Резюме. In present work the parametric interaction of two optical waves presented by their slowly varying amplitude functions \(A_{1}(z)\) and \(A_{2}(z)\) in isotropic nonlinear medium is investigated. The corresponding system of nonlinear amplitude differential equations describes the processes of third-harmonic generation and THz generation in femtosecond region. It is found analytical solution of this system, characterizing the periodic exchange of energy between the two waves. The obtained result can be used for investigating different parametric processes in nonlinear dielectric medium, depending on the value of the wavenumber mismatch \(D k\). These effects are directly related to modern communication systems and optical sensors.

Ключови думи: parametric interaction; narrow-band optical pulses; third-harmonic generation; four-wave mixing

Physics is an Ever Young Science, Varna, October, 27 – 29, 2017

Физиката – вечно млада наука, Варна, 27 – 29 октомври 2017 г.

Introduction

High harmonic generation (HHG) is a nonlinear effect that is able to produces light in visible and ultraviolet regions of the spectrum. In standard silica single mode fibers, the efficiency of Third-Harmonic Generation (THG) is typically low, because of the difficulties in the realization of phase-matching conditions. Special kinds of fibers are needed, e.g. \(\mathrm{SiO}_{2}\) waveguides doped with germanium, erbium and nitrogen. In this case the wave number mismatch \(\Delta k\) is low and THG is easily observed. There are different techniques for reducing wave number mismatch \(\Delta k\) presented by Agrawal (2007), Boyd (2007) and Dmitriev & Tarasov (1982). This phenomenon has been observed through the years in many experiments (Gabriagues, 1983); Österberg, 1990; Nicácio et al. , 1993; Thøgersen & Mark,1994); Hickmann et al.,1995; De Araujo et al.,1996; Bufetov et al. , 1997). The third-order susceptibility \(\chi^{(3)}\) of the medium arises not only parametric THG effect but also leads to others nonlinear parametric effect such as Four-Wave Mixing (FWM). IF in addition in interaction between the waves we get in mind the carrier to envelope frequency, the wave number mismatch \(\Delta k\) becomes minimal and new spectral components with THz spectral shift \((22-26 \mathrm{~nm})\) to the short wavelengths, insteadTHG, are generated. Thus, by similar process the four-photon parametric mixing process between the waves generates coherent THz emission in glasses. In quasi-cw regime, when time derivatives can be set to zero, the corresponding system of evolutionary equations for THz generation is the same as that for THG one. We solve this system analytically. The obtained solutions, in the form of Jacobi elliptic functions, describe exchange of energy between the pulses.

Basic equation

The parametric evolution of two waves on different frequencies, for which the THG condition

(1) \[ \omega_{2}=3 \omega_{1}, \]

or the THz generation conditions

(2) \[ \omega_{2}=3 k_{1}\left(v_{p h}-v_{g r}\right), \]

where \(k_{1}\) is the wave number, \(v_{p h}\) and \(v_{g r}\) are the phase and group velocity of the basic (\(A_{l}\) ) wave, are satisfied can be govern by the following system of differential equations for their amplitude functions \(A_{1}(z)\) and \(A_{2}(z)\) :

(3) \[ i \tfrac{\partial A_{1}}{\partial z}=\gamma_{1} A_{1}^{* 2} A_{2} e^{-i \Delta k z} ; i \tfrac{\partial A_{2}}{\partial z}=\gamma_{2} A_{1}^{3} e^{i \Delta k z} \]

In the equations above \(g_{1}, g_{2}\) and \(D k\) are respectively nonlinearity of the medium for the two pulses and the wave number mismatch. In order to find a solution of the system of equation (3) we use a mathematical method similar to that described by Dmitriev & Tarasov (1982). We make the substitution below in order to find an analytical solution of system (3):

(4) \[ A_{1}(z)=a_{1}(z) e^{i \varphi_{1}(z)} ; A_{2}(z)=a_{2}(z) e^{i \varphi_{2}(z)} \]

where \(a_{1}(z)=\left|A_{1}(z)\right|, a_{2}(z)=\left|A_{2}(z)\right|, j_{1}(z)\) a and \(j_{2}(z)\) are real functions, describing the amplitude functions and phases of the two pulses. By replacing Eq. (4) in Eq. (3), the system of differential equations can be presented in the form of:

