1. Introduction

A dielectric slab layer with grating coupler propagating at least one guided mode is the simplest form of resonant grating. A free space incident wave satisfying the k-vector synchronism condition between the in-plane projection ki of the incident k-vector, the grating Kg-vector and the propagation constant β of a guided mode couples to the said mode and experiences a quite complex redistribution of field in the zeroth transmitted and reflected diffraction orders, the coupling being usually made via the + or -1st order of the grating so as to avoid power losses via higher diffraction orders.

In the early days of integrated optics such corrugated slab waveguide structure was envisaged as a way to couple light in/out. Waveguide gratings were also considered as possible functional waveguide elements capable of performing reflection, polarization conversion, filtering, etc, well before the concept of photonics crystals became popular [1, 2, 3]. Later on in the eighties it was discovered that so simple a structure possibly exhibits remarkable features such as anomalous reflection [4, 5] where 100% of the incident beam can theoretically be reflected in the neighbourhood of the mode coupling synchronism condition. This resonant effect has since then received much attention from both integrated optics and diffractive optics communities [6, 7], and has been used in a number of applications such as biosensors [8, 9, 10], polarization independent WDM filters [11], laser mirrors [12, 13], and active devices have also been demonstrated [14]. Other resonance effects involving guided modes have been revealed such as resonant diffraction from a mirrored grating slab where use is made of a leaky mode resonance [15], and resonant transmission across a thin undulated metal film by plasmon mode excitation [16, 17].

The present paper is devoted to resonant reflection which is known to be related with the grating excitation of a guided mode of the slab waveguide. The physically most vivid representation of this reflection process is given by the coupled wave formalism inherited from the integrated optics community [18] where the structure properties are all accounted for by three phenomenological parameters: the propagation constant β of the guided mode, the radiation coefficient α of the said mode in the presence of the grating, and the coupling coefficient κ between an incident plane wave and the guided mode. α, β and κ are intrinsic parameters of the structure which do not, or only weakly, depend on the excitation conditions; α and β are real parameters. There is an extrinsic parameter which describes the excitation conditions: it is the spatial frequency parameter k which is the sum of the in-plane projection of the incident k_i-vector ± the grating K_g-vector depending on whether the coupling between incident and guided wave is co- or contra-directional; k is real.

The mechanism of resonant reflection is often described in the diffractive optics community by considering the inhomogeneous problem where waveguide mode excitation appears as a pole in the reflection function. The presence of a pole in the reflection function encompasses the following parameters: the off-resonance reflection coefficient, the complex pole coordinates, and the strength of the pole which will be clarified in the next section. In what follows the same real optogeometrical parameter k will be used which contains the experimental quantities: k=k₀n_csinθi±K_g where k₀=2π/λ is the free space wavenumber at vacuum wavelength λ, n_c is the incidence medium refractive index and K_g=2π/Λ is the spatial frequency of the grating of period Λ, θ_i is the incidence angle in the cover medium.

The condensed set of three phenomenological parameters of the coupled wave representation is very useful for all problems involving grating coupling. They not only give a physical understanding of the coupling and resonance effects involved, they also and most importantly enable the designer to solve the inverse problem quasi-mentally. One important remaining problem which the present paper addresses is the experimental access to these parameters in a given structure. α, β and κ are difficult to measure directly. Therefore, access to them will be found first by bridging the coupled wave vision and the pole representation of the reflection coefficient r(k), then by analysing the modulus and phase of r(k), and identifying the experimental situation giving an easy and accurate access to the needed parameters. The proposed methodology provides a powerful and physically intelligible tool for characterizing and also for designing resonant diffractive phenomena.

