1 Introduction

Semiconductor crystals are widely used in photonics [1, 2], nonlinear [3] and waveguide optics [4, 5]. Semiconductor crystals exhibit different types of non-linear effects in different wavelength ranges [6]. The most common are the Kerr effect [7, 8] and the photorefractive effect [9, 10]. Semiconductor crystals such as lithium niobate (LiNbO3), barium titanate (BaTiO3), gallium arsenide (GaAs), bismuth silicon oxide (BSO, Bi12SiO20), bismuth germanium oxide (BGO, Bi12GeO20) could be alternatives to photorefractive crystals [11,12,13].

Many semiconductor heterostructures in advanced optoelectronics are characterized by the refractive index or dielectric function depending on spatial distance (they usually called the graded-index media) [14]. The refractive index of semiconductor photonic crystals GaAs/GaAlAs [15, 16], InGaAs/InAlAs [17], InGaAsP/InP [18] are described by the graded-index profiles.

Of interest are different types of the waves propagating along the waveguide structures of such crystals [19,20,21,22,23]. Usually, waves propagating along the boundaries of nonlinear optical crystals [24,25,26,27] and along media with a refractive index gradient [28,29,30,31,32] are described separately. However, the authors of [33] described theoretically the dispersion properties of slab waveguides with a linear graded-index film and a nonlinear substrate, and dispersion curves of a slab waveguide with a nonlinear covering medium and an exponential graded-index thin film [34, 35].

Recently, we describe theoretically the surface waves in linearly graded-index and intensity-dependent index layered structure [36], in a crystal with the constant gradient of refractive index and photorefractive crystal [37], composite planar waveguide structure consisting of the linearly graded-index layer and the nonlinear layer formed with an increasing the electric field [38], in a linearly graded-index substrate covered by intensity dependent nonlinear self-focusing cladding [39], in a graded-index substrate covered by an intensity-dependent defocusing nonlinear medium [40]. We proposed also the application of the new form of the Schrödinger equation for design a model of an optical waveguide with a refractive index gradient in semiconductor photonic crystals [41], where the waveguide properties of a thin film with a nonlinear optical response in a gradient medium were described analytically.

The properties on nonlinear surface and guided waves could be controlled by different characteristics of the nonlinear and photonic crystals [42,43,44,45,46,47] including the photorefractive crystals [48,49,50]. The most common control parameters are optical parameters such as the refractive index and geometric parameters such as the thickness of the interlayer or coating [51,52,53,54]. The possibility of controlling the field localization with the help of the temperature of the waveguide on the border with a photorefractive crystal was also noted [55, 56]. The temperature dependence of wave characteristics is due to the fact that many properties of semiconductor crystals depend on temperature. In particular, is the authors of [57] observe the temperature dependent magnetization of GaMnAs, GaMnAs0.94P0.06 as well as GaMnAs0.91P0.09. By analogy with the London penetration depth in superconductors [58, 59], the oscillating magnetic field profile and its penetration depth in semiconductor crystals with photorefractive nonlinearity can be controlled by temperature. In this connection, interest arises in studying the possibility of temperature control of the features of field localization along composite waveguides based on semiconductors with different optical properties.

In this paper, we describe theoretically for the first time the nonlinear guided waves propagating along the interfaces between the film with linearly graded-index profile and photorefractive crystal with diffusion type of nonlinearity. We find the exact solutions to the boundary-value problem formulated corresponding to new types of the guided waves propagating in the different ranges of the effective refractive index. We show the possibility of temperature controlling the field localization along the film interfaces.

2 Equations and boundary conditions

We consider the composite planar waveguide based on semiconductor crystals with the different optical properties. The waveguide is characterized by the following structure. Between two semiconductor crystals with diffusion type of photorefractive nonlinearity locates the semiconductor film of 2a thickness characterized by linearly graded-index profile. The plane-parallel interfaces separated the graded-index film and photorefractive crystals are placed at x =  ± a. The graded-index film locates at |x| < a, and the photorefractive crystals locate at |x| > a. The direction of the optical axis of the photorefractive crystal coincides with the direction of the z-axis along the right film boundary x = a, while it is opposite along the left film boundary x =  − a.

