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Approximate solution of diffraction integral equation of resonator

Shengbo Song, Jianing Li, Wenbin Liao, Bingxuan Li, and Ge Zhang

Open Access Open Access

Abstract

Utilizing the diffraction integral equation and the principle of slow amplitude approximation, we obtain a novel approximate solution of the transverse mode including the cavity parameters a (a is the section size of the resonator) and g = 1-L/R (L is the cavity length, R is the radius of curvature of the cavity). With this approximate solution, we can explore the influence of the resonator parameters a and g on the transverse mode. The theoretical analysis demonstrates that a and g have a certain influence on the shape and quality of the transverse mode, and selecting the appropriate a and g can effectively improve the quality of the transverse pattern. Moreover, laser experiments are conducted to validate analysis conclusion.

© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement

1. Introduction

In the field of laser, the research on transverse mode has always been an important branch [1–10]. In order to study the transverse mode in the resonant cavity, the corresponding theoretical models are proposed, in which the diffraction integral equation is the basic equation for studying transverse mode in the cavity.

Based on the resonator’s geometry, it categorizes into parallel plane cavity (Fabry-Perot resonator), curved cavity [11–13] and other cavity types [8,14]. KOTIK explored the diffraction integral equation of a resonator with finite width D in the x direction and infinite width in the y direction. The intensity and phase distribution of transverse modes in the resonant cavity can be calculated by iterating the diffraction integral equation. The disadvantage of the iterative method is that the calculation result is always close to the basic mode, which is not conducive to the study of higher-order modes [15]. Theoretical derivation proved that the eigenfunction solution of the diffraction integral equation exists [16–18]. In 1961, BOYD obtained the solution of diffraction integral equation of confocal cavity, it can be expressed as Hermitian Gauss function (HG) or Laguerre Gauss function (LG) with the paraxial approximation [19]. Although the theory of confocal cavity can be extended to other cavity types via the principle of cavity equivalence [20]. Most of the research uses numerical calculation or approximation approaches to calculate the cavity mode [13,21,22]. These methods generally ignore the influence of the aperture of the resonator on the transverse mode. In addition to calculating the mode and phase of the cavity, diffraction integral equation is also applicable in analyzing the frequency shift and energy loss, as well as the influence of the inclination and deformation of the cavity [23–26]. In 1970, COLLIN combined the ABCD matrix with the diffraction integral equation to obtain the Collins integral formula [27].

In addition to the diffraction integral equation, Maxwell equation stemming from electromagnetism and wave optics, along with the complex Swift-Hohenberg (CSH) equation [28–33], have been incorporated to compute transverse modes within resonant cavities. However, the existing calculation results can not reflect the influence of cavity parameters on the transverse mode.

In order to investigate the influence of resonator parameters on transverse mode. In this paper, using the slow amplitude approximation principle, we obtain a transverse mode approximation solution involving the resonator parameters a (a is the section size of the resonator) and g = 1-L/R (L is the cavity length, R is the radius of curvature of the cavity). Combined with theoretical analysis and experimental results, we systematically analyze the influence of resonator parameters on the transverse mode. We find that the transverse mode described by the approximate solution is more consistent with the experiment.

2. Theory

Let us consider the cavity of Fig. 1, M1 and M2 are two identical spherical mirrors, $F(x,y)$ and $F(x^{\prime},y^{\prime})$ have the same functional form. They are the transverse modes of M1 and M2 cavity mirrors respectively.

 figure: Fig. 1.

Fig. 1. Mirrors M1 and M2 are square mirrors with radius of curvature equal to R, L is cavity length. $F(x,y)$ and $F(x^{\prime},y^{\prime})$ are the transverse modes of mirrors M1 and M2, respectively.

