Annular resonant cavity analysis method based on transfer matrix and reformed q parameter

Abstract

The invention discloses an annular resonant cavity analysis method based on a transfer matrix and a reformed q parameter. The annular resonant cavity analysis method comprises the following steps: setting system parameters in a program code; representing a q parameter at an end face II in positive and negative directions by using the q parameter at the end face I according to system parameters to obtain an equation set about the q parameter at the position I; solving an equation set about the q parameter at the position I to obtain a beam curvature parameter and a beam diameter parameter of the q parameter at the position I, and further calculating the q parameter at the position I of the end face; representing an ABCD transmission matrix from the end face I to a specified position according to system parameters; system parameters are set as variables, and then a q parameter at an end face I and an ABCD transmission rule are utilized to calculate laser beam data at a specified position in the resonant cavity or output laser data so as to analyze the working state of the laser resonant cavity. The method is beneficial to analysis of the performance condition of the annular resonant cavity and analysis of the astigmatism degree of light beams in the cavity so as to optimize the design.

Description

Technical Field

[0001] The invention belongs to the technical field of laser production and manufacturing, and in particular relates to a ring resonator analysis method based on a transfer matrix and a renormalized q parameter. Background Art

[0002] As countries around the world pay more and more attention to high-tech industries, the competition in the field of ultrafast lasers in high-end industries is becoming increasingly fierce. Ultrafast lasers play an important role in cutting-edge science and advanced manufacturing due to their advantages such as short pulses, high intensity, wide spectrum, and ultra-high repetition rate. They are indispensable scientific research tools and manufacturing tools.

[0003] Compared with the ultrafast laser field abroad, although the ultrafast laser field in my country is developing rapidly and has great market potential, the theoretical development is slow and there are certain manufacturing difficulties. Especially for mode-locked lasers, complex theoretical verification and numerical analysis are required during design. Although various analyses of mode-locked laser resonators have been published abroad, including numerical calculation methods using Fresnel diffraction integrals and Fox-Li methods, or methods of approximating self-consistent fields through Gaussian beams and using numerical iterative programs for calculation, the iterative numerical programs required for these methods are difficult to provide general design guidance. In addition, various resonant cavity simulation and analysis systems have been developed on the market in the past decade, but these systems have high entry barriers and often have calculation errors. For those who do not know how to use or are not proficient in using them, it does not bring much convenience to the design of laser resonators.

[0004] When designing the internal structure of a laser resonant cavity, in addition to considering which components need to be added, it is often necessary to consider various factors such as the position and posture of the components, the shape of the gain medium, and environmental influences to achieve the best output state. However, these considerations are difficult to analyze through experimental operations, so computer simulation analysis is required. Summary of the invention

[0005] The present invention aims to solve the problem raised in the background technology. By programming in MATLAB, the ABCD transmission matrix and the renormalized q parameter are used to conveniently and quickly calculate the variation curve of the intracavity beam data of the KLM laser as the system parameters change.

[0006] In order to achieve the purpose of the present invention, the present invention discloses a ring resonator analysis method based on a transfer matrix and a renormalized q parameter, comprising the following steps:

[0007] Step 1: Set system parameters in the program code;

[0008] Step 2: For the two end faces of the gain medium in the laser resonant cavity, they are marked as I and II respectively. According to the system parameters, the q parameter at I is used to express the q parameter at II in both positive and negative directions, and the equation group about the q parameter at I is obtained;

[0009] Step 3, solve the equation group about the q parameter at position I, obtain the beam curvature parameter and beam diameter parameter of the q parameter at position I, and then calculate the q parameter at position I of the end face;

[0010] Step 4, express the ABCD transmission matrix from end face I to the specified position according to the system parameters;

[0011] Step 5: By setting the system parameters in step 1 as variables, and then using the q parameters at end face I and the ABCD transmission rule, calculate the laser beam data at a specified position in the resonant cavity or the output laser data to analyze the working state of the laser resonant cavity.

[0012] Furthermore, in step 1, the parameters needed in the simulation analysis process include the refractive index of the Kerr medium, the ratio K of the intracavity power to the self-focusing critical power, the position of the Kerr medium, the distance between the curved mirrors M1 and M2, the focal length of the curved mirror, the tilt angle of the curved mirror, and the position distance between the plane mirrors M3 and M4, so as to provide basic data and variables for the simulation analysis of the laser resonant cavity.

