Magneto-optical Goos-Hänchen effect in a prism-waveguide coupling structure

Tingting Tang; Jun Qin; Jianliang Xie; Longjiang Deng; Lei Bi

doi:10.1364/OE.22.027042

1. Introduction

GH shift is a lateral shift of the reflected light beam occurs from the position predicted by geometrical optics when a light beam is totally reflected at a dielectric interface [1,2], which is of the order of the wavelength in an ordinary case. Recently the enhancement of GH shift has attracted much attention of researchers for its potential applications in integrated optics, optical storage and optical sensors [3]. Different structures are employed to realize large lateral shift [4], such as using the leaky guided mode in a thin dielectric layer [5–7], or the Krestschmann configuration with long-range surface plasmon resonance (SPR) and a prism-waveguide system with gold layer [3]. In the prism-waveguide system, the positive and negative lateral shift can be widely used in the detection of surface irregularities, roughness because of its high sensitivity [4]. In addition, Chen reported their observation of large positive and negative lateral shifts on a symmetrical metal-cladding waveguide in their experiment [8,9]. And the experimental results help to realize it applications in precise processing and sensors. It was reported that the GH shift was also sensitive to the change of refractive index and could be used as a different mechanism for sensing [3,10]. As tunable giant lateral shift is significant for further applications in flexible optical beam steering and optical devices in information processing, the modulation of giant GH shift was realized by two-dimensional photonic crystals [11,12] or electro-optic nanostructures [13,14]. As another kind of functional material, the modulation of GH shift by magneto-optic (MO) materials, namely the MOGH effect, was rarely studied. The marriage and interplay between GH shift and magneto-optics is very prosperous. Magneto-optics can be used to realize propagation constant modulations to light with different polarizations, which brings special phenomenons to the GH shift.

Compared to GH shift modulation in a waveguide with isotropic optical medium, the MOGH effect shows several novel features, which is essentially due to the gyrotropic nature of the magneto-optical material. (1) The applied magnetic field direction allows controlling of the GH shift of light with different polarizations, such as linear, circular or elliptical polarizations. (2) A nonreciprocal phase shift (NRPS) is introduced to only one of the TE or TM polarizations when the magnetic field is applied perpendicular to the wavevector direction in the waveguide, allowing separate control of the GH shift of TE or TM modes. (3) The MOGH effect is nonreciprocal, which means the backward incident light will show different GH effect under the same applied magnetic field.

In this paper, we propose a prism-waveguide coupling system with a MO material layer: Ce doped Y₃Fe₅O₁₂ (CeYIG), in which the GH shift can be enhanced and modulated by the applied magnetic field. We study the magnetic field direction effect on the GH shift of incident light with various polarizations. To evaluate the MOGH effect quantitatively, we also present the calculation results of both a dielectric waveguide system (prism/air/ CeYIG /SiO₂) and a plasmonic waveguide system (prism/Au/ CeYIG /SiO₂) using the stationary-phase approach with a magneto-optical NRPS perturbation. Finally, the potential applications of MOGH effects are discussed.

2. Theoretical model and mechanism of MOGH effect

2.1 Proposed material and structure

The proposed structure is a classic prism-waveguide system as shown in Fig. 1, where the relative permittivities of the prism (layer 1), coupling layer (layer 2), MO material (layer 3) and dielectric substrate (layer 4) are defined as $ε_{1}^{}$ , $ε_{2}^{}$ , ${\hat{ε}}_{3}$ and $ε_{4}^{}$ respectively, where

{\hat{ε}}_{3} = ε_{0} (\begin{matrix} ε_{x x} & ε_{x y} & - ε_{x z} \\ - ε_{x y} & ε_{y y} & ε_{y z} \\ ε_{x z} & - ε_{y z} & ε_{z z} \end{matrix})

and the relative permeabilities are

μ_{1}^{} = μ_{2}^{} = μ_{3}^{} = μ_{4}^{} = 1

. For MO material with cubic crystal lattice, such as YIG or doped YIG, we assume

ε_{x x} = ε_{y y} = ε_{z z} = ε_{d}

. The off diagonal parts are related to the magnetization of the MO thin film by

ε_{x y} = K M_{z}

,

ε_{x z} = K M_{y}

and

ε_{y z} = K M_{x}

, where

K = K^{'} + i K^{"}

is a material constant related to the Faraday rotation constant

Θ_{F}

by

Θ_{F} = - k_{0} \frac{K^{"} M_{s}}{2 \sqrt{ε_{d}}}

, where M_s is the saturation magnetization of the magneto-optical material and

k_{0}

is the vacuum wavevector. The Faraday ellipticity can be neglected in the transparent region of the magneto-optical material [15]. When the magnetization of the CeYIG material is saturated along one direction, the off diagonal permittivity component is defined as

ε_{o d}

Fig. 1 Schematic of the prism-waveguide coupling structure.

