Analytical model for transverse mode conversion at all-optically induced, transient long-period gratings: from continuous-wave to ultrafast

Abstract

Transverse mode conversion at an index grating, all-optically induced by multi-mode interference and the optical Kerr effect, is commonly studied by numerical simulations relying on either multi-mode implementations of the generalized nonlinear Schrödinger equation or beam propagation methods. Here, we present and discuss an analytical model describing the directed energy exchange between two probe modes moderated by two control modes. The analytical model can be derived in a four-wave mixing representation as well as in a material representation in analogy to the different numerical approaches demonstrating their equivalence. The analytical nature of the model is used to provide general insight into the conversion process in dependence on phase mismatch as well as induced coupling strength. While being a continuous-wave model, the applicability of the model for mode conversion using ultrashort pulses is discussed and guidelines for using the model as a first estimate for experiments or for more precise but time-consuming numerical simulations are given.

Appendices

Appendix 1: Material representation

In this section, coupling coefficients of an optically induced long-period grating are derived in detail in the material representation, i.e., describing the mode conversion process with coupled mode theory. The coupling of transverse modes by a sinusoidal perturbation of the refractive index, as it is the case by a periodic long-period grating, has been described already by Snyder in 1972 [24] before first experimental results have been discussed. The resulting so-called coupled mode theory is very well described in several textbooks [26, 27] as well as papers [28] and in this section only the necessary modifications to describe all-optically induced long-period gratings are given. The nomenclature and mode normalization is the same as in Snyder [26] so that the analytical signal of the electrical field $\mathbf {E}(x,y,z)$ of the forward-propagating bound modes of a translation invariant optical waveguide is given by:

$$\begin{aligned} \mathbf {E}(x,y,z)&= \sum _j A_j \mathbf {E}_j(x,y,z), \end{aligned}$$

(7)

$$\begin{aligned} \mathbf {E}_j(x,y,z)&= {\hat{\mathbf {e}}}_j(x,y) \exp (i\beta _jz). \end{aligned}$$

(8)

Here, the frequency-dependent propagation constant or eigenvalue of the jth mode is called $\beta _j(\omega )$, whereas the kth coefficient of its Taylor series centered at $\omega _0$ will be given as $\beta _j^{(k)}$. The transverse electric field distributions ${\hat{\mathbf {e}}}_j(x,y)$ and magnetic field distributions ${\hat{\mathbf {h}}}_j(x,y)$ contain the implicit time dependence $\exp (-i\omega _0t)$ and are considered orthonormal with regards to the definition given in reference [26]:

$$\begin{aligned} \frac{1}{2} \int _{A_\infty } {\hat{\mathbf {e}}}_j\times {\hat{\mathbf {h}}}^{*}_k \cdot {\hat{\mathbf {z}}} \mathrm{d}A&= {\left\{ \begin{array}{ll} \begin{array}{ll}1,&{}\quad \text {if } j=k\\ 0, &{} \quad \text {if } j\ne k. \end{array}\end{array}\right. } \end{aligned}$$

(9)

The total power $P_j$ of one mode propagating in the z-direction is, therefore, given by the modal expansion coefficient $A_j$:

$$\begin{aligned} P_j&=\vert A_j\vert ^2. \end{aligned}$$

(10)

The electrical field of the multi-mode interference of two control modes can be described as:

$$\begin{aligned} \mathbf {E}(x,y,z)&= A_1{\hat{\mathbf {e}}}_1(x,y)\exp (i\beta _1z)\nonumber \\&\quad \,+A_2{\hat{\mathbf {e}}}_2(x,y)\exp \left( i\left( \beta _2z+\phi _0\right) \right) , \end{aligned}$$

(11)

with $A_1,A_2 \in \mathbb {R}$ and with a relative phase of $\phi _0$ between the two modes. The Kerr-induced local refractive index experienced by the probe beam is proportional to the control beam intensity I(x, y, z) and given by

$$\begin{aligned} n(x,y,z)=n_0+ n_2^{\mathrm{XPM}} \cdot I (x,y,z). \end{aligned}$$

