Double wavefront reversal during six-wave interaction on Kerr nonlinearity in a waveguide with infinitely conducting surfaces

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Introduction

The presence of fifth-order nonlinear susceptibility in the medium makes it possible, when realizing six-wave interaction in the form of ${\omega }_1+{\omega }_1-{\omega }_1-$${\omega }_1+{\omega }_2={\omega }_2,$ to obtain a wave whose complex amplitude is proportional to the square of the complex-conjugate amplitude of the signal wave (a wave with a Doubled Reversed Wavefront (DRWF)) [1–2]. Wave generation with DRWF expands the possibilities of using six-wave radiation converters in the tasks of phase distortion correction, conversion of complex spatially inhomogeneous waves, etc. [3].

In order to increase the efficiency of multiwavelength converters, it is reasonable to switch from the consideration of interaction in transverse dimensionally unbounded media to the consideration of interaction in waveguides [4–6]. At the same time, when it comes to the transformation of complex spatially inhomogeneous fields, there is always the problem of the quality of the transformation, i.e. about the correspondence between the complex amplitudes of the signal wave and the wave with a doubled reversed wavefront [7–8]

Herein, we analyze the effect on the quality of the DRWF at six-wave interaction in a waveguide with infinite conducting surfaces filled with a medium with Kerr nonlinearity, the spatial structure of the pump waves, provided that the ratio of the wave numbers of the pump waves is a multiple of integer or semi-integer.

1. Principal Part

We assume that two pump waves with complex amplitudes $A_1$ and $A_2,$ frequencies ${\omega }_1$ and ${\omega }_2,$ and a signal wave with complex amplitude $A_3$ at frequency ${\omega }_1$ propagate in a waveguide located between the planes towards each other. Nonlinear polarization ${\left(A_1{A}_3^{\ast }\right)}^2{A}_2$ is induced in the medium, which is the source of an object wave with a complex amplitude $A_6$ at the frequency ${\omega }_2.$ The phase of the object wave is proportional to the doubled phase of the signal wave taken with a minus sign (a wave with a Doubled Reversed Wavefront).

In the approximation of a given field by pump waves, at a small conversion factor, without taking into account the change of the refractive index due to the self-interaction of pump waves, the amplitude of the object wave with DRWF at the front edge of the waveguide is [9].

$A_6(x,z=0)=\sum _{r=0}^{N_2}{a}_{6r}{\tilde{f}}_r(x,{\omega }_2)=$ (1)

$=-i\frac{g\mathcal{l}}{2}\sum _{r=0}^{N_2}\frac{{\tilde{f}}_r(x,{\omega }_2)}{{\beta }_r}\sum _{p=0}^{N_1}\sum _{p^{\text{'}}=0}^{N_1}\sum _{m=0}^{N_2}\sum _{s=0}^{N_1}\sum _{s^{\text{'}}=0}^{N_1}{a}_{1p}^{\left(0\right)}{a}_{1p^{\text{'}}}^{\left(0\right)}\times$

$\times a_{2m}^{\left(0\right)}{a_{3s}^{\left(0\right)}}^{\ast }{a_{3s^{\text{'}}}^{\left(0\right)}}^{\ast }{\gamma }_{p{p}^{\text{'}}ms{s}^{\text{'}}r}\sin с\left(\frac{{\Delta }_{p{p}^{\text{'}}ms{s}^{\text{'}}r}\mathcal{l}}{2}\right)\times$

$\times \exp \left(-i\frac{{\Delta }_{p{p}^{\text{'}}ms{s}^{\text{'}}r}\mathcal{l}}{2}\right).$

Here $g=\frac{30\pi {\omega }_2^2}{c^2}{\chi }^{\left(5\right)},$ ${\chi }^{\left(5\right)}$ is the nonlinear susceptibility of the fifth order; $a_{1p}^{\left(0\right)},$ $a_{3s}^{\left(0\right)}$ are the coefficients in the decomposition of the amplitudes of the first pump wave, the signal wave on the leading edge according to the modes of the waveguide; $a_{2m}^{\left(0\right)}$ is the coefficients in the decomposition of the amplitude of the second pump wave on the back face according to the modes of the waveguide; ${\tilde{f}}_r(x,{\omega }_{\mathrm{1,2}})$ and ${\beta }_r\left({\omega }_{\mathrm{1,2}}\right)$ are the transverse component and the propagation constant of the r-th mode of the waveguide; $N_1$ and $N_2$ are the number of waveguide modes at frequencies ${\omega }_1$ and ${\omega }_2,$ and taken into account when calculating the amplitude of a wave with a DRWF;

