Arrayed Waveguide Gratings, Fiber Bragg Gratings, and Photonic Crystals: An Isomorphic Fourier Transform Light Propagation Analysis

Transcription

1 1158 IEEE JOURNAL OF SELECTED TOPICS IN QUANTUM ELECTRONICS, VOL. 8, NO. 6, NOVEMBER/DECEMBER 2002 Arrayed Waveguide Gratings, Fiber Bragg Gratings, Photonic Crystals: An Isomorphic Fourier Transform Light Propagation Analysis Michael C. Parker, Associate Member, IEEE, Stuart D. Walker Abstract In this paper, we present a unified Fourier transform (FT) approach to the study of arrayed waveguide grating (AWG), fiber Bragg grating (FBG), photonic crystal (PC) devices. This methodology allows the design characteristics of transverse (AWG) geometries to be mapped on to longitudinal (or distributed) FBG PC structures, due to a comprehensive isomorphism between these important passive optical grating-based devices. The unified approach presented here, which is based upon a modified Debye Waller approach to the analytical solution of the coupled-mode equations, allows intuitive, yet accurate appraisal of arbitrary strength coupled structures. Exploiting this isomorphism, we relate our theoretical work to a number of practical cases. For example, we describe an FBG carousel configuration (analogous to a chirped AWG cascade) offering virtually ripple-free third-order dispersion compensation of 6.61 ps/nm 2 over a 100-GHz bwidth. Index Terms Arrayed waveguide gratings, dispersion compensation, fiber Bragg gratings, filter theory, Fourier transforms, gratings, photonic crystals, wavelength division multiplexing. I. INTRODUCTION DEVICES such as the AWG [1] or the FBG [2] are assuming increasing importance in the areas of fiber point-to-point communication networking [3]. In the particular context of dense wavelength-division multiplexing (DWDM), these devices play a well-established role as wavelength-selective elements [4]. Recently, photonic bgap effects their embodiment in a new class of optical device, PCs, have also received considerable attention [5], because they also feature wavelength selectivity, but may be realized in a more compact form on photonic integrated circuits (PICs) [6]. It is well known that the AWG succumbs to a Fourier transform (FT) treatment, in the limit of the paraxial approximation combined with the far-field Fraunhofer diffraction [7]. It is also well known, in the case of a weakly coupled FBG, that the first Born approximation offers a direct FT relationship between the reflection spectrum the spatial distribution of scatterers for Manuscript received August 26, 2002; revised September 23, M. C. Parker is with Fujitsu Network Communications Inc., Photonics Networking Laboratory, Richardson, TX USA. He is also with Fujitsu Telecom Europe Ltd. Research, Colchester, Essex CO3 4HG, U.K. ( M.Parker@ftel.co.uk). S. D. Walker is with the Department of Electronic Systems Engineering, University of Essex, Colchester, Essex CO4 3SQ, U.K. ( stuwal@essex.ac.uk). Digital Object Identifier /JSTQE a weakly scattering medium [8]. However, this is in contrast to the strong scattering effects (i.e., high reflectivity) present in photonic crystals the equivalent strongly coupled FBG case. In this paper, we identify a comprehensive isomorphism between AWG FBG/PC devices, which allows the design characteristics of the former transverse geometry to be mapped on to the latter longitudinal structure [9], even when the distributed grating is strongly coupled [10]. Hence, whether a grating is transverse to the light or rotated by 90 to form a longitudinal grating, as indicated in Fig. 1, equivalent behavior of light propagation is observed. From filter theory it is also well known that the AWG is a finite-impulse response (FIR) filter, in contrast to FBGs PCs, which are infinite-impulse response (IIR) devices [11]. Our analysis therefore appears to indicate that a passive IIR device can be related to a passive FIR device by making the FIR transfer function the argument of an appropriate hyperbolic function. In proceeding to the appropriate analysis for a strongly coupled grating or photonic crystal, we make use of the following modifications to conventional coupled-mode theory (CMT), described in detail in the Appendix: use of optical reciprocal space; spatial spectral definition of coupling coefficient via a FT integral; assumption of a finite length grating; employment of a modified Debye Waller approximation. We structure the paper as follows. In Section II, we employ Fresnel Kirchoff diffraction theory to analyze the propagation of light through a transverse grating, such as an AWG. Longitudinal or distributed gratings (e.g., FBGs) are studied in Section III using the CMT, which is shown to be isomorphic with the theory of Section II. The theory is applied to linearly chirped grating structures for dispersion compensation, we demonstrate the power of this isomorph by presenting a novel third-order dispersion compensation FBG configuration, based on an equivalent parabolically chirped AWG design. Strongly coupled devices such as PCs are considered in Section IV, with the appropriate modifications to the underlying assumptions of CMT described, in order to achieve a Fourier transform solution to the coupled-mode equations. A one-dimensional (1-D) PC is analyzed with the new theory, the results for the reflection spectrum, bgap dispersion diagram, propagation constant compared with the solutions from the conventional coupled-mode equations X/02$ IEEE

