Optimization and control of energy transmission across photonic crystal slabs

Transcription

1 Optimization and control of energy transmission across photonic crystal slabs Robert Lipton, Stephen P. Shipman Department of Mathematics Louisiana State University Baton Rouge, LA 783 and Stephanos Venakides Department of Mathematics Duke University Durham, NC 2778 August 3, 26 Abstract A variational approach is developed for the design of defects within a two-dimensional lossless photonic crystal slab to create and manipulate the location of high Q transmission spikes within band gaps. This phenomena is connected to the appearance of resonant behavior within the slab for certain crystal defects. The methodology is applied to design crystals constructed from circular dielectric rods embedded in a contrasting dielectric medium. PACS: 42.7.Qs, s 1 ntroduction We present a method for designing periodic photonic crystal slabs that have prescribed energy transmission properties. Our motivation is based upon the connection between the resonant electromagnetic scattering behavior in photonic crystal slabs as sheets, films, or walls and the anomalous behavior of the transmission of energy through them. t has been observed empirically and through numerical simulations that leaky waveguide modes, Fabry-Perot bandgap resonances, and other resonant phenomena coincide in the frequency domain with narrow regions of enhanced transmission or reflection of incident waves see Refs. [1], [2], [4], and [5]. Here we develop a variational calculus for the transmitted energy as a function of the defect geometry. The variational calculus provides the means to manipulate the resonant properties of crystals constructed from circular dielectric rods embedded in a contrasting dielectric medium. n this paper we illustrate our approach by designing defects within a two-dimensional lossless photonic crystal slab to create and manipulate the location of high Q transmission spikes within a band gap. The numerical implementation is based on the pseudoperiodic free space Green s functions developed Submitted to Phys. Rev. B lipton@math.lsu.edu, Telephone: ; supported by grants AFOSR FA and NSF DMS shipman@math.lsu.edu, Telephone: ; supported by grants NSF DMS and LA Board of Regents grant LEQSF23-6-RD-A-14 ven@math.duke.edu, Telephone ; supported by grant NSF DMS

2 in [3] and [4]. Here the transmitted electric field is given in terms of the solution of the associated boundary integral equations [4] for crystals whose period consists of a finite number of disjoint scatterers with smooth boundaries. The design variables used in this aproach describe the geometry of the defect and the basic lattice parameters of the crystal. Earlier work of the authors focused on the creation of transmission spikes within band gaps, see [6]. n related work Cox and Dobson Refs. [7] and [8] devise a gradient-search method for optimizing and creating bandgaps for two-dimensional photonic crystals that are infinite in both directions. This is based on the generalized gradient of the bandwidth derived from the variational calculus of the dispersion relation of the periodic structure. t is also known that resonant scattering behavior in photonic crystal slabs is related to complex dispersion relations Ref. [5]: source fields at real frequencies near a complex eigenvalue produce resonant scattering in the crystal and the associated enhanced or inhibited transmission. A method based on the manipulation of these dispersion relations holds promise for engineering crystal slabs with desired resonant properties. t is pointed out that the afrementioned approaches focus on the manipulation of the dispersion relations associated with the crystal. This is in contrast to the approach taken here where we manipulate the transmitted energy across the slab as a function of the structure and material properties of the dielectric crystal. 2 Energy transmission across photonic crystal slabs n this section we provide a semi explicit formula for the transmission coefficient that is used in the design of the photinic crystal slab. The photonic crystal slab consists of an array of lossless rods perpendicular to the xy-plane embedded in a lossless medium of contrasting dielectric permittivity ɛ and magnetic permeability µ. The array is periodic in the y direction and is finite in the x direction Fig. 1. The period cell for the structure is made up of a finite number of rod cross sections D j, j = 1,..., n, with boundaries D j, and we put D = n j=1 D j. Each rod as well as the exterior medium is taken to be composed of homogeneous material. The slab is illuminated from the left by a time-harmonic electromagnetic field that is polarized so that either the electric field points out of the plane and the magnetic field is transverse to the rods the transverse magnetic, or TM, case or the magnetic field points out of the plane the TE case. n either case, we denote the out-of-plane component by ψx, ye iωt, where ω is the reduced frequency. Here the reduced frequency is defined by ω = fl/c, where c is the speed of light, L is the period of the structure, and f is the frequency in cycles per time. The reduced spatial variables x, y are related to the coordinates X, Y through x, y = 2πX, Y /L. t is pointed out that the cases of TM or TE illumination result in the same mathematical formulation so to fix ideas we consider the case of TM illumination and suppose the source field is incident upon the slab at an angle θ from the left. The incident electric field is of the form where E inc = ψ inc x, ye iωt ψ inc = e iκ mx e i m+κy. Here m is an integer such that m + κ 2 < ɛµω 2 and κ m = ɛµω 2 m + κ 2. This is a plane wave whose angle of incidence with the slab is θ =arcsin m+κ ɛµω. κ can always be taken to lie in the interval [ 1/2, 1/2, and m = and κ = corresponds to normal incidence. n view of the y periodicity it suffices to consider the energy transmission along the strip S containing one period as shown in Figure 1. The time averaged energy flux of the incident field into S is given by F inc = 2πɛµ 1/2 ω cos θ. The time averaged energy flux passing through the strip to the right of the slab is given by F out = R{ e 1 Sdl} Γ + 2