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The operation principle of MOGH effect can be described as follows: The eigenmodes of a magneto-optical waveguide depends on the film magnetization direction. In Fig. 1, the incident light is coupled through a prism coupler into the waveguide mode propagating along x direction. With the magnetic field applied along x or y (z) directions, the magneto-optical waveguide is under a Faraday or Voigt geometry, which leads to eigenmodes with different polarization of the waveguide. Modulating the magnetic field direction will modulate the propagation constants of these eigenmodes, and cause a change of their GH shift. According to stationary phase approach [4,11], (see appendix for derivations) the GH shift can be expressed as a function of the propagation constants of a waveguide mode by:

L = - \frac{2 Im (Δ β^{r a d})}{Im {(β^{0})}^{2} - Im {(Δ β^{r a d})}^{2}} \cos θ_{r}

where

β_{0}

is the eigenpropagation constant of a guided mode of the three layer waveguide (layer 2, layer 3 and layer 4) in which the thickness of layer 2 is semi-infinite. The polarization state of this mode depends on the magnetization direction.

Δ β^{r a d}

is the difference between the eigenpropagation constants of the three-layer waveguide and the prism-waveguide system. The MOGH effect is due to the modulation of

β_{0}

by the applied magnetic field, which will be discussed in detail as follows.

2.2 Effect of magnetic field direction on MOGH

To analyze how the applied magnetic field influences the MOGH effect, we consider the H field along x, y and z directions respectively. We use a three-layer (layer 2, 3 and 4) dielectric slab waveguide with CeYIG as the guiding layer for example. In all cases the magnetic field is assumed to saturate the magnetization of the CeYIG material.

(1) H//z

In this case, the MO waveguide structure is in the Voigt geometry. The eigenmodes of the MO waveguide are TE and TM modes. The permittivity tensor of CeYIG reduced to:

{\hat{ε}}_{3} = ε_{0} (\begin{matrix} ε_{d} & ε_{o d} & 0 \\ - ε_{o d} & ε_{d} & 0 \\ 0 & 0 & ε_{d} \end{matrix})

The eigenpropagation constant is now $β^{0} \pm Δ β^{T E}$ . The off-diagonal component of this tensor leads to an additional nonreciprocal phase shift (NRPS) to the TE polarized light, which can be expressed by [16,17]:

Δ β^{T E} = \frac{2 ω ε_{0}}{β^{T E} N} \iint ε_{o d} E_{y} \partial_{y} E_{y} d y d z

The integral is across the magneto-optical thin film. Due to the structural symmetry along the y axis, the integral of the $E_{y} \partial_{y} E_{y}$ term will be 0 for all TE modes, therefore there will be no change of the TE mode propagation constant with respect to the magnetic field. The TM mode also shows no field induced NRPS in this case, and the MOGH shift is 0.

(2) H//y

When the magnetic field is applied along the y axis, the structure is also under the Voigt geometry with TE and TM modes as eigenmodes. The permittivity tensor is now:

{\hat{ε}}_{3} = ε_{0} (\begin{matrix} ε_{d} & 0 & - ε_{o d} \\ 0 & ε_{d} & 0 \\ ε_{o d} & 0 & ε_{d} \end{matrix})

The eigenpropagation constant of the TM polarized light can be expressed using perturbation theory as $β^{0} \pm Δ β^{T M}$ , where the NRPS induced by the magnetic field is now [17]:

Δ β^{T M} = - \frac{2 β^{T M}}{ω ε_{0} N} \iint ε_{o d} / ε_{d}^{2} H_{y} \partial_{x} H_{y} d y d z

Similar to the H//z case, the integral is all over the magneto-optical thin film. Nontrivial NRPS of the above expression can be achieved in an asymmetric slab waveguide with CeYIG guiding layer, or a plasmonic waveguide with CeYIG as the dielectric layer. By reversing the magnetic field along + y or –y directions, both $ε_{o d}$ and $Δ β^{T M}$ show reversed sign, and the MOGH effect is observed. Notice that in this case, the TE modes do not show any NRPS induced by the magnetic field, therefore there is no MOGH effect for the TE modes.

