Fresnel's laws in discrete optical media

We present the discrete analog of Snell's refraction law at the interface between two dissimilar waveguide arrays. Additionally, reflection and transmission coefficients for the Bloch waves are derived analytically. The theoretical findings were confirmed by analyzing the evolution of light in femtosecond laser written waveguide arrays.


Introduction
When light crosses the interface between two dissimilar media, it experiences refraction as well as partial reflection. In dielectric media, these phenomena are described by the famous laws of Snell and Fresnel. When deriving these mathematical expressions, it is implicitly assumed that the media on both sides of the interface are continuous. Hence, assuming a fixed wavelength, they are characterized by a single quantity: the refractive index. However, in modern physics, a variety of more sophisticated systems such as periodically structured media, e.g. photonic crystals (Yablonovitch 1993), constitute an important aspect in many research fields (Busch et al 2007). During recent years, a subclass of such periodic media has gained much interest: arrays of evanescently coupled waveguides, which are a representation of so-called discrete systems (Christodoulides et al 2003). During recent years, surfaces of and interfaces between waveguide arrays have attracted rapidly growing interest, due to their strong impact on the evolution of light. Light propagation in finite waveguide lattices was modeled in one-(1D)  and two-dimensional (2D)  geometries. The tunneling escape from a boundary waveguide was investigated (Longhi 2006) in close analogy to quantum mechanics. Furthermore, the conditions for existence of localized linear modes at defects (Trompeter et al 2003) and interfaces  were analyzed in detail. The research on discrete propagation in waveguide arrays gained further stimulus, when the existence of localized nonlinear modes at surfaces (Kartashov et al 2006, Kominis et al 2007, Makris et al 2005 and interfaces (Makris et al 2006, Molina and in waveguide arrays was predicted. These phenomena were experimentally demonstrated in 1D (Iwanow et al 2004, Rosberg et al 2006, Smirnov et al 2006, Suntsov et al 2006 and only recently in 2D , Wang et al 2007. It turned out that light propagation at the interface of two waveguide lattices exhibits a variety of peculiarities as a direct consequence of the discrete nature of the medium. However, to date the transition of light through an interface between two dissimilar waveguide arrays was not analyzed. This is somewhat surprising since one can expect that interfaces also have a strong impact on the refraction, reflection and transmission properties of light, culminating in modifications of the laws of Snell and Fresnel with respect to the continuous case.
In this paper, we bridge this gap and present the discrete version of Snell's law of refraction as well as reflection and transmission coefficients for plane waves passing an interface between two dissimilar waveguide arrays. It will be shown that reflection and refraction are determined by the change of the coupling and the propagation constants at the interface.