3 where e 1 is the unit vector directed along the positive real axis, S is the Poynting vector and the line integral is taken along a line segment parallel Γ+ to the y axis. Here F out is independent of the location of Γ +. The transmission coefficient for an incident electric field of frequency ω is T ω = F out /F inc. The out of plane electric field is given by ψx, ye iωt and the Maxwell equations reduce to the Helmholtz equation 2 ψ + ɛµω 2 ψ =. This equation holds in the interiors of the regions D j and in the exterior medium, where the values of ɛ and µ take on different constant values. For the TM case, the Maxwell equations imply the following matching conditions for the interior and exterior limiting values of the electric field component ψ on D. ψ int = ψ ext µ ext n ψ int = µ int n ψ ext n view of the symmetry provided by the y periodicity the the field ψ is pseudoperiodic in y, and we write it as a Bloch wave ψx, y = ψx, ye iκy, where ψx, y has period 2π in the variable y. The pseudo-periodicity allows us to restrict the analysis to the strip S = {x, y : <x<, y 2π} Fig. 1. A pseudo-periodic radiating fundamental solution of the Helmholtz equation in a homogeneous medium ɛ and µ constant is given by Gx ˆx, y ŷ, where Gx, y satisfies 2 G + ɛµω 2 G = δx, y 2πke 2πkiκ. k= For values of κ and ω such that ɛµω 2 m + κ 2 for all integers m, its Fourier form is Gx, y = 1 1 exp ν m x + im + κy, 4π ν m m= where ν m = iκ m and κ 2 m = ɛµω 2 m + κ 2. Here the Green s function is built of a finite number of propagating plane-wave modes, for which κ m >, m = m 1,..., m 2, and infinitely many decaying evanescent modes, for which ν m <. The Green s function, together with Green s identities shows that the asymptotic behavior of bounded solutions in the strip S as x is a superposition of plane waves traveling to the right and left: ψ m 2 a ± m e iκmx + b ± me iκmx e im+κy The solution approaches the asymptotic form exponentially as x. The total field ψ has the asymptotic form ψ ψ inc + ψ m 2 m 2 r m e iκmx e im+κy t m e iκmx e im+κy x, x ± x in which r m and t m are the coefficients of the modes of the fields reflected by the slab and transmitted through it, respectively. The transmission coefficient is expressed in terms of the total field ψ as T = 1 Γ + ψ n ψ ds. 1 3

4 Figure 1: A cross-section of a photonic crystal slab consisting of an array of homogeneous dielectric rods standing perpendicular to the xy-plane. The rod structure is periodic in the y-direction, with period 2π, and extends indefinitely as y ±. The rod structure is finite in the x-direction. Exterior to the rods is a homogeneous material of contrasting dielectric permittivity extending to infinity to the right and left. Pseudo-periodic fields in the plane can be analyzed in the strip S = {x, y : < x <, y 2π} consisting of a single period of the dielectric permittivity function. 4