${\Delta }_{p{p}^{\text{'}}ms{s}^{\text{'}}r}={\beta }_p+{\beta }_{p^{\text{'}}}-$${\beta }_m-{{\beta }_s}^{\ast }-{{\beta }_{s^{\text{'}}}}^{\ast }+{\beta }_r$ is the wave detuning;

${\gamma }_{p{p}^{\text{'}}ms{s}^{\text{'}}r}=\int {\tilde{f}}_p\left(x,{\omega }_1\right){\tilde{f}}_{p^{\text{'}}}\left(x,{\omega }_1\right){\tilde{f}}_m\left(x,{\omega }_2\right)$${\tilde{f}}_s\left(x,{\omega }_1\right){\tilde{f}}_{s^{\text{'}}}\left(x,{\omega }_1\right){\tilde{f}}_r\left(x,{\omega }_2\right)dx$ is the overlap integral.

For a signal wave from a point source, $\left(A_3\right(x,z=0)=$$\delta (x-x_0),$ $x_0$ is the coordinate that determines the displacement of the point source relative to the axis of the waveguide) in the case of long waveguides, that is, we consider the condition $\mathrm{Re}\left({\Delta }_{p{p}^{\text{'}}ms{s}^{\text{'}}r}\right)\mathcal{l}\gg$ $1$ to be fulfilled if $\mathrm{Re}\left({\Delta }_{p{p}^{\text{'}}ms{s}^{\text{'}}r}\right)\ne$$\mathrm{0,}$ the expression for the amplitude of the object wave takes the form of

$G(x,x_0,z=0)=-i\frac{g\mathcal{l}}{2}\sum _{r=0}^{N_2}\frac{{\tilde{f}}_r(x,{\omega }_2)}{{\beta }_r}\times$ (2)

$\times \sum _{p=0}^{N_1}\sum _{p^{\text{'}}=0}^{N_1}\sum _{m=0}^{N_2}\sum _{s=0}^{N_1}{a}_{1p}^{\left(0\right)}{a}_{1p^{\text{'}}}^{\left(0\right)}{a}_{2m}^{\left(0\right)}{\tilde{f}}_s(x_0,{\omega }_1)\times$

$\times {\tilde{f}}_{s^{\text{'}}}(x_0,{\omega }_1){\gamma }_{p{p}^{\text{'}}ms{s}^{\text{'}}r}.$

As in the case of considering the quality of wavefront reversal in four-wave interaction [10–11], we will refer to the function $G(x,x_0,z=0)$ as the Point-Spread Function (PSF).

Note that while at four-wave interaction the knowledge of PSF fully describes the accuracy of RWF, the nonlinear nature of the relationship between the complex amplitude of the object wave and the complex amplitude of the signal wave allows using PSF to speak only qualitatively about the accuracy of DRWF.

As a waveguide we shall consider a two-dimensional waveguide with infinitely conducting surfaces located at a distance $2a$ from each other, filled with a medium with refractive index $n_1.$ The modes of such a waveguide are functions [12]

${\tilde{f}}_r\left(x\right)=\frac{1}{\sqrt{a}}\sin \left[\frac{\pi \left(r+1\right)}{2a}(x+a)\right].$ (3)

For the near-axis modes of the waveguide the propagation constant of the r-th mode is

${\beta }_r\left({\omega }_{\mathrm{1,2}}\right)\approx k_{\mathrm{1,2}}-\frac{1}{2k_{\mathrm{1,2}}}{\left[\frac{\pi \left(r+1\right)}{2a}\right]}^2,$ (4)

where $k_{\mathrm{1,2}}=\frac{{\omega }_{\mathrm{1,2}}}{с}{n}_1\left({\omega }_{\mathrm{1,2}}\right)$ is the wave number.

The condition of the six-wave interaction with DRWF in a long waveguide with infinite conducting surfaces is written as follows

$h\left[{\left(p+1\right)}^2+{\left(p^{\text{'}}+1\right)}^2-{\left(s+1\right)}^2-{\left(s^{\text{'}}+1\right)}^2\right]-$ (5)

$-\left[{\left(m+1\right)}^2-{\left(r+1\right)}^2\right]=0.$

Here $h=\frac{k_2}{k_1}.$

In the case when the ratio of the pump wave numbers is not equal to an integer or a ratio of integers, condition (5) is satisfied only when the mode number of the objective wave coincides with the mode number of the second pump wave $(r=m).$ This reduces the number of combinations of waveguide modes involved in the formation of the wave with the DRWF.