[0013] Furthermore, in step 2, the equation group for the q parameter at end face I is obtained based on the ABCD transfer matrix and the renormalized q parameter; the expression for the q parameter at end face I of the gain medium is as follows:

[0014]

[0015] Among them, q I represents the q parameter at the end face I of the gain medium; j is an imaginary unit; R I is the curvature radius parameter, which describes the curvature radius of the beam in the laser resonant cavity; It is the beam radius parameter, which describes the spot size of the beam in the laser resonant cavity, w is the radius of the beam, and λ is the wavelength of the beam;

[0016] The equations for the q parameter at end face I are as follows:

[0017]

[0018] The direction from end face I to curved mirror M1 is defined as the positive direction, where A, B, C, and D are the paraxial ABCD transmission matrices from end face I to end face II along the positive direction, L is the length of the gain medium in the resonant cavity, and K is the normalized power, which is equal to the ratio of the intracavity power to the critical power. The numerical solution is performed using a computer program. For example, the solve() function of MATLAB. The first equation is about R I Solve the quadratic equation of R I Then solve the second equation for y I .

[0019] Furthermore, in step 3, R I and Y I Afterwards, according to The q parameter at the end face I is obtained; the working state of the laser resonant cavity in the Kerr medium is obtained by the renormalized q parameter at the end face I, and the renormalized q parameter is defined as follows:

[0020]

[0021] Among them, q' represents the renormalized form of the q parameter, Re() represents taking the real part, and Im() represents taking the imaginary part.

[0022] Furthermore, in step 4, the working data of the laser resonant cavity at the specified position is obtained by calculating the corresponding ABCD transmission matrix at the specified position; the nonlinear ABCD transmission matrix inside the gain medium is simplified to the form of a free propagation matrix by using the renormalized q parameter:

[0023]

[0024] Where L represents the length of the gain medium.

[0025] Furthermore, in step 5, after the ABCD transmission matrix at a specified position in the resonant cavity is obtained, the working state of the laser resonant cavity at the position is obtained from the q parameter at the end face I:

[0026]

[0027] Among them, A, B, C, and D represent the four matrix elements of the ABCD transmission matrix from the upper left to the lower right.

[0028] If the working state of the laser resonant cavity at other positions except the gain medium is studied, the q parameter can be used directly; if the working state of the laser resonant cavity in the gain medium is studied, the renormalized form of the q parameter is used. At this time, the result obtained by the ABCD transfer matrix is ​​also in the form of the renormalized q parameter. According to the definition of the renormalized q parameter, it is restored to the corresponding q parameter to obtain accurate laser resonant cavity working data.

[0029] Compared with the prior art, the significant progress of the present invention is that: 1) the method integrates multiple angles to be considered in analyzing a four-mirror "figure-8" Kerr mode-locked laser ring resonator, providing a basis for designing such a ring cavity, including the spot size at the output end, the spot size at various locations in the cavity, the KLM intensity on the meridian plane and the sagittal plane, etc.; it is not only conducive to analyzing the performance of such a ring resonator, but also conducive to analyzing the degree of astigmatism of the light beam in the cavity, so as to optimize the design, compensate for astigmatism, etc.; 2) the method is based on a four-mirror "figure-8" Kerr mode-locked laser ring resonator model, and by changing the distance between the curved mirrors, the change in the spot size including the output end and the inside of the medium can be analyzed; by changing parameters such as the tilt angle of the curved mirror and the power in the cavity, the astigmatism of the light beam can also be analyzed.

[0030] In order to more clearly illustrate the functional characteristics and structural parameters of the present invention, further description is given below in conjunction with the accompanying drawings and specific implementation methods. BRIEF DESCRIPTION OF THE DRAWINGS

[0031] The drawings described herein are used to provide a further understanding of the present invention and constitute a part of this application. The exemplary embodiments of the present invention and their descriptions are used to explain the present invention and do not constitute an improper limitation of the present invention. In the drawings:

[0032] Figure 1 This is a diagram of the structure of a four-mirror ring resonator.