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The diffraction integral equation [22] corresponding to the resonator shown in Fig. 1 is

$$\gamma F(x,y) = \frac{i}{{\lambda L}}\exp ( - 2i\pi \frac{L}{\lambda })\int\limits_{ - a}^{ + a} {\int\limits_{ - a}^{ + a} {F(x^{\prime},y^{\prime})} } \exp \{ - \frac{{i\pi }}{{\lambda L}}[g({x^{\prime 2}} + {y^{\prime 2}}) + g({x^2} + {y^2}) - 2(xx^{\prime} + yy^{\prime})]\} dx^{\prime}dy^{\prime}$$
Where $\gamma = {\gamma _x}{\gamma _y}$ is a constant, λ is the wavelength, g = 1-L/R. Equation (1) can be further simplified as
$$\gamma F(x,y) = \eta \int\limits_{ - a}^{ + a} {\int\limits_{ - a}^{ + a} {F(x^{\prime},y^{\prime})} } \exp \{ - i[g({x^{\prime 2}} + {y^{\prime 2}}) + g({x^2} + {y^2}) - 2(xx^{\prime} + yy^{\prime})]\} dx^{\prime}dy^{\prime}$$
Where $\eta = \frac{i}{{\lambda L}}\exp ( - 2i\frac{{\pi L}}{\lambda })$, $\frac{\pi }{{\lambda L}} = 1$. Separate the variables of Eq. (2) to obtain:
$${\gamma _x}F(x) = \sqrt \eta \int\limits_{ - a}^{ + a} {F(x^{\prime})} \exp [ - i(g{x^{\prime 2}} + g{x^2} - 2xx^{\prime})]dx^{\prime}$$
$${\gamma _y}F(y) = \sqrt \eta \int\limits_{ - a}^{ + a} {F(y^{\prime})\exp [ - i(g{{y^{\prime}}^2} + g{y^2} - 2yy^{\prime})]dy^{\prime}} $$

Equations (3a) and (3b) have the same form. We only need to discuss one equation. To solve the approximate solution of Eq. (3a), it can be written as:

$$\begin{aligned} {\gamma _x}F(x) &= \sqrt \eta \int\limits_{ - \infty }^{ + \infty } {F(x^{\prime})} \exp [ - i(g{{x^{\prime}}^2}{\,+\,}g{x^2}{\,-\,}2xx^{\prime})]dx^{\prime}{\,-\,}\sqrt \eta \int\limits_{ + a}^{ + \infty } {F(x^{\prime})} \exp [ - i(g{{x^{\prime}}^2}{\,+\,}g{x^2}{\,-\,}2xx^{\prime})]dx^{\prime}\\ &- \sqrt \eta \int\limits_{ - \infty }^{ - a} {F(x^{\prime})} \exp [ - i(g{{x^{\prime}}^2} + g{x^2} - 2xx^{\prime})]dx^{\prime} \end{aligned}$$

The first term on the right of Eq. (4) is an eigenequation with solution, it can be written as:

$${E_m}F(x) = \sqrt \eta \int\limits_{ - \infty }^{ + \infty } {F(x^{\prime})} \exp [ - i(g{x^{\prime 2}} + g{x^2} - 2xx^{\prime})]dx^{\prime}$$
$${E_n}F(y) = \sqrt \eta \int\limits_{ - \infty }^{ + \infty } {F(y^{\prime})\exp [ - i(g{{y^{\prime}}^2} + g{y^2} - 2yy^{\prime})]dy^{\prime}} $$

According to the recurrence formula of Hermite polynomial, the solution of Eq. (5) is Hermite Gaussian function (HG), and it can be expressed as:

$$F(x) = \exp ( - \frac{{{x^2}}}{{{w^2}}}){H_m}(\sqrt 2 \frac{x}{w}),F(y) = \exp ( - \frac{{{y^2}}}{{{w^2}}}){H_n}(\sqrt 2 \frac{y}{w})$$

The eigenvalue is

$${E_m} = \exp [ - \frac{{2i\pi }}{\lambda }L + i\psi (m + 0.5)],{E_n} = \exp [ - \frac{{2i\pi }}{\lambda }L + i\psi (n + 0.5)]$$
Where w is a constant, $\cos (\frac{\psi }{2}) = g$.Replace the first item on the right of Eq. (4) with HGm(x) to obtain:
$$\begin{aligned} {\gamma _x}F(x) = &{E_m}H{G_m}(x) - \sqrt \eta \int\limits_{ + a}^{ + \infty } {F(x^{\prime})} \exp [ - i(g{{x^{\prime}}^2} + g{x^2} - 2xx^{\prime})]dx^{\prime}\\ & - \sqrt \eta \int\limits_{ - \infty }^{ - a} {F(x^{\prime})} \exp [ - i(g{{x^{\prime}}^2} + g{x^2} - 2xx^{\prime})]dx^{\prime} \end{aligned}$$