2 PARKER: AWGs, FBGs, AND PCs: AN ISOMORPHIC FT LIGHT PROPAGATION ANALYSIS 1159 Fig. 1. Schematic diagrams for (a) transverse (b) longitudinal optical filters. II. TRANSVERSE GRATING (AWG) ANALYSIS Fig. 2 shows an AWG, which can be considered as a planar 4f lens-relay system with the central waveguide array acting as a Fourier plane [3]. We have previously shown that, using Fresnel Kirchoff diffraction theory, the transfer function for the whole AWG device is as follows [9]: (1) where we have defined the grating phase as, the detuning parameter, with propagation constant, corresponding with the wavelength at the bpass center of the AWG filter response. With reference to Fig. 2, we have also defined the amplitude function, which describes the variation of electric field amplitude across the waveguide array, the direction cosine, where is the free-propagation region (FPR) length. In (1) we have adopted a spectral filter transmission for a single fixed output waveguide at a position. A different output waveguide (with a different spatial position, hence different value for ) will yield a filter transmission that has been shifted in the spectral domain to a different central wavelength, i.e., the demultiplexing functionality of a conventional AWG. In addition, we have also defined the parameter, which has units m equivalent to a coupling coefficient. We note at this point that will tend to be very small, since the length of the FPR tends to be of the order of hundreds of micrometers, so that is much greater than or the mode spot size. The AWG free spectral range (FSR) is closely given by, where is the incremental geometric pathlength difference between neighboring waveguides in the array. Fig. 2. Schematic of a regular AWG. III. COUPLED-MODE ANALYSIS FOR DISTRIBUTED (LONGITUDINAL) GRATINGS A. Theory CMT offers a physically intuitive approach to the mathematical 1-D modeling of distributed or longitudinal gratings. We use the grating parameters as defined by Kogelnik [8], but the coupled-mode formalism of Yariv [12] (2a) (2b) where are, respectively, the forward backward propagating electric-field amplitudes of the light. The spatially varying coupling coefficient is given by, where is the longitudinal space variable, shown in Fig. 3. Spatial variation in the coupling coefficient can be expressed by the function, where is the maximum coupling strength along the length of the grating, is the purely real, normalized spatial variation in grating strength, is the spatially

3 1160 IEEE JOURNAL OF SELECTED TOPICS IN QUANTUM ELECTRONICS, VOL. 8, NO. 6, NOVEMBER/DECEMBER 2002 Fig. 3. Schematic of a regular FBG, with an apodized (windowed) coupling strength g(z). varying phase of the grating. Conventionally, the detuning parameter is defined as, where the propagation constant is again given by, is the free-space wavelength of the propagating wave, is the geometric period of the grating, the average refractive index is, where we assume a binary grating structure of refractive indices [12]. The angle that the geometric ray analog to the single-mode makes with the waveguide optical axis is defined by ; almost equal to zero, such that we can assume, where the propagation constant at the Bragg resonance is given by.for an FBG of length centered at, using the result from Jaggard et al. [13], the coupled-mode equations can be solved to yield the reflectivity at the front face of the grating (3) In contrast to [13] we have also introduced a phase shift to the overall reflectivity, i.e., multiplied the right-h side of (3) by, to maintain consistency with [8]. In the case of strong coupling, it has been noted that (3), whilst being exact at the b center, becomes inaccurate due to a phase error accumulation [14], [15], so that an overly narrow passb width (or photonic bgap) is predicted. We discuss the appropriate modifications to CMT for the strong-coupling case (e.g., as applied to PCs) in Section IV the Appendix. For a grating of overall length, we can assume that the (apodization) function for, such that we can rewrite (3) as a generalized transfer function (4) Equation (4) thus indicates that the transfer function consists of an FT relationship within a hyperbolic tangent function. Hence, we can consider the Fourier plane of an FBG to lie along its length. For a weakly coupled distributed grating, where is small, we can ignore the function, simply write the transfer function in its first Born approximation form (5) Immediately apparent is the functional isomorphism between (5) (1) for the AWG. Both are FT integrals, with