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Z-scan analytical solutions for thin media with high order refractive nonlinearity and multiphoton absorption

J. J. Mendez Rodriguez, M. L. Arroyo Carrasco, R. Torres Romero, M. M. Mendez Otero, B. A. Martinez Irivas, and M. D. Iturbe Castillo

Open Access Open Access

Abstract

Considering that a thin media presents solely one type of nonlinearity, we obtain an analytical solution for far field on-axis Z-scan detection; a second solution is obtained for open aperture case. To find the analytical solutions, we propose a form for the nonlinear high order refractive and multiphoton absorption of the media. Z-scan curves are presented where the nonlinear high order refractive and multiphoton absorption are varied.

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

1. Introduction

The Z-scan technique, proposed by Sheik-Bahae et al. [1], is widely used to obtain the nonlinear refractive and absorptive index of different media. The first theoretical analysis of the technique [2] was limited to thin samples with small nonlinear refractive or absorptive phase changes (< 0.1 radian) where the electric field at the position of the aperture was calculated using the Gaussian decomposition method [3]. Since then many improvements have been done to the theory in order to eliminate the main restriction of the first proposal. A very complete review of different theoretical treatments of the Z-scan technique can be found in Ref. [4]. Additional to this review, it is important to mention two on axis detection works: in [5] an analytical formula based on the incomplete Gamma function was obtained for materials with solely refractive nonlinearity and in [6] another analytical formula was obtained based on the hypergeometric function but for materials with both refractive and absorptive nonlinearity. For the open aperture Z-scan technique different formulas has been presented using different functions [7–12]. The most general formula was given by Gu et al. [13], where in particular they present the functions that describe two and three photon absorption.

In this paper we propose a form for the simultaneous nonlinear high order refractive and multiphoton absorption of the same order, that can present a thin media which allows to obtain analytical solutions for the on-axis normalized transmittance and the total transmittance of the Z-scan technique (close and open Z-scan, respectively).

2. Model

We are going to consider a Gaussian beam of wavelength λ, amplitude E0, beam waist w0 and ignoring the phase changes independent of transversal coordinate, of the form:

$$E(z )= {E_0}\frac{{{W_0}}}{{W(z )}}\exp \left[ { - \frac{{{r^2}}}{{{W^2}(z )}} - i\frac{{k{r^2}}}{{2R(z )}}} \right], $$
where;
$${w^2}(z )= w_0^2\left( {1 + {{\left( {\frac{z}{{{z_0}}}} \right)}^2}} \right),\;R(z )= z\left( {1 + {{\left( {\frac{{{z_0}}}{z}} \right)}^2}} \right)\quad \textrm{and}\quad {z_0} = \frac{{\pi W_0^2}}{\lambda }.$$

This Gaussian beam illuminates a thin sample of thickness $L < < {z_0}$, with linear absorption coefficient α0, nonlinear refractive coefficient γ and nonlinear absorptive coefficient β. Under the slowly varying envelope approximation, the intensity and the phase inside the nonlinear media must fulfill the following dependences:

$$\frac{{\partial \Delta \phi }}{{\partial z^{\prime}}} = {k_0}\gamma {I^{m/2}}, $$
$$\frac{{\partial I}}{{\partial z^{\prime}}} ={-} {\alpha _0}I - \beta {I^{\left( {\frac{m}{2} + 1} \right)}}. $$
where k0 is the wavenumber in the vacuum, z’ is the propagation distance inside the media and $m \ne 0$ is an integer even number related to the order of the nonlinearity. Solving Eq. (4) we obtain that:
$$I({z^{\prime}} )= \frac{{{I_{in}}\textrm{exp}({ - {\alpha_0}z^{\prime}} )}}{{{{[{1 + \beta I_{in}^{m/2}{L_{efm}}} ]}^{2/m}}}}, $$
where Iin is the incident intensity on the sample, m/2 + 1 is the number of absorbed photons, β is the multiphoton absorption coefficient, and
$${L_{efm}} = \frac{{1 - \textrm{exp}\left( { - {\alpha_0}\frac{m}{2}z^{\prime}} \right)}}{{{\alpha _0}}}. $$