(12)

As the index change is induced by cross-phase modulation (XPM), the nonlinear index $n_2^{\mathrm{XPM}}$ differs from the regular nonlinear index $n_2$ known from, e.g., self-phase modulation by a factor of $n_2^{\mathrm{XPM}}=2n_2$ for two beams with the same polarization and $n_2^{\mathrm{XPM}}=\tfrac{2}{3}n_2$ for two cross-polarized beams in cylindrically symmetric optical fibers [29]. The induced index change is proportional to the multi-mode interference intensity or power density pattern that can be calculated from the z-component of the Poynting vector $I = \langle S_z\rangle$ [26]. If the longitudinal component of the electrical field $e_z$ can be neglected in comparison with its transverse components, as it is the case in the weakly guiding regime, the mode fields ${\hat{\mathbf {e}}}_j$ under investigation in this section can be transformed to be completely real valued. When, furthermore, assuming $\beta _1\approx \beta _2\approx \tfrac{2\pi n_0}{\lambda _0}$ for calculating the magnitude of the modal intensities (both are of the order of magnitude $10^6/m$), the power density reduces to

$$\begin{aligned} \langle S_z\rangle&= \left\langle \frac{1}{2} \text {Re} \left( \mathbf {E}\times \mathbf {{H}^{*}}\right) {\hat{\mathbf {z}}}\right\rangle \nonumber \\&= \frac{n_0}{2} \sqrt{\frac{\epsilon _0}{\mu _0}} \quad \left\{ \vert A_1\vert ^2 \vert {\hat{\mathbf {e}}}_1(x,y)\vert ^2+\vert A_2\vert ^2 \vert {\hat{\mathbf {e}}}_2(x,y)\vert ^2 \right. \nonumber \\&\quad \left. + \,\, 2A_1A_2{\hat{\mathbf {e}}}_1(x,y)\cdot {\hat{\mathbf {e}}}_2(x,y) \cos (\varOmega z-\phi _0)\right\} , \end{aligned}$$

(13)

with $n_0$ being the refractive index of the core of the fiber, $\lambda _0$ the wavelength of the light in vacuum and the angular wave number $\varOmega = \beta _1 - \beta _2$, here, the approximation used for the magnitude of the modulation is not valid as for the phase of the cosine the small difference between $\beta _1$ and $\beta _2$ is of significance. The resulting changed refractive index,

$$\begin{aligned} n(x,y,z)&= n_0^\prime (x,y)+ \Delta n(x,y) \cos \left( \varOmega z-\phi _0\right) , \end{aligned}$$

(14)

can be interpreted as a z-independent change of the nominal refractive index $n_0(x,y)$ due to an added constant offset

$$\begin{aligned} n_0^\prime (x,y)&=n_0(x,y)+ \frac{1}{2} n_2^{\mathrm{XPM}}\cdot n_0 \sqrt{\frac{\epsilon _0}{\mu _0}}\nonumber \\&\quad \cdot \, \left\{ \vert A_1\vert ^2 \vert {\hat{\mathbf {e}}}_1(x,y)\vert ^2+\vert A_2\vert ^2\vert {\hat{\mathbf {e}}}_2(x,y)\vert ^2\right\} \end{aligned}$$

(15)

and an interference term that leads to a cosinusoidal modulation of the refractive index with magnitude

$$\begin{aligned} \Delta n(x,y)&= n_2^{\mathrm{XPM}} \cdot n_0 \sqrt{\frac{\epsilon _0}{\mu _0}} A_1A_2{\hat{\mathbf {e}}}_1(x,y)\cdot {\hat{\mathbf {e}}}_2(x,y). \end{aligned}$$

(16)