similar grating function variables. The spatial variation of the distributed grating is equivalent to the envelope amplitude in the central array of waveguides in the AWG, while the spatially varying grating phase is equivalent to the phase variation across the AWG waveguide array. Likewise, the strength of the grating is isomorphic to the parameter for the AWG. Thus, somewhat paradoxically, the AWG, whilst being a FIR filter, can also be regarded as an asymptotically uncoupled IIR device. Of interest is the implication that the diffraction of light in free space can therefore also be considered to be a weak coupled-mode problem. Table I shows the correspondence between the equivalent physical geometric parameters of these transverse longitudinal grating devices. The incremental pathlength difference in the arrayed waveguide section of the AWG is equivalent to twice the geometric period of an FBG due to its reflective geometry. We also note that the length of an FBG is isomorphic with both (i.e., half the length of an AWG) (the AWG width ). This arises due to the 2-D planar geometry of an AWG, in contrast to the essentially 1-D linear structure of an FBG. Thus, there is a factor of two multiplying the coordinate, due to the reflective nature of the FBG, which relates only to the length of the grating, but not to its width. The angle of an AWG is equivalent to the angle in an FBG, we find that the grating order FSR for each type of device follow the same form. Overall, the isomorphism allows an understing of the behavior of a longitudinal grating device to be carried over completely to a transverse one, e.g., the functionality of a FBG can easily be translated to an AWG. The term at the front of (5) does not appear in (1) can be understood to arise from the fact that (5) is a reflection transfer function, while (1) is a transmission transfer function, these two quantities need to be 90 out of phase due to power conservation considerations for these two modes. Likewise, there is also an extra phase delay term in (5) to take into account the fact that the transfer function of the FBG is at the front (input) facet of the device, such that reflected light has not actually travelled any distance. The associated transmission function for a FBG is of the form with energy conservation being obeyed. Since (4) (6) contain FT integrals, we can also take the inverse FT of them to yield an expression for the required distributed coupling coefficient with a particular reflectivity or transmissivity: a solution to the inverse problem. However, Parseval s Theorem (similar to that for the AWG above [9]) must still be obeyed, placing a joint constraint on the form that the grating function may take the maximum value of. B. Uniqueness Analyticity of Solutions We note that the transfer response is unique obeys the Kramers Krönig dispersion relations for a given longitudinal grating. This is because the FT of a finite (i.e., spatially bounded) grating structure yields an (6)

4 PARKER: AWGs, FBGs, AND PCs: AN ISOMORPHIC FT LIGHT PROPAGATION ANALYSIS 1161 TABLE I ISOMORPHIC PHYSICAL AND GEOMETRIC PARAMETERS FOR AWGs AND FBGs For an AWG, in (8b) is appropriately given by. The normalized parameters are given by (9a) (9b) Fig. 4. Parabolically chirped AWG cascade equivalent FBG carousel for third-order dispersion compensation. Analysis of the dispersion characteristic of the grating requires knowledge of the phase response associated with. The overall IIR phase response is a function of the -modified phase characteristic associated with the FT integral as defined above. can be physically understood to be the phase response of a weakly coupled grating (i.e., the phase response for an FIR filter, or an AWG), by inspection of (8b) is simply given by analytic (or holomorphic) function (i.e., one that obeys the Cauchy Riemann equations [16]). This can be understood from its equivalence to the FT of a causal function, which is again analytic [17]. As analyticity is preserved in a Taylor power series, we find that both the first Born approximation the exact solution (i.e., FT, hyperbolic function of an FT, respectively) are unique, analytic, obey the dispersion relations. C. Chirped Grating Structures for Dispersion Compensation We have taken advantage of the analogy between AWGs FBGs to apply a known AWG design for third-order dispersion compensation [18] to a suitable FBG configuration. Thus, an AWG cascade is equivalent to a carousel of FBGs around an optical circulator, as indicated in Fig. 4. In general, analytic solutions to polynomial chirped gratings consist of confluent hypergeometric Whittaker functions [19], where the linearly parabolically chirped grating phase functions are respectively given by [8] (10) However, the hyperbolic function operating on in equation (8a) modifies the phase characteristic, so that the overall IIR phase response is given by: (11) Filter group-delay is defined as the angular-frequency derivative of the phase, whilst filter dispersion is the additional derivative of the group-delay with respect to wavelength. Thus filter dispersion as a function of wavelength is closely given by, the second- At the center of the Bragg resonance, where order dispersion is found to be exactly given by (12) (7a) (7b) where is the chirping strength. However, the case of a linear chirp lends itself to an intuitive solution consisting of a summation of Fresnel cosine sine integrals [7] where (8a) The factor Bragg resonance such that (13a) arises from evaluating (8b) at the (8b) For an AWG, where, we can assume that the factor. For values of linear-chirping parameter