By integrating Eq. (3), we obtain;

$$\mathrm{\Delta }\phi ({z^{\prime}} )= \frac{{{k_0}\gamma }}{{\frac{m}{2}\beta }}\ln ({1 + \beta I_{in}^{m/2}{L_{efm}}} ).$$

Then the electric field at the output face of the thin nonlinear sample can be written as:

$${E_{out}} = {E_{in}}\exp \left( { - \frac{{{\alpha_0}L}}{2}} \right){[{1 + \beta I_{in}^{m/2}{L_{efm}}} ]^{\left( { - i\left( {\frac{{2{k_0}\gamma }}{{m\beta }}} \right)\; - \; \frac{1}{m}} \right)}}, $$
where ${E_{in}}$ is the electric field at the input face of the sample. Note that Eq. (8) reproduce that already reported in Ref. [2], when m = 2.

Using the same formalism described in [6], it is calculated the on axis field at the position of the aperture, at a distance d, as the following integral;

$${E_{ap}}(z )= ik/dexp ({ - ikd} )\mathop \smallint \nolimits_0^\infty {E_{out}}({z,r} )\exp \left( {\frac{{ik}}{{2d}}{r^2}} \right)rdr, $$
substituting Eqs. (1) and (2) in Eq. (9) and with changes of variable, the on axis electric field at the aperture plane can be written as:
$${E_{ap}}(z )= \mathrm{\Lambda }\mathop \smallint \nolimits_0^\infty \exp ({ - a{r^2}} ){[{1 + {q_0}\exp ({ - b{r^2}} )} ]^{\left( {\left( { - i\frac{{2\mathrm{\Delta }{\mathrm{\Phi }_0}}}{{m\mathrm{\Delta }{\psi_0}}}} \right) - \frac{1}{m}} \right)}}rdr, $$
where
$$a = \frac{1}{{{w^2}(z )}} + i\frac{k}{2}\left[ {\frac{1}{{R(z )}} + \frac{1}{d}} \right], $$
$$b = \frac{m}{{{w^2}(z )}}, $$
$$\mathrm{\Lambda } = \frac{{i{k_0}{E_0}{w_0}}}{{dw(z )}}\exp ({ - {\alpha_0}L/2} )\exp ({ - i{k_0}d} ), $$
$${q_0}(z )= \frac{{\mathrm{\Delta }{\mathrm{\Psi }_0}}}{{{{\left( {1 + {{\left( {\frac{z}{{{z_0}}}} \right)}^2}} \right)}^{\frac{m}{2}}}}}, $$
$$\mathrm{\Delta }{\mathrm{\Phi }_0} = \mathrm{k\gamma }I_0^{m/2}{L_{efm}}, $$
$$\mathrm{\Delta }{\mathrm{\Psi }_0} = \beta I_0^{m/2}{L_{efm}}, $$
ΔΦ0 and ΔΨ0 are the maximum on-axis nonlinear refractive and absorptive changes, respectively. With the aid of the hypergeometric function F [6,14], the on axis electric field at the aperture plane can be expressed as;
$${E_a}p(z )= \frac{\mathrm{\Lambda }}{{2a}}F\left[ {i\frac{{2\mathrm{\Delta }{\mathrm{\Phi }_0}}}{{m\mathrm{\Delta }{\mathrm{\Psi }_0}}} + \frac{1}{m},\frac{a}{b};\frac{a}{b} + 1; - {q_0}(z )} \right], $$
then, the on axis normalized transmittance is given by:
$$T(z )= {\left|{F\left[ {i\frac{{2\mathrm{\Delta }{\mathrm{\Phi }_0}}}{{m\mathrm{\Delta }{\mathrm{\Psi }_0}}} + \frac{1}{m},\frac{a}{b};\frac{a}{b} + 1; - {q_0}(z )} \right]} \right|^2}. $$

Note that, when m = 2 this expression is the same to that of Ref. [6]. In that paper they also made a comparison with other models demonstrating that under the small phase variation approximation this expression reproduces the known formulas. In this paper we are going to solve directly Eq. (18) for different values of the m parameter without restrictions in the magnitude of the nonlinear phase variations.