Derive Eq. (8) to obtain:

$$\begin{aligned} {\gamma _x}F^{\prime}(x) &= {E_m}H{{G^{\prime}}_m}(x) + \sqrt \eta \int\limits_{ + a}^{ + \infty } {2i(gx - x^{\prime})F(x^{\prime})} \exp [ - i(g{{x^{\prime}}^2} + g{x^2} - 2xx^{\prime})]dx^{\prime}\\ & + \sqrt \eta \int\limits_{ - \infty }^{ - a} {2i(gx - x^{\prime})F(x^{\prime})} \exp [ - i(g{{x^{\prime}}^2} + g{x^2} - 2xx^{\prime})]dx^{\prime} \end{aligned}$$

In addition, from Eq. (8), we can deduce a relationship.

$$\begin{aligned} 2igx[{E_m}H{G_m}(x) - {\gamma _x}F(x)] &= \sqrt \eta 2igx\int\limits_{ + a}^{ + \infty } {F(x^{\prime})} \exp [ - i(g{{x^{\prime}}^2} + g{x^2} - 2xx^{\prime})]dx^{\prime}\\ & + \sqrt \eta 2igx\int\limits_{ - \infty }^{ - a} {F(x^{\prime})} \exp [ - i(g{{x^{\prime}}^2} + g{x^2} - 2xx^{\prime})]dx^{\prime} \end{aligned}$$

Substitute Eq. (10) into Eq. (9), and Eq. (9) can be expressed as:

$$\scalebox{0.9}{$\begin{aligned} {E_m}H{G_m}^\prime (x) - {\gamma _x}F^{\prime}(x) &={-} 2igx[{E_m}H{G_m}(x) - {\gamma _x}F(x)] + \sqrt \eta \int\limits_{ + a}^{ + \infty } {2ix^{\prime}F(x^{\prime})} \exp [ - i(g{{x^{\prime}}^2} + g{x^2} - 2xx^{\prime})]dx^{\prime}\\ & + \sqrt \eta \int\limits_{ - \infty }^{ - a} {2ix^{\prime}F(x^{\prime})} \exp [ - i(g{{x^{\prime}}^2} + g{x^2} - 2xx^{\prime})]dx^{\prime} \end{aligned}$}$$

For the last two terms on the right side of Eq. (11), the integration range is (-∞, -a) and (+a, +∞), where a is the cross-sectional dimension of the cavity. Since the laser energy is mainly concentrated near the optical axis of the cavity, the sum of these two terms is a small quantity. Using the principle of slow amplitude approximation,

$$\sqrt \eta \int\limits_{ + a}^{ + \infty } {2ix^{\prime}F(x^{\prime})} \exp [ - i(g{x^{\prime 2}} + g{x^2} - 2xx^{\prime})]dx^{\prime} + \sqrt \eta \int\limits_{ - \infty }^{ - a} {2ix^{\prime}F(x^{\prime})} \exp [ - i(g{x^{\prime 2}} + g{x^2} - 2xx^{\prime})]dx^{\prime} \Rightarrow 0$$
we can get
$$[{E_m}H{G^{\prime}_m}(x) - {\gamma _x}F^{\prime}(x)] ={-} 2igx[{E_m}H{G_m}(x) - {\gamma _x}F(x)]$$

The solution of Eq. (13) is

$$F(x) = H{G_m}(x) - {A_m}\exp ( - ig{x^2})$$

Compared with HGm(x) in Eq. (6), Eq. (14a) has one more term Amexp(-igx2). Am is determined by the boundary condition F(a)⇒0, and exp(-igx2) represents the phase. Amexp(-igx2) can be regarded as a loss term, which is related to the transverse dimensions a and g. The reason for the loss term is that the beam energy in the rectangular section with the linearity of (-∞, +∞) is larger than that in the rectangular section with the linearity of (-a, +a). The smaller the value of a, the greater the loss, and the greater the value of Am. Combining the y component and adding subscript to F (x, y), we can get:

$${F_{mn}}(x,y) = [H{G_m}(x) - {A_m}\exp ( - ig{x^2})][H{G_n}(y) - {A_n}\exp ( - ig{y^2})]$$

By using (14b), we can intuitively analyze the transverse mode in the cavity and avoid complex calculation [13,15,20]. It can also be used to study the influence of Am and g on the transverse mode.