(5) \[ i \tfrac{\partial a_{1}}{\partial z}-a_{1} \tfrac{\partial \varphi_{1}}{\partial z}=\gamma_{1} a_{1}^{2} a_{2} e^{-i \Psi(z)} ; i \tfrac{\partial a_{2}}{\partial z}-a_{2} \tfrac{\partial \varphi_{2}}{\partial z}=\gamma_{2} a_{1}^{3} e^{i \Psi(z)} \]

where the generalized phase is of the kind:

(6) \[ \Psi(z)=3 \varphi_{1}-\varphi_{2}+\Delta k z \]

As a next step, we equalize the real and imaginary parts on both sides of the equalities. The following systems of ordinary differential equations are obtained:

(7a) \(\operatorname{Re}: \quad \tfrac{\partial \varphi_{1}}{\partial z}=-\gamma_{1} a_{1} a_{2} \cos \Psi \)

(7b) \(\cfrac{\partial \varphi_{2}}{\partial z}=-\gamma_{2} \cfrac{a_{1}^{3}}{a_{2}} \cos \Psi \)

(8a) \(\operatorname{Im}: \quad \tfrac{\partial a_{1}}{\partial z}=-\gamma_{1} a_{1}^{2} a_{2} \sin \Psi \)

(8b) \(\cfrac{\partial a_{2}}{\partial z}=\gamma_{2} a_{1}^{3} \sin \Psi \)

Let us consider first the system of Eqs. (8a) and (8b), obtained by equalizing the imaginary parts of Eq. (5). It can be presented in the form of one equation as follow: we multiply the first Eq. (8a) by \(\gamma_{2} a_{1}\) and the second Eq. (8b) by \(\gamma_{1} a_{2}\). Moreover, we collect the obtained expressions. The result is presented as follow:

(9) \[ \tfrac{\gamma_{2}}{2} \tfrac{\partial\left(a_{1}^{2}\right)}{\partial z}+\tfrac{\gamma_{1}}{2} \tfrac{\partial\left(a_{2}^{2}\right)}{\partial z}=0 \]

where \(a_{1}^{2}\) and \(a_{2}^{2}\) are the intensities of the two pulses. Eq. (9) can be written as well in the form:

(10) \[ \tfrac{1}{2} \tfrac{\partial}{\partial z}\left[\gamma_{2} a_{1}^{2}+\gamma_{1} a_{2}^{2}\right]=0 \]

As it can be clearly seen:

(11) \[ \gamma_{2} a_{1}^{2}+\gamma_{1} a_{2}^{2}=C=\text { const. } \]

The Eq. (11) presents the conservation law for the sum of the intensities of two pulses. As a next step, we will consider the system of Eqs. (7a) and (7b), obtained by equalizing the real parts of Eq. (5).

We multiply the first Eq.(7a) by \(\gamma_{2} a_{1}^{2}\) and the second Eq. (7b) by \(\gamma_{1} a_{2}^{2}\). Thus, we deduct the two expressions. The result is presented as follow.

(12) \[ \gamma_{2} a_{1}^{2} \tfrac{\partial \varphi_{1}}{\partial z}-\gamma_{1} a_{2}^{2} \tfrac{\partial \varphi_{2}}{\partial z} \]

Having in mind the conservation law for the sum of the intensities of two pulses, Eq. (12) can be presented in the form:

(13) \[ \tfrac{\partial \varphi_{1}}{\partial z}=\tfrac{\gamma_{1} a_{2}^{2}}{C-\gamma_{1} a_{2}^{2}} \tfrac{\partial \varphi_{2}}{\partial z} \]

The expression above gives the connection between the derivatives of the pulses’ phases. Let us now go back to the expression for the generalized Eq. (6). By using it, we can find a link between functions, characterizing the generalized phrase Y and the intensity of one of the pulses \(a_{2}\). We differentiate Eq. (6) by the independent variable \(z\). As a result, the next equation is obtained:

(14) \[ \tfrac{d \Psi(z)}{d z}=3 \tfrac{d \varphi_{1}}{d z}-\tfrac{d \varphi_{2}}{d z}+\Delta k \]

In the equation above if we substitute the derivatives \(\tfrac{d \varphi_{1}}{d z}\) and \(\tfrac{d \varphi_{2}}{d z}\) respectively with expressions (7a) and (7b), Eq.(14) can be presented in the form of:

(15) \[ \tfrac{\mathrm{d} \Psi(z)}{d z}=\tfrac{a_{1}}{a_{2}}\left[\gamma_{2} a_{1}^{2}-3 \gamma_{1} a_{2}^{2}\right] \cos \Psi+\Delta k \]

The unknown functions in this equation are Y, a re \(\mathrm{Y}, a_{1}\) and \(a_{2}\). However, there is a connection between the functions \(a_{1}\) a and \(a_{2}\), given by Eq. (11). Thus, it is found that:

(16) \[ a_{1}=\tfrac{1}{\sqrt{\gamma_{2}}} \sqrt{C-\gamma_{1} a_{2}^{2}} \]

By substituting Eq. (16) in Eq. (15), it is obtained an equation, in which are included the unknown functions Y and \(a_{2}\) :

(17) \[ \tfrac{\mathrm{d} \Psi(z)}{d z}=\tfrac{1}{a_{2} \sqrt{\gamma_{2}}} \sqrt{C-\gamma_{1} a_{2}^{2}}\left[C-4 \gamma_{1} a_{2}^{2}\right] \cos \Psi+\Delta k \]

For this reason, by using Eqs. (17), (16) and (8b), it is obtained the following system:

(18a) \(\tfrac{\mathrm{d} \Psi(z)}{d z}=\tfrac{1}{a_{2} \sqrt{\gamma_{2}}} \sqrt{C-\gamma_{1} a_{2}^{2}}\left[C-4 \gamma_{1} a_{2}^{2}\right] \cos \Psi+\Delta k \)

(18b) \(\cfrac{\partial a_{1}}{\partial z}=\tfrac{\left[C-\gamma_{1} a_{2}^{2}\right]^{3 / 2}}{\sqrt{\gamma_{2}}} \sin \Psi \)

This system of equations can be reduced to one by multiplying Eq. (18a) with \(\sin \mathrm{Y}\) and divide Eq. (18a) to Eq. (18b)

(19) \[ \tfrac{\mathrm{d} \Psi(z)}{d a_{2}}=-\tfrac{\left[C-4 \gamma_{1} a_{2}^{2}\right]}{a_{2}\left[C-\gamma_{1} a_{2}^{2}\right]} \cos \Psi-\tfrac{\Delta k \sqrt{\gamma_{2}}}{\left[C-4 \gamma_{1} a_{2}^{2}\right]^{3 / 2}} \]

The equation above is a linear ordinary differential equation of first order, in which the unknown function is \(\cos \mathrm{Y}\) and the unknown variable is \(a_{2}\). It can be solved by using well-known methods. Its solution is of the kind:

(20) \[ \cos \Psi=\tfrac{2 \mathrm{~B}-\Delta k a_{2}^{2} \sqrt{\gamma_{2}}}{2 a_{2}\left[C-\gamma_{1} a_{2}^{2}\right]^{3 / 2}} \]

where \(B\) is an integration constant. The Eq. (20) can be written in the form:

(21) \[ 2 a_{2}\left[C-\gamma_{1} a_{2}^{2}\right]^{3 / 2} \cos \Psi+\Delta k a_{2}^{2} \sqrt{\gamma_{2}}=2 \mathrm{~B}=\text { const } \]

The expression (21) gives the connection between intensity \(a_{2}\) a and thegeneralized phase Y for arbitrary value of the variable \(z\). It can be considered as conservation law for these quantity. Thus, we have found a connection between \(a_{1}\) a and \(a_{2}\) (Eq. (11)), the derivatives \(\tfrac{d \varphi_{1}}{d z}\) and \(\tfrac{d \varphi_{2}}{d z}\) (Eq. (13)) and intensity \(a_{2}\) a2 and the generalized phrase Y (Eq. (21)). By using these results, our next step will be defining the form of the functions: \(a_{2}^{2}=a_{2}^{2}(z), a_{1}^{2}=a_{1}^{2}(z), \varphi_{1}=\varphi_{1}(z)\) and \(\varphi_{2}=\varphi_{2}(z)\), characterizing respectively the intensities and phases of the pulses.