3. Analytical results

As it was mention in the last section when m = 2 then Eq. (18) is the same to that obtained in [6]. The graphs presented in that paper are also obtained by us and we are not going to repeat here. However, in order to demonstrate that the obtained expression can reproduce results for large phase shifts, in Fig. 1(a) we plot Eq. (18) for a medium with pure third order nonlinearity (m = 2) and refractive phase change ΔΦ0, as large as 4π rads. The results reproduce recently reported analytical expressions [4]. In Figs. 1(b)–1(d), Eq. (18) is plotted considering the influence of a positive nonlinear absorption for different magnitudes of the nonlinear refraction. We can observe that as the ratio between refractive/absorptive nonlinearity increases the influence of the nonlinear absorption is larger in the peak.

 figure: Fig. 1.

Fig. 1. Z-scan on axis normalized transmittance for a medium with a pure third order nonlinearity (m = 2), a) phase change ΔΦ0 of π/2 (green), 3π/2 (red) and 4π (blue). Influence of the nonlinear absorption ΔΨ0 with magnitudes of 0 (black), 0.3 (blue), 0.6 (red) and 0.9 (green) for b) ΔΦ0= π/2, c) ΔΦ0= 3π/2 and d) ΔΦ0= 4π.

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In order to illustrate the Z-scan curve changes for materials with larger nonlinearity, in Fig. 2 we plot the results obtained for a media with a fifth order refractive nonlinearity (m = 4) and the same magnitude of the refractive and absorptive nonlinearity as in the previous case. We can observe in Fig. 2(a) that the amplitude of the Z-scan curve is reduced even when the same magnitudes, of the phase changes, for the third order nonlinearity were used. Then as the order of the refractive nonlinearity is increased the amplitude of the Z-scan curve is reduced for the same magnitude of the nonlinear phase change. The influence of the nonlinear absorption in the Z-scan curves follows a similar behavior that in the previous case.

 figure: Fig. 2.

Fig. 2. Z-scan on axis normalized transmittance for a medium with a fifth order nonlinearity (m = 4), a) phase change ΔΦ0 of π/2 (green), 3π/2 (red) and 4π (blue). Influence of the nonlinear absorption ΔΨ0 with magnitudes of 0 (black), 0.3 (blue), 0.6 (red) and 0.9 (green) for b) ΔΦ0= π/2, c) ΔΦ0= 3π/2 and d) ΔΦ0= 4π.

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The peak-valley normalized transmittance difference (ΔTP-V) behavior as a function of the nonlinear phase shift ΔΦ0 is plotted in Fig. 3(a), for different values of m. ΔTP-V increases linearly for small ΔΦ0 and increases smoothly for large ΔΦ0. For m = 2 (black), third order nonlinearity case, it is observed that for ΔΦ0 < π, ΔTP-V increases linearly and for ΔΦ0 > π, ΔTP-V increases smoothly. For m = 4 (blue), fifth order nonlinearity case and m = 6 (red), seventh order nonlinearity case, both curves tend to behave with a very slow increase of ΔTP-V as ΔΦ0 increases. This occurs because the effect of the nonlinear optical response decreases as the order of the optical nonlinearity increases, which is reasonable considering that for high-order effects to occur, a greater amount of energy is involved in these non-linear optical processes. In Fig. 3(b) it is noted that the peak–valley separation distance ΔzP-V is almost constant when ΔΦ0 is small and increases smoothly for ΔΦ0 > 1.5 π, the increase being slower when the number of absorbed photons is greater. The curves presented in Fig. 3 were obtained for ΔΨ0 = 0.

 figure: Fig. 3.

Fig. 3. a) Peak-valley normalized transmittance difference ΔTP-V as function of the nonlinear phase shift ΔΦ0/π, b) peak–valley separation distance ΔzP-V/z0 as function of the nonlinear phase shift ΔΦ0/π for m: 2 (black), 4 (blue) and 6 (red); for ΔΨ0 = 0.