5 1162 IEEE JOURNAL OF SELECTED TOPICS IN QUANTUM ELECTRONICS, VOL. 8, NO. 6, NOVEMBER/DECEMBER 2002 IV. PCS AND STRONG COUPLED DEVICES CMT can also be employed to describe light propagation in a PC or strong-coupled longitudinal grating, since the slowly varying approximation has been shown to be redundant [21]. However, it is known that the solution (4) for a strong-coupled device suffers from a phase accumulation error [14], [15], so that the reflection bwidth (or photonic bgap) is narrower than is to be expected. In the case of strong coupling, detuning from the Bragg resonance is reduced for a given, which directly leads to the observed broader bwidth of the main lobe. Thus, the detuning parameter needs to be appropriately modified in order to take strong coupling into account avoid the phase accumulation error. We introduce a modified detuning parameter, derived in the Appendix (A9), which is related to the original detuning parameter via (14) Fig. 5. Characteristics of a parabolically chirped FBG carousel. (a) Normalized transmission (db). (b) Group delay (ps). (c) Second-order dispersion D (ps/nm). (d) Third-order dispersion S (ps/nm ). where is the spectral coupling coefficient also defined in the Appendix. Thus, we modify the coupled-mode equations to read, the dispersion at the passb center increases in a linear fashion (13a) can be very closely approximated by (15a) (15b) (13b) However, as exceeds 17.6, it is straightforward to show that (13a) becomes sinusoidal in nature (of period ), with the amplitude decreasing asymptotically toward zero [18] (13c) The general characteristics with may be explained heuristically as Moiré fringe-like behavior, as the increasing chirp traverses the fixed periodicity of the incident light. A parabolically chirped FBG carousel, using (7b) is shown in Fig. 4, similar to a 2-FBG dispersion compensating carousel device described elsewhere [20]. Shown in Fig. 4, we have detuned the FBGs by with respect to each other [18], to achieve a flat third-order dispersion of ps/nm over a 100-GHz passb, centered at nm, as shown in Fig. 5. The detuning parameters from the central wavelength are pm pm, parabolic chirping strength is. The individual FBG length is m, with a coupling strength of, the refractive index is. Although the behavior of an AWG cascade is straightforward, the reflective nature of the FBGs requires a different construction for their concatenation. The combination of a six-port circulator four FBGs allows sequential addition of the FBG dispersion compensation properties, which is isomorphic with the AWG cascade. In both cases, the overall transfer response is given by the product of the individual device characteristics. with the reflection spectrum associated with the solution to the coupled-mode equations (15) given by (16) We make the reflectivity an explicit function of the geometrically averaged detuning parameter, since, from a phenomenological point of view, the use of to express an observed detuning of the wave field from the Bragg condition is more appropriate than. This is because direct measurement or assessment of the wavelength associated with the field is generally performed in low refractive index contrast (RIC) conditions, such as free-space vacuum or air, for which is the appropriate quantity. In addition, one normally assumes that the reflection spectrum for a grating is for light entering the structure either from vacuum or a medium of uniform refractive index. Either way, tends to be the wave quantity which is experimentally measured, the theory is therefore expressed in terms of measurable quantities. A. Solution to the Inverse Problem We can now take the inverse Fourier transform of (16) to yield an expression for the required coupling coefficient as a function of space, for a desired reflectivity photonic b gap structure. Thus, we find (17)