(3) H//x

When the magnetic field is applied along the x axis, the device is under the Faraday geometry. In this case the eigenmodes of the waveguide is changed to elliptical polarizations. The TE and TM modes are subjected to a nonreciprocal mode conversion [15]

(\begin{matrix} A (x) \\ B (x) \end{matrix}) = (\begin{matrix} \cos ν (x - x_{0}) - i \cos θ \sin ν (x - x_{0}) & - \sin θ \sin ν (x - x_{0}) \\ - \sin θ \sin ν (x - x_{0}) & \cos ν (x - x_{0}) + i \cos θ \sin ν (x - x_{0}) \end{matrix}) (\begin{matrix} A (x_{0}) \\ B (x_{0}) \end{matrix})

where A(x), B(x), A(x₀), B(x₀) are the electric field amplitude along y and z directions in Fig. 1 at the position of x and x₀ respectively. The modal mismatch between TE and TM modes is defined as

Δ β = β_{T E} - β_{T M}

. We have

ν = {[{(Δ β / 2)}^{2} + κ^{2}]}^{1 / 2}

,

\cos θ = (1 / 2) ν^{- 1} Δ β

,

\sin θ = κ / ν

.

κ

is the Faraday rotation per length of the magneto-optical waveguide, which can be expressed as

κ = \iint Θ_{F} E_{T E} E_{T M} d A

[15], where

Θ_{F}

is the Faraday rotation per length of the CeYIG material. The modal conversion efficiency is

R = \sin^{2} θ \sin^{2} ν x

. In a special case if

β_{T E} = β_{T M} = β^{0}

, the TE and TM modes are phase matched, and the maximum value of R can reach 1. The eigenmode of the waveguide is circular polarized. Depending on the chirality of the circular polarized light, the eigenpropagation constants are

β_{\pm} = β^{0} \pm κ

. According to Eq. (2), the GH shift of these two modes will be different and also be dependent on the magnetic field directions. On the other hand if

Δ β > > κ

, the eigenmode of the waveguide remains to almost linearly polarized and unchanged by the applied magnetic field. In between these two cases, the eigenmode is generally elliptically polarized, which shows different GH shift under the applied magnetic field depending on its chirality.

In a more general case, if the magnetic field is applied along an arbitrary direction of the CeYIG film, the permittivity tensor takes the form of Eq. (1). The eigenmode of the magneto-optical waveguide is elliptically polarized in general. They will show MOGH effect induced by the magnetic field both due to the Faraday effect and NRPS effect as discussed above. Note in this case, it is the magnetization vector of the CeYIG thin film that determines the permittivity tensor, which may not align collinearly with the applied magnetic field direction due to the magnetic anisotropies. We summarize the above discussions of MOGH effect in Table 1 below:

Table 1. Summarize of the MOGH Effect in a Three Layer Dielectric Waveguide Shown in Fig. 1

View Table

Experimentally, the above MOGH effect can be studied using a quarter wave plate (QWP) and a two dimensional position sensitive detector [10]. By changing the applied magnetic field direction and rotating the wave plate directions, the above mentioned MOGH effect can be explored for different polarizations of incident light.

To quantitatively evaluate the MOGH effect in the proposed structure, we focus our discussion on the H//y case, where the TM mode of the waveguide will show an NRPS and MOGH effect. For magnetic field applied along the + y or –y directions, the propagation constant β⁰ can be expressed as

β_{y +}^{0} = β_{}^{0} + Δ β a n d β_{y -}^{0} = β_{}^{0} - Δ β

where Δβ is the nonreciprocal phase shift (NRPS) caused by the magneto-optical material. According to Eq. (2), this nonreciprocal phase shift also leads to different GH shift as follows,

Δ L = \frac{1}{k_{0} n_{1}} \cdot \frac{d ϕ_{2}^{y -}}{d θ} - \frac{1}{k_{0} n_{1}} \cdot \frac{d ϕ_{2}^{y +}}{d θ}

where

ϕ_{2}^{y +}

and

ϕ_{2}^{y -}

are calculated by Eq. (11) in which

β_{}^{0}

is substituted by

β_{y +}^{0}

and

β_{y -}^{0}

, respectively. Here

Δ L

is called the MOGH shift of the prism-waveguide coupling structure.