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4. Open aperture Z-scan

In order to obtain an analytical expression for the case of purely multi-photon absorption in this section we will calculate the total transmittance for the open aperture Z-scan technique and we will see that the proposed model reproduce alredy published results. For a medium of length L it is possible to calculate the total transmitted power PT, through the thin nonlinear sample as:

$${P_T}(z )= 2\pi \mathop \smallint \nolimits_0^\infty {I_{out}}({z,r} )rdr, $$
substituting Iout from Eq. (5), evaluated in z’=L, the total transmitted power can be written as:
$${P_T}(z )= {\mathrm{\Lambda }_1} \mathop \smallint \nolimits_0^\infty \exp ({ - {a_1}{r^2}} ){[{1 + {q_0}(z )\exp ({ - b{r^2}} )} ]^{\left( { - \frac{2}{m}} \right)}}rdr, $$
where
$${a_1} = \frac{2}{{{w^2}(z )}},$$
$${\mathrm{\Lambda }_1} = \frac{{2\pi {I_0}w_0^2}}{{{w^2}(z )}}\exp ({ - {\alpha_0}L} ). $$
with b, q0(z) and $\Delta {\Psi _0}$ as in Eqs. (12), (14) and (16). Then the total transmitted power through the thin nonlinear sample can be expressed as:
$${P_T}(z )= \frac{{{\mathrm{\Lambda }_1}}}{{2{a_1}}}F\left[ {\frac{2}{m},\frac{2}{m};\frac{2}{m} + 1; - {q_0}(z )} \right]. $$

Thus, the open aperture Z-scan normalized transmittance for (m/2 + 1)-photon absorption is given by:

$$T(z )= F\left[ {\frac{2}{m},\frac{2}{m};\frac{2}{m} + 1; - {q_0}(z )} \right]. $$

Note that Eq. (24) is the same to that obtained in [13], if our parameter m/2 is identify with n of that paper. Depending on the values of m/2 Eq. (24) can take the form of more simple functions, as it was demonstrated in [13].

In Fig. 4 we show the behavior of the open aperture Z-scan normalized transmittance for different values of the parameter m/2. For two photon absorption (black curve), three photon absorption (red curve) and four photon absorption (blue curve), can be observed that the width and normalized transmittance decrease as the number of absorbed photons increases. This behavior indicates that the nonlinear optical response due to multiphoton absorption is strongly diminished when the number of absorbed photons is large. Similar behavior was obtained in [9].

 figure: Fig. 4.

Fig. 4. Open aperture Z-scan normalized transmittance with ΔΨ0 = 0.9 and the following values of the parameter m: 2 (black), 4 (red) and 6 (blue).

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In Fig. 5(a) we plot the open aperture Z-scan normalized transmittance with m = 2 and different values of ΔΨ0 and in Fig. 5(b) how the normalized transmittance at z = 0 changes as a function of ΔΨ0.

 figure: Fig. 5.

Fig. 5. Open aperture Z-scan normalized transmittance for a medium with two photon absorption (m = 2), a) nonlinear absorption magnitude ΔΨ0 of: 0.3 (black), 0.6(blue), 0.9 (red) and 1.2 (green), b) normalized transmittance changes at z = 0, as a function of ΔΨ0.

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In Fig. 6(a) we plot the open aperture Z-scan normalized transmittance with m = 4 and different values of ΔΨ0 and in Fig. 6(b) how the normalized transmittance at z = 0 changes as a function of ΔΨ0.

 figure: Fig. 6.

Fig. 6. Open aperture Z-scan normalized transmittance for a medium with three photon absorption (m = 4), a) nonlinear absorption magnitude ΔΨ0 of: 0.3 (black), 0.6(blue), 0.9 (red) and 1.2 (green), b) normalized transmittance changes at z = 0, as a function of ΔΨ0.

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The behavior obtained in Figs. 5 and 6 is the same to that reported in [13]. The advantage to considers any absorption process separately is that one can evaluate, by comparison with experimental results, what process has the main contribution. As it was mentioned in that paper, it is possible to identify the number of absorbed photons by the analyzed sample making a comparison with the reported expression.