3. Theoretical analysis

Via numerical simulation, We can discuss the effect of Am and g on Fmn(x, y), and compare Fmn(x, y) and HGmn(x, y).

First, we compare the similarities and differences between HGmn(x, y) and Fmn(x, y). The expression of HGmn(x, y) is

$$H{G_{mn}}(x,y) = {H_m}(\sqrt 2 \frac{x}{w}){H_n}(\sqrt 2 \frac{y}{w})\exp ( - \frac{{{x^2} + {y^2}}}{{{w^2}}})$$

For Fmn(x, y), when g = -0.9, Fmn(x, y) can be written as:

$${F_{mn}}(x,y) = [H{G_m}(x) - {A_m}\exp (0.9i{x^2})][H{G_n}(y) - {A_n}\exp (0.9i{y^2})]$$

We simulate the patterns of F00, F11, and F22 when A0= A1 = A2 = 0.1, g = -0.9, and compare them with HG00,HG11,and HG22. The results are shown in Fig. 2.

 figure: Fig. 2.

Fig. 2. Mode patterns of HG00,HG11,HG22 and F00, F11, F22 when A0= A1 = A2 = 0.1, g = -0.9.

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It can be seen from the Fig. 2 that the edges of F00, F11, and F22 are slightly larger and blurred compared with HG00,HG11,and HG22. The physical reason for this phenomenon can be explained as that the integral limit of the integral Eq. (5) corresponding to HGmn(x, y) is (-∞, +∞), while the integral limit (-a, +a) corresponding to Fmn(x, y) is limited. Therefore, the enhancement of edge diffraction effect leads to a slight deformation and blurring of the edges.

In addition, for the same order transverse mode Fmm(x, y), different Am values should have different effects on Fmm(x, y). Because the smaller the transverse dimension (-a, +a) of the resonator, the greater the loss, and the greater the value of Am. We calculate F22 with A2 = 0.05, 0.1 and 0.15, and the results are shown in Fig. 3.

 figure: Fig. 3.

Fig. 3. Numerical simulations of F22 when A2 = 0.05, 0.1, 0.15, g = -0.9.

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Figure 3 shows that with the increase of A2, the more obvious the change in the edge of F22. In addition, there is another parameter g in the expression of Fmn(x, y). To study the influence of g on Fmn(x, y), we choose g = -0.9, -0.5, 0.5, 0.9, and A2 = 0.15 for simulation, the results are shown in Fig. 4. The expression of g = 1-L/R indicates that when the R value is constant, the change of g value means the change of cavity length L.

 figure: Fig. 4.

Fig. 4. Simulation results of F22 when g = -0.9, -0.5, 0.5, 0.9, A2 = 0.15.

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As illustrated in Fig. 4, when g = -0.9, the edge of the pattern is convex outward, while when g = 0.9, the edge of the pattern is concave inward, they have opposite effects on the edge of the pattern. We further reduce the value of g to study the influence of g on Fmn(x, y), take g = -0.1, 0.1, and A2 = 0.15, and compare it with HG22. The simulation results are shown in Fig. 5.

 figure: Fig. 5.

Fig. 5. Simulation results of F22 when g = -0.1, 0.1, A2 = 0.15, and HG22.

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Combined with Figs. 4 and 5, It can be seen that the deformation of F22 gradually decreases with the decrease of g value. When g→0 (L→R), the laser pattern of F22 approaches the ideal HG22, the spot quality of F22 is the best. The expression g = 1-L/R indicates that when g→0, the resonator is close to the confocal cavity. While the solution of diffraction integral equation of confocal cavity obtained by BOYD in paraxial approximation is HGmn(x, y) [19,34]. The fact that the simulation results of F22 and HG22 are almost the same when the resonator is close to the confocal cavity proves the rationality of our calculation.

From the above discussion, we know the influence of Am and g on the transverse mode. The positive and negative values of g determine the change trend of the mode, and Am determines the change amplitude of the mode. Under the condition of the same Am value, the quality of light spot is the best when g→0.

4. Experimental results and discussion

We carried out the experiment with the experimental setup shown in Fig. 6. The two mirrors of the resonator we used have the same size, the radius of curvature of two mirrors R = 60 mm.

 figure: Fig. 6.

