an optical sampling oscilloscope. 2 (t) and (b) show the square roots of these values ​​converted to amplitude a(t), and (c) shows the optical spectrum A(ω) observed with an optical spectrum analyzer, displayed in dB (20 log|A(ω)|). The solid line shows the experimental results, and the dashed line shows the computer analysis results shown in Figures 12(a) and (c). The two lines match well, demonstrating that the optical filter in Figure 28 can generate triangular pulses defined by intensity as designed.

[0057] The shape of the optical filter used to generate a parabolic pulse defined by the electric field amplitude in the sixth embodiment of the present invention is shown in Fig. 30. (a) of the figure shows the amplitude characteristic, and (b) shows the phase characteristic. The solid line shows the F F The filter (ω) is band-limited at 640 GHz (Fig. 13(c)). The dashed line shows the filter shape implemented on the LCoS element by approximating this with a step function every 10 GHz.

[0058] The waveform and optical spectrum of the parabolic pulse defined by the electric field amplitude generated using this filter are shown in Figure 31. Figure 31 (a) shows the intensity waveform a observed with an optical sampling oscilloscope. 2 (t) and (b) show the square roots of these values ​​converted to amplitude a(t), and (c) shows the optical spectrum A(ω) observed with an optical spectrum analyzer, displayed in dB (20 log|A(ω)|). The solid line shows the experimental results, and the dashed line shows the computer analysis results shown in Figures 14(a) and (c). The two lines match well, demonstrating that the optical filter in Figure 30 can generate parabolic pulses defined by the electric field amplitude as designed.

[0059] The shape of the optical filter used to generate a parabolic pulse defined by the intensity (square of the electric field) in the eighth embodiment of the present invention is shown in Fig. 32. (a) of the figure shows the amplitude characteristic, and (b) shows the phase characteristic. The solid line shows the F F The filter (ω) is band-limited at 640 GHz (Fig. 15(c)). The dashed line shows the filter shape implemented on the LCoS element by approximating this with a step function every 10 GHz.

[0060] The waveform and optical spectrum of the parabolic pulse generated using this filter, defined by the intensity (square of the electric field), are shown in Figure 33. Figure 33 (a) shows the intensity waveform a observed with an optical sampling oscilloscope. 2 (t) and (b) show the square roots of these values ​​converted to amplitude a(t), and (c) shows the optical spectrum A(ω) observed with an optical spectrum analyzer, displayed in dB (20 log|A(ω)|). The solid line shows the experimental results, and the dashed line shows the computer analysis results shown in Figures 16(a) and (c). The two lines match well, demonstrating that the optical filter in Figure 32 can generate parabolic pulses defined by the electric field intensity as designed.

[0061] The shape of the optical filter used to generate a rectangular pulse in the eighth embodiment of the present invention is shown in Fig. 34. (a) in the figure shows the amplitude characteristic, and (b) shows the phase characteristic. The solid line indicates F in equation (30). FThe filter (ω) is band-limited at 640 GHz (Fig. 17(c)), and the dashed line shows the filter shape implemented on the LCoS element by approximating this with a step function every 10 GHz.

[0062] The waveform and optical spectrum of a rectangular pulse generated using this filter are shown in Figure 35. Figure 35 (a) shows the intensity waveform a observed with an optical sampling oscilloscope. 2 (t) and (b) show the square roots of these values ​​converted to amplitude a(t), and (c) shows the optical spectrum A(ω) observed with an optical spectrum analyzer, displayed in dB (20 log|A(ω)|). The solid line shows the experimental results, and the dashed line shows the computer analysis results shown in Figures 18(a) and (c). The two lines match well, demonstrating that the optical filter in Figure 34 can generate bi-exponential pulses as designed. [Industrial Applicability]

[0063] As explained in detail above, the present invention makes it possible to easily generate pulses with any time waveform by appropriately designing the amplitude and phase characteristics of an optical filter inserted into a laser resonator. Pulse trains with time waveforms such as exponential, triangular, rectangular, and parabolic waveforms obtained by the present invention can be used in a wide range of applications, including signal pulses for ultrafast time-division multiplexed optical communications and sampling pulses for ultrafast measurement. [Explanation of symbols]

[0064] 1. Optical fiber 2. Optical amplifier 3 Optical phase modulator 4 Optical Filter 5. Polarization-maintaining erbium fiber 6 Pump LD 7 WDM coupler 8 Phase Modulator 9 Etalon 10 PZT element 11 Coupler 12 Isolator 13 LCoS element 14 Amplifier 15 Phase Shifter 16 10 GHz synthesizers