Except for GH shift, there is a lateral shift of reflected light in circular or elliptical polarization after total internal reflection named Fedorov–Imbert (FI) effect [18,19]. This effect is the circular polarization analog of the Goos–Hänchen effect [20,21]. As we discussed before, when the applied magnetic field is along the x-axis, eigenmode of the waveguide is circularly or elliptically polarized. In this case, the FI effect is associated with incident light width, the circular or elliptical polarization and the effective propagation constant. By controlling the magnetic field direction, the eigenpropagation constant of circularly or elliptically polarized light can be modulated as well as the FI shift. This phenomenon can also be detected by a two dimensional position sensitive detector experimentally [20], and will be discussed in more detail in another following work. In this paper, we mainly focus on the discussion of MOGH effect, with the applied magnetic field is along the y-axis in Fig. 1.

3. Results and discussion

The MOGH effect from two example structures of prism/air/CeYIG/SiO₂ and prism/Au/CeYIG/SiO₂ are calculated in this section. The influence of multilayer thicknesses, material dielectric constants on the MOGH effect is systematically studied. For the p-polarized incident light, we follow all the definitions of terms in section 2. The MOGH is calculated using Eq. (9) in all structures, while the nonreciprocal phase shift Δβ is simulated by applying a finite element method (FEM) using the COMSOL Multiphysics software [22].

3.1 Prism/air/CeYIG/SiO₂

In Fig. 2(a), the black, red and blue curves show the GH shifts for different applied magnetic fields, respectively. There is a difference between the GH shifts with opposite magnetic fields which is the MOGH shift in this case. In order to obtain clear comparison between them, we show the calculated GH shift and the MOGH shift of the prism/air/CeYIG/SiO₂ structure in Fig. 2(b). The inset shows the excited guiding mode in the three-layer waveguide (air/CeYIG/SiO₂) when the phase-matching condition is satisfied. From this figure, the MOGH shift can be within ± 300 μm range, tuned under a saturation magnetic field of 100 Oe along + y and –y directions for materials such as CeYIG [23], demonstrating the high sensitivity of reflected light beam position by applied magnetic fields.

Fig. 2 (a) GH shifts with different applied magnetic fields along + y ( + H), -y(-H) and no field applied (0H) case (b) Comparison of reflectivity, GH shift and MOGH shift in the prism/air/CeYIG/SiO₂ waveguide. The wavelength of the incident light is set at 1.24 μm. The material dielectric constants and layer thicknesses are taken from experiment values as: ε₁ = 5.29, ε₂ = 1, ε_d = 5.1529 + 0.02707i, ε_od = 0.0047 + 0.02086i, ε₄ = 2.1, d₂ = 0.575 μm, d₃ = 0.3 μm [23,25,26].

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According to Eq. (2), the relative amplitude of Im(β⁰) and Im(Δβ^rad) determines the sign of GH shift, causing by the interference between the directly reflected light. For each GH shift versus incident angle curve, there is an incident angle satisfying the phase-matching condition which corresponds to the largest absolute value of GH shift. The sign of GH shift depends on the relative values between the intrinsic damping and the radiative damping. When the intrinsic damping is larger than the radiative damping, GH shift is negative, while when the intrinsic damping is smaller than the radiative damping, GH shift is positive. Due to the competition of these two damping factors, hereafter, we name the negative GH shift as “over-coupling” and the positive GH shift as “under-coupling” conditions respectively, in analogous to a resonator system. This coupling condition as well as GH shift and MOGH effect can be tuned by controlling the layer thickness and indices, achieving different signs or amplitudes of MOGH effect controlled by the applied magnetic field.

We firstly consider the influence of indices of layers. Figure 3(a)–3(d) show the GH shifts and MOGH shifts in a prism/air/CeYIG/SiO₂ structure for different n₁ and n₂ respectively. In Fig. 3(a) and 3(b), with increasing n₁ from 2.28 to 2.33, the minimum value of reflectivity has a variation less than 1 dB (data not shown), whereas the phase-matching angle θ_r moves toward smaller incident angles according to Eq. (A6). Except the variation of θ_r, the GH shift line-shape and maximum value showed little change with n₁. This leads to similar MOGH effects shifting along the θ as shown in Fig. 3(b). The weak influence of n₁ on MOGH effect can be understood by the discussion in section 2. Due to the weak field confinement of the guiding mode in the prism layer, varying the prism index will cause little disturbance on the field distribution and propagation loss Im(β⁰), radiative loss Im(Δβ^rad) of the guided mode except the phase-matching angle θ_r shift, leading to similar NRPS and MOGH spectrum.