Let us first consider Eq. (8b), describing the behavior of the amplitude function \(a_{2}^{2}(z)\) of the second pulse. In the expression it is included \(\sin \mathrm{Y}\). Having in mind Eqs. (16) and (20) and that \(\sin \Psi=\sqrt{1-\cos ^{2} \Psi}\), equation (8b) can be presented as follow:

(22) \[ \tfrac{\partial a_{2}}{\partial z}=\tfrac{\left[C-\gamma_{1} a_{2}^{2}\right]^{3 / 2}}{\sqrt{\gamma_{2}}} \sqrt{1-\tfrac{\left(2 B-\Delta k a_{2}^{2} \sqrt{\gamma_{2}}\right)^{2}}{4 a_{2}^{2}\left[C-\gamma_{1} a_{2}^{2}\right]^{3}}} \]

It is useful in the equation above to make the following substitution:

(23) \[ a_{2}^{2}(z)=p(z) \]

After a couple of transformations, it can be presented in the form:

(24) \[ \tfrac{d p}{d z}=2 \gamma_{1} \sqrt{\tfrac{\gamma_{1}}{\gamma_{2}}} \sqrt{p(\alpha-p)^{3}-\eta(\beta-p)^{2}} \]

where for \(\Delta k \neq 0\) :

(25) \[ \alpha=\tfrac{c}{\gamma_{1}}=\text { const } \gt 0, \beta=\tfrac{2 B}{\Delta k \sqrt{\gamma_{2}}}=\text { const } \gt 0, \eta=\tfrac{\gamma_{2} \Delta k^{2}}{4 \gamma_{1}^{8}} \]

As a next step, in order to remove the coefficient in front of the square root on the right side, we change the variable \(z\) with \(\xi\) by the substitution:

(26) \[ \xi=2 \gamma_{1} \sqrt{\tfrac{\gamma_{1}}{\gamma_{2}} z} \]

The function on the right side of Eq. (24) is complicated - fourth degree polynomial of \(p\), in which are included all degrees:

(27) \[ F(p)=p(\alpha-p)^{3}-\eta(\beta-p)^{2}=-p^{4}+3 \alpha p^{2}-\left(3 \alpha^{2}+\eta\right) p^{2}-\eta \beta^{2} \]

Using the substitution:

(28) \[ p=y+\tfrac{3}{4} \alpha, \]

where \(y\) is new unknown function; the function under square root can be presented as follow:

(29) \[ F(y)=\bar{A} y^{4}+\bar{B} y^{2}+\bar{C} y+\bar{D} \]

where

(30) \[ \begin{gathered} \bar{A}=-1 \bar{A}=-1, \quad \bar{B}=\tfrac{3}{2}\left(\tfrac{\alpha}{2}\right)^{2}-\eta=\mathrm{const} \\ \bar{C}=2 \beta \eta-3 \eta\left(\tfrac{\alpha}{2}\right)-\left(\tfrac{\alpha}{2}\right)^{3}=\mathrm{const} \\ \bar{D}=\tfrac{3}{16}\left(\tfrac{\alpha}{2}\right)^{4}-\tfrac{9}{4}\left(\tfrac{\alpha}{2}\right)^{2}+3 \beta \eta\left(\tfrac{\alpha}{2}\right)-\eta \beta^{2}=\mathrm{const} \end{gathered} \]

We replace Eqs. (28), (29) and (30) in Eq. (24). As result it is written in the form:

(31) \[ \tfrac{d y}{d \xi}=\sqrt{\bar{A} y^{4}+\bar{B} y^{2}+\bar{C} y+\bar{D}} \]

We raise to power 2 the both sides of the equation above. As a next step we differentiate it by the unknown variable \(\xi . \xi\). After a couple of transformations it can be presented as follow:

(32) \[ \tfrac{d^{2} y}{d \xi^{2}}-2 \bar{A} y^{3}-\bar{B} y-\tfrac{\bar{C}}{2}=0 \]

Eq. (32) is of the same kind as the following equation:

(33) \[ \tfrac{d^{2} y}{d \xi^{2}}-\left(2-\bar{\kappa}^{2}\right) y+2 y^{3}=0, \quad 0 \lt \bar{\kappa} \lt 1 \]

Under the initial condition \(y(0)=1\), the solution of Eq. (33) is the elliptic delta function. Having in mind the Eq. (30) for the constants and making the substitution:

(34) \[ \bar{C}=0 \]

Eq. (32) can be presented in the form:

(35) \[ \tfrac{d^{2} y}{d \xi^{2}}-\left(\tfrac{3}{2}\left(\tfrac{\alpha}{2}\right)^{2}-\eta\right) y+2 y^{3}=0 \]

When

(36) \[ \bar{\kappa}^{2}=2+\eta-\tfrac{3}{2}\left(\tfrac{\alpha}{2}\right)^{2} \]

Eq. (35) takes exactly the form of Eq. (33).