2. The relationship between the pole and coupled wave representations

It is known that resonant reflection from a corrugated slab waveguide is related with the excitation of a guided mode of the slab [5, 18]. Under definite conditions of symmetry, and in the presence of a single pole, resonant reflection can always reach 100% [19]. Under collinear incidence, the resonance condition is defined as the matching in the corrugated waveguide plane of the k-vectors of the incident and the guided waves, k_i=k₀n_csinθ_i and β=-n_ek₀ respectively, via the grating spatial frequency K_g: β=k_i-K_g where the excitation of the waveguide mode is made collinearly via the -1^st diffraction order (i.e. contradirectional coupling) for the sake of preventing radiation losses via other orders. The waveguide slab has substrate, guide and cover refractive index n_s, n_g, n_c with n_g>n_s>n_c; the cover is usually air with n_c=1; n_e is the effective index of the excited mode. The grating corrugation can be at either or both sides of the waveguide layer. The reflection from such coupled structure in the resonance domain is a quite complex mechanism involving two contributions: the first one is the non-resonant Fresnel reflection from the structure interfaces which will here be called r₀, and the second one, r_g, is the contribution of the coupled mode which is re-radiated away by the grating into the substrate and cover. The sum of these two complex contributions is a priori difficult to foresee, and the variation in modulus and phase of the resulting reflection coefficient across resonance versus θ_i, Λ or λ can have a wide variety of behaviours. Figure 1 represents the different waves involved in the mechanism of grating waveguide reflection.

A clarification can however be made by considering the so-called inhomogeneous problem where the occurrence of grating mode excitation is expressed as a pole of the reflection function. Numerical and curve fitting approaches were presented for the retrieval of phenomenological parameters [20, 21] from such polar function. We write here the polar function in terms of observable physical quantities, provide a clear picture of the resonant reflection phenomenon and give analytical relationships between phenomenological parameters.

 

Fig. 1. Contradirectional grating mode coupling of a plane wave into a slab waveguide

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As a result of the theory of analytical functions, the reflection at and around the resonance can be expressed from the pole-zero model of Ref. [22] as the sum of two terms: an essentially constant regular term r₀ and a term r_g resulting from the presence of a single pole in the k-vector space:

where k_p represents the complex pole coordinates, and a_p is a complex constant coefficient which must somehow be determined. The reflection coefficient r(k) is a complex function of the real variable k. Since k describes the incidence conditions in the concrete case of waveguide grating coupling, it is real. Therefore, the scan of k will be experimentally made on the real axis by varying the incidence angle θ_i or the period Λ or the wavelength λ:

The algebraic expression (1) will now be given a physical meaning. The parameters a_p and k_p will first be connected to the phenomenological parameters of the coupled wave formalism.

The differential equation governing the complex amplitude a_g(x) of the field of a guided mode propagating along x in the slab waveguide simply states that the longitudinal variation of the modal field is the sum of two terms [23]: a field feeding term proportional to the field q(x) of the incident free space wave, and a field leakage term proportional to the local modal field a_g(x) with the complex propagation constant of the modal field as a coefficient

where α is the radiation coefficient, β the propagation constant of the mode, κ the coupling coefficient, and q(x) the profile of the incident beam taken equal to 1 here as we are restricting our interest to an incident wave and grating of infinite extent. (k-β), with k=k_i - K_g, is the detuning factor expressing how far the coupling process is from perfect resonance synchronism (k=β). In k, k_i is taken with the + sign to express contradirectional first order coupling. Under these hypotheses the steady state solution a_g of Eq. (3) is simply:

It is worth remarking at this stage that k=β is the condition for maximum field in the slab waveguide.

Expression (4) gives the amplitude of the modal field under plane wave incidence in the region of coupling synchronism. However, a comparison with the pole representation (expression (1)) requires an expression for the resonant part r_g of the reflection. As resonant reflection relies upon waveguide mode excitation it is natural to assume that r_g is proportional to a_g, thus, by using (4):

where η_c is the proportionality coefficient whose meaning is that of a coupling diffraction efficiency of the guided mode towards the cover.

Comparing expressions (1) and (5) establishes the relationship between the pole coefficients and the phenomenological coefficients of the coupled wave representation:

which states that the real and imaginary pole coordinates are the propagation constant β and the radiation coefficient α respectively, and that the coefficient a_p is the product of the coefficient κ describing the guided mode field feeding by the incident wave by the coefficient η_c describing the contribution of the modal field to the resonant reflection. This doesn’t indicate how to easily quantify κ and η_c. However, the main interest of the present paper is in the reflection as expressed in (5) where the κ and η_c parameters are involved in the form of their product. The separation of the roles of κ and ηc is only needed when discussing the modal field which then has to be normalized. This is beyond the scope of the present paper; the interested reader will find in ref. [19, 24, 25] the detail of the analytical results obtained by Sychugov et al. for the radiation coefficient and for the coupling coefficient under the Rayleigh-Fourier approximation; this approximation holds well since in most cases there is no need for deep grooves for giving rise to the resonant reflection of an incident beam of usual diameter, 1 mm for instance.

