Diffraction Gratings
Transcription
Diffraction Gratings
2/23/2016 ECE 5322 21st Century Electromagnetics Instructor: Office: Phone: E‐Mail: Dr. Raymond C. Rumpf A‐337 (915) 747‐6958 [email protected] Lecture #9 Diffraction Gratings Lecture 9 1 Lecture Outline • • • • • • • • Lecture 9 Fourier series Diffraction from gratings The plane wave spectrum Plane wave spectrum for crossed gratings The grating spectrometer Littrow gratings Patterned fanout gratings Diffractive optical elements Slide 2 1 2/23/2016 Fourier Series Born: March 21, 1768 in Yonne, France. Died: May 16, 1830 in Paris, France. Jean Baptiste Joseph Fourier 1D Complex Fourier Series If a function f(x) is periodic with period x, it can be expanded into a complex Fourier series. f x a m e j 2 mx m a m 2 2 mx j 1 f x e dx 2 Typically, we retain only a finite number of terms in the expansion. f x M a m e j 2 mx m M Lecture 9 Slide 4 2 2/23/2016 2D Complex Fourier Series For 2D periodic functions, the complex Fourier series generalizes to f x, y a p, q e p q Lecture 9 2 px 2 qy j x y 2 px 2 qy x y j 1 a p, q f x, y e A A dA Slide 5 Diffraction from Gratings 3 2/23/2016 Fields in Periodic Structures Waves in periodic structures take on the same periodicity as their host. k Lecture 9 Slide 7 Diffraction Orders The field must be continuous so only discrete directions are allowed. The allowed directions are called the diffraction orders. The allowed angles are calculated using the famous grating equation. Allowed Lecture 9 Not Allowed Allowed Slide 8 4 2/23/2016 Field in a Periodic Structure The dielectric function of a sinusoidal grating can be written as r r r ,avg cos K r A wave propagating through this grating takes on the same symmetry. E r A r e jkinc r A r ,avg cos K r e jkinc r A j kinc K r A j kinc K r A r ,avg e e e 2 2 jkinc r wave 1 wave 2 wave 3 Lecture 9 9 Grating Produces New Waves The applied wave splits into three waves. e jkinc r e e e jkinc r j kinc K r j kinc K r Each of those splits into three waves as well. e jkinc r e e e jkinc r j kinc K r j kinc K r e e j kinc K r e j kinc K r j kinc 2 K r e jkinc r And each of these split, and so on. k m kinc mK m ,..., 2, 1, 0,1, 2,..., Lecture 9 e j kinc K r e j kinc K r jkinc r e e j kinc 2 K r This equation describes the total set of allowed harmonics. 10 5 2/23/2016 Wave Incident on a Grating Boundary conditions required the tangential component of the wave vector be continuous. inc ? k x ,trn k x ,inc The wave is entering a grating, so the phase matching condition is k x m k x ,inc mK x The longitudinal vector component is calculated from the dispersion relation. K k z2 m k0 navg k x2 m 2 For large m, kz,m can actually become imaginary. This indicates that the highest order spatial harmonics are evanescent. Lecture 9 11 The Grating Equation kinc inc +1 +2 0 x ‐1 +1 0 K Lecture 9 0 sin Note, this really is just k x m k x ,inc mK x Proof: k x m k x ,inc mK x k0 navg sin m k0 ninc sin inc m 2 0 ntrn z navg sin m ninc sin inc m ‐1 +2 navg The Grating Equation ninc navg sin m 2 0 2 x ninc sin inc m navg sin m ninc sin inc m navg sin m ninc sin inc m 2 x 0 x 0 sin 12 6 2/23/2016 Grating Equation in Different Regions The angles of the diffracted modes are related to the wavelength and grating through the grating equation. nref ninc The grating equation only predicts the directions of the modes, not how much power is in them. Reflection Region nref sin m ninc sin inc m Transmission Region ntrn sin m ninc sin inc m 0 x 0 x x ntrn Lecture 9 Slide 13 Diffraction in Two Dimensions • We know everything about the direction of diffracted waves just from the grating period. Diffraction tends to occur along the lattice planes. • The grating equation says nothing about how much power is in the diffracted modes. – We need to solve Maxwell’s equations for that! Lecture 9 Slide 14 7 2/23/2016 Effect of Grating Periodicity Subwavelength Grating x “Subwavelength” Grating