5 3 Sensitivity analysis of transmission coefficient n this section we develop the variational method for computing the sensitivity of the transmission coefficient with respect to changes in the rod configuration. Consider first the truncated strip = {x, y : x < x < x, y 2π}; and let Γ and Γ + denote the left and right boundaries of : Γ ± = {±x, y : y 2π}; put Γ = Γ Γ + ; and let n denote the outward-pointing normal vector to Γ. The strip is chosen large enough to contain all the rods D. n what follows it is advantageous to formulate the scattering problem for ψ in its weak form given by µ 1 ψ λ + ɛω 2 ψλ da + µ 1 λ n ψ ds =, Γ for all pseudoperiodic, smooth trial fields λ and over all truncated strips ; ψ ψ inc + 2 m 2 r m e iκmx e im+κy x ψ m 2 t m e iκmx e im+κy x Here over bars denote complex conjugation and the complex numbers r m and t m are the coefficients of the reflected and transmitted radiating plane waves. Next we purterb the locations and shape of the rod cross sections and deduce the required expression for the associated change in the Transmission coefficient. The change in material properties ɛ, µ 1 associated with a purturbation in the rod shapes and location are denoted by ˆɛ and µ 1. The associated changes in transmission coefficient and field are denoted by ˆT and ˆψ. The purturbed field ψ + ˆψ solves [ µ 1 ψ + ˆψ + λ + ˆɛω 2 ψ + ˆψ ] λ µ 1 ˆψ λ + ɛω 2 ˆψλ da + da + Γ µ 1 λ n ˆψ ds =, for all pseudoperiodic, smooth trial fields λ and over all truncated strips ; 3 ˆψ m 2 ˆr m e iκmx e im+κy ˆψ m 2 ˆt m e iκmx e im+κy x x, and ˆT = 1 ψ n ˆψ + ˆψ n ψ + 2πκ ˆψ n ˆψ ds. 4 m Γ + The next step is to calculate the gradient of T with respect to ɛ and µ 1 and to express ˆT to first order as a linear function of ˆɛ and µ 1. To this end we seek an adjoint function λ such that, for all perturbations ˆψ satisfying the asymptotic conditions in 3, [ µ 1 ˆψ ] λ + ɛω 2 ˆψλ da + µ 1 λ n ˆψ ds = µ 1 ext ψ n ˆψ + ˆψ n ψ ds. 5 Γ Γ + We have multiplied ˆT by the constant µ 1 ext for convenience. Once we have found such a function λ, we use 3 and 4 to deduce that ˆT = µ ext [ µ 1 ψ λ ˆɛω ψλ] 2 da + h.o.t., 6 where h.o.t. refers to the higher order terms that are quadratic in the perturbation. This equation gives ˆT in terms of ˆɛ and µ 1. We remark that both sides of 5 are independent of the truncation 5