3. Modulus and phase of the reflection coefficient

A more general and synthetic modelling of the reflection across a waveguide mode resonance will now be given by resorting to a graphic representation. Now as expression (1) for the reflection of a plane wave from a grating waveguide has received a physical content, we will proceed to the scan of the free real parameter k across resonance and give a representation of the effect of the scan on r_g(k) and identify all meaningful k-values.

3.1 Complex circle representation of r(k)

r_g(k) is a complex function having a first order pole. The locus of r_g(k) in the complex plane upon the scan of k on the real axis is a circle as it is known from the theory of analytical functions. This is shown by finding out that there exists a complex point z₀ relative to which the modulus of (r(k)-z₀) is constant for all k. The expression of the circle in the complex plane is given by substituting the identities (6) into expression (5):

Removing the (k-β) parameter from the two real equations which the complex expression (7) corresponds to gives the equation of a circle:

 

Fig. 2. Representation of the reflection coefficient r(k) in the complex plane.

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As illustrated in Fig. 2, the circle r_g(k) is centered at the complex point z₀=ja_p/2α and its radius is |a_p|/2α. The location of the running point on the r_g(k) circle for every value of the real parameter k is determined by the argument (9) of the complex phasor r_g(k)-z₀ :

where ϕ_p is the phase of the complex constant a_p. Upon a scan along k from -∞ to +∞, the argument varies from π to -π. This means that the r_g apex describes only one turn of the circle, and in the clockwise direction.

It is noteworthy that well outside resonance (k→±∞) there is no mode excitation, therefore the resonant part of reflection, r_g(k), tends to zero. Thus the rg(k) circle passes through the origin of the complex plane and the origin is the start (k→-∞) and end point (k→+∞) of the scan.

Stating now that the reflection coefficient r(k) of the resonant grating is the sum of the regular part r₀ and the resonant part r_g(k) just amounts to translating the described circle by the complex quantity r₀. The graphic location of the r(k) circle is easy: knowing that the reflection modulus is 1 at the condition of resonant reflection r_M=r(k_M), the r(k) circle is tangent at this point to the circle of radius 1 centered at the origin of the complex plane (k_M is the value taken by k at the resonant reflection point). Therefore, the latter, the r(k) circle center and the complex unit reflection point rM are on the same straight line. On the same straight line is also the point of minimum reflection r_m. The r(k) circle is thus centered at point:

and its radius is still $R=\frac{\mid a_p\mid }{2\alpha }$. Expression (11) hereunder is the polar expression (1) for r(k) with all phenomenological parameters substituted:

3.2 Relationships between radiation coefficient, coupling and propagation constants, and reflection coefficients

Let us write r(k) at resonant waveguide mode excitation, i.e. at k=β where r(β) is denoted rβ. Setting k=β in (11) yields:

Substituting ja_p/2α from (10) into (12) gives:

Complex equality (13) states in phase that r_β, C and r₀ are on the same straight line, therefore on a diameter of the circle, and, in modulus, that r_β and r₀ are equidistant from C. This means that the Fresnel reflection and the reflection at resonance are diametrically opposed on the circle. Now extracting a_p from expression (12) of r_β yields:

This states that the coefficient a_p is proportional to the product of the radiation coefficient by the difference r_β-r₀.

Another interesting expression derived at synchronous waveguide mode excitation is obtained by setting k=β in expression (4) for a_g which is the condition for maximum modal field excitation:

which states that the coupling coefficient is equal to the product of the radiation coefficient α by the amplitude of the guided wave field a_g at its maximum.