[0033] Figure 2 It is a schematic diagram of the simulation analysis process of the ring resonator;

[0034] Figure 3 This is a schematic diagram of the output spot size change curve in mode 1;

[0035] Figure 4 This is a schematic diagram of the output spot size change curve under mode 2;

[0036] Figure 5 This is a schematic diagram of the output spot size change curve under mode three;

[0037] Figure 6 This is a schematic diagram of the variation curve of the locking intensity of the Kerr lens;

[0038] Figure 7 is a schematic diagram of the change curve of the light spot in the cavity in the x direction;

[0039] Figure 8 It is a schematic diagram of the change curve of the light spot in the cavity in the y direction;

[0040] Fig. 9 It is a schematic diagram of the light spot variation curve in the medium in the x direction;

[0041] Fig.10It is a schematic diagram of the change curve of the light spot in the medium in the y direction;

[0042] Fig.11 This is a schematic diagram of the output spot change curve under the following conditions;

[0043] Fig.12 is a schematic diagram of the output spot change curve under condition 2;

[0044] Fig.13 This is a schematic diagram of the change curve of the light spot in the cavity in case 1;

[0045] Fig.14 This is a schematic diagram of the change curve of the light spot in the cavity in case 2;

[0046] Fig.15 This is a schematic diagram of the light spot change curve in the medium in case 1;

[0047] Fig.16 This is a schematic diagram of the light spot change curve in the medium in case 2. DETAILED DESCRIPTION

[0048] The technical solutions in the embodiments of the present invention will be clearly and completely described below in conjunction with the drawings in the embodiments of the present invention. Obviously, the described embodiments are only part of the embodiments of the present invention, rather than all the embodiments; based on the embodiments of the present invention, all other embodiments obtained by ordinary technicians in this field without making creative work are within the scope of protection of the present invention.

[0049] like Figure 2 As shown, the present invention discloses a ring resonator analysis method based on transfer matrix and renormalized q parameters, comprising the following steps:

[0050] Step 1: Set system parameters in the program code;

[0051] Step 2: For the two end faces of the gain medium in the laser resonant cavity, they are marked as I and II respectively. According to the system parameters, the q parameter at I is used to express the q parameter at II in both positive and negative directions, and the equation group about the q parameter at I is obtained;

[0052] Step 3, solve the equation group about the q parameter at position I, obtain the beam curvature parameter and beam diameter parameter of the q parameter at position I, and then calculate the q parameter at position I of the end face;

[0053] Step 4, expressing the ABCD transmission matrix from end face I to the specified position according to the system parameters of the system;

[0054] Step 5: By setting the system parameters in step 1 as variables, and then using the q parameters at end face I and the ABCD transmission rule, calculate the laser beam data at a specified position in the resonant cavity or the output laser data to analyze the working state of the laser resonant cavity.

[0055] Specifically, in step 1, the parameters needed in the simulation analysis process include the refractive index of the Kerr medium, the ratio K of the intracavity power to the self-focusing critical power, the position of the Kerr medium, the distance between the curved mirrors M1 and M2, the focal length of the curved mirror, the tilt angle of the curved mirror, and the position distance between the plane mirrors M3 and M4, so as to provide basic data and variables for the simulation analysis of the laser resonant cavity.

[0056] Specifically, in step 2, the equation group for the q parameter at end face I is obtained based on the ABCD transmission matrix and the renormalized q parameter; the expression of the q parameter at end face I of the gain medium is as follows:

[0057]

[0058] Among them, q I represents the q parameter at the end face I of the gain medium; j is an imaginary unit; R I is the curvature radius parameter, which describes the curvature radius of the beam in the laser resonant cavity; It is the beam radius parameter, which describes the spot size of the beam in the laser resonant cavity, w is the radius of the beam, and λ is the wavelength of the beam;

[0059] The equations for the q parameter at end face I are as follows:

[0060]

[0061] The direction from end face I to curved mirror M1 is defined as the positive direction, where A, B, C, and D are the paraxial ABCD transmission matrices from end face I to end face II along the positive direction, L is the length of the gain medium in the resonant cavity, and K is the normalized power, which is equal to the ratio of the intracavity power to the critical power. The numerical solution is performed using a computer program. For example, the solve() function of MATLAB. The first equation is about R I Solve the quadratic equation of R I Then solve the second equation for y I .