Let us describe the stationary transverse magnetic field H = (0, Hy, 0) distributed along to the z axis [60]. We suppose that \(H_{y} (x,z) = e^{{ik_{0} nz}} H(x)\), where k0 is the wave number, n is the unknown effective refractive index, and the transverse field distribution H(x) obeys the stationary equation [61,62,63,64]:

$$H^{\prime\prime}(x) + k_{0}^{2} \{ n^{2} (x,\left| H \right|) - n^{2} \} H(x) = 0,$$
(1)

where the refractive index \(n(x,\left| H \right|)\) can be written as

$$n(x,\left| H \right|) = \left\{ \begin{gathered} n_{G} (x),\;\left| x \right| < a \hfill \\ n_{P} (\left| H \right|),\;\left| x \right| > a \hfill \\ \end{gathered} \right..$$
(2)

Transverse distribution of the refractive index Eq. (2) in the linearly graded-index film nG is given by

$$n_{G}^{2} (x) = n_{G0}^{2} \{ 1 - \Delta \left| x \right|/a\} ,$$
(3)

where nG0 is the unperturbed refractive index in the film center, \(a\) is the film thickness, \(\Delta = \left( {n_{G0}^{2} - n_{Ga}^{2} } \right)/n_{G0}^{2}\) is the relative change in the film refractive index, and nG0 is the unperturbed refractive index at the film boundaries x =  ± a.

The refractive index in the semiconductor crystal with photorefractive nonlinearity of diffusion type can be presented as [61,62,63,64]

$$n_{P}^{2} (x) = n_{P0}^{2} \left( {1 \pm n_{P0}^{2} r_{eff} \frac{{k_{B} T}}{e}\frac{{I^{\prime}}}{{I + I_{d} }}} \right),$$
(4)

where nP0 is the unperturbed refractive index of semiconductor photorefractive crystal, reff is the effective electro-optic coefficient, kB is the Boltzmann constant, T is the temperature, e is the electron charge modulus, I ~ \(\left| {H_{y} } \right|^{2}\) is the intensity of the magnetic field, Id is the dark intensity. Here we assume that the addition to the unperturbed refractive index of the photorefractive crystal with only diffusion mechanism of nonlinearity formation is small and the dark intensity can be neglected (I >  > Id) [65,66,67]. In Eq. (4) the “ + ” sign is chosen for the right half-space (at x > a), and the “ − ” sign is chosen for the left one (at x <  − a).

The substitution of Eqs. (2)–(4) into Eq. (1) leads to the following equations:

$$H^{\prime\prime}(x) + k_{0}^{2} \{ n_{G0}^{2} (1 - \Delta \left| x \right|/a) - n^{2} \} H(x) = 0,\;\;\left| x \right| < a,$$
(5)
$$H^{\prime\prime}_{{}} \pm 2\mu H^{\prime}_{{}} + k_{0}^{2} (n_{P0}^{2} - n^{2} )H = 0,\,\;\left| x \right| > a,$$
(6)

where \(\mu = k_{0}^{{}} T/T_{P}\) is the attenuation factor in the photorefractive crystal, and \(T_{P} = e/k_{0}^{{}} k_{B} r_{{{\text{eff}}}} n_{P0}^{4}\) is the its characteristic temperature.

We use the boundary conditions at the interfaces:

$$H( \pm a + 0) = H( \pm a - 0) = H_{a} ,$$
(7)
$$H^{\prime}_{{}} ( \pm a \mp 0)/n_{G0}^{2} = H^{\prime}_{{}} ( \pm a \pm 0)/n_{P0}^{2} ,$$
(8)

where Ha is the amplitude of the magnetic field at the film interfaces. We suppose that the magnetic field disappears at infinity: |H(x)| → 0 at |x| → ∞.