Low Order Grating High Order Grating ninc ninc ninc ninc navg navg navg navg 0 0 navg navg x 0 ninc x 0 ninc Lecture 9 x 0 ninc Slide 15 Animation of Grating Diffraction at Normal Incidence Lecture 9 Slide 16 8 2/23/2016 Animation of Grating Diffraction at an Angle of Incidence Lecture 9 Slide 17 Wood’s Anomalies Robert W. Wood observed rapid variations in the spectrum of light diffracted by gratings which he could not explain. Type 1 – Rayleigh Singularities Rapid variation in the amplitudes of the diffracted modes the correspond to the onset or disappearance of other diffracted modes. R. W. Wood, Phil. Mag. 4, 396 (1902) Lecture 9 Type 2 – Resonance Effects A resonance condition arising from leaky waves supported by the grating. Today, we call this guided‐mode resonance. Robert Williams Wood 1868 ‐ 1955 A. Hessel, A. A. Oliner, “A New Theory of Wood’s Anomalies on Optical Gratings,” Appl. Opt., Vol. 4, No. 10, 1275 (1965). 18 9 2/23/2016 Grating Cutoff Wavelength When m becomes imaginary, the mode is evanescent and cut off. Assuming normal incidence (i.e. inc = 0°), the grating equation reduces to n sin m m 0 x The first diffracted modes to appear are m = 1. The cutoff for the first‐order modes happens when (±1) = 90°. 1 90 sin 90 1 x To prevent the first‐order modes, we need 0 x n x 0 0 n To ensure we have first‐order modes, we need n x 0 n Lecture 9 Slide 19 Total Number of Diffracted Modes Given the grating period x and the wavelength 0, we can determine how many diffracted modes exist. Again, assuming normal incidence, sin m m0 navg x sin m m0 1 navg x Therefore, a maximum value for m is mmax navg x 0 The total number of possible diffracted modes M is then 2mmax+1 M Lecture 9 2navg x 0 1 Slide 20 10 2/23/2016 Determining Grating Cutoff Conditions Condition 0‐order mode No 1st‐order modes Ensure 1st‐order modes No 2nd‐order modes Ensure 2nd‐order modes No mth‐order modes Requirements Always exists Grating period must be shorter than what causes 1) = 90° Grating period must be larger than what causes 1) = 90° Grating period must be shorter than what causes 2) = 90° Grating period must be larger than what causes 2) = 90° Grating period must be shorter than what causes m) = 90° Ensure mth‐order modes Grating period must be larger than what causes m) = 90° Lecture 9 Slide 21 Three Modes of Operation for 1D Gratings Bragg Grating Diffraction Grating Long Period Grating Couples energy between counter‐propagating waves. Couples energy between waves at different angles. Couples energy between co‐ propagating waves. Applications • Thin film optical filters • Fiber optic gratings • Wavelength division multiplexing • Dielectric mirrors • Photonic crystal waveguides Lecture 9 Applications • Beam splitters • Patterned fanout gratings • Laser locking • Spectrometry • Sensing • Anti‐reflection • Frequency selective surfaces • Grating couplers Applications • Sensing • Directional coupling Slide 22 11 2/23/2016 Analysis of Diffraction Gratings Direction of the Diffracted Modes Diffraction Efficiency and Polarization of the Diffracted Modes We must obtain a rigorous solution to Maxwell’s equations to determine amplitude and polarization of the diffracted modes. E j H H j E E 0 H 0 n sin m ninc sin inc m 0 sin Lecture 9 Slide 23 Applications of Gratings Subwavelength Gratings Littrow Gratings Patterned Fanout Gratings Only the zero‐order modes may exist. Gratings in the littrow configuration are a spectrally selective retroreflector. Gratings diffract laser light to form images. Holograms Holograms are stored as gratings. Applications • Polarizers • Artificial birefringence • Form birefringence • Anti‐reflection • Effective index media Lecture 9 Applications • Sensors • Lasers Spectrometry Gratings separate broadband light into its component colors. Slide 24 12 2/23/2016 The Plane Wave Spectrum Periodic Functions Can Be Expanded into a Fourier Series x Waves in periodic structures