[0062] Specifically, in step 3, find R I and I Afterwards, according to The q parameter at the end face I is obtained; the working state of the laser resonant cavity in the Kerr medium is obtained by the renormalized q parameter at the end face I, and the renormalized q parameter is defined as follows:

[0063]

[0064] Among them, q' represents the renormalized form of the q parameter, Re() represents taking the real part, and Im() represents taking the imaginary part.

[0065] Specifically, in step 4, the working data of the laser resonant cavity at the specified position is obtained by calculating the corresponding ABCD transmission matrix at the specified position; the nonlinear ABCD transmission matrix inside the gain medium is simplified to the form of a free propagation matrix by using the renormalized q parameter:

[0066]

[0067] Where L represents the length of the gain medium.

[0068] Specifically, in step 5, after obtaining the ABCD transmission matrix at a specified position in the resonant cavity, the working state of the laser resonant cavity at this position is obtained by the q parameter at the end face I:

[0069]

[0070] Among them, A, B, C, and D represent the four matrix elements of the ABCD transmission matrix from the upper left to the lower right.

[0071] If the working state of the laser resonant cavity at other positions except the gain medium is studied, the q parameter can be used directly; if the working state of the laser resonant cavity in the gain medium is studied, the renormalized form of the q parameter is used. At this time, the result obtained by the ABCD transfer matrix is ​​also in the form of the renormalized q parameter. According to the definition of the renormalized q parameter, it is restored to the corresponding q parameter to obtain accurate laser resonant cavity working data.

[0072] Example 1

[0073] The four-mirror "figure-8" ring resonator is a common structure of the mode-locked laser resonator. It has the advantages of low noise and good frequency characteristics, and can also maintain the output power while compressing the output spectrum.

[0074] However, this type of resonant cavity will produce serious astigmatism when working. The method of analyzing the change of astigmatism is to analyze the spot diameter in the x and y directions at the output coupler. The method of compensating astigmatism can be to change the distance between the curved mirrors, change the mirror inclination angle θ, change the K parameter (equivalent to changing the power in the cavity), etc.

[0075] like Figure 1 As shown, assuming Figure 1The crystal in the image is a titanium-doped sapphire crystal with a length of 20 mm. The total distance from M1 to M2 via M3 and M4 is 1700 mm. The entire structure is symmetrically distributed. Let the distance between the curved mirrors vary between 108 mm and 124 mm, and under the following three astigmatism compensation conditions: ①θ=0°, K=0; ②θ=15.8°, K=0; ③θ=15.8°, K=0.5, the x and y direction spot diameter change curves at the output coupler are obtained as follows: Figure 3 It can be seen that the spot diameters in the x and y directions are highly inconsistent, which means that there is severe astigmatism in the laser resonator at this time, which has a great impact on the output beam and needs to be compensated for. Figure 4 and Figure 5 These are the output results under two different astigmatism compensation measures. Figure 4 Only by tilting the curved mirror to compensate for the angle, we can see that the spot diameters in the x and y directions are basically the same, and the astigmatism is significantly suppressed. Figure 5 The power in the cavity is increased on the basis of angle compensation, so that the spot diameter curves in two directions basically overlap, further compensating for astigmatism, and the output beam is closer to a circular spot.

[0076] Example 2

[0077] like Figure 6 As shown, under the assumptions in Example 1, the distance between the curved mirrors is changed between 112.8 mm and 120 mm, and the Kerr lens mode locking strength (KLM strength) at M1 can be calculated: It can be seen that the mode locking strength is significantly different in the x and y directions, which indicates that if a slit is used instead of an aperture in the laser resonator, inserting a vertical slit is more effective than inserting a horizontal slit, and Figure 6 The curve shown is a guide as to where to insert the slit in the resonant cavity.

[0078] Example 3

[0079] like Figure 7 , Figure 8 As shown, the change process of the spot diameter in the x and y directions in the entire cavity is analyzed. Under the assumptions in Example 1, the spacing between the curved mirrors is set to 115 mm. When K = 0 and K = 0.5, the spot curve changes from the positive direction of end face I to end face II as shown below: Under such a curved mirror spacing distance, the spot size of the high-intensity light field (K = 0.5) is always larger than that of the continuous wave operation (K = 0), which means that the horizontal slit cannot be used for the hard aperture KLM. However, as Figure 8 As shown, when the distance from the crystal end face I is less than 280 mm, the spot size of the high-intensity field is smaller than the spot size of the continuous wave operation in the y direction. Therefore, the vertical slit may still be used for the hard aperture KLM.