6 PARKER: AWGs, FBGs, AND PCs: AN ISOMORPHIC FT LIGHT PROPAGATION ANALYSIS 1163 where the spectral reflectivity is a function of the modified detuning parameter. However, when designing a filter we are interested in the spectral reflectivity as a function of the observed exterior average detuning,. To convert the spectral reflectivity into a function of the modified detuning requires (14), which already assumes a value for the coupling coefficient. Hence, (17), which implicitly has a coupling coefficient dependency within the Fourier integral, cannot be solved analytically. Instead, since the theory requires a perturbation about the Bragg wavelength, it is reasonable to assume that at the Bragg frequency the spectral coupling coefficient will be of the order of, where is the greatest relative permittivity contrast, which we can use in (14) to a good approximation. Given that we have the distributed coupling coefficient the reflection-spectrum expression related to each other by the FT, then it immediately follows that their integrated power spectra are related by Parseval s Theorem such that Fig. 6. High-dielectric-contrast 1-D regular PC grating structure. (18) This can be considered to be a distributed version of the Fresnel formulas. We now have two FT expressions for the localized coupling coefficient, (17) (A2b), they can be equated to yield an alternative expression for the spectral reflectivity at the front facet (19) Fig. 7. Reflection spectra using linear logarithmic scales. where is the angular frequency, is the speed of light, we have assumed that provided the RIC is not too high,. Thus, we find that the spectral response of a grating is equivalent to the response of a uniform material of spatially average permittivity, but with an appropriate frequency-dependent dielectric permittivity [22], i.e., an effective index [23], [24]. We also note that (19) contains no information relating to the spatial distribution of the permittivity, or a grating structure. Thus it is applicable to any material with a frequency-dependent refractive index. B. Application of Theory to a 1-D PC Fig. 6 shows a 1-D PC with a photonic bgap centered on frequency, corresponding to a Bragg wavelength. We can assume a constant coupling strength along the length of the PC, such that the conventional CMT equation describing the reflection spectrum for the structure is [12] where (20a) (20b) (20c) From the definition of spectral coupling coefficient given in (A4), the same regular square grating of finite length consisting of periods of period, with dielectric contrast, has a spectral coupling coefficient closely given by with as (21a) (21b) In the Appendix, we define as a vector in optical reciprocal space, we also define a detuning vector from the Bragg condition,, where the optical spatial frequency at the Bragg condition is given by. At the Bragg resonant frequency (i.e., ), the spectral coupling coefficient has the same magnitude as the coupling coefficient used in conventional CMT. As the number of periods tends to infinity (i.e., the implicit assumption of an infinitely repeating grating in conventional CMT) the limiting function for the coupling coefficient becomes a delta function in the spectral domain, as indicated by (21b) with the appropriate weight of. As the number of periods tends to infinity, we find that (14) can be closely approximated to (22)

7 1164 IEEE JOURNAL OF SELECTED TOPICS IN QUANTUM ELECTRONICS, VOL. 8, NO. 6, NOVEMBER/DECEMBER 2002 in (A8a) (A8b), the propagation constant within the PC is given by of the wave (24) Thus, as would be expected, the propagation parameter of the photon wave within the PC has both real imaginary components, with the imaginary component maximized at the Bragg resonance ( ), when the wave is at its most evanescent. It is straightforward to verify that, as the number of periods in the crystal tends to infinity,, (24) can be simplified to yield (a) (25) Hence for an infinitely periodic grating, the general expression (24) for the propagation constant tends toward the conventional CMT expression for the propagation constant (25). The dispersion diagrams for a finite grating an infinitely periodic structure, described by (24) (25), respectively, are shown in Fig. 8(a) in exped detail in Fig. 8(b). They show the photonic bgap structure associated with the PC of Fig. 6, with the same parameters as the reflection spectra of Fig. 7. The greatest discrepancy between the two curves occurs at the edge of the Brillouin zone, where the slope of the dispersion curve for (25) tends toward zero, according to conventional solid-state theory [26], whereas (24) shows an increase. (b) Fig. 8. (a) Photonic b structure associated with (b) an exped view of the photonic bgap structure of (a). Assuming that the parameter from conventional CMT can be defined as, we find that (22) is equivalent to (20b). From (16), the reflection spectrum for a PC with constant coupling strength along its length is (23) Both (20a) (23) yield the same reflectivity at the bgap center. Fig. 7 plots them against each other for a strong coupling example [25] where,, with the grating having periods. The figure shows excellent agreement between the two equations, with similar bgap widths positions of the sidelobes zeros. Since we have that, by substituting V. CONCLUSION Optical devices such as the AWG FBG discussed in this paper play an important role in current DWDM photonic networking. More recently, PC research has shown that these devices may play a pivotal role in future device integration. In