Fig. 6. Experimental setup for studying the relationship between laser mode and g parameter.

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The gain medium was Nd:YVO4 (3 × 3 × 2 mm). The output coupler was mounted on a translation stage to precisely control the g (g = 1-L/R). The pump source was an 808 nm fiber-coupled laser diode with a core diameter of 200 µm. The CCD camera was used to detect the lasing patterns.

In the experiment, resonator generated single mode Fm0 and Fmn by off-axis pumping, and then obtained different values of g parameter by adjusting the cavity length L. When the pump power was 50 mW and the off-axis displacement was (Δx = 0.04 mm, Δy = 0.0 mm), F50 is obtained. Figure 7 (a) displays the experimental results of the output mode varying with g at four different values of 0.35, 0.1, -0.35 and -0.5. The theoretical results corresponding to the experimental observations were calculated with Eq. (14b) and the parameters of A5 = 0.12, A0= 0.1. For comparison, the theoretical results are shown in Fig. 7(b).

 figure: Fig. 7.

Fig. 7. (a) and (b) show the experimental results and theoretical simulation results of the output mode varying with g at four different values of 0.35, 0.1, -0.35 and -0.5, respectively.

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By analyzing the experimental results and theoretical simulation results in Fig. 7, it can be seen that when g = -0.35, the spot of F50 is convex and blurred, while when g = 0.35, the change trend of the light spot is opposite to that of g=-035. When g decreases to 0.1, the resonator is close to the confocal cavity (g→0, L→R), the spot quality of F50 is obviously better than that of F50 when g = 0.35, -0.35 and -0.5. This conclusion is consistent with the analysis results in section 3: the quality of light spot is the best when g→0.

To further investigate the influence of g parameter on lasing patterns, F21 was obtained at pump power of 70 mW. The off-axis pump distributions corresponding to F21 was (Δx = 0.02 mm, Δy = 0.01 mm). Figure 8 shows the experimental and theoretical simulation results of F21 (A2 = 0.11, A1 = 0.1).

As depicted in Fig. 8, it can be seen that as the g parameter changes from -0.1 to -0.9, the deformation of F21 gradually increases and the spot quality gradually decreases. Although the transverse mode F21 is different from F50, the variation trend of transverse mode F21 and F50 is the same with the change of g. The smaller the |g| value, the better the quality of the transverse mode.

 figure: Fig. 8.

Fig. 8. (a) Experimental output patterns when g = -0.1, -0.75, -0.9. (b) Calculated patterns with A2=0.12, A1 = 0.1 when g = -0.1, -0.75, -0.9.

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The good agreement between experimental results and theoretical simulations not only confirms the theoretical analysis but also validates rationality of approximate solution. Using this approximate solution, we can obtain the results which are more consistent with the experiment.

5. Conclusion

In summary, we use the principle of slow amplitude approximation to obtain the approximate solution of the diffraction integral equation. Utilizing this approximate solution, we systematically analyze the influence of the resonator parameters on the transverse mode. Theoretical analysis shows that the resonant parameters have a certain influence on the quality of transverse mode, and the analysis conclusion is proved by experiments. Compared with the numerical calculation method, the advantage of this method is that it can reduce a lot of calculation while ensuring the calculation accuracy. In addition, the diffraction integral equation is the basic equation in physics and mathematics. The method for solving the equation proposed in this paper can also be extended to other studies.

Funding

National Key Research and Development Program of China (2021YFB3601504); Scientific Instrument Developing Project of the Chinese Academy of Sciences (YZLY202001); National Natural Science Foundation of China (62105334, 61975208); Youth Innovation Promotion Association of the Chinese Academy of Sciences (2022303); Fujian Science & Technology Innovation Laboratory for Optoelectronic Information of China (2021ZR203, 2020ZZ108, 2021ZZ118); Project of Science and Technology of Fujian Province (2021H0047).

Disclosures

The authors declare that there are no conflicts of interest related to this article.

Data availability

The data of this paper are available from the corresponding authors upon reasonable request.

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The data of this paper are available from the corresponding authors upon reasonable request.

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Figures (8)
Fig. 1.
Fig. 1. Mirrors M1 and M2 are square mirrors with radius of curvature equal to R, L is cavity length. $F(x,y)$ and $F(x^{\prime},y^{\prime})$ are the transverse modes of mirrors M1 and M2, respectively.

