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ECE 5322 21st Century Electromagnetics

Instructor:Office:Phone:E‐Mail:

Dr. Raymond C. RumpfA‐337(915) 747‐6958rcrumpf@utep.edu

Diffraction Gratings

Lecture #9

Lecture 9 1

Lecture Outline

• 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

Lecture 9 Slide 2

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Fourier Series

Jean Baptiste Joseph Fourier

Born:

Died:

March 21, 1768in Yonne, France.

May 16, 1830in Paris, France.

1D Complex Fourier Series

Lecture 9 Slide 4

If a function f(x) is periodic with period x, it can be expanded into a complex Fourier series.

2

2 2

2

1

mxj

m

mxj

f x a m e

a m f x e dx

Typically, we retain only a finite number of terms in the expansion.

2 mxM j

m M

f x a m e

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2D Complex Fourier Series

Lecture 9 Slide 5

For 2D periodic functions, the complex Fourier series generalizes to

2 2 2 2

1, , , ,x y x y

px qy px qyj j

p q A

f x y a p q e a p q f x y e dAA

Diffraction from Gratings

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Fields in Periodic Structures

Lecture 9 Slide 7

Waves in periodic structures take on the same periodicity as their host.

k

Diffraction Orders

Lecture 9 Slide 8

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 Not Allowed Allowed

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inc incincr,avg

wave 1wave 2 wave 3

2 2

j k K r j k K rjk r A AA e e e

Field in a Periodic Structure

Lecture 9 9

r r,avg cosr K r

The dielectric function of a sinusoidal grating can be written as

A wave propagating through this grating takes on the same symmetry.

incjk rAE r er

incr,avg cos jk rA K r e

Grating Produces New Waves

Lecture 9 10

The applied wave splits into three waves.

inc

incinc

inc

jk r

j k K rjk r

j k K r

e

e e

e

Each of those splits into three waves as well.

inc

incinc

inc

jk r

j k K rjk r

j k K r

e

e e

e

inc

inc inc

inc

2

j k K r

j k K r j k K r

jk r

e

e e

e

inc

inc inc

inc 2

j k K r

j k K r jk r

j k K r

e

e e

e

And each of these split, and so on.

inc ,..., 2, 1,0,1,2,...,k m k mK m This equation describes 

the total set of allowed diffraction orders.

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Wave Incident on a Grating

Lecture 9 11

inc

K

Boundary conditions required the tangential component of the wave vector be continuous.

?

,trn ,incx xk kThe wave is entering a grating, so the phase matching condition is

,incx x xk m k mK The longitudinal vector component is calculated from the dispersion relation.

22 20 avgz xk m k n k m

For large m, kz,m can actually become imaginary.  This indicates that the highest diffraction‐orders are evanescent.

The Grating Equation

Lecture 9 12

The Grating Equation

0avg inc incsin sin sinn m n m

Note, this really is just

,incx x xk m k mK

Proof:

,inc

0 avg 0 inc inc

avg inc inc0 0

0avg inc inc

0avg inc inc

2sin sin

2 2 2sin sin

sin sin

sin sin sin

x x x

x

x

x

k m k mK

k n m k n m

n m n m

n m n m

n m n m

inc

K

inck

0

‐1

+1

+2

+10

‐1+2

x

z

incn

avgn

trnn

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Grating Equation in Different Regions

Lecture 9 Slide 13

The angles of the diffracted modes are related to the wavelength 0, refractive index, and grating period x through the grating equation.

The grating equation only predicts the directions of the modes, not how muchpower is in them.

Reflection Region

0trn inc incsin sin

x

n m n m

0ref inc incsin sin

x

n m n m

Transmission Region

ref incn n

trnn

x

Diffraction in Two Dimensions

• We know everything about the direction of diffracted waves just from the grating period.

• 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

Diffraction tends to occur along the lattice planes.

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Effect of Grating Periodicity

Lecture 9 Slide 15

0

avgx n

0 0

avg incxn n

0

incx n

0

incx n

SubwavelengthGrating

“Subwavelength”Grating

Low Order Grating High Order Grating

navg navg navg navg

ninc ninc ninc ninc

Animation of Grating Diffraction at Normal Incidence

Lecture 9 Slide 16

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Animation of Grating Diffraction at an Angle of Incidence

Lecture 9 Slide 17

Wood’s Anomalies

Lecture 9 18

Type 1 – Rayleigh Singularities Type 2 – Resonance Effects

Robert W. Wood observed rapid variations in the spectrum of light diffracted by gratings which he could not explain.  Later, Hesselexplained them an identified two different types.

Rapid variation in the amplitudes of the diffracted modes that correspond to the onset or disappearance of other diffracted modes.

A resonance condition arising from leaky waves supported by the grating.  Today, we call this guided‐mode resonance.

R. W. Wood, Phil. Mag. 4, 396 (1902)A. Hessel, A. A. Oliner, “A New Theory of Wood’s Anomalies on Optical Gratings,” Appl. Opt., Vol. 4, No. 10, 1275 (1965).

Robert Williams Wood1868 ‐ 1955

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Grating Cutoff Wavelength

Lecture 9 Slide 19

0sinx

n m m

When m becomes imaginary, the mth mode is evanescent and cut off.