this paper, we have shown that all three classes of device may be studied using a unified FT analysis. This not only aids in understing the characterization of currently available devices, but offers a powerful tool for the prediction of new modes of operation the design of novel optical components. We have demonstrated that AWGs FBGs can be considered to be equivalent devices, since their transfer characteristics both contain a functionally identical FT equation. This is despite the fact that the AWG is a FIR filter, whilst the FBG is an IIR device. This powerful relationship between the two devices can be exploited to apply design rules originally intended for the AWG to the FBG, vice versa. APPENDIX MODIFICATIONS TO CMT FOR STRONGLY COUPLED GRATINGS A. Optical Reciprocal Space For a weak to moderately strong coupled device, the geometric distance optical pathlength have an almost linear relationship, so that we can assume ; however, for a grating with a high RIC, are strongly divergent. The

8 PARKER: AWGs, FBGs, AND PCs: AN ISOMORPHIC FT LIGHT PROPAGATION ANALYSIS 1165 divergence means that the relationship between geometric (or optical) space reciprocal space, which is conventionally related via, also requires modification. Instead, to achieve the correct phase relationships, we need to define vectors in optical space optical reciprocal space as the appropriate Fourier conjugate variables [27]. This means that the optical spatial frequency at the Bragg condition is given by (as opposed to the conventional ) where remains the geometric period of the grating structure. Thus, we define to be a vector in optical reciprocal space, we also define a detuning vector from the Bragg condition. We can relate optical reciprocal space to the detuning parameter propagation constant, since we have that, we know that. We also know that are both zero at the Bragg condition, we can assume that they are linearly related via. Thus from the phase-matching condition, we can write a general resonance condition, equivalent to Bragg s law (A1) B. Spatial Spectral Definition of the Coupling Coefficient Conventionally, the coupling coefficient is defined in terms of the Fourier series coefficient associated with the permittivity of an infinitely periodic grating [12]. However, localized reflection (i.e., local coupling) is simply due to changes in refractive index with an appropriate phase change, i.e., equivalent to a Fresnel reflection, so that we can define the spatially varying coupling coefficient as (A2a) (A2b) Since the refractive index is related to the relative permittivity via, both (A2a) (A2b) are appropriate to describe the localized coupling coefficient. We note that scattering is phenomenologically due to the differential of the refractive index (permittivity), that the quantity, using the absolute permittivity, where is the vacuum permittivity, would also be appropriate in equation (A2b). Akin to the quantum mechanical paradox of the twin-slit experiment of a transverse grating, a photon can be considered both as a localized particle being scattered at localized positions within the grating, i.e., passing through only one of the twin slits, as well as a wave of infinite extent interacting with the grating simultaneously along its whole length (i.e., passing through both slits simultaneously) interfering with itself. Hence, we can assume that a photon also therefore has knowledge of a globally averaged coupling coefficient associated with the grating, as opposed to only the localized coupling coefficient. The global spatially averaged coupling coefficient appropriate for a wave extending over the whole grating is simply given by (A3) By substituting (A2b) for the localized coupling coefficient into (A3), we find that (A3) now functions as an FT integral. Thus, is equivalent to the FT of the spatial structure of the grating permittivity, so describes the spectral dependency (in optical reciprocal space ) of the coupling coefficient. In which case, a more appropriate symbol for is, with the tilda symbol indicating that it is closely related to the FT of the spatial coupling coefficient is variable in optical reciprocal space (A 4) Equation (A4) indicates that the spectral coupling coefficient associated with a grating structure, while having the same dimensionality as, can also be considered as the FT analog of the spatial coupling coefficient. The two coupling coefficients are complementary to each other. Up to now, no distinction has been made between the types of coupling coefficient (spatial spectral), they have tended to be treated as one. However, by making the distinction, we gain a better understing of the CMT describing how light is scattered. Equation (A4) can also be derived by taking the stard Fourier series definition of the coupling coefficient, re-expressing it in terms of a FT analysis [27]. The Fourier series definition is appropriate for a grating structure which is infinitely periodic (i.e., the grating is equivalent to a Born von