Fig. 1 shows characteristic normalized graphs of the dependence of the moduli of the blur functions of a point located on the axis of the waveguide $(\tilde{G}=\left|\frac{G\left(x,x_0=\mathrm{0,}z=0\right)}{G_{\max }}\right|,$ $G_{\max }$ – is the maximum

value of the function) on the normalized transverse coordinate, provided that the first pump wave is single–mode with the mode number $p=\mathrm{0,}$ and the amplitude of the second pump wave on the back face of the waveguide is described by the Gaussian function $A_2(x,z=\mathcal{l})=$$\exp (-x^2/b^2).$ Here $b$ is the pump wavelength.

 

Fig. 1. Dependence of the PSF modulus on the transverse coordinate at $h=2$ (a, b), 0,5 (c, d): the first pump wave is single-mode with the mode number $p=\mathrm{0,}$ the amplitude of the second pump wave changes according to the Gaussian law, $a/b=1$ (a, c), 8 (b, d)

Рис. 1. Зависимость модуля ФРТ от поперечной координаты при $h=2$ (а, б), 0,5 (в, г): первая волна накачки одномодовая с номером моды $p=\mathrm{0,}$ амплитуда второй волны накачки меняется по гауссову закону, $a/b=1$ (а, в), 8 (б, г)

 

Similar graphs of the dependence of the moduli of the point blur functions are observed when the second pump wave is single-mode with the mode number $m=\mathrm{0,}$ and the amplitude of the first pump wave on the leading edge of the waveguide is described by the Gaussian function $A_1(x,z=0)=$$\exp (-x^2/b^2).$

In calculating the PSF modulus, 20 modes of the signal wave, the first pump wave, and 20h modes of the object wave, the second pump wave, were considered.

Both when considering the Gaussian structure of the first pumping wave and when considering the Gaussian structure of the second pumping wave, the PSF modulus consists of central and side maxima. At a fixed frequency of the first pump wave, increasing the frequency of the second pump wave increases the number of side maxima, decreases the width of the central maximum. Changing the width of the Gaussian pump wave changes the ratio between the central and side maxima, affecting the width of the central maximum.

The quantitative parameters characterizing the quality of the DRWF may consist of the half-width of the central maximum of the PSF modulus $\left(\Delta x\right),$ determined from the solution of the equation

$\left|G\left(x=\Delta x,x_0=\mathrm{0,}z=0\right)\right|=$ (6)

$=\frac{1}{2}\left|G\left(x=a,x_0=\mathrm{0,}z=0\right)\right|,$

and the contrast of the PSF modulus, defined as the ratio of the value of the central maximum to the largest value of one of the side maximums (K).

It is obvious that the use of the half-width of the central maximum of the PSF modulus for the analysis of the quality of the DRWF assumes the presence of a pronounced central maximum. Otherwise, which is observed, for example, at $h=\mathrm{0,5}$ and $b>a,$ it is not possible to use this parameter to analyze the quality of DRWF.

Improving the quality of the DRWF involves both narrowing the central maximum and increasing the contrast of the PSF modulus.

In the case of $h=\mathrm{2,}$ with a decrease in the width of the second Gaussian pump wave under the condition of a single-mode first pump wave with the mode number $p=\mathrm{0,}$ the half-width of the central maximum of the PSF modulus changes slightly and increases monotonously with a decrease in the half-width of the first Gaussian pump wave under the condition of a single-mode second pump wave with the mode number $m=0$ (Fig. 2, a). In the range of variation of the width of the second Gaussian pump wave from $b=\mathrm{0,125}a$ to $b=\mathrm{1,25}a,$ the half-width of the central maximum of the PSF modulus varies from $\mathrm{0,043}a$ to $\mathrm{0,047}a,$ and when the width of the first pump wave is varied in this range, it increases from $\mathrm{0,041}a$ to $\mathrm{0,052}a.$

 

Fig. 2. Dependence of the half-width of the central maximum of the PSF module on the width of the Gaussian pump wave: the second pump wave is Gaussian, the first pump wave is single-mode (1); the first pump wave is Gaussian, the second pump wave is single-mode (2) at $h=2$ (a), 0,5 (b)

Рис. 2. Зависимость полуширины центрального максимума модуля ФРТ от ширины гауссовой волны накачки: вторая волна накачки гауссова, первая волна накачки одномодовая (1); первая волна накачки гауссова, вторая волна накачки одномодовая (2) при $h=2$ (а), 0,5 (б)

 

The contrast of the PSF modulus increases with decreasing width of the second Gaussian pump wave under the condition of single-mode first pump wave and changes weakly with changing width of the first Gaussian pump wave under the condition of single-mode second pump wave (Fig. 3, a).

