Nonlinear Optical Ring-Cavity Model Driven by a Gas Laser

Abstract

The development of spontaneous stationary equilibrium patterns induced by the injection of a laser pump field into a purely absorptive two-level atomic sodium vapor ring cavity is investigated by means of a hexagonal planform nonlinear stability analysis applied to the appropriate governing evolution equation for this optical phenomenon. In the quasi-equilibrium limit for its atomic variables, the mathematical system modelling that phenomenon can be reduced to a single modified Swift-Hohenberg nonlinear partial differential time-evolution equation describing the intracavity field on an unbounded two-dimensional spatial domain. Diffraction of radiation can induce transverse patterns consisting of stripes and hexagonal arrays of bright spots or honeycombs in an initially uniform plane-wave configuration. Then, these theoretical predictions are compared with both relevant experimental evidence and existing numerical simulations from some recent nonlinear optical pattern formation studies. There are four problems: The first two fill in some details of this analysis while the last two examine bistability for a related nonlinear optical phenomenon and hexagonal pattern formation for the relevant amplitude-phase equations with a hypothetical growth rate and set of Landau constants.

Problems

Problems 17.1 and 17.2 depend upon the function $F(A, A^*)$ defined in Section 17.1 when $\Delta = 0$ given by

$$\begin{aligned} G(A, A^*) = -(1 + A)\left[ 1 + i\theta + \frac{\beta }{1 + \alpha (1 + A)(1 + A^*)}\right] . \end{aligned}$$

Define

$$\begin{aligned} \gamma _{n\ell } = \frac{1}{(n-\ell )!\ell !}\frac{\partial ^n G(0,0)}{\partial A^{n-\ell }\partial {A^*}^\ell }. \end{aligned}$$

Then

$$\begin{aligned} \gamma _{10} = -\left( 1 + \frac{\beta }{D^2} + i\theta \right) , \, \gamma _{11} = \frac{\alpha \beta }{D^2} \text{ with } D = 1 + \alpha ; \end{aligned}$$

$$\begin{aligned} \gamma _{20} = \frac{\alpha \beta }{D^3}, \, \gamma _{21} = \frac{2\alpha \beta }{D^3}, \, \gamma _{22} = -\frac{\beta \alpha ^2}{D^3}; \end{aligned}$$

$$\begin{aligned} \gamma _{30} = -\frac{\beta \alpha ^2}{D^4}, \, \gamma _{31} = \frac{\alpha \beta (1 - 2\alpha )}{D^4}, \, \gamma _{32} = -\frac{3\beta \alpha ^2}{D^4}, \, \gamma _{33} = \frac{\beta \alpha ^3}{D^4}. \end{aligned}$$

17.1.

LetA of (17.1.9) be of the form

$$\begin{aligned} A(x,y, t) = \varepsilon _1 \mathcal {A}_1(x,y, t) + O(\varepsilon _1^2) \text{ where } |\varepsilon _1|<< 1. \end{aligned}$$

• (a) Show that the linear problem discussed in Section 17.1 satisfies

  $$\begin{aligned} \frac{\partial \mathcal {A}_1}{\partial t} = \gamma _{10}\mathcal {A}_1 + \gamma _{11}\mathcal {A}_1^*+ i\chi \nabla _2^2 \mathcal {A}_1. \end{aligned}$$

  (P1.1a)

• (b) Taking the complex conjugate of (P1.1a) obtain

  $$\begin{aligned} \frac{\partial \mathcal {A}_1^*}{\partial t} = \gamma _{10}^*\mathcal {A}_1^*+ \gamma _{11}\mathcal {A}_1 - i \chi \nabla _2^2 \mathcal {A}_1^*. \end{aligned}$$

  (P1.1b)

• (c) Conclude that seeking a normal-mode solution of (P1.1) of the form (17.1.10a) yields the following linear homogeneous system of equations for $k_1$ and $k_2$:

  $$\begin{aligned} \left[ \sigma + 1 + \frac{\beta }{D^2} + i(\theta + \chi q^2)\right] k_1 - \frac{\alpha \beta k_2}{D^2} = 0, \end{aligned}$$

  (P1.2a)
  $$\begin{aligned} -\frac{\alpha \beta k_1}{D^2} + \left[ \sigma + 1 + \frac{\beta }{D^2} - i(\theta + \chi q^2)\right] k_2 = 0. \end{aligned}$$

  (P1.2b)

• (d) Deduce that the vanishing of the determinant of the matrix of coefficients of (P1.2) required by the nontriviality condition of (17.1.10b) results in the secular equation

  $$\begin{aligned} \sigma ^2 + 2\left[ 1+ \frac{\beta }{(\alpha + 1)^2}\right] \sigma + \left[ 1 + \frac{\beta }{(\alpha + 1)^2}\right] ^2 - \left[ \frac{\alpha \beta }{(\alpha + 1)^2}\right] ^2 + (\theta + \chi q^2)^2 = 0. \end{aligned}$$

  (P1.3a)

• (e) Finally, demonstrate that (P1.3a) is equivalent to (17.1.11a) by employing the difference of squares

  $$\begin{aligned} \left[ 1 + \frac{\beta }{(\alpha + 1)^2}\right] ^2 - \left[ \frac{\alpha \beta }{(\alpha + 1)^2}\right] ^2 = \left[ 1 + \frac{\beta (1 - \alpha )}{(\alpha + 1)^2}\right] \left[ 1 + \frac{\beta }{\alpha + 1}\right] . \end{aligned}$$

  (P1.3b)

17.2.