From an experimental point of view, r_β is not easy to measure as the propagation constant β of the grating waveguide mode is unknown. Writing the circle center coordinate as:

permits to express r_β in terms of easily measurable quantities:

Three cases must be considered in the phase dependence of r(k) depending on the position of the circle relative to the origin. In the first case |a_p|<α, the origin is outside the circle and the phase of r(k) varies as an oscillation around the phase ϕ₀ of r₀. The phase variation is smaller than 2 arctan $\left(\frac{\mid r_M\mid -\mid r_m\mid }{\mid r_M\mid +\mid r_m\mid }\right)$, i.e. smaller or much smaller than π. In the second case |a_p|>α, the origin is contained in the circle and the phase experiences a 2π variation. In a third case the origin is on the circle (|a_p|=α) and the phase variation is π. This is not an exceptional situation; it is often desired as for instance in biosensors [8] to have zero reflection outside resonance (r₀=0).

Without measuring the phase of the reflection coefficients it is difficult to find out whether the origin is contained in the r(k) circle or not. But the absolute phase of a reflection coefficient is difficult to measure. We can bypass this difficulty by noting that α does not depend on ϕ_M, but only on the relative phases ϕ₀-ϕ_M and ϕ_m-ϕ_M. Once the modulus |r₀| is known, its relative phase ϕ₀-ϕ_M is easily determined by finding graphically the intersection of the circle of radius |r₀| and center O with the part of the r(k) circle corresponding to k values off the resonance domain. Less easy however is the determination of φ_m-φ_M because of the unknown sign of φ_m relative to φ_M. This phase difference can be straightforwardly determined by a simple polarimetric arrangement. After the quantities |r_m| and |r_M| have been measured for the polarization exhibiting resonance, a further power measurement is made in the resonance domain of the reflection coefficient of the orthogonal polarization |r_p|. |r_p| is essentially constant in the resonance domain, its unknown phase also. Then, the measurement of |r_m| and |r_M| is resumed with the incident polarization at 45 degrees from the incidence plane. An analyzer at 45 degrees placed after the reflection point projects the two polarizations in the same direction where they interfere. Let us first consider the interference product behind the analyser in the situation of resonant reflection r_M: regardless of the origin being in the circle or not, the interference product is generally constructive when r_p points in the same general direction as r_M in the complex plane of Fig. 2, i.e. the measured power is larger than the quadratic mean of |r_M| and |r_p| corresponding to the case when rp is orthogonal to r_M ; it is generally destructive when r_p points in the opposite general direction and its value is smaller than the quadratic mean. Considering now the interference product with the origin outside the circle with the parameter k set at k=k_m, i.e. at r_m, |r_m| increases when r_p generally points towards r_M and decreases when r_p points opposite. When the origin is contained in the circle, the converse applies.

To summarize, if the interferential contribution of r_p in the direction of the analyser is for both r_m and r_M constructive or destructive, the origin is not in the circle. If it is constructive for r_m and destructive for r_M, or conversely, the circle contains the origin and the phase change across resonance is 2π. Whether the phase of the reflection undergoes an oscillation or a jump across resonance is a critical issue for all applications dealing with femtosecond pulse temporal control as analysed in ref [26].

The mode propagation constant β is also a phenomenological parameter which is difficult to determine in a grating waveguide. Applying the Pythagoras theorem in triangle r_β r_M r₀ and using the property r₀+r_β=rm+r_M (since r₀ and r_β are diametrically opposed) in the complex plane lead to the important relationships between reflection moduli:

ϕ_M=ϕ_m if the origin O is outside the circle, and ϕ_M=ϕ_m+π if O is contained in the circle. The above relationships permit a concise expression for the modulus of r(k):

By setting the first derivative of |r(k)| with respect to the parameter k to zero it is possible to obtain an expression the k-values k_M and k_m for which the resonant reflection r_M and the minimum reflection r_m take place. This yields:

where $\rho =\sqrt{\frac{{\mid r_M\mid }^2-{\mid r_0\mid }^2}{{\mid r_0\mid }^2-{\mid r_m\mid }^2}}$

Theoretically |r_M| is always 1 under plane wave excitation, therefore ρ is always larger than 1. From expressions (20) α (therefore a_p from (14)) can be expressed from easily measurable reflection coefficients and incidence parameter values k:

Expressions (20) also reveal a very interesting property of the resonance domain which can be expressed as:

Therefore, the mode propagation constant β can be expressed as:

Expression (23) is a result in itself in that it gives the possibility to measure the propagation constant, or effective index, of a corrugated waveguide mode without ambiguity. As a matter of fact the effective index of a mode is usually measured by finding the so-called “m-line” observed in the reflected beam; the internal features of the m-line is generally overlooked which either leads to a low accuracy estimate of the effective index or to an error because there has so far been no criterion to identify what feature of the resonant reflection domain bears the stamp of the coupling synchronism condition: the reflection maximum, or the minimum, or half-way between? Expression (23) can also be used to measure |r_β| and check on with the first expression of (17). Now knowing β, r₀, α and r_β the coupling constant a_p can be retrieved using (14). All the phenomenological parameters are then known and the polar approximated function describing the resonant reflection can be calculated.

By using the circle representation of the resonant reflection, expressions have been found to link the phenomenological parameters and pole parameters to the modulus of three specific reflection coefficients: the Fresnel reflection |r₀|, the resonant reflection |r_M| and the minimum reflection |r_m|, and their locations k_M and k_m. The scan of k can be performed by varying the incidence angle, the grating pitch, or the wavelength. These results are valid regardless of the polarization and of the side(s) at which the corrugation or index modulation is made. Furthermore, although resonant reflection can in a lossless structure always be 100% (|r_M|=1) by a proper choice of the beam and grating parameters, the results of the present analysis are also valid when |r_M| < 1, in the presence of scattering and absorption losses for instance; the retrieved α in such case is the sum of the radiation coefficient and of a loss coefficient. More generally, the above analysis also brings a clarification of the complex behaviour of the phase of reflection of a grating slab waveguide in the resonance domain.

4. Exact numerical simulation

The described methodology of parameter retrieval will now be tested against the exact modelling of a grating coupled structure. The chosen structure is that of an evanescent wave biosensor for non-labelled species where the bioreaction takes place at the surface of a waveguide under the monitoring of the evanescent field tail of a guided mode. A number of grating excitation/readout schemes have been proposed [27, 28]. The scheme which best illustrates the usefulness of the present phenomenological approach is that which makes use of resonant reflection from a grating coupled slab waveguide with biospecies immobilized at its surface and excited from the substrate [8, 9, 10].

 

Fig. 3. (a). Amplitude and b) phase variations of r(k) across resonance. Solid line: exact numerical modelling. Crosses: polar function.

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The structure is made of a pyrex glass substrate coated with an ion plated hafnium oxide layer of 2.12 refractive index and excited with a 780 nm laser beam of TE polarization. The optimum sensitivity conditions for the detection of nanometer sized biospecies at the corrugated HfO₂ surface in a water based solution are found to be obtained with a waveguide thickness h_opt=70 nm [29].

In the chosen example of a sensing waveguide with grating coupler, the thickness of the non-corrugated HfO₂ layer is 56 nm and the depth of the binary HfO₂ corrugation is 30 nm so that the effective thickness of the resonant grating is close to the optimal thickness h_opt needed for maximum surface sensing sensitivity. Such grating depth was chosen to satisfy the condition for high, close to 100% resonant reflection wα≫1 [30], w being the length of the impact zone of the incident beam in the grating plane. The incidence angle in the substrate of index 1.49 is chosen to be θ_i=30 degrees which implies a grating period of Λ=339 nm to excite the TE₀ mode. The scan of the k-parameter is made by varying the incidence angle θi in the substrate using the exact grating code of Lyndin [31] based on the modal method. As sketched in Fig. 1, the mode excitation is contradirectional. The reflection modulus is 1 at k_M=125300 cm^-1and |r_m|=0.3616 at k_m=124818 cm^-1. The large number of digits retained here is irrelevant practically; it is only intended to permit a comparison with the polar model and check on the self-consistency of the exact and polar representations of r(k). Figure 3 represents the reflection coefficient modulus and phase versus θ_i in the resonance domain. The phase of the maximum and minimum reflection coefficients is ϕ_M=72 degrees, and ϕ_m=-108 degrees respectively; this by the way means that the origin is contained in the circle. Representing the numerical results of Fig. 3 in the complex plane with k as a parameter gives the solid curve of Fig. 4 which is the exact representation of the complex function r(k).