The discrete Snell's law
In the so-called coupled mode approximation the transverse profile of the propagating modes inside the individual guides is assumed to remain constant, so that only the amplitudes evolve along z. Then one can derive a set of coupled ordinary homogeneous first-order differential equations Here, ϕ m is the amplitude in the mth waveguide and c is the coupling constant, denoting the coupling strength between the individual lattice sites. For reasons of simplicity, the propagation constant β 0 has been removed by a phase transformation. The normalized eigenfunctions of this periodic system are the so-called discrete Bloch waves: The quantity β is the longitudinal propagation constant and κ denotes the transverse propagation constant multiplied by the waveguide spacing corresponding to a phase difference between adjacent waveguides. Inserting equation (2) into (1), one finds β entirely defined by the dispersion relation describing a band structure due to the system's periodicity. All Bloch waves have to fulfill equation (3), otherwise they cannot propagate and decay exponentially. However, equation (1) describes an infinite system with identical and equally spaced waveguides. Hence, in the presence of an interface, one has to modify these equations. Assuming an interface as depicted in figure 1, the coupled mode equations take the form where m = 0 is the position of the interface. The quantities c 1 and c 2 are the coupling constants on the two sides of the interface, whereas δβ is the difference of the propagation constants in the isolated waveguides between these two parts of the array (figure 1). If the guide m = 0 exhibits a detuning δβ, the relation = δβ holds, otherwise = 0. For = 0, one obtains two different dispersion relations for both regions with n < 0 and n > 0, respectively, which read (m < 0) : β 1 = 2c 1 cos κ 1 , (m > 0) : β 2 = δβ + 2c 2 cos κ 2 .
Due to translational invariance in the z-direction, the propagation constant β must be conserved in the whole array. Hence, the relation β 1 = β 2 = β holds, yielding the expression cos κ 2 cos κ 1 = c 1 c 2 − δβ 2c 2 cos κ 1 . This is the discrete equivalent of the well-known Snell's law, which describes the refraction of a propagating plane wave at an interface between two discrete media. Following this analogy, in the discrete case in equation (7) one has two parameters describing the refraction: the ratio of the coupling constants c 1 /c 2 as well as the ratio δβ/c 2 , which is determined by an additional difference of the waveguide profiles. This is in strong contrast to continuous media, where only the ratio of the refractive indices of both materials affects the refraction at the interface. The actual angle α 1,2 of the propagation direction of the individual Bloch waves in the respective array region can be easily calculated by (Eisenberg et al 2000, Pertsch et al 2002 tan α 1,2 = 1,2 ∂β ∂κ 1,2 = −2 1,2 c 1,2 sin κ 1,2 . Furthermore, it is evident that only such waves can pass the interface for which |cos κ 2 | 1 holds. Otherwise the Bloch waves experience total internal reflection, since they cannot penetrate the region behind the interface and decay exponentially in this region in the form of evanescent waves. As a peculiar feature of discrete systems it follows from equation (7) that all incoming Bloch waves are reflected totally when |δβ| 2(c 1 + c 2 ).
In this particular case, the interface acts as a perfect mirror. Since the coupling constants and the detuning are a function of the wavelength, condition (9) will only be satisfied for a certain range of wavelengths. Hence, this feature may find an application as a spectral filter, where only certain wavelengths can penetrate the interface. The occurrence of total reflection can be illustrated by means of the band structure. We consider the bands on both sides of the interface according to equations (5) and (6). Since only waves with a positive value of the Bloch vector reach the interface, we examine the bands in the interval [0; π]. Two examples are shown in figure 2. In figure 2(a) the band is shifted up behind the interface because a positive value for δβ is assumed. In this case, no propagation constant exists inside the right part of the array for Bloch waves from the bottom of the band of the left part. Therefore, such a wave would be completely reflected. An analogous situation occurs for a shrinking of the band due to a decrease of the coupling constant (see figure 2(b)), where total reflection appears if the incoming wave travels at the top or bottom of the respective band.

Reflection and transmission coefficients
When the Bloch waves reach the interface, they split into a reflected and transmitted part. The resulting amplitudes of the Bloch waves can be characterized by reflection and transmission coefficients, in close analogy to the Fresnel's formulae at interfaces between continuous media. To calculate these coefficients, for the propagating Bloch waves one uses the ansatz The coefficient ρ denotes the reflection strength, whereas τ and τ are the transmission coefficients. Since the guide at m = 0 can be assumed as a singular defect, one has to consider the transmission into this defect and the m > 0-region separately. Inserting equations (10) into (4), one obtains Equation (11) When inserting this into equation (12), one obtains Using equations (14) and (15), one can resolve equation (13), resulting in where β has to fulfill the dispersion relations (5) and (6). Equations (15) and (16)  An interesting issue is the symmetry of the reflection and transmission coefficients around κ 1 = π 2 , since due to equation (8) this is the maximum angle at which Bloch waves propagate through the array (Pertsch et al 2002). In the particular case of a vanishing detuning δβ = 0, the reflection coefficient simplifies to This expression, which is the analog of Fresnel's formula for reflection at an interface between two continuous media, is always symmetric around κ 1 = π 2 , since the reflection is only dependent on the direction of the incoming wave. Hence, the absolute value |ρ|, accounting for the reflected amplitude, as well as the argument arg(ρ), representing the phase shift due to the reflection, are symmetric, as shown in figure 3(a). This symmetry is broken when δβ = 0. In this case, it becomes apparent that the phase as well as the amplitude of the reflection (and, hence, also of the transmission) are dependent on the transverse wave vector of the incoming wave rather than on the direction. This is due to the periodic character of waveguide arrays, where different Bloch waves travel in the same direction (see equation (8)).
Concerning the occurrence of total internal reflection, the results are in agreement with the explanation given to figure 2. It might be astonishing that the transmitted wave does not vanish in this case. However, this becomes clear keeping in mind that the transmitted wave has to be either a Bloch wave (for real κ 2 ) or an evanescent wave decaying exponentially in the case of an imaginary κ 2 . As both cases are included, the transmission coefficient τ does not vanish even for total reflection |ρ| = 1. To derive a quantity that represents the power conservation, it is useful to examine the energy flow inside the array. For an infinite array, it follows from equation (1) iϕ Subtracting the second from the first equation yields d dz which denotes the power exchange between a respective guide and its neighbors. The power inside a cluster of guides extending from −M to M is given by Power can escape the cluster only via the two outermost waveguides. This is reflected in the corresponding equation where only the amplitudes of the outermost guides and their neighbors outside the cluster contribute. To determine the power flow one assumes a cluster including all guides from m = −∞ to m = 0. The power change of this area, which corresponds to the (m 0)-region of the array, is given by Assuming plane waves ϕ m = e iκ 2 m , this equation becomes d dz Q m 0 = −2c 2 sin κ 2 (if κ 2 is real), 0 (if κ 2 is imaginary).
As expected, no power flow exists for evanescent waves (κ 2 imaginary). Assuming that the power transported by the reflected and transmitted wave equals the power of the incoming wave, the coefficients have to fulfill c 1 sin κ 1 = |ρ| 2 c 1 sin κ 1 + |τ | 2 c 2 sin κ 2 .
Using equation (7), one finally arrives at which represents the conservation of power inside the array. Hence, the quantities represent the reflection and transmission coefficients for the power. As expected, the transmission T grows with decreasing reflection R and vice versa, whereas their sum is constant.
The corresponding values for R and T from figure 3 are depicted in figure 4.