obey Bloch’s equation E x, y A x e j r The envelop A(x) is periodic with period x so it can be expanded into a Fourier series. E x, y e j r S m; y e j 2 mx x m S m; y A x, y e j 2 mx x dx Lecture 9 Slide 26 13 2/23/2016 Rearrange the Fourier Series A periodic field can be expanded into a Fourier series. E x, y e j r S m; y e j 2 mx x m Let’s define the transverse wave vector component of the mth term. kx m x 2 m x This has the form of a sum of plane waves all at different angles So the field can be written as E x, y S m e jk x ,m x e E x, y j y y S m e jkm r m m Lecture 9 Slide 27 The Plane Wave Spectrum E x, y S m e j k x m x k y m y m We rearranged terms and saw that a periodic field can also be thought of as an infinite sum of plane waves at different angles. This is the “plane wave spectrum” of a field. Lecture 9 Slide 28 14 2/23/2016 Longitudinal Wave Vector Components of the Plane Wave Spectrum The wave incident on a grating can be written as Einc x, y E0 e j k xinc x k inc y y k xinc k0 ninc sin inc k yinc k0 ninc cos inc Phase matching into the grating leads to k xgrat m k xinc m 2 x m , 2, 1, 0,1, 2, Note: kx is always real. Each wave must satisfy the dispersion relation. k xgrat m k ygrat m k0 ngrat 2 k ygrat m 2 k n 0 grat 2 k xgrat m 2 We have two possible solutions here. 1. Purely real ky 2. Purely imaginary ky. 2 Lecture 9 Slide 29 Visualizing Phase Matching into the Grating The wave vector expansion for the first 11 modes can be visualized as… kinc k x 5 k x 4 k x 3 k x 2 k x 1 kx 0 k x 1 k x 2 k x 3 k x 4 k x 5 Each of these is phase matched into material 2. The longitudinal component of the wave vector is calculated using the dispersion relation in material 2. n1 n2 ky is imaginary. The field in material 2 is evanescent. ky is real. A wave propagates into material 2. Note: The “evanescent” fields in material 2 are not completely evanescent. They have a purely real kx so they do flow energy in the transverse direction. Lecture 9 ky is imaginary. The field in material 2 is evanescent. x y Slide 30 15 2/23/2016 Conclusions • Fields in periodic media take on the same periodicity as the media they are in. • Periodic fields can be expanded into a Fourier series. • Each term of the Fourier series represents a spatial harmonic (plane wave). • Since there are in infinite number of terms in the Fourier series, there are an infinite number of spatial harmonics. • Only a few of the spatial harmonics are propagating waves. Only these can carry energy away from a device. Lecture 9 Slide 31 Plane Wave Spectrum from Crossed Gratings 16 2/23/2016 Grating Terminology 1D grating Ruled grating 2D grating Crossed grating Requires a 2D simulation Requires a 3D simulation Lecture 9 Slide 33 Diffraction from Crossed Gratings Doubly‐periodic gratings, also called crossed gratings, can diffract waves into many directions. They are described by two grating vectors, Kx and Ky. Two boundary conditions are necessary here. k x m k x ,inc mK x m ..., 2, 1, 0,1, 2,... k y n k y ,inc nK y n ..., 2, 1, 0,1, 2,... 2 Kx xˆ x 2 Ky yˆ y Lecture 9 kinc Kx Ky Slide 34 17 2/23/2016 Visualizing the Transverse Wave Vector Expansion ky n kx m k tran m, n k x m xˆ k y n yˆ Lecture 9 Slide 35 Longitudinal Wave Vector Expansion The longitudinal components of the wave vectors are computed as k zref m, n k0 nref k x2 m k y2 n k ztrn m, n k0 ntrn k x2 m k y2 n The center few modes will have real kz’s. These correspond to propagating waves. The others will have imaginary kz’s and correspond to evanescent waves that do not transport energy. k tran m, n Lecture 9 k z m, n Slide 36 18 2/23/2016 Visualizing the Overall Wave Vector Expansion k tran m, n kref m, n k zref m, n k trn m, n Lecture 9 Slide 37 The Grating Spectrometer 19 2/23/2016 What is a Grating Spectrometer Diffraction Grating Separated Colors Input Light Lecture 9 39 Spectral Sensitivity We start with the grating equation. navg sin m ninc sin inc m 0 x We define spectral sensitivity as how much the diffracted angle changes with respect to wavelength . m 0 m m 0 x navg cos m m m 0 x navg cos m This equation tells us how to maximize sensitivity. 