The algebraic polar expression of resonant reflection from a grating waveguide excited by a free space wave is formulated in terms of the physically meaningful phenomenological parameters of the coupled wave formalism. The reflection coefficient is simply represented as a circle in the complex plane which sheds light on the behaviour of the modulus and phase of anomalous reflection. Analytical expressions are derived for the phenomenological parameters that can now be calculated from optogeometrical quantities which are simple to measure. The relevance and usefulness of bridging the two formalisms is shown in the example of the design of an evanescent wave biosensor.
©2007 Optical Society of America
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 , laser mirrors [12, 13], and active devices have also been demonstrated . 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 , 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  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 ki-vector ± the grating Kg-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=k0ncsinθi±Kg where k0=2π/λ is the free space wavenumber at vacuum wavelength λ, nc is the incidence medium refractive index and Kg=2π/Λ is the spatial frequency of the grating of period Λ, θi 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% . 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, ki=k0ncsinθi and β=-nek0 respectively, via the grating spatial frequency Kg: β=ki-Kg where the excitation of the waveguide mode is made collinearly via the -1st 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 ns, ng, nc with ng>ns>nc; the cover is usually air with nc=1; ne 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, rg, 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 θi, 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.
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.  as the sum of two terms: an essentially constant regular term r0 and a term rg resulting from the presence of a single pole in the k-vector space:
where kp represents the complex pole coordinates, and ap 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 θi or the period Λ or the wavelength λ:
The algebraic expression (1) will now be given a physical meaning. The parameters ap and kp will first be connected to the phenomenological parameters of the coupled wave formalism.
The differential equation governing the complex amplitude ag(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 : 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 ag(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=ki - Kg, is the detuning factor expressing how far the coupling process is from perfect resonance synchronism (k=β). In k, ki is taken with the + sign to express contradirectional first order coupling. Under these hypotheses the steady state solution ag 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 rg of the reflection. As resonant reflection relies upon waveguide mode excitation it is natural to assume that rg is proportional to ag, thus, by using (4):
where ηc 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 ap is the product of the coefficient κ describing the guided mode field feeding by the incident wave by the coefficient ηc describing the contribution of the modal field to the resonant reflection. This doesn’t indicate how to easily quantify κ and ηc. However, the main interest of the present paper is in the reflection as expressed in (5) where the κ and ηc 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 rg(k) and identify all meaningful k-values.
3.1 Complex circle representation of r(k)
rg(k) is a complex function having a first order pole. The locus of rg(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:
As illustrated in Fig. 2, the circle rg(k) is centered at the complex point z0=jap/2α and its radius is |ap|/2α. The location of the running point on the rg(k) circle for every value of the real parameter k is determined by the argument (9) of the complex phasor rg(k)-z0 :
where ϕp is the phase of the complex constant ap. Upon a scan along k from -∞ to +∞, the argument varies from π to -π. This means that the rg 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, rg(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 rg(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 rM=r(kM), the r(k) circle is tangent at this point to the circle of radius 1 centered at the origin of the complex plane (kM 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 rm. The r(k) circle is thus centered at point:
and its radius is still . 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 jap/2α from (10) into (12) gives:
Complex equality (13) states in phase that rβ, C and r0 are on the same straight line, therefore on a diameter of the circle, and, in modulus, that rβ 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 ap from expression (12) of rβ yields:
This states that the coefficient ap is proportional to the product of the radiation coefficient by the difference rβ-r0.
Another interesting expression derived at synchronous waveguide mode excitation is obtained by setting k=β in expression (4) for ag 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 ag at its maximum.
From an experimental point of view, rβ 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 rβ 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 |ap|<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 , i.e. smaller or much smaller than π. In the second case |ap|>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 (|ap|=α) and the phase variation is π. This is not an exceptional situation; it is often desired as for instance in biosensors  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 ϕM, but only on the relative phases ϕ0-ϕM and ϕm-ϕM. Once the modulus |r0| is known, its relative phase ϕ0-ϕM 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 φm-φM because of the unknown sign of φm relative to φM. This phase difference can be straightforwardly determined by a simple polarimetric arrangement. After the quantities |rm| and |rM| 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 |rp|. |rp| is essentially constant in the resonance domain, its unknown phase also. Then, the measurement of |rm| and |rM| 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 rM: regardless of the origin being in the circle or not, the interference product is generally constructive when rp points in the same general direction as rM in the complex plane of Fig. 2, i.e. the measured power is larger than the quadratic mean of |rM| and |rp| corresponding to the case when rp is orthogonal to rM ; it is generally destructive when rp 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=km, i.e. at rm, |rm| increases when rp generally points towards rM and decreases when rp points opposite. When the origin is contained in the circle, the converse applies.
To summarize, if the interferential contribution of rp in the direction of the analyser is for both rm and rM constructive or destructive, the origin is not in the circle. If it is constructive for rm and destructive for rM, 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 .
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 rβ rM r0 and using the property r0+rβ=rm+rM (since r0 and rβ are diametrically opposed) in the complex plane lead to the important relationships between reflection moduli:
ϕM=ϕm if the origin O is outside the circle, and ϕM=ϕm+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 kM and km for which the resonant reflection rM and the minimum reflection rm take place. This yields:
Theoretically |rM| is always 1 under plane wave excitation, therefore ρ is always larger than 1. From expressions (20) α (therefore ap 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 |rβ| and check on with the first expression of (17). Now knowing β, r0, α and rβ the coupling constant ap 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 |rM| and the minimum reflection |rm|, and their locations kM and km. 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% (|rM|=1) by a proper choice of the beam and grating parameters, the results of the present analysis are also valid when |rM| < 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].
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 hopt=70 nm .
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 hopt needed for maximum surface sensing sensitivity. Such grating depth was chosen to satisfy the condition for high, close to 100% resonant reflection wα≫1 , 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 θi=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  based on the modal method. As sketched in Fig. 1, the mode excitation is contradirectional. The reflection modulus is 1 at kM=125300 cm-1and |rm|=0.3616 at km=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 θi in the resonance domain. The phase of the maximum and minimum reflection coefficients is ϕM=72 degrees, and ϕm=-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).
The code referred to in ref.  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|, |rm|, km and kM 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 -1st 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, rβ can be retrieved exactly: its amplitude is 0.9754 and its phase 54 degrees. Using kM, km, as well as |rm| and the phases φM 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 ap 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 Δθi of the incidence angle in the substrate with Δθi=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.
The effect of 0th 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 .
This work is a contribution of the TSI laboratory (now Laboratoire Hubert Curien) on waveguide characterization and resonant gratings in the Network of Excellence of the European Community on Microoptics NEMO.
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