Discussion of the results
To test the analytical results of the previous section, the coupled mode equations (4) were solved numerically. The first point under investigation will be the discrete Snell's law (7). Since this expression is valid for every single Bloch wave, it is useful to analyze this for a broad excitation covering at least five waveguides. In this case, the Fourier spectrum is narrow, so that only a small number of Bloch waves will be excited (Pertsch et al 2002). Investigating the evolution of the light inside the array around κ = π/2 diffraction is small due to the significantly reduced curvature of the band in this area . Therefore almost no broadening of the beam appears (see figure 5).
In the array under consideration, on the left side (m < 0) the coupling was set to c 1 = 1 cm −1 for a spacing of 1 = 15 µm and on the right side (m > 0) to c 2 = 2 cm −1 , according to 2 = 11 µm . For these values, in region 1 one obtains a propagation angle of α 1 = 0.18 • , whereas in the other region α 2 = 0.25 • , which is shown in Figure 5. In the vicinity of the normalized transverse wave vector κ = π 2 the second derivative of the longitudinal wave vector vanishes, which is equivalent to diffraction-free propagation.
figure 6(a). The picture changes for an additional detuning of the refractive index in the m > 0 region. For a detuning of δn = −1.5 × 10 −4 , the angle of propagation inside this region reduces to α 2 = 0.23 • (see figure 6(b)), whereas for an index detuning of δn = −2.7 × 10 −4 , the angle of propagation reduces to α 2 = 0.18 • , so that no refraction occurs (but still a reflection), which is shown in figure 6(c).
The next peculiar feature of refraction between two discrete media is the fact that the ratio between transmission and reflection is not unique for a specific angle of incidence. In periodic media, the transverse component of the individual Bloch waves range within the first Brillouin zone from κ min = 0 to κ max = π , whereas the angle of propagation (see equation (8)) ranges only from α min = 0 • to α max = arctan {−2 c}, so that two different values of κ yield the same α. Due to equations (16) and (15), in general the reflection and transmission is not symmetric around κ = π 2 . Hence, for the same angle α, two different reflection and transmission coefficients exist for two different Bloch waves residing at different positions in the first Brillouin zone. This feature is depicted in figure 7. In an array with c 1 = 1.05 cm −1 , c 2 = 1.45 cm −1 and a detuning of the refractive index of δn = −2.0 × 10 −4 , two Gaussian excitations with κ (1) 1 = π 3 (figure 7(a)) and κ (2) 1 = 2π 3 (figure 7(b)) propagate at α = 0.18 • but exhibit different reflection and transmission coefficients (figure 7(c)).