[0080] like Fig. 9 , Fig.10 As shown, keeping the spacing between the curved mirrors at 115mm, the change in the spot diameter in the x and y directions in the medium can also be calculated by renormalizing the q parameter: The results show that the beam waist in the x-axis direction is located at 1.8mm from the Kerr medium to the M2 mirror and as the power in the cavity increases to K=0.5, the position of the x-axis waist moves to the right by 0.2mm. However, the beam waist in the y-axis direction is displaced by 5.4mm from the center of the medium to the M2 mirror when K=0, and by 5mm when K=0.5. The displacement of the x and y waists at K=0 and K=0.5 is the result of astigmatism caused by multiple factors. Fig. 9 , Fig.10 The curve can provide a basis for achieving the best pump-cavity field matching.

[0081] Example 4

[0082] like Fig.11 , Fig.12 As shown in the figure, the ABCD transmission matrix and q parameter transfer method can also be used to realize the intracavity z-scan analysis. The specific method is: keep the distance between the two curved mirrors at 114.6mm unchanged, and change the distance between the two mirrors. As shown in the figure below, the spot change diagram at OC is obtained when the crystal thickness is 1mm and 20mm, and the intracavity power is K=0, 0.5, and 0.99 respectively: Fig.11 It is observed that when K = 0, the output spot size is constant. When the cavity power is greater than 0, the peak and valley values ​​on the spot size curve begin to increase near r = 51 and r = 53 mm, respectively, which is similar to the external z-scan curve. As the cavity laser power increases, the peak-valley value increases. Fig.12 In the figure, a common z-scan phenomenon was found for a 2 cm thick crystal under low cavity power conditions (K = 0 and K = 0.5), but the distance between the peak (r = 41.7 mm) and the trough (r = 52.5 mm) was larger than that of a thinner crystal. As the cavity laser power increases, two peaks are observed near r = 43.3 mm and r = 51.3 mm superimposed on the spot size curve, and the peaks tend to increase the output spot size. Fig.11 and Fig.12 The results help to understand the effect of the displacement of the sapphire crystal in the resonant cavity on the working state of the resonant cavity.

[0083] like Fig.13 , Fig.14 , Fig.15 , Fig.16 As shown, for Fig.12The two peaks appearing in the cavity can be obtained by setting the distances from the end face I to M1 to r=43.3mm and r=51.3mm respectively while keeping the distance between the two curved mirrors at 114.6mm. Then, the operation method of Example 3 is used to obtain the change diagram of the spot diameter in the cavity and the medium under three different resonant cavity powers: Fig.13 and Fig.14 is the curve of the spot size variation in the entire annular cavity. Note that the spot size variation for high intracavity power K = 0.99 is very different from the spot size variation for low powers K = 0 and K = 0.5. When K = 0 and K = 0.5, there is a Gaussian beam waist in the region from M1 through M3, M4 to M2; however, when K = 0.99, there is no beam waist in this region. This indicates that at such high intracavity power, the focusing power of the 2 cm titanium sapphire crystal should be comparable to that of any curved mirror, resulting in a huge change in the mode distribution in the cavity. In this case, the self-focusing effect causes the Gaussian beam waist to move between the two curved mirrors too close to one of the mirrors. Fig.15 and Fig.16 This is confirmed. Fig.15 and Fig.16 It was observed that at low intracavity powers K = 0 and K = 0.5, the position of the Gaussian beam waist in the crystal hardly changes. However, when the intracavity power K = 0.99, the beam waist moves out of the crystal and becomes too close to one of the curved mirrors. This change in the beam waist can be explained by Fig.12 The reason for the peak.

[0084] It should be noted that, in this article, relational terms such as first and second, etc. are only used to distinguish one entity or operation from another entity or operation, and do not necessarily require or imply any such actual relationship or order between these entities or operations. Moreover, the terms "include", "comprise" or any other variants thereof are intended to cover non-exclusive inclusion, so that a process, method, article or device including a series of elements includes not only those elements, but also other elements not explicitly listed, or also includes elements inherent to such process, method, article or device.