Thus, we propose the model to describe the guided waves propagation along the linearly graded-index film between photorefractive crystals. The guided waves are described by the continuous and limited solutions to Eqs. (5) and (6) satisfying the boundary conditions Eqs. (7) and (8). We consider the different kinds of the guided waves defined by the ranges of the effective refractive index. The magnetic field described in the present paper is localized at the distance x < \(x_{\max } = a/\Delta\), so that the film is supposed to be wide and the value of Δ is small.

We describe symmetric field distribution (even modes), for which H(− x) = H(x), and anti-symmetric field distribution (odd modes), for which H(− x) =  − H(x), due to the symmetry of the waveguide system.

3 Guided waves with an oscillating profile

3.1 Symmetric distribution

In the case of n > nG0 Eq. (5) is solved by Airy function [68, 69] as follows:

$$H(x) = H_{m} {\text{Ai(}}\left| x \right|/x_{G} + \delta {),}$$
(9)

where

$$\delta = - (n_{G0}^{2} - n^{2} )(ak_{0} /n_{G0} \Delta )^{2/3} ,$$
(10)
$$x_{G}^{{}} = \left( {\frac{a}{{k_{0}^{2} n_{G0}^{2} \Delta }}} \right)^{1/3} ,$$
(11)

and the amplitude Hm will be found from the boundary conditions.

We focus on a symmetric solution with H(− x) = H(x). Therefore, the field distribution must have an extremum (maximum or minimum) at the film center x = 0. Thus, we can write the solution to Eq. (5) as

$$H(x) = H_{m} {\text{Ai(}}\left| x \right|/x_{G} + \delta_{j} {),}$$
(12)

where δj are the zeros of the derivative of the Airy function: Ai′(δj) = 0, j = 1, 2,…, which are negative and form an increasing sequence of modules: \(\left| {\delta_{j} } \right| < \left| {\delta_{j + 1} } \right|\) (in particular, δ1 =  − 1.018792972, δ2 =  − 3.248197582, δ3 =  − 4.820099211) [68, 69].

Comparison of Eqs. (9) and (12) allows finding the that the guided waves of considered type can propagate with the discrete spectrum of the effective refractive

$$n_{j}^{2} = n_{G0}^{2} - n_{Sj}^{2} ,$$
(13)

where \(n_{Sj}^{2} = \left| {\delta_{j} } \right|(n_{G0}^{2} \Delta /ak_{0}^{{}} )^{2/3}\), j = 1, 2,…

The discrete spectrum Eq. (13) generates the order j of waveguide mode. The mode order j exited in the film waveguide is determined by the number of the zeros of the derivative of the Airy function δj for which the condition \(\left| {\delta_{j} } \right| < (ak_{0}^{{}} n_{G0}^{2} /\Delta )^{2/3}\) is fulfilled. On the other hand, we can select the film thickness \(a > \Delta \left| {\delta_{j} } \right|^{3/2} /k_{0}^{{}} n_{G0}^{2}\) for exciting a mode of a certain order j.

In the case of \(\;n\; < \sqrt {n_{P0}^{2} - \mu^{2} /k_{0}^{2} }\) Eq. (8) is solved by function:

$$H(x) = H_{a} e^{ \mp \mu (x \mp a)} \cos (p(x \mp a) \pm \phi )/\cos (\phi ),$$
(14)

where the upper sign corresponds to the photorefractive crystal region x > a, and the lower one corresponds to the photorefractive crystal region x <  − a, and

$$p^{2} = k_{0}^{2} (n_{P0}^{2} - n^{2} ) - \mu^{2} ,$$
(15)

the phase φ will be found from the boundary conditions.

Combining Eqs. (13) and (15), we derive

$$p^{2} = k_{0}^{2} (n_{P0}^{2} - n_{G0}^{2} + n_{Sj}^{2} ) - \mu^{2} ,$$
(16)

which determines the period of spatial oscillations in a photorefractive crystal Λ = 2π/p. Using Eq. (16), we get

$$\Lambda = 2\pi /k_{0}^{{}} \sqrt {n_{P0}^{2} - n_{G0}^{2} + n_{Sj}^{2} - (T/T_{P} )^{2} } .$$
(17)

It follows from Eq. (17) that waveguide modes of a certain order with an oscillating profile can be excited at crystal temperatures not exceeding the critical value determined by the optical parameters of the waveguide: \(T < T_{P} (n_{P0}^{2} - n_{G0}^{2} + n_{Sj}^{2} )^{1/2}\). The longest period of oscillations is observed at low temperatures.