From the comparison of our results with other treatments for Z-scan, solely for the refractive case, we notice that the behavior presented in Fig. 3 for a fifth order nonlinearity was the same to that reported for media with a spatial nonlocal response characterized with m = 4 [15]. There is a factor of 2 that it is necessary to consider for ΔΦ0. This coincidence suggested that our formula for high order refractive nonlinearity can be used to calculate Z-scan curves for nonlocal thin media too. In order to probe this assumption, we choose to evaluate Eq. (18) for m = 1 and different values of ΔΦ0 with ΔΨ0 = 0. The results obtained are presented in Figs. 7 and 8.

 figure: Fig. 7.

Fig. 7. Z-scan on axis normalized transmittance obtained evaluating Eq. (1)8 with m = 1, ΔΨ0 = 0 and phase change ΔΦ0 of: π/4 (green), π/2 (red) and π (blue).

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 figure: Fig. 8.

Fig. 8. a) Peak-valley transmittance difference ΔTP-V as function of ΔΦ0/π, b) Normalized peak–valley position difference ΔzP-V/z0 as function of ΔΦ0/π for m = 1.

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Note that, for this case, the amplitude of the peak in the Z-scan curve grows very fast compared with the previous cases. For this reason, the maximum ΔΦ0 used in Fig. 7 was of π rads. The peak-valley transmittance difference and position difference as a function of ΔΦ0 are plotted in Fig. 8. The behavior presented in these two graphs is the same to that reported in [15], where there is a factor of 2 in the x-axis. This difference is due to the definition of Lefm in this paper and Leff in that paper used to define ΔΦ0. A nonlocality characterized by values of m < 2 means that the nonlinear phase changes extend beyond the illuminated area. It is also possible to evaluate Eq. (18) using no integer values of m, however this study is not presented here.

5. Conclusions

In this paper we have presented a form of the output field for a thin medium that can present high order refractive nonlinearity and multiphoton absorption. This form allows to obtain an analytical expression for the far-field on axis Z-scan normalized transmittance and an analytical expression for the open aperture Z-scan normalized transmittance. Evaluations of these formulas were made for a medium with a third or fifth order nonlinearity. The formulas can be used to predict Z-scan curves of thin media with high order nonlinearity, besides the formula for on-axis detection, but for solely nonlinear refraction, can be used to describe the influence of the nonlocality.

Disclosures

The authors declare no conflicts of interest.

Data availability

No data were generated or analyzed in the presented research, only Eqs. (18) and (24) where plotted.

References

1. M. Sheik Bahae, A. A. Said, and E. W. Van Stryland, “High sensitive single beam n2 measurements,” Opt. Lett. 14(17), 955–957 (1989). [CrossRef]  

2. M. Sheik-Bahae, A. A. Said, T. H. Wei, et al., “Sensitive measurement of optical nonlinearities using a single beam,” IEEE J. Quantum Electron. 26(4), 760–769 (1990). [CrossRef]  

3. D. Weaire, B. S. Wherrett, D. A. B. Miller, et al., “Effect of low-power nonlinear refraction on laser beam propagation in InSb,” Opt. Lett. 4(10), 331–333 (1979). [CrossRef]  

4. E. Jaatinen, D. Namarathne, and R. Donaldson, “Analytic solutions to closed and eclipsing aperture Z-scan power transmission,” J. Opt. Soc. Am. B 39(4), 1130–1140 (2022). [CrossRef]  

5. R. E. Samad and N. Diaz Vieira, “Analytical description of z-scan on-axis intensity based on Huygens-Fresnel principle,” J. Opt. Soc. Am. 15(11), 2742 (1998). [CrossRef]  

6. F.-Q. Li, X.-F. Zhang, F. Yang, et al., “Analytical solution of on-axis beam propagation for Z-scan technique,” J. Opt. Soc. Am. B 26(11), 2125–2130 (2009). [CrossRef]  

7. P. B. Chaple, J. Staromlynska, J. A. Hermman, et al., “Single-beam z-scan: Measurements, technique and analysis,” J. Nonlinear Opt. Phys. Mater. 06(03), 251–293 (1997). [CrossRef]  