 

Fig. 4. Polar representation of the reflection coefficient of a resonant grating. Solid line: exact numerical simulations. Crosses: polar function.

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The code referred to in ref. [31] also gives the pole coordinates: α=112 cm^-1 and β=125272 cm^-1 (such value for α implies that a beam of about 0.5 mm cross-section and above exhibits 100% resonant reflection). Returning to the modelling of the reflection coefficient by a polar function, the propagation constant can be calculated from expression (23) and the radiation coefficient from expression (22) by borrowing |r₀|, |r_m|, k_m and k_M from the exact modelling. The resulting phenomenological parameters are β=125275cm^-1, and α=105 cm^-1, β corresponding to an effective index of 1.5552 at wavelength λ=780 nm for the TE₀ mode excited by the -1^st diffraction order. These values are very close to the numerical ones. The remaining discrepancy is small in the present structure and coupling conditions, but it can be larger. The main hypothesis in the polar representation (11) in terms of the phenomenological parameters is the constancy of the latter over the scanning range across resonance. This is only approximately true. For instance, scanning the parameter k by varying the grating period or the incidence angle obviously modifies slightly the interference conditions in the adjacent media between radiated waves, therefore α. Similarly, scanning k by varying the wavelength also slightly changes α(the modal field size changes) and the effective index.

From the knowledge of β, r_β can be retrieved exactly: its amplitude is 0.9754 and its phase 54 degrees. Using k_M, k_m, as well as |r_m| and the phases φ_M and φm given by the exact code permits to determine rβ - r₀ by using (17), therefore ap can be calculated from expression (14): its amplitude is 142 cm^-1 and its phase -45 degrees. Using the polar expression (1) with the calculated parameters α, β and a_p one can represent the r(k) circle as shown by the crosses in Fig. 4. In Figs. 3 and 4 the exact numerical results closely coincide with the polar function calculated by using the retrieved phenomenological parameters; the density of crosses along the solid lines of Figs. 3 and 4 expresses the rate of change of r(k) upon an incremental variation Δθ_i of the incidence angle in the substrate with Δθ_i=0.02 degree. This confirms that the polar algebraic model of resonant reflection closely describes the features of this useful electromagnetic effect and can be used to find out the phenomenological parameters of a given structure as well as to design novel resonant devices. Such statement remains true as long as the coupled mode theory properly accounts for the involved coupling mechanism.

The retrieval of the phenomenological parameters could actually be made directly from an experimental scan of the parameter k and expression (11) by numerical optimisation without analytical development. However, the present development gives much physical insight into this resonant coupling phenomenon as well as meaningful and useful quantitative expressions for β and α.

5. Conclusion

The effect of 0^th order resonant reflection from a slab waveguide grating is clarified by a vivid graphic representation in modulus and phase based on a polar approach of the inhomogeneous problem. This algebraic representation and its coefficients are also related with the physically meaningful phenomenological parameters of the coupled wave formalism. The analytical relationships obtained establish a bridge between the two representations of this useful resonant diffraction effect and therefore permit to use whatever approach is best adapted to solve a given characterization, fabrication or design problem. The three fundamental phenomenological parameters α, β and κ which alone govern the behaviour of any resonant reflection element are difficult to measure experimentally. The obtained relationships permit to express them in terms of easily measurable quantities such as the modulus of reflection coefficients under specific incidence conditions. In particular, the propagation constant β of a waveguide mode in the presence of the grating, which is so important in all grating waveguide sensors, is given an especially simple and meaningful expression.

The outcome of the present analysis has direct implications in resonant structure characterization and design. Beyond this, the synthesis accomplished in the present work is a basis for the exploration of novel applications of this still puzzling effect of anomalous reflection. This effect can be associated with mirrors, filters to give rise to optical functions using its adjustable spectral phase dependence and quasi 100% modulus. One promising field of application is femtosecond laser pulse processing [26].