Rewrite the equation for A in Section 17.1 when $\Delta = 0$ as

$$\begin{aligned} A_t \sim \gamma _{10}A + \gamma _{11}A^*+ \gamma _{20}A^2 +&\gamma _{21}AA^*+ \gamma _{22}{A^*}^2 + \gamma _{30}A^3 + \gamma _{31}A^2 A^*+ \gamma _{32}A{A^*}^2 \end{aligned}$$

(P2.1)

$$\begin{aligned} + \gamma _{33}{A^*}^3 + i\chi \nabla _2^2 A. \end{aligned}$$

(17.4.1)

• (a) Letting $A = R + iI$ with $\theta = -\chi q_c^2 = -1$ and representing $\gamma _{10} = \gamma _{10}^{(r)} + i \chi q_c^2$ where $\gamma _{10}^{(r)} = -(1 + \beta /D^2)$ in (P2.1) obtain

  $$\begin{aligned} R_t + iI_t \sim \gamma _{10}^{(r)}(R +iI) + \gamma _{11}(R - iI) + \gamma _{20}(R^2 + 2iRI - I^2) + \gamma _{21}(R^2 + I^2) \end{aligned}$$

  (P2.2)
  $$\begin{aligned}&+ \gamma _{22}(R^2 - 2iRI - I^2) + \gamma _{30}(R^3 + 3iR^2I - 3RI^2 - iI^3) + \gamma _{31}(R^3 + iR^2I + RI^2 + iI^3)\nonumber \\&+\gamma _{32}(R^3 - iR^2I + RI^2 - iI^3) + \gamma _{33}(R^2 - 3iR^2I - 3RI^2 + iI^3) + i\chi (\nabla _2^2 + q_c^2)(R + iI). \end{aligned}$$

  (17.4.2)

• (b) Separating the real and imaginary parts of (P2.2) and retaining terms through third order on the right-hand side of the former and through second order on the corresponding side of the latter while noting I, in this context, actually represents a second-order effect (see below) show that

  $$\begin{aligned} R_t\sim [\gamma _{10}^{(r)} + \gamma _{11}]R + (\gamma _{20} + \gamma _{21} + \gamma _{22})R^2 + (\gamma _{30} + \gamma _{31} + \gamma _{32} + \gamma _{33})R^3 - \chi (\nabla _2^2 + q_c^2)I, \end{aligned}$$

  (P2.3a)
  $$\begin{aligned} I_t \sim [\gamma _{10}^{(r)} - \gamma _{11}]I + \chi (\nabla _2^2 + q_c^2)R. \end{aligned}$$

  (P2.3b)

• (c) Upon simplifying the coefficients in these equations make the following identifications:

  $$\begin{aligned} \gamma _{10}^{(r)} + \gamma _{11} = -1 + \frac{\beta (\alpha - 1)}{(\alpha + 1)^2} = \sigma _R(\alpha ,\beta ), \end{aligned}$$

  (P2.4a)
  $$\begin{aligned} \gamma _{10}^{(r)} - \gamma _{11} = -1 -\frac{\beta }{\alpha + 1} = \sigma _I(\alpha ,\beta ), \end{aligned}$$

  (P2.4b)
  $$\begin{aligned} \gamma _{20} + \gamma _{21} + \gamma _{22} = \frac{\beta \alpha (3 - \alpha )}{(\alpha + 1)^3} = -\omega _0(\alpha ,\beta ), \end{aligned}$$

  (P2.4c)
  $$\begin{aligned} \gamma _{30} + \gamma _{31} + \gamma _{32} + \gamma _{33} = \frac{\beta \alpha [(\alpha + 1)^2 - 8\alpha ]}{(\alpha + 1)^4} = -\omega _1(\alpha ,\beta ). \end{aligned}$$

  (P2.4d)

• (d) Finally, employing (P2.4) in (P2.3), assuming in addition that (P2.3b) satisfies the quasi-equilibrium condition

  $$\begin{aligned} I \sim - \frac{\chi }{\sigma _I(\alpha ,\beta )}(\nabla _2^2 + q_c^2)R, \end{aligned}$$

  (P2.5)

  and using this asymptotic relation to eliminate I from (P2.3a), deduce the modified Swift–Hohenberg equation (17.1.14)

  $$\begin{aligned} R_t \sim \sigma _R(\alpha ,\beta )R - \omega _0(\alpha ,\beta )R^2 - \omega _1(\alpha ,\beta )R^3 + \frac{\chi ^2}{\sigma _I(\alpha ,\beta )}(\nabla _2^2 + q_c^2)^2 R, \end{aligned}$$

  (P2.6a)

  where

  $$\begin{aligned} \omega _0(\alpha ,\beta ) = \frac{\beta \alpha (\alpha - 3)}{(\alpha + 1)^3}, \, \omega _1(\alpha ,\beta ) = \frac{\beta \alpha [8\alpha ^2 - (\alpha + 1)^2]}{(\alpha + 1)^4}. \end{aligned}$$

  (P2.6b)

17.3.