The substitution of Eqs. (12) and (14) into the boundary conditions Eqs. (7) and (8) allows finding the amplitude

$$H_{m} = H_{a} /{\text{Ai(}}a/x_{G} + \delta_{j} {),}$$
(18)

and the phase

$$\phi = - \tan^{ - 1} \left( {(\mu + \gamma_{S} )/p} \right),$$
(19)

where

$$\gamma _{S} = \frac{{n_{{P0}}^{2} }}{{x_{G} n_{{G0}}^{2} }}\frac{{{\text{Ai'}}\left( {a/x_{G} + \delta _{j} } \right)}}{{{\text{Ai}}\left( {a/x_{G} + \delta _{j} } \right)}}.$$
(20)

It is interesting to note that phase φ can turn to zero at the temperature

$$T = - T_{P} \frac{{n_{{P0}}^{2} }}{{k_{0} x_{G} n_{{G0}}^{2} }}\frac{{{\text{Ai'}}\left( {a/x_{G} + \delta _{j} } \right)}}{{{\text{Ai}}\left( {a/x_{G} + \delta _{j} } \right)}},$$
(21)

which can be controlled by optic and geometric characteristics of the film waveguide system.

Thus, we obtain the guided waves with symmetrical oscillation profile as follows:

$$H(x) = H_{a} \left\{ \begin{gathered} \frac{{{\text{Ai(}}\left| x \right|/x_{G} + \delta_{j} {)}}}{{{\text{Ai(}}a/x_{G} + \delta_{j} {)}}}{,}\;\left| x \right| < a \hfill \\ \left( {\frac{{n_{S}^{2} + n_{TS}^{2} }}{{n_{S}^{2} - (T/T_{P} ){}^{2}}}} \right)^{1/2} e^{{ \mp k_{0} (x \mp a)T/T_{P} }} \cos (p(x \mp a) \pm \phi ),\;\left| x \right| > a \hfill \\ \end{gathered} \right.,$$
(22)

where \(n_{S}^{2} = n_{P0}^{2} - n_{G0}^{2} + n_{Sj}^{2}\), \(n_{TS}^{2} = \gamma_{S} (\gamma_{S} + T/T_{P} )/k_{0}^{2}\), where p is defined by Eq. (16) and φ is defined by Eq. (19).

It is important to note that the main guided wave characteristics such as amplitude, period of spatial oscillations and the depth of the filed penetration can be controlled by the temperature of the film waveguide system.

Figure 1 shows the typical field distributions along x-direction for modes of the order 1 (a), 2 (b), 3 (c), and 4 (d). The field penetration depth increases with an increase in the relative change in the refractive index Δ. It is necessary to increase the film thickness to excite a mode of higher orders.

Fig. 1
figure 1

The relative field H(x)/Ha determined by Eq. (22) along x direction (in conventional units) with k0 = 2, nP0 = 5.1, nG0 = 3.9, μ = 1, Δ = 0.1 (1), Δ = 0.2 (2), a j = 1, a = 1; b j = 2, a = 1.4; c j = 3, a = 2; d j = 4, a = 4

Full size image

The symmetrical distributed modes of odd orders j = 1, 3, … are characterized by the field maximum, and the modes of even orders j = 2, 4, … are characterized by the field minimum placed at the film center x = 0, the values of which are defined by

$$H_{0} = H_{a} {\text{Ai(}}\delta_{j} {\text{)/Ai(}}a/x_{G} + \delta_{j} {)}{\text{.}}$$
(23)

The value of the filed maximum of the first order mode increases monotonically with an increase in the film thickness (Fig. 2a). The filed at the film center does not depend on the temperature. The value of the filed maximum of the order mode greater than the second depends non-monotonic on the film thickness (Fig. 2b, c, d).