Fig. 3 GH shifts and MOGH shifts in a prism/air/CeYIG/SiO₂ structure for different n₁ and n_2.

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This situation is very different when the index of the coupling layer n₂ is changed. Figures 3(c) and 3(d) show the influence of the coupling layer index (n₂) on the GH shift and MOGH shift respectively. With increasing n₂, the GH and MOGH spectra show clear variations. The GH shift in Fig. 3(c) changes from negative to positive, while the MOGH shift line shape also changes accordingly. This is because when increasing the coupling layer index, the guiding TM mode propagates more into the coupling layer along –z direction, causing a higher radiative loss and drive the coupling condition from “under-coupling” to “over-coupling”. When varying n₂, the MOGH spectrum shows symmetric line shape and high sensitivity with respect to the coupling layer index n₂ . The symmetric line shape indicates that the NRPS induced by magnetic field does not influence the coupling status between the incident light and the waveguide mode significantly. This strong dependence of GH and MOGH effect on the coupling layer and substrate index may make them useful for refractometric sensors.

The waveguide layer thickness also controls the radiative and propagation losses and the GH and MOGH effect. This is shown in Figs. 4(a)–4(d) respectively. With increasing d₂, the guiding mode is less penetrated to the prism showing lower radiative loss Im(Δβ^rad). The positive GH shift becomes negative, indicating a transition from “under-coupling” to “over-coupling”. We can find θ_r has very small changes (less than 1°) with the increase of air thickness, meaning guiding mode propagation constant and the phase matching angle is not obviously changed. The MOGH effect is shown in Fig. 4(b). At around d₂ = 0.575 μm, the MOGH shifts shows large positive and negative values in a narrow range of incident angle, indicating the high tunability of GH shift by the applied magnetic field from “under-coupling” to “over-coupling”.

Fig. 4 GH shifts and MOGH shifts in a prism/air/CeYIG/SiO₂ structure for different d₂ and d_3.

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Similar trend of coupling condition, GH and MOGH shift change caused by the magneto-optical layer thickness d₃ can be observed in Figs. 4(c) and 4(d). Meanwhile, the thickness of the magneto-optical layer caused a change of the magneto-optical NRPS effect and MOGH lineshape. From the MOGH shift curves in Fig. 4(d), we can find that for d₃<0.3 μm, the MOGH is symmetric and reaches the largest value at 0.3 μm, for d₃>0.3 μm, the MOGH shift of positive and negative parts are not symmetric anymore. This phenomenon can be explained as that with the increase of CeYIG layer thickness, the NRPS induced Δβ^TM is enhanced. This propagation constant difference is large enough to cause significant change in the coupling condition, leading to a large variation of GH shift under different applied magnetic field and asymmetric MOGH spectrum.

3.2 Prsim/Au/ CeYIG /SiO₂

Another unique structure that supports the TM polarized light is the plasmonic waveguide. Due to the strong field gradient across the metal-dielectric interface, this structure is particularly interesting for enhancing the NRPS effect for nonreciprocal photonic devices [24]. The MOGH effect will also be interesting for theoretical investigations. For this waveguide, we choose a prism/Au/CeYIG/SiO₂ structure (Kretschmann configuration). The spectra of reflectivity, the GH shift and the MOGH shift are shown in Fig. 5 with the inset showing a surface plamon polariton (SPP) mode propagating at the interface of Au and CeYIG. In this structure, the SPP mode is excited and the incident light is coupled into the CeYIG layer. An obvious difference compared to the dielectric waveguide structure in 3.1 is that the MOGH spectrum shows an asymmetric double-peak line shape, which we will soon see it is originated from the enhanced MO effect due to the magneto-plasmonic structure.

Fig. 5 Comparison of reflectivity, GH shift and MOGH shift of plasmonic waveguide with material and dimensional parameters of ε₁ = 5.29, ε₂ = −76.7745 + 6.5249i, ε_d = 5.1529 + 0.02707i, ε_od = 0.0047 + 0.02086i, ε₄ = 2.1, d₂ = 0.021 μm, d₃ = 0.3 μm at λ = 1.24 μm [23,25–27].