For the coupling of transverse modes, the square of the refractive index (or the permittivity) is the relevant material parameter, which can be calculated to

$$\begin{aligned} n^2(x,y,z)&= n_0^{\prime 2}(x,y)+2n_0^\prime \cdot \Delta n(x,y) \cos (\varOmega z-\phi _0)+\mathcal {O}(\Delta n^2) \nonumber \\&\approx n_0^{\prime 2}(x,y)+2n_0^\prime \Delta n \cos (\varOmega z-\phi _0). \end{aligned}$$

(17)

The coupling coefficients for an optically induced long-period grating can, therefore, be reduced to the same form as in the standard coupled mode equations in reference [26] with $\delta n^2= 2n_0^\prime (x,y) \cdot \Delta n (x,y)$, when the initial phase between both control beam modes is $\phi _0=\pi /2$. In order to directly adapt the results from [26], the relative phase between the two control modes is assumed to be $\phi _0=\pi /2$ for the rest of this section. As the all-optically induced refractive index change can then be reduced to the same type of equation as for a conventional sinusoidally shaped long-period grating, the asymptotic solution, describing the energy transfer from one mode to another, as well as the higher-order corrections leading to a modulation of the energy transfer, can be readily taken from reference [26] to describe the cw-limit of all-optical mode conversion. The coupled mode equations in this case are

$$\begin{aligned} \frac{\partial A_3}{\partial z}&=i\beta _3 \cdot A_3 + iD_{34}\cdot A_4+iD_{33} A_3,\end{aligned}$$

(18)

$$\begin{aligned} \frac{\partial A_4}{\partial z}&=i\beta _4 \cdot A_4 +iD_{43}\cdot A_3+iD_{44} A_4. \end{aligned}$$

(19)

The coupling coefficient for two probe modes ($j=3,4$) determined by the index grating induced by control modes ($j=1,2$), which is the main result of this section, is then given by

$$\begin{aligned} D_{34}&= \bar{D}_{34} \cdot \sin (\varOmega z)\nonumber \\ \bar{D}_{34}&=n_2^{\mathrm{XPM}} \cdot \frac{k}{2}n_0^2 \frac{\epsilon _0}{\mu _0} A_1 A_2 \int _{A_\infty } \left( {\hat{\mathbf {e}}}_1\cdot {\hat{\mathbf {e}}}_2\right) \left( {\hat{\mathbf {e}}}_3\cdot {\hat{\mathbf {e}}}_4\right) \, \mathrm{d}A, \end{aligned}$$

(20)

with the integral of the area element dA over the total mode area.

Appendix 2: Four-wave mixing representation

The theoretical description of OLPGs given in the last section can be interpreted as a periodic perturbation of the local probe modes of the waveguide by the refractive index change induced by the control beam. This description is well suited for long pulses in the picosecond regime or ultimately in the cw-limit. In this section, the all-optical mode conversion will be described in detail in the four-wave mixing representation derived from the multi-mode generalized nonlinear Schrödinger equations (MM-NLSE) presented by Poletti and Horak [25]. This set of equations can be used to accurately describe the nonlinear interaction of different transverse modes even for ultrashort pulses. It will be shown that the description of an optically induced grating by the MM-NLSE is equivalent to the material representation of Sect. 1 in the cw-limit and is, therefore, perfectly suited for numerically modeling the conversion process in case ultrashort pulses are used.

The evolution of the complex modal amplitude $A_p(t,z)$ as formulated in reference [25] is given by the following coupled equations:

$$\begin{aligned} \frac{\partial A_p}{\partial z}= & {} i\left( \beta _p^{(0)}-\beta ^{(0)}\right) A_p + D _p (A_p)\nonumber \\&\quad +\, \frac{in_2\omega _0}{c} \sum _{l,m,n} \left\{ \left( 1+i\tau _{plmn}^{(1)}\frac{\partial }{\partial t}\right) \cdot 2Q^{(1)}_{plmn}A_l\right. \nonumber \\&\quad \cdot \int {R(\tau )A_m(t-\tau )A_n^*(t-\tau )\text {d}\tau }+ \left( 1+i\tau _{plmn}^{(2)}\frac{\partial }{\partial t}\right) \cdot Q^{(2)}_{plmn}A_l^*\nonumber \\&\quad \cdot \left. \int {R(\tau )A_m(t-\tau )A_n(t-\tau ){\hbox {e}}^{2i\omega _0\tau }\text {d}\tau }\right\} , \end{aligned}$$