Fig. 2
figure 2

The relative field at the film center H0/Ha determined by Eq. (23) versus the film thickness a (in conventional units) with k0 = 2, nP0 = 5.1, nG0 = 3.9, μ = 1, Δ = 0.1, j = 1 (a), j = 2 (b), j = 3 (c), j = 4 (d)

Full size image

We find the maxima of filed distribution in the photorefractive crystals are located at positions (at x > a):

$$x_{m} = a - \frac{1}{p}\left\{ {{\text{arctan}}\left( {\frac{\mu }{p}} \right) + {\text{arctan}}\left( {\frac{{\mu + \gamma_{S} }}{p}} \right) - \pi m} \right\},$$
(24)

where m = 0, 1, 2…, and the values of the maxima are defined by:

$$H_{m} = H(x_{m} ) = H_{a} \sqrt {1 + \frac{{n_{{{\text{TS}}}}^{2} }}{{n_{S}^{2} }}} {\text{e}}^{{ - k_{0} (x_{m} - a)T/T_{P} }} .$$
(25)

The effective refractive index n is related to the angle of incidence of the beam exciting the guided wave, and the attenuation coefficient μ is directly proportional to the temperature of the photorefractive crystal T. An increase in the temperature leads to an increase in the frequency of oscillation and to a decrease in the field amplitude in the photorefractive crystal. Therefore, the positions xm and heights of the maxima Hm can be controlled by the angle of incidence and by the waveguide temperature. Increase in the temperature leads to decrease in the field intensity in the photorefractive crystal.

3.2 Anti-symmetric distribution

Odd solution to Eq. (5) in the case of n > nG0 can be written as follows:

$$H(x) = \pm H_{m} {\text{Ai(}}\left| x \right|/x_{G} + \delta {),}$$
(26)

where the upper sign corresponds to the film region 0 < x < a, and the lower one corresponds to film region − a < x < 0.

Here we focus on an anti-symmetrical solution with H(− x) =  − H(x). Therefore, the field distribution must turn to zero at the film center x = 0. Thus, we can write the solution to Eq. (5) as

$$H(x) = \pm H_{m} {\text{Ai(}}\left| x \right|/x_{G} + \xi_{j} {),}$$
(27)

where ξj are the zeros of the Airy function: Ai(ξj) = 0, j = 1, 2,…, which are negative and form an increasing sequence of modules: \(\left| {\xi_{j} } \right| < \left| {\xi_{j + 1} } \right|\) (in particular, ξ1 =  − 2.33810741, ξ2 =  − 4.08794944, ξ3 =  − 5.52055983) [68, 69].

Comparison of Eqs. (9) and (27) allows finding the that the anti-symmetric guided waves can propagate with the discrete spectrum of the effective refractive

$$n_{j}^{2} = n_{G0}^{2} - n_{Aj}^{2} ,$$
(28)

where \(n_{Aj}^{2} = \left| {\xi_{j} } \right|(n_{G0}^{2} \Delta /ak_{0}^{{}} )^{2/3}\), j = 1, 2,…

The discrete spectrum Eq. (28) generates the order j of odd waveguide mode. The mode order j is determined by the zero of the Airy function ξj for which the condition \(\left| {\xi_{j} } \right| < (ak_{0}^{{}} n_{G0}^{2} /\Delta )^{2/3}\) is fulfilled. On the other hand, we can select the film thickness \(a > \Delta \left| {\xi_{j} } \right|^{3/2} /k_{0}^{{}} n_{G0}^{2}\) for exciting a mode of a certain order j.

Odd solution to Eq. (8) in the case of \(n\; < \sqrt {n_{P0}^{2} - \mu^{2} /k_{0}^{2} }\) can be written as follows

$$H(x) = \pm H_{a} e^{ \mp \mu (x \mp a)} \cos (p(x \mp a) \pm \phi )/\cos (\phi ),$$
(29)

where

$$p^{2} = k_{0}^{2} (n_{P0}^{2} - n_{G0}^{2} + n_{Aj}^{2} ) - \mu^{2} ,$$
(30)

We find the period of spatial oscillations in a photorefractive crystal using Eq. (16),

$$\Lambda = 2\pi /k_{0}^{{}} \sqrt {n_{P0}^{2} - n_{G0}^{2} + n_{Aj}^{2} - (T/T_{P} )^{2} } .$$
(31)

It follows from Eq. (31) that waveguide modes of considered type can be excited at crystal temperatures not exceeding the critical value determined by the optical parameters of the waveguide: \(T < T_{P} (n_{P0}^{2} - n_{G0}^{2} + n_{Aj}^{2} )^{1/2}\).