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We calculated the layer index and thickness effect on the MOGH shift of the plasmonic waveguide structure. Two examples of the prism index and the magneto-optical layer thickness are shown in Fig. 6. For all structural parameters we explored including the the substrate index, the coupling layer thickness and the CeYIG layer thickness, an “under-coupling” to “over-coupling” feature is observed, similar to the dielectric waveguide case. However there are several new features of the MOGH effect observed in the plasmonic waveguide. Figures 6(a) and 6(b) show the influence of prism refractive index on the GH shift and MOGH shift respectively. An obvious difference compared to Fig. 3(a) is that for the plasmonic waveguide structure, with increasing the prism index, the coupling state changes from “over-coupling” to “under-coupling”. This phenomenon can be understood by comparing the mode distributions between the two waveguides. In the dielectric waveguide, the guided mode is confined in the middle of the guiding layer; while in the plasmonic waveguide, the SPP mode is generated on the interface of the gold layer and the CeYIG layer. We notice that compared with the coupling layer in the dielectric waveguide (air), the gold layer is rather thin which is about 20 nm, with the SPP mode closely located to the prism. Therefore the prism index shows a stronger influence on the mode propagation and coupling efficiency, which leads to an obvious influence on the GH shift.

Fig. 6 GH shifts and MOGH shifts in a prism/Au/CeYIG/SiO₂ structure for different n₁ and n_4.

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Another difference is that the MOGH spectrum becomes significantly non-symmetric featuring two peaks with different amplitudes. This is due to the strongly enhanced MO NRPS effect caused by the field localization at the Au/CeYIG interface [24]. For example, the typical NRPS in the plasmonic waveguide is calculated to be 167 rad/cm, compared with the value of the 19.5 rad/cm in the dielectric waveguide. The propagation constant of TM polarized light is therefore significantly altered by the applied magnetic field. The sign and amplitude of GH shift is also strongly altered, leading to the two-peak spectrum observed in Fig. 6(b).

This large NRPS can be further enhanced by increasing the CeYIG magneto-optical layer thickness. Figures 6(c) and 6(d) show the influence of the thicknesses of CeYIG layer on the GH and MOGH shifts. The critical angle θ_r also increases to satisfy the phase-matching condition as shown in Fig. 6(a). The corresponding MOGH shift spectrum changes from symmetric (d₃ = 0.31 μm) to asymmetric (d₃ = 0.32 μm) as shown in Fig. 6(d). Then the two-peak MOGH spectrum appears at d₃ = 0.32 μm. With the increase of d₃, one of these peaks becomes increasingly larger, and the other one becomes increasingly smaller. The largest MOGH shift appears when d₃ = 0.35 μm, which is due to the thicker MO layer induced stronger NRPS and MOGH effect. Notice in this case in one of the magnetization directions the GH effect is totally suppressed, indicating that the prism-waveguide structure is tuned out of phase-matching condition due to the applied magnetic field. This observation shows that the SPP mode coupling can be switched on and off by the applied magnetic field, leading to large GH variation and a unique single-peaked MOGH spectrum.

The apparent difference between the MOGH effect in the dielectric waveguide and the plasmonic waveguide described above may lead to different applications. In the dielectric waveguide, the MOGH shift is very sensitive to the refractive index change of the coupling layer as shown in Fig. 3(d). Based on this property, refractive index sensors with high sensitivity can be realized by replacing the air with samples to be detected. Moreover incident-angle selectors or mode-selectors can also be realized as MOGH is also sensitive to the angles and propagation modes of incident light. In the plasmonic waveguide, significant modulation of the GH shift can be achieved with magnetic field applied along different directions(from positive to negative saturation magnetization). The reflected light spot has obvious shifts with different magnetic field distribution, which can be used to design MO switches and magnetic field detectors. The unique feature of MOGH to switch on and off TM mode polarizations coupled to the waveguide makes it very interesting for reflection polarization control. These unique properties of the MOGH effect make them potentially useful in photonic sensors and polarization dependent reflection manipulations.