(21)

with $\beta ^{(j)}_p$ being the jth Taylor series coefficient of the propagation constant $\beta$ of the pth mode, $D _p$ the dispersion operator, $Q^{(1,2)}_{plmn}$ the coupling coefficients between the transverse modes, $\tau _{plmn}$ the shock-time coefficients and $R(\tau )$ the delayed Raman response of the waveguide. In order to derive the mode conversion efficiency in the long-pulse or quasi-cw approximation, dispersion as well as self-steepening or the Raman effect can be neglected reducing the coupled equations to the simpler form

$$\begin{aligned} \frac{\partial A_p}{\partial z}&= i\left( \beta _p^{(0)}-\beta ^{(0)}\right) A_p + \frac{in_2\omega _0}{c}\cdot \sum _{l,m,n}\left\{ 2Q^{(1)}_{plmn}A_l A_m A_n^* + Q^{(2)}_{plmn}A_l^*A_m A_n \right\} . \end{aligned}$$

(22)

At first, the case of a strong control beam being distributed between two modes ($p=1,2$) and also being cross-polarized to the probe beam is considered. With the power of the control beam being much higher than that of the two probe modes ($p=3,4$, $P_1\approx P_2 \gg P_3\approx P_4$), changes of the control beam induced by the probe beam can be easily disregarded. This treatment can be compared to the case of an undepleted pump beam in conventional four-wave mixing and leads to an unperturbed control beam and to a straightforward description of the modal coefficients of the control beam

$$\begin{aligned} A_p(z,t)=\vert A_p(t)\vert \cdot \exp \left( i\beta _p^{(0)}\cdot z\right) , \end{aligned}$$

(23)

with $p=1,2$. In contrast to four-wave mixing amplification, the self- and cross-phase modulation of the control beam itself can be neglected in a first approximation as well as only phase differences between the modes are of interest, and the difference in acquired nonlinear phase between the two control modes is usually small. The small probe beam power (usually two to three orders of magnitude smaller than the control beam), furthermore, allows to disregard all terms in the sum over l, m and n in Eq. (22) which contain any product of $A_3$ and $A_4$, as it will be small in comparison with a product containing at least two of the control beam modal coefficients ($A_1$ and/or $A_2$). Finally, noting that in case of a cross-polarized control and probe beam $Q_{3lmn}=0$ for $l\ne 3,4$ [25], we end up with the following equation for the change in amplitude of the probe mode 3 ($A_3(z,t)$, omitting the superscript index at $\beta _j^{(0)}$ for better readability):

$$\begin{aligned} \frac{\partial A_3}{\partial z}= & {} i\beta _1 A_3 + \frac{in_2\omega _0}{c}\left\{ A_4 \left( 2Q^{(1)}_{3412} \cdot \vert A_1\vert \vert A_2\vert \cdot {\text{ e }}^{i(\beta _1-\beta _2) z} \right. \right. \nonumber \\&\qquad \qquad \qquad \qquad \qquad + \left. 2Q^{(1)}_{3421} \cdot \vert A_2\vert \vert A_1\vert \cdot {\text{ e }}^{i(\beta _2-\beta _1) z} \right) \end{aligned}$$

(24a)

$$\begin{aligned}&\qquad \qquad \qquad \qquad + \,A_3 \left( 2Q^{(1)}_{3312} \cdot \vert A_1\vert \vert A_2\vert \cdot {\text{ e }}^{i(\beta _1-\beta _2) z} \right. \nonumber \\&\qquad \qquad \qquad \qquad \qquad +\,\left. 2Q^{(1)}_{3321} \cdot \vert A_2\vert \vert A_1\vert \cdot {\text{ e }}^{i(\beta _2-\beta _1) z} \right) \end{aligned}$$