The substitution of Eqs. (27) and (29) into the boundary conditions Eqs. (7) and (8) allows finding the amplitude

$$H_{m} = H_{a} /{\text{Ai(}}a/x_{G} + \xi_{j} {),}$$
(32)

and the phase

$$\phi = - \tan^{ - 1} \left( {(\mu + \gamma_{A} )/p} \right),$$
(33)

where

$$\gamma _{A} = \frac{{n_{{P0}}^{2} }}{{x_{G} n_{{G0}}^{2} }}\frac{{{\text{Ai'}}\left( {a/x_{G} + \xi _{j} } \right)}}{{{\text{Ai}}\left( {a/x_{G} + \xi _{j} } \right)}}.$$
(34)

Thus, we obtain the guided waves with anti-symmetrical oscillation profile as follows:

$$H(x) = \pm H_{a} \left\{ \begin{gathered} \frac{{{\text{Ai(}}\left| x \right|/x_{G} + \xi_{j} {)}}}{{{\text{Ai(}}a/x_{G} + \xi_{j} {)}}}{,}\;\left| x \right| < a \hfill \\ \left( {\frac{{n_{A}^{2} + n_{TA}^{2} }}{{n_{A}^{2} - (T/T_{P} ){}^{2}}}} \right)^{1/2} {\text{e}}^{{ \mp k_{0} (x \mp a)T/T_{P} }} \cos \,(p(x \mp a) \pm \phi ),\;\left| x \right| > a \hfill \\ \end{gathered} \right.,$$
(35)

where \(n_{A}^{2} = n_{P0}^{2} - n_{G0}^{2} + n_{Aj}^{2}\), \(n_{TA}^{2} = \gamma_{A} (\gamma_{A} + T/T_{P} )/k_{0}^{2}\), where p is defined by Eq. (30) and φ is defined by Eq. (33).

Figure 3 shows the typical field distributions of anti-symmetric guided waves defined by Eq. (35) along x-direction for modes of the order 1 (a), 2 (b), 3 (c), and 4 (d).

Fig. 3
figure 3

The relative field H(x)/Ha determined by Eq. (35) along x direction (in conventional units) with k0 = 2, nP0 = 5.1, nG0 = 3.9, μ = 1, Δ = 0.1 (1), Δ = 0.2 (2), a j = 1, a = 1; b j = 2, a = 1.4; c j = 3, a = 3; d j = 4, a = 5

Full size image

The anti-symmetrical distributed modes are characterized by the field maximum located inside the film at positions

$$x_{mj} = x_{G} {(}\delta_{m} - \xi_{j} {)}$$
(36)

the values of which are defined by

$$H_{mj} = H_{a} {\text{Ai(}}\delta_{m} {\text{)/Ai(}}a/x_{G} + \xi_{j} {),}$$
(37)

where m is the number of the maximum, and j is the number of the mode order.

We find that the maxima of filed distribution in the photorefractive crystals are located at positions:

$$x_{m} = a - \frac{1}{p}\left\{ {{\text{arctan}}\left( {\frac{\mu }{p}} \right) + {\text{arctan}}\left( {\frac{{\mu + \gamma_{A} }}{p}} \right) - \pi m} \right\},$$
(38)

where m = 0, 1, 2… is the number of maximum, p is defined by Eq. (30), and the values of the maxima are defined by:

$$H_{m} = H_{a} \sqrt {1 + \frac{{n_{TA}^{2} }}{{n_{A}^{2} }}} {\text{e}}^{{ - k_{0} (x_{m} - a)T/T_{P} }} .$$
(39)

The influence of the optical and geometrical parameters of the waveguide system on the guided wave profile with an anti-symmetric distribution is similar to that described in Sec. 3.1 for the guided waves with a symmetric distribution.