Claims

1. an FM mode-locked laser including an optical phase modulator, an optical amplifier, and an optical filter in a laser resonator; The optical phase modulator has a repetition rate of Ω m and The optical filter has variable amplitude and phase characteristics, and the amplitude and phase characteristics are determined by the modulation depth M PM of the optical phase modulator and the spectrum A(ω) of the optical pulse to be output, and by the conversion of A(ω) into Ω m A function A(ω-nΩ) shifted positively or negatively by an integer multiple of m ) (n: integer), the transfer function FF(ω) is given by the following equation (1), [Equation 1] (where J n (x) is the nth-order Bessel function of the first kind.) By setting the amplitude and phase characteristics according to the shape of the optical pulse to be output, it is possible to generate an optical pulse having any shape, including an optical pulse that cannot be generated by an AM mode-locked laser. An optical function generator characterized by:

2. The optical function generator according to claim 1, characterized in that the optical filter is configured to set a band limit on the amplitude and phase characteristics according to the spectral width of the optical pulse to be output, thereby obtaining a TL (Transform-limited) pulse.

3. The optical filter calculates the amplitude and phase characteristics by a Gaussian function A(ω) and repeating A(ω) at an angular frequency Ω m A function A(ω-nΩ) shifted positively or negatively by an integer multiple of m 2. An optical function generator according to claim 1, wherein said optical function generator is capable of generating a Gaussian pulse by applying a voltage to said optical function generator.

4. The optical filter calculates the amplitude and phase characteristics by a sech function A(ω) and a function A(ω) which is repeated at an angular frequency Ω m A function A(ω-nΩ) shifted positively or negatively by an integer multiple of m 2. An optical function generator according to claim 1, wherein the optical function generator is capable of generating a sech pulse by applying a signal to the optical function generator.

5. The optical filter calculates the amplitude and phase characteristics by a Lorentz function A(ω) and a function A(ω) repeated at an angular frequency Ω m A function A(ω-nΩ) shifted positively or negatively by an integer multiple of m 2. An optical function generator according to claim 1, wherein said optical function generator is capable of generating a pulse having a shape of a double exponential function by applying a voltage to said optical function.

6. The optical filter calculates the amplitude and phase characteristics by a squared sinc function A(ω) and a function A(ω) that repeats at an angular frequency Ω m A function A(ω-nΩ) shifted positively or negatively by an integer multiple of m 2. The optical function generator according to claim 1, wherein said optical function generator is capable of generating a pulse having a triangular electric field amplitude by applying a voltage of 100 V to said optical function generator.

7. The optical filter calculates the amplitude and phase characteristics using a Bessel function of the first kind J 2k (ω) (k: integer), and A(ω) is given by the series, and A(ω) is repeated at angular frequency Ω m A function A(ω-nΩ) shifted positively or negatively by an integer multiple of m 2. An optical function generator according to claim 1, wherein said optical function generator is capable of generating a pulse having a triangular intensity by applying a voltage to said optical function generator.

8. The optical filter calculates the amplitude and phase characteristics as a function sin ω / ω 3 and cos ω / ω 2 A(ω) is given by the sum of A(ω) and A(ω) is repeated at angular frequency Ω m A function A(ω-nΩ) shifted positively or negatively by an integer multiple of m 2. An optical function generator according to claim 1, wherein said optical function generator is configured to be capable of generating a pulse having a parabolic electric field amplitude by applying said electric field amplitude to said optical function generator.

9. The optical filter calculates the amplitude and phase characteristics using a function J 1 (ω) / ω(J 1 (ω): First-order Bessel function of the first kind) and A(ω) are given by the following equation: m A function A(ω-nΩ) shifted positively or negatively by an integer multiple of m 2. An optical function generator according to claim 1, wherein said optical function generator is capable of generating a pulse having a parabolic intensity by applying a pulse having a parabolic intensity.

10. The optical filter calculates the amplitude and phase characteristics by a sinc function A(ω) and a function A(ω) repeated at an angular frequency Ω m A function A(ω-nΩ) shifted positively or negatively by an integer multiple of m 2. An optical function generator according to claim 1, wherein said optical function generator is capable of generating a pulse having a rectangular shape by applying a voltage to said optical function generator.

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