(24b)

$$\begin{aligned}&\qquad \qquad \qquad \qquad + \,A_3 \left( 2Q^{(1)}_{3311}\cdot \vert A_1\vert ^2 + 2Q^{(1)}_{3322} \cdot \vert A_2\vert ^2\right) \end{aligned}$$

(24c)

$$\begin{aligned}&\qquad \qquad \qquad \qquad + \,A_4 \left( 2Q^{(1)}_{3411}\cdot \vert A_1\vert ^2 + 2Q^{(1)}_{3422} \cdot \vert A_2\vert ^2\right) \nonumber \\&\qquad \qquad \qquad \qquad +\, A_3^* \left( Q^{(2)}_{3311} \cdot \vert A_1\vert ^2 \cdot {\text{ e }}^{i2\beta _1 z}\right. + Q^{(2)}_{3322} \cdot \vert A_2\vert ^2 \cdot {\text{ e }}^{i2\beta _2z} \nonumber \\&\qquad \qquad \qquad \qquad \qquad +\, Q^{(2)}_{3312} \cdot \vert A_1\vert \vert A_2\vert \cdot {\text{ e }}^{i(\beta _1+\beta _2) z} + \left. Q^{(2)}_{3321} \cdot \vert A_2\vert \vert A_1\vert \cdot {\text{ e }}^{i(\beta _2+\beta _1) z}\right) \nonumber \\&\qquad \qquad \qquad \qquad + \,A_4^* \left( Q^{(2)}_{3411} \cdot \vert A_1\vert ^2 \cdot {\text{ e }}^{i2\beta _1 z}\right. + Q^{(2)}_{3422} \cdot \vert A_2\vert ^2 \cdot {\text{ e }}^{i2\beta _2z} \nonumber \\&\qquad \qquad \qquad \qquad \qquad \left. + Q^{(2)}_{3412} \cdot \vert A_1\vert \vert A_2\vert \cdot {\text{ e }}^{i(\beta _1+\beta _2) z} +\left. Q^{(2)}_{3421} \cdot A_4^* \vert A_2\vert \vert A_1\vert \cdot {\text{ e }}^{i(\beta _2+\beta _1) z} \right) \right\} .\nonumber \\ \end{aligned}$$

(24d)

The according equation for $A_4$ can be obtained by exchanging the index 3 with the index 4 and vice versa for the terms $A,\beta$ and Q. The two terms in Eq. (24c), containing the coupling coefficients $Q^{(1)}_{3311}$ and $Q^{(1)}_{3322}$, constitute the XPM acting from the control beam on the probe beam that can be identified with the constant refractive index offset introduced in Eq. (15). In the considered scenario, XPM as well as SPM can be approximated as being of the same strength for both control beam modes, except for a slight dependence on the difference in transverse mode shape. As for phase matching only relative phase differences matter, we neglect these terms in the following as a constant phase contribution during propagation. The term labeled (24d) and the following terms are considered non-phase-matched terms. Assuming a nearly phase-matched mode conversion scenario with $\beta _1 -\beta _2 \approx \beta _3-\beta _4$, in general, these terms oscillate with a frequency relative to $A_3$ that is very fast in comparison with the phase-matched contributions from Eq. (24a). Therefore, the terms following Eq. (24d) do not contribute to the energy exchange, resulting in a description of the energy transfer as an asymptotic solution governed by the $Q^{(1)}$ coupling coefficients of Eq. (24a). While this assumption cannot exactly be justified easily a priori, it will be shown that it leads to the correct description of the all-optically induced mode conversion. The remaining terms denote a coupling of the amplitude of probe mode 3 ($A_3$) to probe mode 4 ($A_4$) described by the terms in (24a) as well as a self-coupling (24b), both induced by the magnitude of the two control modes ($\vert A_1\vert$ and $\vert A_2\vert$). With these assumptions and recognizing that

$$\begin{aligned} Q_{34}&:=Q^{(1)}_{3412}=Q^{(1)}_{3421}=Q_{43}:=Q^{(1)}_{4312}=Q^{(1)}_{4321} \end{aligned}$$