8. C. H. Kwak, Y. L. Lee, and S. G. Kim, “Analysis of asymmetric z-scan measurements for large optical nonlinearities in an amorphous As 2 S 3 thin film,” J. Opt. Soc. Am. B 16(4), 600 (1999). [CrossRef]  

9. D. S. Corrêa, L. De Boni, L. Misoguti, et al., “Z-scan theoretical analysis for tree-, four- and five-photon absorption,” Opt. Commun. 277(2), 440–445 (2007). [CrossRef]  

10. B. Gu, X. Q. Huang, S. Q. Tam, et al., “Z-scan analytical theories, for characterizing multiphoton absorbers,” Appl. Phys. B: Lasers Opt. 95(2), 375–381 (2009). [CrossRef]  

11. B. Gu, K. Lou, J. Chen, et al., “Determination of the nonlinear refractive index in multiphoton absorbers by Z-scan measurements,” J. Opt. Soc. Am. B 27(11), 2438–2442 (2010). [CrossRef]  

12. F. Kessi and N. Hamadouche, “Z-scan theoretical model for simultaneous n and (n+1)-photon absorption,” J. Nonlinear Opt. Phys. Mater. 28(03), 1950022 (2019). [CrossRef]  

13. B. Gu, J. Wang, J. Chen, et al., “Z-scan theory for material with two- and three- photon absorption,” Opt. Express 13(23), 9230–9234 (2005). [CrossRef]  

14. M. Abramowitz and I. A. Stegun, “Handbook of mathematical functions,” 555–565 (1972).

15. E. V. Garcia Ramirez, M. L. Arroyo Carrasco, M. M. Mendez Otero, et al., “Z-scan and spatial self-phase modulation of a Gaussian beam in a thin nonlocal nonlinear media,” J. Opt. 13(8), 085203 (2011). [CrossRef]  

Data availability

No data were generated or analyzed in the presented research, only Eqs. (18) and (24) where plotted.

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Figures (8)
Fig. 1.
Fig. 1. Z-scan on axis normalized transmittance for a medium with a pure third order nonlinearity (m = 2), a) phase change ΔΦ0 of π/2 (green), 3π/2 (red) and 4π (blue). Influence of the nonlinear absorption ΔΨ0 with magnitudes of 0 (black), 0.3 (blue), 0.6 (red) and 0.9 (green) for b) ΔΦ0= π/2, c) ΔΦ0= 3π/2 and d) ΔΦ0= 4π.

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Fig. 2.
Fig. 2. Z-scan on axis normalized transmittance for a medium with a fifth order nonlinearity (m = 4), a) phase change ΔΦ0 of π/2 (green), 3π/2 (red) and 4π (blue). Influence of the nonlinear absorption ΔΨ0 with magnitudes of 0 (black), 0.3 (blue), 0.6 (red) and 0.9 (green) for b) ΔΦ0= π/2, c) ΔΦ0= 3π/2 and d) ΔΦ0= 4π.

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Fig. 3.
Fig. 3. a) Peak-valley normalized transmittance difference ΔTP-V as function of the nonlinear phase shift ΔΦ0/π, b) peak–valley separation distance ΔzP-V/z0 as function of the nonlinear phase shift ΔΦ0/π for m: 2 (black), 4 (blue) and 6 (red); for ΔΨ0 = 0.

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Fig. 4.
Fig. 4. Open aperture Z-scan normalized transmittance with ΔΨ0 = 0.9 and the following values of the parameter m: 2 (black), 4 (red) and 6 (blue).

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Fig. 5.
Fig. 5. Open aperture Z-scan normalized transmittance for a medium with two photon absorption (m = 2), a) nonlinear absorption magnitude ΔΨ0 of: 0.3 (black), 0.6(blue), 0.9 (red) and 1.2 (green), b) normalized transmittance changes at z = 0, as a function of ΔΨ0.