4. Conclusion

In this paper, we study the enhancement of Goos-Hänchen (GH) effect MOGH effect in a prism-waveguide coupling system with magneto-optical layer. The MOGH effect as a function of the applied magnetic field direction and incident light polarization is discussed. Both dielectric waveguides (prism/Air/CeYIG/SiO₂) and plasmonic waveguides (prism/Au/CeYIG/SiO₂) are simulated and discussed. Theoretical analysis is derived by stationary-phase method and calculation results are presented. Giant GH shift are observed in both structures. By applying opposite magnetic fields across the MO layer, the GH shift can be controlled resulting in the MOGH effect. The dependence of GH and MOGH effect on the multilayer thickness and indices are simulated. Calculated results demonstrate that this MOGH effect possesses high sensitivity, making them very promising for MO switches or modulators, as well as ultra-sensitive chemical and biomedical sensor applications.

Appendix

For the GH effect calculation, we follow a similar derivation in previous reports [4,8]. The reflection coefficient of the four-layer optical system can be written as

r_{1234} = \frac{r_{12} + r_{12} r_{23} r_{34} \exp (2 i k_{3 z} d_{3}) + [r_{23} + r_{34} \exp (2 i k_{3 z} d_{3})] \exp (2 i k_{2 z} d_{2})}{1 + r_{23} r_{34} \exp (2 i k_{3 z} d_{3}) + r_{12} [r_{23} + r_{34} \exp (2 i k_{3 z} d_{3})] \exp (2 i k_{2 z} d_{2})}

with

r_{i j} = {\begin{matrix} \frac{k_{i z} / ε_{i} - k_{j z} / ε_{j}}{k_{i z} / ε_{i} + k_{j z} / ε_{j}} T M w a v e \\ \frac{k_{i z} - k_{j z}}{k_{i z} + k_{j z}} T E w a v e \end{matrix}

where

r_{i j}

is the Fresnel reflection coefficient at the interface of layer i and layer j;

k_{i z}

is the z component of the wave vectors in layer i, which can be expressed as

k_{i z} = k_{0} \sqrt{ε_{i} μ_{i} - k_{x}^{2}}

_{, where} _kx is the x component of the wavevector and k0 is the vacuum wave-vector of the incident light.

The reflection coefficient of the four-layer optical system can be approximated by a Lorentzian-type relation [4] around the resonance angle and written as [4]

r_{1234} = r_{12} \frac{k_{x} - Re (β^{0}) - Re (β^{r a d}) - i [Im (β^{0}) - Im (Δ β^{r a d})]}{k_{x} - Re (β^{0}) - Re (β^{r a d}) - i [Im (β^{0}) + Im (Δ β^{r a d})]}

where

β^{0}

is the eigenpropagation constant of a guided mode of the three layer waveguide (layer 2, layer 3 and layer 4) in which the thickness of layer 2 is semi-infinite.

Δ β^{r a d}

is the difference between the eigenpropagation constants of the three-layer waveguide and the prism-waveguide system. The imaginary parts of

β^{0}

and

Δ β^{r a d}

are called the intrinsic and radiative dampings, respectively. Meanwhile

β^{0}

can be derived by solving the dispersion relation [16]

γ_{3} d_{3} = m π + arc \tan T_{23} + arc \tan T_{43}, m = 0, 1, 2, 3 \dots

where

γ_{3} = k_{0} \sqrt{ε_{3} μ_{3} - {(β^{0})}^{2}}, γ_{2} = k_{0} \sqrt{{(β^{0})}^{2} - ε_{2} μ_{2}}, γ_{4} = k_{0} \sqrt{{(β^{0})}^{2} - ε_{4} μ_{4}},

T_{23} = \frac{ε_{3}}{ε_{2}} \frac{γ_{2}}{γ_{3}}

and

T_{43} = \frac{ε_{3}}{ε_{4}} \frac{γ_{4}}{γ_{3}}

.

For the Au/ CeYIG /SiO₂ waveguide, the fundamental TM mode (TM₀) has two sets of dispersion relations. When the propagation constant of the TM₀ mode satisfies $\sqrt{ε_{2} ε_{4} / (ε_{2} + ε_{4})} < β^{0} / k_{0} < \sqrt{ε_{3}}$ , it propagates as a guided mode with most energy confined in the CeYIG layer, while if $\sqrt{ε_{3}} < β^{0} / k_{0} < \sqrt{ε_{2} ε_{3} / (ε_{2} + ε_{3})}$ , the TM₀ mode propagates as a SPP mode with most energy confined on the interface of Au and CeYIG.