1. 2. 3. 4. Lecture 9 Diffract into higher order modes ( m). Use short period gratings ( x). Diffract into large angles ( m)). Diffract into air ( navg). 40 20 2/23/2016 Ocean Optics HR4000 Grating Spectrometer http://www.oceanoptics.com/Products/benchoptions_hr.asp Fiber Optic Input 1 SMA Connector 4 Collimating Mirror 7 Collection Lenses 2 Entrance Slit 5 Diffraction Grating 8 Detector Array 3 Optional Filter 6 Focusing Mirror 9 Optional Filter 10 Optional UV Detector Lecture 9 41 Littrow Gratings 21 2/23/2016 Littrow Configuration In the littrow configuration, the +1‐order reflected mode is parallel to the incident wave vector. This forms a spectrally selective mirror. Incident 0 +1 Lecture 9 43 Conditions for the Littrow Configuration The grating equation is n sin m n sin inc m 0 x The littrow configuration occurs when 1 inc The condition for the littrow configuration is found by substituting this into the grating equation. 2n sin inc Lecture 9 0 x 44 22 2/23/2016 Spectral Selectivity Typically only a cone of angles reflected from a grating is detected. We wish to find d/d by differentiating our last equation. d 0 2n x cos d Typically this is used to calculate the reflected bandwidth. 0 2n x cos f 2n x f 2 cos c0 Linewidth (optics and photonics) Bandwidth (RF and microwave) Lecture 9 45 Example (1 of 2) Design a metallic grating in air that is to be operated in the littrow configuration at 10 GHz at an angle of 45. Solution Right away, we know that n 1.0 inc 45 0 c0 3 108 ms 3.00 cm f 10 GHz The grating period is then found to be x Lecture 9 0 2n sin inc 3.00 cm 2.12 cm 2 1.0 sin 45 46 23 2/23/2016 Example (2 of 2) Solution continued Assuming a 5 cone of angles is detected upon reflection, the bandwidth is 2 1.0 2.12 cm 10 GHz cos 45 f 5 0.87 GHz 8 m 3 10 s 180 2 Lecture 9 47 Patterned Fanout Gratings 24 2/23/2016 Near-Field to Far-Field After propagating a long distance, the field within a plane tends toward the Fourier transform of the initial field. L FFT E x, y , 0 E x, y , L Lecture 9 49 What is a Patterned Fanout Grating? Diffraction grating forces the field to take on the profile of the inverse Fourier transform of an image. After propagating very far, the field takes on the profile of the image. Lecture 9 50 25 2/23/2016 Gerchberg-Saxton Algorithm: Initialization Near‐Field amplitude Far‐Field phase amplitude phase FFT Step 3 – Replace amplitude amplitude Step 4 – Calculate far‐field phase amplitude phase FFT-1 Step 2 – Calculate near‐field Step 1 – Start with desired far‐field image. Lecture 9 51 Gerchberg-Saxton Algorithm: Iteration Near‐Field amplitude Far‐Field phase amplitude phase FFT Step 7 – Replace amplitude amplitude Step 8 – Calculate far‐field phase FFT-1 Step 6 – Calculate near‐field Lecture 9 Step 5 – Replace amplitude with desired image. 52 26 2/23/2016 Gerchberg-Saxton Algorithm: End Near‐Field amplitude Far‐Field phase amplitude phase FFT This is the phase function of the diffractive optical element. This is what the final image will look like. After several dozen iterations… Lecture 9 53 The Final Fanout Grating Phase Function (rad) A surface relief pattern is etched into glass to induce the phase function onto the beam of light. We could also print an amplitude mask using a high resolution laser printer. Lecture 9 54 27 2/23/2016 Diffractive Optical Elements What is a Diffractive Optical Element Conventional Lens Diffractive Optical Lens (Fresnel Zone Plate) If the device is only required to operate over a narrow band, devices can be “flattened.” The flattened device is called a diffractive optical element (DOE). Lukas Chrostowski, “Optical gratings: Nano-engineered lenses,” Nature Photonics 4, 413-415 (2010). Lecture 9 56 28