Karman unit cell in solid state analysis [28]). However, for a finite grating, bounded in space, a FT integral is the most appropriate mathematical tool to analyze such an essentially aperiodic structure. The result of the FT is a continuous distribution in reciprocal -space, rather than the discontinuous, deltafunction distributions associated with a Fourier series representation. Thus we find that the spectral coupling coefficient has a continuous distribution in optical reciprocal space, which also needs to be taken into account. C. Modified Debye Waller Approximation The local phase of the propagating wave with respect to the grating phase (i.e., the detuning from the Bragg condition) can be understood to be the ratio. The detuning parameter has a spatial dependence, since the local propagation constant depends on the value of the local refractive index. However, similar to the local global coupling coefficient, the overall phase of the wave with respect to grating can be averaged over the length of the grating to yield where (A5) (A6)

9 1166 IEEE JOURNAL OF SELECTED TOPICS IN QUANTUM ELECTRONICS, VOL. 8, NO. 6, NOVEMBER/DECEMBER 2002 so that the angled brackets mean the spatial average over the length of the large RIC grating, rather than a simple geometric average (appropriate for low RIC gratings) conventionally denoted by a bar across the top of the variable. In a somewhat recursive manner, globally-averaged geometrically-averaged quantities are related by the phase, via the following modified Debye Waller factor [27]:q (A7) A Debye Waller factor [29] can be recognized within the curly brackets, the latter symbol refers to the fact that we need to take the appropriate sign for into account, since the sign information is lost when it is squared,, as it appears in the exponential. This is in contrast to the symbol, which is related to the two phase conjugate terms which are required from symmetry considerations, equivalent to the upper lower bs in the Brillouin zone. The globally averaged detuning parameter is now simply given by, where the geometrically averaged (i.e., conventional) detuning parameter is simply given by. The refractive index quantities are similarly related, as are etc. We can decompose the modified detuning parameter into its quadrature components,, such that by appropriate substitution of (A5) into (A7) we find (A8a) (A8b) We have made the spectral coupling coefficient a function of the detuning parameter, by assuming that [see (A1)]. For the coupled mode equations, we are only interested in the real part of the modified detuning parameter [30],, which corresponds to a real phase-detuning from the resonant Bragg phase-matching condition. Multiplying the two conjugate solutions for together, we find that (A9) The substitution of for in Sections III IV is all that is required to correct for strong-coupled devices avoid the phase accumulation problem, with (A9) used to relate to the experimentally observed detuning parameter. 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10 PARKER: AWGs, FBGs, AND PCs: AN ISOMORPHIC FT LIGHT PROPAGATION ANALYSIS 1167 Michael C. Parker (S 96 A 96) was born in London, U.K., in He received the first class B.A. degree in electrical information sciences in 1992 the Ph.D. degree in optical communications from Cambridge University, Cambridge, U.K., in , respectively. Prior to beginning his doctoral studies, he spent six months working for Carl Zeiss, Germany, developing large-field objectives for photolithography. During his doctoral work, he conducted research into holographic WDM switching using programmable liquid crystal spatial light modulators. Since 1997, he has been working for Fujitsu, based in Colchester, U.K., conducting research into AWG design theory, photonic bgap structures, chromatic dispersion PMD compensation techniques, WDM optical access networks. Since 2000, he has also been appointed a Principal Researcher of the Photonics Networking Laboratory, Fujitsu Network Communications, Richardson, TX, has also been accorded the title Visiting Fujitsu Senior Research Fellow by the University of Essex, U.K. He has filed 11 patents authored over 70 publications. Dr. Parker is a Member of the Optical Society of America an Associate Member of the IEE. Stuart D. Walker was born in Dover, U.K., in He received the B.Sc. degree in physics from Manchester University, Manchester, U.K., in 1973 the M.Sc. Ph.D. degrees from the University of Essex, Essex, U.K., in , respectively. After completing a period of contractual work for British Telecom Laboratories between , he joined as a Staff Member in He worked on various aspects of optical system design was promoted to Head of the Regenerator Design Group in In 1988 he took up his present position of Senior Lecturer in Optical Communication at the University of Essex. His current research interests are concerned with the modeling analysis of advanced optical network components systems. He has filed eight patents authored over 100 publications.