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Fig. 2.
Fig. 2. Mode patterns of HG00,HG11,HG22 and F00, F11, F22 when A0= A1 = A2 = 0.1, g = -0.9.

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Fig. 3.
Fig. 3. Numerical simulations of F22 when A2 = 0.05, 0.1, 0.15, g = -0.9.

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Fig. 4.
Fig. 4. Simulation results of F22 when g = -0.9, -0.5, 0.5, 0.9, A2 = 0.15.

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Fig. 5.
Fig. 5. Simulation results of F22 when g = -0.1, 0.1, A2 = 0.15, and HG22.

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Fig. 6.
Fig. 6. Experimental setup for studying the relationship between laser mode and g parameter.

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Fig. 7.
Fig. 7. (a) and (b) show the experimental results and theoretical simulation results of the output mode varying with g at four different values of 0.35, 0.1, -0.35 and -0.5, respectively.

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Fig. 8.
Fig. 8. (a) Experimental output patterns when g = -0.1, -0.75, -0.9. (b) Calculated patterns with A2=0.12, A1 = 0.1 when g = -0.1, -0.75, -0.9.

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Equations (19)

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$$\gamma F(x,y) = \frac{i}{{\lambda L}}\exp ( - 2i\pi \frac{L}{\lambda })\int\limits_{ - a}^{ + a} {\int\limits_{ - a}^{ + a} {F(x^{\prime},y^{\prime})} } \exp \{ - \frac{{i\pi }}{{\lambda L}}[g({x^{\prime 2}} + {y^{\prime 2}}) + g({x^2} + {y^2}) - 2(xx^{\prime} + yy^{\prime})]\} dx^{\prime}dy^{\prime}$$
$$\gamma F(x,y) = \eta \int\limits_{ - a}^{ + a} {\int\limits_{ - a}^{ + a} {F(x^{\prime},y^{\prime})} } \exp \{ - i[g({x^{\prime 2}} + {y^{\prime 2}}) + g({x^2} + {y^2}) - 2(xx^{\prime} + yy^{\prime})]\} dx^{\prime}dy^{\prime}$$
$${\gamma _x}F(x) = \sqrt \eta \int\limits_{ - a}^{ + a} {F(x^{\prime})} \exp [ - i(g{x^{\prime 2}} + g{x^2} - 2xx^{\prime})]dx^{\prime}$$
$${\gamma _y}F(y) = \sqrt \eta \int\limits_{ - a}^{ + a} {F(y^{\prime})\exp [ - i(g{{y^{\prime}}^2} + g{y^2} - 2yy^{\prime})]dy^{\prime}} $$
$$\begin{aligned} {\gamma _x}F(x) &= \sqrt \eta \int\limits_{ - \infty }^{ + \infty } {F(x^{\prime})} \exp [ - i(g{{x^{\prime}}^2}{\,+\,}g{x^2}{\,-\,}2xx^{\prime})]dx^{\prime}{\,-\,}\sqrt \eta \int\limits_{ + a}^{ + \infty } {F(x^{\prime})} \exp [ - i(g{{x^{\prime}}^2}{\,+\,}g{x^2}{\,-\,}2xx^{\prime})]dx^{\prime}\\ &- \sqrt \eta \int\limits_{ - \infty }^{ - a} {F(x^{\prime})} \exp [ - i(g{{x^{\prime}}^2} + g{x^2} - 2xx^{\prime})]dx^{\prime} \end{aligned}$$
$${E_m}F(x) = \sqrt \eta \int\limits_{ - \infty }^{ + \infty } {F(x^{\prime})} \exp [ - i(g{x^{\prime 2}} + g{x^2} - 2xx^{\prime})]dx^{\prime}$$