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Fig. 6.
Fig. 6. Open aperture Z-scan normalized transmittance for a medium with three photon absorption (m = 4), a) nonlinear absorption magnitude ΔΨ0 of: 0.3 (black), 0.6(blue), 0.9 (red) and 1.2 (green), b) normalized transmittance changes at z = 0, as a function of ΔΨ0.

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Fig. 7.
Fig. 7. Z-scan on axis normalized transmittance obtained evaluating Eq. (1)8 with m = 1, ΔΨ0 = 0 and phase change ΔΦ0 of: π/4 (green), π/2 (red) and π (blue).

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Fig. 8.
Fig. 8. a) Peak-valley transmittance difference ΔTP-V as function of ΔΦ0/π, b) Normalized peak–valley position difference ΔzP-V/z0 as function of ΔΦ0/π for m = 1.

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Equations (24)

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E ( z ) = E 0 W 0 W ( z ) exp ⁡ [ − r 2 W 2 ( z ) − i k r 2 2 R ( z ) ] ,
w 2 ( z ) = w 0 2 ( 1 + ( z z 0 ) 2 ) , R ( z ) = z ( 1 + ( z 0 z ) 2 ) and z 0 = π W 0 2 λ .
∂ Δ ϕ ∂ z ′ = k 0 γ I m / 2 ,
∂ I ∂ z ′ = − α 0 I − β I ( m 2 + 1 ) .
I ( z ′ ) = I i n exp ( − α 0 z ′ ) [ 1 + β I i n m / 2 L e f m ] 2 / m ,
L e f m = 1 − exp ( − α 0 m 2 z ′ ) α 0 .
Δ ϕ ( z ′ ) = k 0 γ m 2 β ln ⁡ ( 1 + β I i n m / 2 L e f m ) .
E o u t = E i n exp ⁡ ( − α 0 L 2 ) [ 1 + β I i n m / 2 L e f m ] ( − i ( 2 k 0 γ m β ) − 1 m ) ,
E a p ( z ) = i k / d e x p ( − i k d ) ∫ 0 ∞ ⁡ E o u t ( z , r ) exp ⁡ ( i k 2 d r 2 ) r d r ,
E a p ( z ) = Λ ∫ 0 ∞ ⁡ exp ⁡ ( − a r 2 ) [ 1 + q 0 exp ⁡ ( − b r 2 ) ] ( ( − i 2 Δ Φ 0 m Δ ψ 0 ) − 1 m ) r d r ,
a = 1 w 2 ( z ) + i k 2 [ 1 R ( z ) + 1 d ] ,
b = m w 2 ( z ) ,
Λ = i k 0 E 0 w 0 d w ( z ) exp ⁡ ( − α 0 L / 2 ) exp ⁡ ( − i k 0 d ) ,
q 0 ( z ) = Δ Ψ 0 ( 1 + ( z z 0 ) 2 ) m 2 ,
Δ Φ 0 = k γ I 0 m / 2 L e f m ,
Δ Ψ 0 = β I 0 m / 2 L e f m ,
E a p ( z ) = Λ 2 a F [ i 2 Δ Φ 0 m Δ Ψ 0 + 1 m , a b ; a b + 1 ; − q 0 ( z ) ] ,
T ( z ) = | F [ i 2 Δ Φ 0 m Δ Ψ 0 + 1 m , a b ; a b + 1 ; − q 0 ( z ) ] | 2 .
P T ( z ) = 2 π ∫ 0 ∞ ⁡ I o u t ( z , r ) r d r ,
P T ( z ) = Λ 1 ∫ 0 ∞ ⁡ exp ⁡ ( − a 1 r 2 ) [ 1 + q 0 ( z ) exp ⁡ ( − b r 2 ) ] ( − 2 m ) r d r ,
a 1 = 2 w 2 ( z ) ,
Λ 1 = 2 π I 0 w 0 2 w 2 ( z ) exp ⁡ ( − α 0 L ) .
P T ( z ) = Λ 1 2 a 1 F [ 2 m , 2 m ; 2 m + 1 ; − q 0 ( z ) ] .
T ( z ) = F [ 2 m , 2 m ; 2 m + 1 ; − q 0 ( z ) ] .
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