 

Fig. 3. Dependence of the contrast of the PSF module on the width of the Gaussian pump wave: the second pump wave is Gaussian, the first pump wave is single-mode (1); the first pump wave is Gaussian, the second pump wave is single-mode (2) at $h=2$ (a), 0,5 (b)

Рис. 3. Зависимость контраста модуля ФРТ от ширины гауссовой волны накачки: вторая волна накачки гауссова, первая волна накачки одномодовая (1); первая волна накачки гауссова, вторая волна накачки одномодовая (2) при $h=2$ (а), 0,5 (б)

 

In the case of $h=\mathrm{0,5,}$ as the width of either the first or second Gaussian pump waves decreases, the half-width of the central maximum of the PSF modulus decreases (Fig. 2, b). In the range of variation $b$ from $\mathrm{0,125}a$ to $\mathrm{1,25}a,$ the half-width of the central maximum of the PSF modulus decreases from $\mathrm{0,26}a$ to $\mathrm{0,13}a,$ provided that the amplitude of the first pump wave varies according to the Gaussian law, from $\mathrm{0,26}a$ to $\mathrm{0,10}a,$ provided that the amplitude of the second pump wave varies according to the Gaussian law.

The contrast of the PSF modulus in the range of variation $b$ from $\mathrm{0,125}a$ to $\mathrm{1,25}a,$ with decreasing width of the second Gaussian pump wave, increases and comes to a constant value equal to 3,46. As the width of the first Gaussian pump wave decreases, the contrast of the PSF modulus increases at first, reaches a maximum value of 5,41, and then slowly decreases (Fig. 3, b).

The maximum value of the contrast change rate, the half-width of the central maximum of the PSF modulus for both $h=\mathrm{2,}$ and $h=\mathrm{0,5,}$ is observed in the range of variation in the width of Gaussian pump waves from $b=\mathrm{0,3}a$ to $b=2a.$ With a further decrease in the width of the Gaussian pump wave, the rates of change of both the half-width of the central maximum and the contrast of the PSF modulus decrease significantly and, in some cases, tend to zero. Thus, the main influence of the Gaussian pump wave width on the quality of the DRWF is observed in the range of pump wave widths from $b=\mathrm{0,3}a$ to $b=2a.$

Conclusion

At the ratio of wave numbers of the second and first pump waves equal to 2 and 0,5, the quality of the doubled wavefront reversal of a signal wave from a point source at six-wave interaction on the Kerr nonlinearity in a waveguide with infinite conducting walls is analyzed.

We demonstrated that the maximum change in the characteristics of the wave with DRWF is observed when changing the width of Gaussian pump waves from $b=\mathrm{0,3}a$ to $b=2a.$ In this range, at $h=\mathrm{0,5,}$ a decrease in the widths of the Gaussian pump waves leads both to a decrease in the half-width of the central maximum and to an increase in the contrast of the PSF modulus. At $h=\mathrm{2,}$ a decrease in the width of the first Gaussian pump wave under the condition of a single-mode second pump wave increases the half-width of the central maximum and weakly affects the contrast of the PSF modulus. The quality of the doubled wavefront reversal of the signal wave from a point source located at the leading edge of the waveguide deteriorates with decreasing width of the first Gaussian pump wave.

About the authors

Valery V. Ivakhnik

Samara National Research University

Email: ivakhnik@ssau.ru
SPIN-code: 2791-2471
ResearcherId: AAH-3584-2019

Doctor of Physical and Mathematical Sciences, professor, head of the Department of Optics and Spectroscopy

Russian Federation, 34, Moskovskoye shosse, Samara, 443086

Darkhan R. Kapizov

Samara National Research University

Email: darkhankapizov@gmail.com
SPIN-code: 6771-2032

4th year postgraduate of the Department of Optics and Spectroscopy

Russian Federation, 34, Moskovskoye shosse, Samara, 443086

Vladimir I. Nikonov

Samara National Research University

Author for correspondence.
Email: nikon5919@mail.ru
SPIN-code: 1965-3523
ResearcherId: S-8516-2016

Candidate of Physical and Mathematical Sciences, associate professor of the Department of Optics and Spectroscopy

Russian Federation, 34, Moskovskoye shosse, Samara, 443086

References

Supplementary files