(25)

$$\begin{aligned} Q_{33}&:=Q^{(1)}_{3312}=Q^{(1)}_{3321}, \end{aligned}$$

(26)

we end up with the following two coupled equations for the modal coefficients of both probe modes $A_3$ and $A_4$:

$$\begin{aligned} \frac{\partial A_3}{\partial z}&=i\beta _3 \cdot A_3 \nonumber \\&\quad + \,4 \frac{in_2\omega _0}{c} \vert A_1\vert \vert A_2 \vert \nonumber \\&\quad \times \,\cos \left[ (\beta _1-\beta _2) z\right] \cdot \left( Q_{34} A_4+Q_{33}A_3\right) \nonumber \\&= i\beta _3 \cdot A_3 + iD^\prime _{34}\cdot A_4+iD^\prime _{33} A_3 \end{aligned}$$

(27)

$$\begin{aligned} \frac{\partial A_4}{\partial z}&=i\beta _4 \cdot A_4 +iD^\prime _{43}\cdot A_3+iD^\prime _{44} A_4 \end{aligned}$$

(28)

These two equations are of identical form to the coupled mode equations derived in reference [26]. Using the definition for the Q’s of reference [25], the coupling coefficients can be expressed as:

$$\begin{aligned} D^\prime _{34}&= 4 \frac{n_2\omega _0}{c} \vert A_1\vert \vert A_2 \vert \cos \left( \beta _1-\beta _2\right) z \nonumber \\&\quad \cdot \frac{n_0^2 \epsilon _0 }{12 \mu _0 } \int _{A_\infty } \left( \hat{\mathbf {e}}_3\hat{\mathbf {e}}_4\right) \cdot \left( \hat{\mathbf {e}}_1\hat{\mathbf {e}}_2\right) \,\mathrm{d}A \end{aligned}$$

(29)

$$\begin{aligned}&\quad \quad= \bar{D}^\prime _{34} \cdot \sin (\beta _1-\beta _2) , \end{aligned}$$

(30)

with

$$\begin{aligned} \bar{D}^\prime _{34}&= \frac{2 n_2}{3} \frac{k}{2} n_0^2 \frac{\epsilon _0}{\mu _0} \\ &\quad\cdot \vert A_1\vert \vert A_2 \vert \int _{A_\infty } \left( \hat{\mathbf {e}}_3\hat{\mathbf {e}}_4\right) \cdot \left( \hat{\mathbf {e}}_1\hat{\mathbf {e}}_2\right) \, \mathrm{d}A. \end{aligned}$$

(31)

Comparing the coefficients $D^\prime _{34}$ and $D^\prime _{43}$ with the coupling coefficients $D_{34}$ and $D_{43}$ introduced in Eq. (20), furthermore, reveals that they are indeed identical to the cross-polarized case in the material representation. Extending the analysis described above for the co-polarized case is easily done by acknowledging that $Q_{3lmn}\ne 0$ also for $l\ne 3,4$ is leading to additional nonzero contributions that change the value of the fractional in front of the nonlinear index in Eq. (31) to $2\cdot n_2$. Thereby, it is shown that describing the OLPG by the four-wave mixing interaction that can be numerically modeled by solving the multi-mode coupled nonlinear Schrödinger equations is equivalent to the solution of the coupled mode perturbation theory in the long-pulse or quasi-cw regime. This equivalence analytically verifies that indeed the numerically observed transverse mode conversion during nonlinear interaction of probe and control beams physically originates from an optically induced long-period grating.