[0085] Although embodiments of the present invention have been shown and described, it will be appreciated by those skilled in the art that various changes, modifications, substitutions and variations may be made to the embodiments without departing from the principles and spirit of the present invention, and that the scope of the present invention is defined by the appended claims and their equivalents.

Claims

1. A ring resonator analysis method based on transfer matrix and renormalized q-parameters, characterized in that: The following steps are involved: Step 1: Set system parameters in the program code; Step 2: For the two end faces of the gain medium in the laser resonant cavity, they are marked as I and II respectively. According to the system parameters, the q parameter at I is used to express the q parameter at II in both positive and negative directions, and the equation group about the q parameter at I is obtained; Step 3, solve the equation group about the q parameter at position I, obtain the beam curvature parameter and beam diameter parameter of the q parameter at position I, and then calculate the q parameter at position I of the end face; Step 4, express the ABCD transmission matrix from end face I to the specified position according to the system parameters; Step 5: By setting the system parameters in step 1 as variables, and then using the q parameters at end face I and the ABCD transmission rule, calculate the laser beam data at a specified position in the resonant cavity or the output laser data to analyze the working state of the laser resonant cavity.

2. A ring resonator analysis method based on transfer matrix and renormalized q-parameters according to claim 1, characterized in that: In step 1, the parameters needed in the simulation analysis process include the refractive index of the Kerr medium, the ratio K of the intracavity power to the self-focusing critical power, the position of the Kerr medium, the distance between the curved mirrors M1 and M2, the focal length of the curved mirror, the tilt angle of the curved mirror, and the position distance of the plane mirrors M3 and M4.

3. The ring resonator analysis method based on transfer matrix and renormalized q-parameter according to claim 1, characterized in that: In step 2, the equations for the q parameter at end face I are obtained based on the ABCD transfer matrix and the renormalized q parameter; the expression for the q parameter at end face I of the gain medium is as follows: Among them, q I represents the q parameter at the end face I of the gain medium; j is an imaginary unit; R I is the curvature radius parameter, which describes the curvature radius of the beam in the laser resonant cavity; It is the beam radius parameter, which describes the spot size of the beam in the laser resonant cavity, w is the radius of the beam, and λ is the wavelength of the beam; The equations for the q parameter at end face I are as follows: The direction from end face I to the curved mirror M1 is defined as the positive direction, where A, B, C, and D are the paraxial ABCD transmission matrices from end face I to end face II along the positive direction, L is the length of the gain medium in the resonant cavity, and K is the normalized power, which is equal to the ratio of the intracavity power to the critical power. The numerical solution is performed using a computer program.

4. The ring resonator analysis method based on transfer matrix and renormalized q-parameter according to claim 3, characterized in that: In step 3, find R I and I Afterwards, according to The q parameter at the end face I is obtained; the working state of the laser resonant cavity in the Kerr medium is obtained by the renormalized q parameter at the end face I, and the renormalized q parameter is defined as follows: Among them, q' represents the renormalized form of the q parameter, Re() represents taking the real part, and Im() represents taking the imaginary part.

5. The ring resonator analysis method based on transfer matrix and renormalized q-parameter according to claim 1, characterized in that: In step 4, the working data of the laser resonant cavity at the specified position is obtained by calculating the corresponding ABCD transmission matrix at the specified position; the nonlinear ABCD transmission matrix inside the gain medium is simplified to the form of a free propagation matrix by using the renormalized q parameter: Where L represents the length of the gain medium.

6. The ring resonator analysis method based on transfer matrix and renormalized q-parameter according to claim 1, characterized in that: In step 5, after obtaining the ABCD transmission matrix at a specified position in the resonant cavity, the working state of the laser resonant cavity at this position is obtained by the q parameter at end face I: Among them, A, B, C, and D represent the four matrix elements of the ABCD transmission matrix from the upper left to the lower right. If the working state of the laser resonant cavity at other positions except the gain medium is studied, the q parameter can be used directly; if the working state of the laser resonant cavity in the gain medium is studied, the renormalized form of the q parameter is used. At this time, the result obtained by the ABCD transfer matrix is ​​also in the form of the renormalized q parameter. According to the definition of the renormalized q parameter, it is restored to the corresponding q parameter to obtain accurate laser resonant cavity working data.

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