$${E_n}F(y) = \sqrt \eta \int\limits_{ - \infty }^{ + \infty } {F(y^{\prime})\exp [ - i(g{{y^{\prime}}^2} + g{y^2} - 2yy^{\prime})]dy^{\prime}} $$
$$F(x) = \exp ( - \frac{{{x^2}}}{{{w^2}}}){H_m}(\sqrt 2 \frac{x}{w}),F(y) = \exp ( - \frac{{{y^2}}}{{{w^2}}}){H_n}(\sqrt 2 \frac{y}{w})$$
$${E_m} = \exp [ - \frac{{2i\pi }}{\lambda }L + i\psi (m + 0.5)],{E_n} = \exp [ - \frac{{2i\pi }}{\lambda }L + i\psi (n + 0.5)]$$
$$\begin{aligned} {\gamma _x}F(x) = &{E_m}H{G_m}(x) - \sqrt \eta \int\limits_{ + a}^{ + \infty } {F(x^{\prime})} \exp [ - i(g{{x^{\prime}}^2} + g{x^2} - 2xx^{\prime})]dx^{\prime}\\ & - \sqrt \eta \int\limits_{ - \infty }^{ - a} {F(x^{\prime})} \exp [ - i(g{{x^{\prime}}^2} + g{x^2} - 2xx^{\prime})]dx^{\prime} \end{aligned}$$
$$\begin{aligned} {\gamma _x}F^{\prime}(x) &= {E_m}H{{G^{\prime}}_m}(x) + \sqrt \eta \int\limits_{ + a}^{ + \infty } {2i(gx - x^{\prime})F(x^{\prime})} \exp [ - i(g{{x^{\prime}}^2} + g{x^2} - 2xx^{\prime})]dx^{\prime}\\ & + \sqrt \eta \int\limits_{ - \infty }^{ - a} {2i(gx - x^{\prime})F(x^{\prime})} \exp [ - i(g{{x^{\prime}}^2} + g{x^2} - 2xx^{\prime})]dx^{\prime} \end{aligned}$$
$$\begin{aligned} 2igx[{E_m}H{G_m}(x) - {\gamma _x}F(x)] &= \sqrt \eta 2igx\int\limits_{ + a}^{ + \infty } {F(x^{\prime})} \exp [ - i(g{{x^{\prime}}^2} + g{x^2} - 2xx^{\prime})]dx^{\prime}\\ & + \sqrt \eta 2igx\int\limits_{ - \infty }^{ - a} {F(x^{\prime})} \exp [ - i(g{{x^{\prime}}^2} + g{x^2} - 2xx^{\prime})]dx^{\prime} \end{aligned}$$
$$\scalebox{0.9}{$\begin{aligned} {E_m}H{G_m}^\prime (x) - {\gamma _x}F^{\prime}(x) &={-} 2igx[{E_m}H{G_m}(x) - {\gamma _x}F(x)] + \sqrt \eta \int\limits_{ + a}^{ + \infty } {2ix^{\prime}F(x^{\prime})} \exp [ - i(g{{x^{\prime}}^2} + g{x^2} - 2xx^{\prime})]dx^{\prime}\\ & + \sqrt \eta \int\limits_{ - \infty }^{ - a} {2ix^{\prime}F(x^{\prime})} \exp [ - i(g{{x^{\prime}}^2} + g{x^2} - 2xx^{\prime})]dx^{\prime} \end{aligned}$}$$
$$\sqrt \eta \int\limits_{ + a}^{ + \infty } {2ix^{\prime}F(x^{\prime})} \exp [ - i(g{x^{\prime 2}} + g{x^2} - 2xx^{\prime})]dx^{\prime} + \sqrt \eta \int\limits_{ - \infty }^{ - a} {2ix^{\prime}F(x^{\prime})} \exp [ - i(g{x^{\prime 2}} + g{x^2} - 2xx^{\prime})]dx^{\prime} \Rightarrow 0$$
$$[{E_m}H{G^{\prime}_m}(x) - {\gamma _x}F^{\prime}(x)] ={-} 2igx[{E_m}H{G_m}(x) - {\gamma _x}F(x)]$$
$$F(x) = H{G_m}(x) - {A_m}\exp ( - ig{x^2})$$
$${F_{mn}}(x,y) = [H{G_m}(x) - {A_m}\exp ( - ig{x^2})][H{G_n}(y) - {A_n}\exp ( - ig{y^2})]$$
$$H{G_{mn}}(x,y) = {H_m}(\sqrt 2 \frac{x}{w}){H_n}(\sqrt 2 \frac{y}{w})\exp ( - \frac{{{x^2} + {y^2}}}{{{w^2}}})$$
$${F_{mn}}(x,y) = [H{G_m}(x) - {A_m}\exp (0.9i{x^2})][H{G_n}(y) - {A_n}\exp (0.9i{y^2})]$$
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