6 boundary Γ, assuming that the perturbations ˆɛ and ˆµ are contained in. That the left-hand side is independent of Γ is evident from 3. That the right-hand side is independent of Γ is seen as follows: ˆT in equation 4 is a constant multiple of the difference of the energy flows of ψ and ψ + ˆψ through Γ +, which is independent of the x-value of Γ +. The quadratic term ˆψ n Γ + ˆψ ds is also independent of Γ + : ˆψ is a radiating pseudo-periodic field; it has a Fourier expansion ˆψ = m= a m e νmx e im+κy, where ν m = iκ m for the finite number of propagating modes and ν m < for the rest, and we compute m 2 ˆψ n ˆψ ds = κ m a m 2, Γ + which is independent of x. We now show that a solution λ to the following adjoint problem satisfies 5: µ 1 φ λ + ɛω 2 φλ da + µ 1 φ n λ s = Γ for all pseudoperiodic φ and truncated strips λ m 2 τ m e iκmx e im+κy x λ m 2 t m e iκmx e im+κy + m 2 ρ m e iκmx e im+κy x This is an antiradiating scattering problem, in which the source field is a sum of plane waves traveling to the right, equal to the transmitted field in the scattering problem for ψ, and the scattered field is antiradiating. The problem can also be interpreted in a physical sense if we use the time harmonic factor e iωt instead of e iωt. Then the problem for λ is just the scattering problem in which the transmitted field of ψ is reversed and sent back toward the slab from the right. The left-hand side of 5 can now be rewritten and calculated using 7: [ µ 1 ˆψ ] λ + ɛω 2 ˆψλ da + µ 1 λ n ˆψ ds = Γ m 2 = µ 1 λ n ˆψ ˆψ n λ ds = 4πiµ 1 ext κ m t mˆt m. 8 Γ On the other hand, we compute the right-hand side of 5 using the asymptotics in 2 and 3: m 2 µ 1 ext ψ n ˆψ + ˆψ n ψ ds = µ 1 ext ψ n ˆψ ˆψ n ψ ds = 4πiµ 1 ext κ m t mˆt m. Γ + Γ + Thus, we have shown that if λ satisfies problem 7, then 5 holds for all perturbations ˆψ. A word about the calculations of the integrals over Γ and Γ + that we have just performed: Each is a combination of integrals of the form 2π φ1 x φ 2 φ 2 x φ 1 dy, 7 in which φ 1 and φ 2 are of the form φ = c m e νmx e im+κy, m= 6

7 x 8 x 8 incident r m ψ tm r m τ m ψ λ tm t m ρ m Figure 2: The far-field asymptotic behavior of the solution ψ to the scattering problem 2, the perturbed field ˆψ, and the solution λ to the adjoint scattering problem 7. where ν m = iκ m if the oscillatory modes propagate is to the right and ν m = iκ m if they propagate to the left, and ν m is real for the rest of the modes. Let a m be the coefficients for φ 1 and b m the coefficients for φ 2. Simple computation yields 2π φ1 x φ 2 φ 2 x φ 1 dy = = { m2 ±4πi κ m a m b m if φ 1 and φ 2 propagate in the same direction, if φ 1 and φ 2 propagate in opposite directions. Using this rule and the diagram of asymptotics in Figure 2, the calculations above follow easily. 3.1 Variation of Circular Components We take the D j to be circles and compute the gradient of T with respect to their radii r j and centers x j, y j. Calculation and application of 6 gives and T r j = µ ext T x j = µ ext T y j = µ ext n j=1 n j=1 n j=1 [ µ 1 j D j [ µ 1 j D j [ µ 1 j D j where the integration is taken counter-clockwise. 3.2 Variation of the Values of ɛ and µ µ 1 ext ψ λ ɛj ɛ ext ω 2 ψλ ] ds 9 µ 1 ext ψ λ ɛj ɛ ext ω 2 ψλ ] dy, 1 µ 1 ext ψ λ ɛj ɛ ext ω 2 ψλ ] dx, 11 For completeness we provide the formula for ˆT when the rod geometry is fixed and the values of ɛ and µ are allowed to change. Suppose that a period of the crystal slab consists of components {D j } n j=1 with dielectric and magnetic constants ɛ j and µ j and exterior constants ɛ ext and µ ext. Let 7