In their study of pattern formation driven by a pump field in a unidirectional ring cavity containing a so-called Kerr medium Scroggie et al. [113] deduced the following relationship between $X_e$ and Y analogous to (17.1.5a) but with $\theta > 0$

$$\begin{aligned} Y = X_e[1 + i\eta (|X_e|^2 - \theta )^2] \end{aligned}$$

(P3.1a)

where the parameter $\eta$ equals $+1$ for a self-focusing Kerr medium and $-1$ for a self-defocusing one. Hence $|\eta | = 1$. A Kerr medium is a material the refractive index of which changes with an applied electric field. This change is proportional to the square of the intensity of the applied field and is positive for a self-focusing medium while it is negative for a self-defocusing one.

• (a) Taking the complex conjugate of (P1.1a) deduce that

  $$\begin{aligned} Y^*= Y = X_e^*[1 - i\eta (|X_e|^2 - \theta )^2]. \end{aligned}$$

  (P3.1b)

• (b) Computing the product of (P3.1a) and (P3.1b) derive the analogous relationship to (17.1.5b)

  $$\begin{aligned} YY^*= Y^2 = \alpha [1 + (\alpha - \theta )^2] = \alpha ^3 - 2\theta \alpha ^2 + (1 + \theta ^2)\alpha \text{ where } \alpha = X_eX_e^*= |X_e|^2. \end{aligned}$$

  (P3.2)

  Note that this relationship is independent of $\eta$ by virtue of $|\eta | = 1$.

• (c) Taking the derivative of (P3.2) with respect to $\alpha$ obtain that

  $$\begin{aligned} \frac{dY^2}{d\alpha } = 3\alpha ^2 - 4\theta \alpha + 1 + \theta ^2. \end{aligned}$$

  (P3.3)

• (d) Conclude from (P3.3) that $dY^2/d\alpha > 0$ for $0< \theta < \theta _{\text {crit}}$ where $\alpha$ satisfies

  $$\begin{aligned} 3\alpha ^2 - 4\theta \alpha + 1 + \theta ^2 > 0, \end{aligned}$$

  (P3.4a)

  and $\theta _{\text {crit}}$ corresponds to that $\theta$-value such that the discriminant of the related quadratic equation

  $$\begin{aligned} a\alpha ^2 + b\alpha + c = 0 \text{ with } a = 3, \, b = -4\theta , \, c = 1 + \theta ^2 \end{aligned}$$

  (P3.4b)

  vanishes or

  $$\begin{aligned} \mathcal {D} = b^2 - 4ac = 0. \end{aligned}$$

  (P3.4c)

• (e) Calculating the discriminant of (P3.4) show that

  $$\begin{aligned} \mathcal {D} = 4(\theta _{\text {crit}}^2 - 3) = 0 \text{ or } \theta _{\text {crit}} = \sqrt{3}. \end{aligned}$$

  (P3.5)

• (f) Finally, from these results deduce that (P3.2) is single-valued in $Y^2$ for $0< \theta < \sqrt{3}$ and S-shaped (exhibits optical bistability) for $\theta > \sqrt{3}$.

17.4.

Consider an interfacial perturbation $\zeta (x,y, t)$ of an originally planar surface where $t\equiv$ time and $(x, y)\equiv$ a transverse Cartesian coordinate system, all of which are dimensionless. Upon seeking a real hexagonal planform solution for $\zeta (x,y, t)$ of the form of (17.3.1), assume it is found that the coefficients of the amplitude–phase equations satisfy

$$\begin{aligned} \sigma = 2(1 - \beta ), \, a_0 = -\alpha , \, a_1 = a_2 = 4(1 + \alpha ^2), \end{aligned}$$

(P4.1a)

for $\beta > 0$ and $\alpha \in \mathbb {R}$, an experimentally controllable and a material parameter, respectively. Here the equivalence classes of potentially stable critical points of these equations defined in (17.3.11) have the following morphological identifications: I represents a planar interface; II, parallel ridges; and III$^\pm$, hexagonal arrays of elevated dots or circular holes, respectively. The orbital stability conditions for those critical points can be posed in terms of $\sigma$. Thus critical point I is stable in this sense for $\sigma < 0$ while the orbital stability behavior of II and III$^\pm$ which depends only on the sign of $a_0$ (since $2a_2-a_1 = a_1 > 0$) and on the quantities $\sigma _j$ for $j=-1,1$, and 2 defined in (17.3.10) has been summarized in Table 17.3. Construct a morphological stability diagram analogous to Fig. 17.9 in the physically relevant portion of the $\alpha$-$\beta$ plane consistent with these predictions identifying those regions corresponding to stable dots, ridges, and holes, respectively.