4 Guided waves with a non-oscillating profile

4.1 Symmetric distribution

The guided wave of symmetric type in the film is defined by even solution to Eq. (5) which is given by Eq. (12) and it propagates with the discrete spectrum of the effective refractive defined by Eq. (13).

In the case of \(\sqrt {n_{P0}^{2} - \mu^{2} /k_{0}^{2} } < n\; < n_{P0}\) Eq. (8) is solved by function:

$$H(x) = H_{a} {\text{e}}^{ \mp \mu (x \mp a)} \cosh (\nu (x \mp a) \pm \alpha )/\cosh (\alpha ),$$
(40)

where

$$\nu^{2} = \mu^{2} - k_{0}^{2} (n_{P0}^{2} - n^{2} ),$$
(41)

and α will be found from the boundary conditions.

Combining Eqs. (13) and (41), we derive

$$\nu^{2} = \mu^{2} - k_{0}^{2} (n_{P0}^{2} - n_{G0}^{2} + n_{Sj}^{2} ).$$
(42)

After substitution Eqs. (12) and (40) into the boundary conditions Eqs. (7) and (8) we find that the amplitude Hm is defined by Eq. (18), and the value of α is given by

$$\alpha = \tanh^{ - 1} \left( {(\mu + \gamma_{S} )/\nu } \right).$$
(43)

Thus, we obtain the guided waves with symmetrical non-oscillation profile as follows:

$$H(x) = H_{a} \left\{ \begin{gathered} \frac{{{\text{Ai(}}\left| x \right|/x_{G} + \delta_{j} {)}}}{{{\text{Ai(}}a/x_{G} + \delta_{j} {)}}}{,}\;\left| x \right| < a \hfill \\ \frac{{\sqrt {\nu^{2} - (\mu + \gamma_{s} )^{2} } }}{\nu }{\text{e}}^{ \mp \mu (x \mp a)} \cosh (\nu (x \mp a) \pm \alpha ),\;\left| x \right| > a \hfill \\ \end{gathered} \right.,$$
(44)

where ν is defined by Eq. (42) and α is defined by Eq. (43).

Figure 4 shows the typical field profiles of the guided wave defined by Eq. (44) along x-direction for modes of the order 1 (a), 2 (b), 3 (c), and 4 (d). Note that the guided waves with a non-oscillating profile defined by Eq. (44) can be observed at small incidence angles of the beam that excites this wave [62]. The influence of the optical and geometrical parameters of the waveguide system on the guided wave with a non-oscillating profile is similar to that described in Sec. 3.1 for the guided waves with an oscillating profile.

Fig. 4
figure 4

The relative field H(x)/Ha determined by Eq. (44) along x direction (in conventional units) with k0 = 3, nP0 = 2, nG0 = 1.7, μ = 5, Δ = 0.2 (1), Δ = 0.25 (2), a j = 1, a = 2; b j = 2, a = 4; c j = 3, a = 7; d j = 4, a = 9

Full size image

Note that the guided waves with oscillating profile exist in the range n < nc and the guided waves with non-oscillating profile exist in the range nc < n < nP0 where \(n_{c} = \sqrt {n_{P0}^{2} - (T/T_{P}^{{}} )^{2} }\) is the critical value of the effective refractive index. The value of nc determines the boundary in the discrete spectrum of the refractive index, when passing through which the oscillating mode of the field decay changes to non-oscillating one. Changing the critical value of the effective refractive index is possible by varying the temperature. The maximum temperature at which such a transition between wave types is possible is defined as Tm = TPnP0. Thus, the transition boundary between the guided wave types can be controlled by temperature.

4.2 Anti-symmetric distribution

The guided wave of anti-symmetric type in the film is defined by odd solution to Eq. (5) which is given by Eq. (27) and it propagates with the discrete spectrum of the effective refractive defined by Eq. (28).