By defining $W = k_{x} - Re (β^{0}) - Re (Δ β^{r a d})$ , the reflectivity of the prism-coupling waveguide system can be expressed as

R = {| r_{12} |}^{2} (1 - \frac{4 Im (β^{0}) Im (Δ β^{r a d})}{W^{2} + (Im {(β^{0})}^{2} + Im {(Δ β^{r a d})}^{2})}) .

When

W = 0

, the phase-matching condition

k_{x} = k_{0} \sin θ = Re (β^{0}) + Re (Δ β^{r a d})

is satisfied, where θ is the incident angle. In this case the incident light is strongly coupled into the guiding layer and guided modes are excited. The reflectivity reaches the minimum and the GH shift reaches the maximum value. Here we use

θ_{r}

to denote the incident angle under the phase-matching condition.

According to the stationary-phase approach, the lateral beam shift is given by [4]

L = - \frac{1}{k} \cdot \frac{d ϕ}{d θ}

where

ϕ = a \tan (\frac{Im (N D^{*})}{Re (N D^{*})})

,

N

and

D^{*}

represent the numerator and the complex conjugate of denominator of

r_{1234}

. Per Eq. (A7), the phase shift

ϕ

is composed of two terms. We use

ϕ_{1}

and

ϕ_{2}

to denote the phase shift induced by

r_{12}

and the second term in Eq. (A3), respectively, where

ϕ_{1}

is the phase shift of light directly reflected at the interface of prism and coupling layer and

ϕ_{2}

is phase shift induced by the coupling of light into the waveguide. Therefore

L = - \frac{1}{k_{0} n_{1}} \cdot \frac{d ϕ}{d θ} = - \frac{1}{k_{0} n_{1}} (\frac{d ϕ_{1}}{d θ} + \frac{d ϕ_{2}}{d θ})

where

ϕ_{2} = arc \tan (\frac{2 W Im (Δ β^{r a d})}{W^{2} + [Im {(β^{0})}^{2} - Im {(Δ β^{r a d})}^{2}]})

Calculations show that

\frac{d ϕ_{1}}{d θ} ≪ \frac{d ϕ_{2}}{d θ}

in which

\frac{d ϕ_{1}}{d θ}

is about 3 orders smaller than

\frac{d ϕ_{2}}{d θ}

, therefore

\frac{d ϕ_{1}}{d θ}

can be ignored. The GH shift can be rewritten as

L = - \frac{1}{k_{0} n_{1}} \cdot \frac{d ϕ_{2}}{d θ}

The lateral shift under phase-matching condition can be simplified as

L = - \frac{2 Im (Δ β^{r a d})}{Im {(β^{0})}^{2} - Im (Δ β^{r a d}) {}^{2}} \cos θ_{r}

Notice that the maximum GH value is achieved when Im(β⁰) = Im(Δβ^rad).

Acknowledgments

This work was supported by the National Natural Science Foundation of China under Grant No. 51302027 and No. 61475031.

References and links

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Magnetic Field Direction	Eigenmodes	MOGH effect
H//z	TE/TM	None
H//x	Circular or Elliptical polarization	Left/right Circular or Elliptical polarization(Faraday Effect)
H//y	TE/TM	TM (NRPS)

Magneto-optical Goos-Hänchen effect in a prism-waveguide coupling structure

Abstract

1. Introduction

2. Theoretical model and mechanism of MOGH effect

2.1 Proposed material and structure

2.2 Effect of magnetic field direction on MOGH

(1) H//z

(2) H//y

(3) H//x

3. Results and discussion

3.1 Prism/air/CeYIG/SiO₂

3.2 Prsim/Au/ CeYIG /SiO₂

4. Conclusion

Appendix

Acknowledgments

References and links

Cited By

Figures (6)

Tables (1)

Equations (20)

Optics Express

Abstract

1. Introduction

2. Theoretical model and mechanism of MOGH effect

2.1 Proposed material and structure

2.2 Effect of magnetic field direction on MOGH

(1) H//z

(2) H//y

(3) H//x

3. Results and discussion

3.1 Prism/air/CeYIG/SiO2

3.2 Prsim/Au/ CeYIG /SiO2

4. Conclusion

Appendix

Acknowledgments

References and links

Cited By

Figures (6)

Tables (1)

Equations (20)

Optics Express

3.1 Prism/air/CeYIG/SiO₂

3.2 Prsim/Au/ CeYIG /SiO₂