Module 2 Spatially-Variant Planar Gratings

Transcription

1 Pioneering 21 st Century Electromagnetics and Photonics Module 2 Spatially-Variant Planar Gratings Course Website: EM Lab Website: Lecture Outline The grating vector Grating phase Generating spatially-variant planar gratings SVL -- Module 2 Slide 2 1

2 The Grating Vector K. The Grating Vector in Two Dimensions y The grating vector K is very similar to a wave vector k. The direction of K is perpendicular to the grating planes. The magnitude of K is 2 divided by the period of the grating. 2 K Given the slant angle, it is calculated as 2 K aˆ cos ˆ aysin It allows a convenient calculation of the analog grating. cos K r SVL -- Module 2 Slide 4 2

3 The Grating Vector in Three Dimensions cos K r r ˆ yyˆzzˆ position vector 2 K K K ˆK yˆkzˆ y z SVL -- Module 2 K Slide 5 The Analog Grating y For simplicity, calculation of the analog grating will be written as r cos K r a or a r Re ep jk r If instead you wish to generate an actual analog grating, it will need to be scaled to convey physical permittivity values. r cos K r avg SVL -- Module 2 Slide 6 3

4 The Binary Grating Analog Grating Binary Grating Controlling the Fill Fraction a r Kr b r cos Kr b r cos Kr.8 cos Generation of the grating using a cosine function gives us a smooth analog profile. This requires functionally grading the material properties to realize this. That is hard. We can use a threshold to convert the analog profile to a binary grating. This us much easier to physically realize. We can adjust the threshold value to control the duty cycle, or fill fraction, of the grating. SVL -- Module 2 7 Grating Phase r 4

5 Concept of Grating Phase Here we make an analogy with a standard wave. A wave propagates in the direction of the wave vector k. E r E cos k r t As the wave propagates, it accumulates phase which is a function that increases in the direction of k. The phase results from the wave propagating with wave vector k and we could alternatively reconstruct the wave just from the phase. E r E cos r r k r t For gratings, the grating vector K serves a similar purpose as the wave vector k does for waves. kr Kr rr E r r a k r or K r or r r E r r or a SVL -- Module 2 9 Definition of Grating Phase K X y We can think of a grating as a wave because it has the same mathematical form. As a wave propagates it accumulates phase. The wave can be calculated from just the phase. Think of the grating phase this way a rcoskrcos r r Kr KKy y r K K y K Y y + = cos Note: Unfortunately, this intuitive definition of grating phase only holds when K is a constant and not a function of position. SVL -- Module 2 1 5

6 Problem of a Chirped Grating Suppose we wish to generate the following chirped grating with period (z). z z 1 z 2 This 1D grating is described by the following grating vector function. 2 2 Kz z z What happens when we try to calculate the grating using cos K r. 2 2 r zcos Kzz cos z cos z 2 The argument in the cosine is a constant so we do not even generate a grating here. SVL -- Module 2 11 Conclusion From the Chirped Grating When the grating vector is a function of position, we can no longer calculate the grating directly from it. a r cos Kr r Instead, we must generate the grating through the intermediate parameter of the grating phase. a r cos r However, we must adopt a more rigorous definition of grating phase. r K r Key equation r K r r SVL -- Module

7 Revised Solution for Chirped Gratings Now let s construct the chirped grating using the grating phase. K r d 2 dz z z 2 2 z dz dz z z z 2 2 z ln z ln z ln1 1 1 The analog grating is then 2 a zcos z cos ln z z 1 2 dz z This term is just a constant. Since it is phase, we are free to choose whatever is convenient. Here we choose zero. SVL -- Module 2 13 Generating Spatially-Variant Planar Gratings 7

8 Procedure for Generating Spatially-Variant Planar Gratings 1. Define the grating vector 2 Kr aˆ ˆ cos r aysin r function Kr, or K function. r 2. Calculate the grating phase r from Kr. r K r 3. Calculate the analog grating from the grating phase. 4. Calculate the binary grating from the analog grating. a r cos r b 1 a r 2 a r r r r SVL -- Module 2 15 Spatially-Variant Planar Gratings No spatial variance r r r r r r Spatially Variant Orientation Spatially Variant Period Spatially Variant Threshold Spatially Variant Everything SVL -- Module

9 Build Input Functions We need two things to build the K function: 1. Orientation of the grating as a function of position, r. 2. Period of the grating as a function of position, r. 1 r 1. r tan y % GRATING PARAMETERS a = 1; er1 = 2.5; er2 = 1.; gth = ; % GRID PARAMETERS S = 5*a; Sy = S; N = 1; Ny = round(n*sy/s); % CALCULATE GRID d = S/N; dy = Sy/Ny; a = [1:N]*d; ya = [1:Ny]*dy; [Y,X] = meshgrid(ya,a); 4 2 % SPATIALLY-VARIANT PARAMETERS PER = a*ones(n,ny); THETA = atan2(y,x); SVL -- Module 2 17 Calculate the K-Function The K-function is calculated directly from the input functions. K r K y r 2 K r cos r r 2 K y r sin r r % CALCULATE K-FUNCTION K = 2*pi./PER.*cos(THETA); Ky = 2*pi./PER.*sin(THETA); 2 a a 2 a a SVL -- Module

10 Calculate Grating Phase We calculate the grating phase from the K-function as a best fit. r r K D k Φ D k y y % COMPUTE GRATING PHASE PHI = svlsolve(k,ky,d,dy); SVL -- Module 2 19 A More Efficient Grid Strategy Observe how smooth the grating phase function is compared to the final grating. r b r We can get away with a much coarser grid up to the point where the planar grating is calculated. Coarse Grid PER, THETA, K, PHI Fine Grid FF, PHI2, ERA, ERB SVL -- Module 2 2 1

11 Calculate Analog Grating The analog grating is calculated directly from the grating phase. a r a r cos r % COMPUTE ANALOG GRATING ERA = cos(phi); SVL -- Module 2 21 Calculate Binary Grating The binary grating is calculated directly from the analog grating using the threshold technique. b r b r 1 a r 2 a r r r % COMPUTE BINARY GRATING ERB = er1*(era <= gth) + er2*(era > gth); SVL -- Module

12 Controlling Fill Fraction Through the Threshold Function Analog Grating a r b r 1 a r 2 a r r r We can estimate the threshold value in order to realize a given fill fraction f of 1. r cos f r f = f =.2 f =.4 f =.6 f =.8 f = = 1. =.8 =.3 = -.3 = -.8 = -1. SVL -- Module