The non-oscillating anti-symmetric filed in the photorefractive crystal in the case of \(\sqrt {n_{P0}^{2} - \mu^{2} /k_{0}^{2} } < n\; < n_{P0}\) is defined by odd solution to Eq. (8) as follows:

$$H(x) = \pm H_{a} {\text{e}}^{ \mp \mu (x \mp a)} \cosh (\nu (x \mp a) \pm \alpha )/\cosh (\alpha ),$$
(45)

where the value of α will be found from the boundary conditions.

Combining Eqs. (28) and (41), we derive

$$\nu^{2} = \mu^{2} - k_{0}^{2} \left( {n_{P0}^{2} - n_{G0}^{2} + n_{Aj}^{2} } \right),$$
(46)

After substitution Eqs. (27) and (45) into the boundary conditions Eqs. (7) and (8) we find that the amplitude Hm is defined by Eq. (32), and the value of α is given by

$$\alpha = \tanh^{ - 1} \left( {(\mu + \gamma_{A} )/\nu } \right).$$
(47)

Thus, we obtain the guided waves with anti-symmetrical non-oscillation profile as follows:

$$H(x) = \pm H_{a} \left\{ \begin{gathered} \frac{{{\text{Ai(}}\left| x \right|/x_{G} + \xi_{j} {)}}}{{{\text{Ai(}}a/x_{G} + \xi_{j} {)}}}{,}\;\left| x \right| < a \hfill \\ \frac{{\sqrt {\nu^{2} - (\mu + \gamma_{A} )^{2} } }}{\nu }{\text{e}}^{ \mp \mu (x \mp a)} \cosh (\nu (x \mp a) \pm \alpha ),\;\left| x \right| > a \hfill \\ \end{gathered} \right.,$$
(48)

where ν is defined by Eq. (46) and α is defined by Eq. (47).

Figure 5 shows the typical field profiles of the guided wave defined by Eq. (48) along x-direction for modes of the order 1 (a), 2 (b), 3 (c), and 4 (d).

Fig. 5
figure 5

The relative field H(x)/Ha determined by Eq. (48) along x direction (in conventional units) with k0 = 3, nP0 = 2, nG0 = 1.7, μ = 6, Δ = 0.2 (1), Δ = 0.3 (2), a j = 1, a = 2; b j = 2, a = 4; c j = 3, a = 6; d j = 4, a = 9

Full size image

Note that the minimum film thickness required to excite a waveguide mode of the chosen order is larger for guided waves with a non-oscillating profile than for guided waves with an oscillating profile. Therefore, if field oscillations along the waveguide are undesirable, then it is necessary to choose a wider interlayer film. On the other hand, the greatest localization and the highest intensity of the field at the center of the waveguide system are characteristic of the first order mode. However, to excite the first order mode, the film thickness is required to be less than for higher order modes. Therefore, to achieve the required distribution of the field profile in the waveguide, it is necessary to choose the optimal film thickness, which, on the one hand, is sufficient to excite the guided wave, and, on the other hand, makes it possible to obtain a non-oscillating decrease in the field. This choice of required film thickness is justified for all four types of the guided waves described in this paper.

5 Conclusion

We proposed the model of the composite planar waveguide based on semiconductor crystals in which the film with a linear profile of the refractive index sandwiched by two photorefractive crystals with diffusion nonlinearity. We described the transverse magnetic stationary waves propagating along such film waveguide. We found four types of guided waves. The guided waves with oscillating and non-oscillating profile and non-oscillating symmetrically and anti-symmetrically distributed across film interfaces are obtained.

We found that the temperature of the semiconductor crystal effects on the profiles and the type of the guided waves. We analyzed the influence of the optical and geometrical parameters of the film waveguide on the guided wave characteristics and the filed intensity distributions. The waves of all types described can propagate with the discrete spectrum of the effective refractive index. We derived that it is necessary to select the optimal film thickness, which is sufficient to excite the guided wave and makes it possible to obtain a non-oscillating decrease in the field.

Described waveguide properties of the composite semiconductor planar structure expand theory of semiconductor electronics and can be applied for design of optic and opto-electronic devises, waveguides, logic gates, filters.