From the assumptions made by now, it is clearly seen that under condition

(37) \[ 2 \beta \eta-3 \eta\left(\tfrac{\alpha}{2}\right)-\left(\tfrac{\alpha}{2}\right)^{3}=0 ; \bar{\kappa}=\sqrt{2+\eta-\tfrac{3}{2}\left(\tfrac{\alpha}{2}\right)^{2}} \lt 1 \]

By using Eqs. (23), (28) and (11) we find that:

(38) \[ a_{2}^{2}=\cfrac{3}{4} \cfrac{C}{\gamma_{1}}+d n(\xi, \bar{\kappa}) \]

(39) \[ a_{1}^{2}=\cfrac{1}{4} \cfrac{C}{\gamma_{2}}-\cfrac{\gamma_{1}}{\gamma_{2}} d n(\bar{\xi}, \bar{\kappa}) \]

The expressions above characterize the intensities of two pulses during their evolution in nonlinear isotropic media.

Conclusions

In the paper, it is presented a mathematical model describing the process of nonlinear interaction of two waves satisfied the THG (Eq, (1)) and THz generation (Eq. (2)) phase mismatch conditions. When the second order of dispersion is neglected the system of shortcut Eq. (3) can be analytically solved. The obtained solutions, in the form of Jacobi elliptic functions, describe the periodic exchange of energy between the pulses.

Acknowledgments: The present work is supported by project FP17-FF-010 to the Scientific Research Fund at Plovdiv University “Paisii Hilendarski”, Plovdiv, Bulgaria.

REFERENCES

Agrawal, G.P. (2007). Nonlinear fiber optics. New York: Academic Press.

Boyd, R.W. (2007). Nonlinear optics. New York: Elsevier.

Bufetov, I.A., Grekov, M.V., Golant, K.M., Dianov, E.M. & Khrapko, R.R. (1997). Ultraviolet-light generation in nitrogen-doped silica fiber. Opt. Lett., 22, 1394 – 1396.

De Araujo, M.T., Vermelho, M.V.D., Hickmann, J.M. & Goueveia-Neto, A.S. (1996). 440 nm light generation by frequency tripling in germanosilicate glasses. J. Appl. Phys., 80, 4196 – 4198.

Dmitriev, V.G. & Tarasov, L.V. (1982). Applied nonlinear optics. Moscow: Radio i svyaz [Дмитриев, В.Г. & Тарасов, Л.В. (1982). Прикладная нелинейная оптика. Москва: Радио и связь].

Gabriagues, J.M. (1983). Third-harmonic and three-wave sum-frequency light generation in an elliptical-core optical fiber. Opt. Lett., 8, 183 – 185.

Hickmann, J.M., Goueveia, E.A., Goueveia-Neto, A.S., Dini, D.C. & Gelaschi, S. (1995). Enhancement of third-harmonic blue-violet light at 440 nm by erbium ions in \(\mathrm{Er}^{3+}-\mathrm{GeO}_{2}\)-doped silica monomode optical fibers pumped at \(1.319 \mu \mu \mathrm{~m}\). Opt. Lett., 20, 1692 – 1694.

Nicácio, D.L., Goueveia, E.A., Birges, N.M. & Goueveia-Neto, A.S. (1993). Third-harmonic generation in \(\mathrm{GeO}_{2}\)-doped silica single-mode optical fibers. Appl. Phys. Lett., 62, 2179 – 2181.

Österberg, U. (1990). Growth of third-harmonic signal in optical glass fiber. Electron. Lett., 26, 103 – 104.

Thøgersen, J. & Mark, J. (1994). Third harmonic generation in standard and erbium-doped fibers. Opt. Comm., 110, 435 – 444.