8 ˆɛ j and 6 gives ˆT µ 1 j be perturbations of ɛ j and µ 1 j, keeping the boundaries of the domains D j fixed. Then µ ext [ n ] µ 1 j ψ λ da ˆɛ j ω 2 ψλ da j=1 D j D j [ n ] µ 1 j µ j ɛ j ω 2 ψλ da λ n ψ ds ˆɛ j ω 2 ψλ da, 12 D j D j D j = µ ext j=1 in which means equal to leading linear order in the variations ˆɛ j and is a result of integration by parts. 1 µ j. The last equality 4 Numerical method for optimization and control n two examples, we describe how we implement computationally our results on the variation of circular components subsection 3.1 to intensify a transmission anomaly and to shift the location of a transmission peak. Example 1. The objective of our first example Fig. 3 is to create a sharp resonance within a transmission gap. We begin with a square lattice of circular dielectric rods with ɛ=12 and µ=1 in air ɛ=µ=1, truncated to a slab three rods thick, with a periodic channel running through it. The channel is created by adding an additional horizontal strip of air space between every three rows of rods. Thus, the new period in the y-direction is the supercell consisting of a 3 3 array of rods plus the additional horizontal strip of air. The reason for beginning with a structure with a periodic channel is that this defect causes transmission anomalies, such as spikes and humps, to appear compare the top and middle graphs in Fig. 3. Numerical tests indicate that it is more effective to begin with a mild anomaly, as that indicated by the arrow in the middle graph, and try to intensify it than to create an anomaly within a flat transmission profile as in the top graph. We have chosen to begin with the shallow hump indicated in the middle graph and to intensify it into a sharp peak by manipulating the centers and radii of the nine rods comprising a unit supercell. n doing so, we retain normal incidence. The field ψ corresponding to the peak of the mild anomaly is the response to a harmonic source field at normal incidence and exhibits a small but significant amount of resonant enhancement within the channel. See Ref. [4] for simulations and discussion of the fields at various transmission anomalies. This resonant enhancement is greatly magnified after optimization; Fig. 4 shows the amplitude of the field at the frequency at the top of the transmission spike indicated in the bottom graph of Fig. 3. The black color to the left and right of the structure indicate that the amplitude of the incident field is extremely small compared to the enhanced amplitude of the steady-state field inside the structure. We use the following algorithm to carry out the optimization. 1. Start with a structure whose transmission coefficient T κ, ω exhibits resonant features to be optimized. We begin with the structure in the middle of Fig. 3. To compute the transmission coefficient at κ =, we must solve the scattering problem 2 numerically. We do this using the method of boundary integrals based on Green s identities and the pseudo-periodic Green s function. Details can be found in Ref. [4]. 2. Choose a set of frequencies and directions in which the transmission or extrema in the transmission coefficient should move. We choose the frequency at which the peak of the transmission anomaly occurs, where we seek to increase the transmission, and two neighboring frequencies, where we seek to decrease or at least not increase the transmission. 8

9 3. Compute the transmission and its variational gradient with respect to the allowed structural parameters at the three frequencies. We choose to restrict our admissible structures to arrays of circular rods with a fixed dielectric contrast of 12. Our design variables are the radii and centers of the rods. We compute the variational gradient by solving the scattering problems 2 and 7 numerically and computing the integrals 9, 1, and 11. We denote the gradient of the transmission with respect to these parameters by p T, where p is the 27-dimensional vector of x, y-coordinates of the centers and the radii of the nine rods in the supercell. 4. Compute the smallest change in the parameters that gives the desired change in the transmission. Let M denote the 3 27 matrix whose rows are equal to p T at the three chosen frequencies, and let b denote the vector of desired increments in T at these frequencies. b has components b 1, b 2, b 3, in which b 1 and b 3 can be taken to be zero and b 2 is a small fraction of the difference between 1 and the transmission at the middle frequency. We compute the smallest vector p that produces the change b by solving the 3 3 system MM t x = b for x and putting p = M t x. 6. ncrement the structure by the computed amount. 7. Update the set of frequencies if necessary. As the structure is modified, the peak of the transmission anomaly may shift to a different frequency. To realign the frequencies so that the middle one is at the peak of transmission, we assume a quadratic shape for the transmission near the peak, which is uniquely determined by the computed transmission values at the three frequencies. These frequencies can then be relocated to place the middle one at the peak. 8. Go back to step 3. Example 2. n the second example Fig. 5, we shift the location of a transmission peak. The rods in the initial structure are indicated in the figure by the smaller circles. Notice that the horizontal distance between adjacent rods is greater than the vertical distance. We are dealing with a rather degenerate case of a Fabry-Perot mirror in which the each wall of the mirror consists merely of one column of rods. See Ref. [3] for simulations and discussion of Fabry-Perot resonances in photonic crystals. The peak we are dealing with separates a region of low transmission due to a complete spectral gap it exists at all angles for the perfect infinite two-dimensional periodic structure from a region corresponding to an incomplete one, although for a structure with only two layers of rods, these gaps are not necessarily well formed. We use the following algorithm to carry out the modification. 1. Start with a structure that has a transmission coefficient exhibiting a peak. n Fig. 5, the peak we seek to shift is indicated by the arrow; the transmission of the original structure is shown with a solid line. 2. Begin with three frequencies: one at the peak and one to each side of the peak. We choose the frequencies to be close enough to each other so that the transmission coefficient appears quadratic in their vicinity and far enough apart to avoid round-off error in the difference quotients. 3. Compute the transmission and its variational gradient with respect to the allowed structural parameters at the three frequencies. This is done as in the first example, except now the parameter space is six dimensiona: there are two rods in a period, whose centers and radii are to be varied. 9