Experimental results
For the experimental verification of these results we used planar waveguide arrays fabricated by the fs laser writing technique (Szameit et al 2005. When ultrashort laser pulses are tightly focused into the bulk material, nonlinear absorption takes place leading to optical breakdown and the formation of a microplasma. Thereby, a permanent change of the material's molecular structure is induced. In the particular case of fused silica, the density is locally increased yielding a positive modulation of the refractive index (Itoh et al 2006). By moving the sample transversely with respect to the beam a continuous modification is obtained and a waveguide is created (Miura et al 1997) (see figure 8(a)). By this method, waveguides can be written along arbitrary paths since the only limiting factor in the placement of the focus is the focal length of the writing objective (Nolte et al 2003). Since the induced change of the refractive index is a function of the writing velocity (Blomer et al 2006), the fs writing technique is in particular well suited to induce defects and interfaces in waveguide arrays. Furthermore, all structural changes are permanent and stable after fabrication (Nolte et al 2002). For the fabrication of our waveguides we used a Ti:sapphire laser system (RegA/Mira, Coherent Inc.) with a repetition rate of 100 kHz, a pulse duration of about 150 fs and 0.3 µJ pulse energy at a laser wavelength of 800 nm. The beam was focused into polished fused-silica samples by a 20× microscope objective with a numerical aperture of 0.35. The average power of the laser beam was 32 mW, while the writing velocity was 1300 µm s −1 resulting in a refractive index change of 6 × 10 −4 . The length of all samples was 70 mm. To directly monitor the light propagation we applied a recently developed method of detecting fluorescent light . For that purpose, we used a particular type of fused silica glass with a high content of OH which leads to a massive formation of non-bridging oxygen hole (NBOH) color centers in the modified volume. These can be excited with a HeNe source at 633 nm, which was launched by fiber butt coupling, so that the fluorescence with its maximum at 650 nm can be detected by a camera from above the sample  (see figure 8(b)). Two different setups were investigated experimentally (figure 9(a)), with a spacing of 13 µm and 15 µm on the left and right side, respectively. Additionally, in a second sample (a) (b) (c) Figure 7. Partial reflection of a broad Gaussian excitation at an interface between two waveguide arrays with c 1 = 1.05 cm −1 , c 2 = 1.45 cm −1 and a refractive index detuning of −2.0 × 10 −4 . In (a) the excitation is centered around κ 1 = π 3 and in (b) around κ 1 = 2π 3 , yielding in both cases α = 0.18 • , but different reflection and transmission coefficients, which are depicted in (c). The propagation direction is from top to bottom. a detuning of the refractive index of δn ≈ −2 × 10 −4 was applied in the region with 13 µm spacing, which was achieved by a slightly higher writing speed of 1800 µm s −1 . The resulting coupling coefficient in the region with = 15 µm is c 1 = 1.09 cm −1 , in the region with = 13 µm is c 2 = 1.56 cm −1 . These values remain valid even in the case of a slightly negative detuning, since the lower refractive index of the waveguides is compensated reflected at the interface. From figure 9(b) it follows that in particular the side lobes at κ = π 2 are almost entirely transmitted and only slightly refracted from α 1 = 0.18 • to α 1 = 0.23 • . In contrast, the Bloch waves in the center and the edge of the Brillouin zone experience strong reflection. This is supported by the observation that a large fraction of the light between the side lobes, which results from these Bloch waves, remains in this semi-array where the excited waveguide is located.
This picture changes when an additional detuning δn ≈ −2 × 10 −4 is introduced in the (m > 0)-region, since in this case the reflection and transmission coefficients are modified ( figure 9(c)). The experimentally observed light evolution from a point source in such an array is depicted in figure 11(b) for the excited waveguide at m = 2 and figure 11(d) with the light launched in waveguide m = 0. The corresponding simulations are shown in figures 11(f) and (h). The experimental propagation pattern of the undetuned array is presented in figures 11(a) and (c) for the excited waveguide at m = 2 and m = 0, respectively. The corresponding simulations are given in figures 11(e) and (g). The main feature is now that in the detuned array the reflection is much stronger for all Bloch waves, so that almost all of the incident light is reflected at the interface (figures 11(b) and (d)). Additionally, due to equation (7), the angle of propagation in the detuned array region changes to α 1 = 0.20 • . This points out again that in discrete media there are two quantities which characterize the light evolution: the coupling strength and an additional detuning of the propagation constant.

Conclusions
We have presented the discrete analog of Snell's refraction law at an interface. It was shown that the refraction is characterized by two quantities: the coupling strength and an additional detuning of the propagation constant, in contrast to continuous media. Additionally, refraction and transmission coefficients for the Bloch waves and the power were introduced as the discrete analog of Fresnel's formulae. The results were experimentally verified in fs laser written waveguide arrays, where the light propagation was directly monitored using fluorescent light.