10 4. Compute the variational gradient of the frequency at which the transmission has a peak. Let W p denote this frequency for the material parameters p. W is defined implicitly by the equation T W p, p =. ω Taking its differential and solving for p W gives 2 1 T p W = W p, p ω2 ω pt. The term in parentheses is calculated as a centered second difference quotient of the transmission, and the second term on the right is calculated as a centered first difference quotient of the gradient of the transmission. 5. Compute the smallest change in the parameters that gives the desired change in the frequency at the peak. Since we are varying a scalar quantity W p, we simply take the appropriate multiple of p W as a vector in the six-dimensional parameter space. 6. ncrement the structure by the computed amount. 7. Update the frequencies if necessary. This is done as in the first example. 8. Go back to step 3. 1

11 1 3pi 2pi pi transmission reduced frequency 1 3pi 2pi pi transmission reduced frequency 1 3pi 2pi pi transmission reduced frequency Figure 3: A normalized frequency of 1 corresponds to a wavelength equal to the distance between adjacent rods. Top: a photonic crystal slab and its transmission coefficient at normal incidence. Middle: the slab with periodically placed channels and its transmission coefficient; two periods of the structure are shown. The channel gives rise to the transmission anomalies. The weak one indicated by the arrow was chosen for optimization. Bottom: the modified slab. The shallow hump has been transformed into a sharp peak. 11

12 Figure 4: A simulation of the field amplitude at the frequency of the peak transmission for the modified structure at the bottom of Fig pi 4pi 2pi transmission reduced frequency Figure 5: The slab indicated by the smaller rods has transmission coefficient given by the solid black line. The slab has been modified to move the peak indicated by the arrow to the left. The transmission coefficient of the final structure dashed line and an intermediate one dotted line are shown. The larger circles indicate the positions of the rods of the final structure. 12

13 References [1] A. Krishnan, T. Thio, T. J. Kim, H. J. Lezec, T. W. Ebbesen, P. A. Wolf, J. Pendry, L. Martin- Moreno, and F. J. Garcia-Vidal, Evanescently coupled resonance in surface plasmon enhanced transmission, Optics Communications 2, pp. 1 7, 21. [2] S. Tikhodeev, A. Yablonskii, E. Muljarov, N. Gippius, and T. shihara, Quasiguided modes and optical properties of photonic crystal slabs, Phys Rev B 66, pp. 4512:1 17, 22. [3] S. Venakides, M. A. Haider, and V. Papanicolaou, Boundary integral calculations of twodimensional electromagnetic scattering by photonic crystal Fabry Perot structures, SAM J. Appl. Math, 6 2, pp [4] M. Haider, S. Shipman, and S. Venakides, Boundary-integral calculations of two-dimensional electromagnetic scattering in infinite photonic crystal slabs: Channel defects and resonances, SAM J Appl Math 62-6, pp , 22. [5] S. Shipman and S. Venakides, Resonance and bound states in photonic crystal slabs, SAM J Appl Math 64-1, pp , 23. [6] R. Lipton, S. Shipman, and S. Venakides, Optimization of Resonances in Photonic Crystal Slabs, Proceedings of SPE, 5184 pp , 23. [7] S. Cox and D. Dobson, Maximizing band gaps in two-dimensional photonic crystals, SAM J Appl Math 59-6, pp , [8] S. Cox and D. Dobson, Band structure optimization of two-dimensional photonic crystals in h-polarization, J Comp Phys 158, pp , 2. [9] O. Pironneau, Optimal Shape Design for Elliptic Systems, Springer-Verlag, New York,