Assuming normal incidence (i.e. inc = 0°), the grating equation reduces to

The first diffracted modes to appear are m = 1.

The cutoff for the first‐order modes happens when (±1) = 90°.

0

0

1 90

sin 90 1x

x

n

n

To prevent the first‐order modes, we need

0x n

To ensure we have first‐order modes, we need

0x n

or x

or x

Total Number of Diffracted Modes

Lecture 9 Slide 20

Given the grating period x and the wavelength 0, we can determine how many diffracted modes exist.

Again, assuming normal incidence, the grating equation becomes

0 0

avg avg

sin sin 1x x

m mm m

n n

avgmax

0

xnm

Therefore, a maximum value for m is 

The total number of possible diffracted modes M is then 2mmax+1

avg

0

21xn

M

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Determining Grating Cutoff Conditions

Lecture 9 Slide 21

Condition Requirements0‐order mode Always exists unless there is total‐internal reflection

No 1st‐order modes Grating period must be shorter than what causes1) = 90°

Ensure 1st‐order modes Grating period must be larger than what causes1) = 90°

No 2nd‐order modes Grating period must be shorter than what causes2) = 90°

Ensure 2nd‐order modes Grating period must be larger than what causes2) = 90°

No mth‐order modes Grating period must be shorter than what causesm) = 90°

Ensure mth‐order modes Grating period must be larger than what causesm) = 90°

Three Modes of Operation for 1D Gratings

Lecture 9 Slide 22

Bragg GratingCouples energy between counter‐propagating waves.

Diffraction GratingCouples energy between waves at different angles.

Long Period GratingCouples energy between co‐propagating waves.

Applications• Thin film optical filters• Fiber optic gratings• Wavelength division multiplexing

• Dielectric mirrors• Photonic crystal waveguides

Applications• Beam splitters• Patterned fanout gratings• Laser locking• Spectrometry• Sensing• Anti‐reflection• Frequency selective surfaces• Grating couplers

Applications• Sensing• Directional coupling

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Analysis of Diffraction Gratings

Lecture 9 Slide 23

Direction of the Diffracted Modes Diffraction Efficiency and Polarizationof the Diffracted Modes

0inc incsin sin sinn m n m

0

0

E j H

H j E

E

H

We must obtain a rigorous solution to Maxwell’s equations to determine amplitude and polarization of the diffracted modes.

Applications of Gratings

Lecture 9 Slide 24

Subwavelength GratingsOnly the zero‐order modes may exist.

Applications• Polarizers• Artificial birefringence• Form birefringence• Anti‐reflection• Effective index media

Littrow GratingsGratings in the littrowconfiguration are a spectrally selective retroreflector.

Applications• Sensors• Lasers

Patterned Fanout GratingsGratings diffract laser light to form images.

HologramsHolograms are stored as gratings.

SpectrometryGratings separate broadband light into its component colors.

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The Plane Wave Spectrum

Lecture 9 Slide 26

Periodic Functions Can Be Expanded into a Fourier Series

2

2

x

x

mxj

m

mxj

A x S m e

S m A x e dx

The envelop A(x) is periodic along x with period x so it can be expanded into a Fourier series.

Waves in periodic structures obey Bloch’s equation

, j rE x y A x e

x A x

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Lecture 9 Slide 27

Rearrange the Fourier Series (1 of 2)

A periodic field can be expanded into a Fourier series.

x

, j rE x y A x e

2

2

2

x

x

yx x

mxj

j r

m

mxj

j r

m

mxj

j yj x

m

S m e e

S m e e

S m e e e

Here the plane wave term ejxx is brought inside of the summation.

Lecture 9 Slide 28

Rearrange the Fourier Series (2 of 2)

x can be combined with the last complex exponential.

x

2

, yx x

mxj

j yj x

m

E x y S m e e e

2

xy x

mj x

j y

m

S m e e

Now let                                and y,m yk ,

2x m x

x

mk

, jk m r

m

E x y S m e

2

ˆ ˆx x y yx

mk m a a

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Lecture 9 Slide 29

The Plane Wave Spectrum

, jk m r

m

E x y S m e

We rearranged terms and now we see 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 periodic field.

FieldPlane Wave Spectrum

Lecture 9 Slide 30

Longitudinal Wave Vector Components of the Plane Wave Spectrum

The wave incident on a grating can be written as

,inc ,inc ,inc 0 inc inc

inc 0,inc 0 inc inc

sin,

cosx yj k x k y x

y

k k nE x y E e

k k n

Phase matching into the grating leads to

,inc

2x x

x

k m k m

, 2, 1,0,1, 2,m

Each wave must satisfy the dispersion relation.

22 20 grat

2 20 grat

x y

y x

k m k m k n

k m k n k m

We have two possible solutions here.1. Purely real ky

2. Purely imaginary ky.

Note: kx is always real.

inc

x

y

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Lecture 9 Slide 31

Visualizing Phase Matching into the Grating

The wave vector expansion for the first 11 modes can be visualized as…

inck

0xk 1xk 2xk 3xk 4xk 5xk 1xk 2xk 3xk 4xk 5xk

2n

1n

ky is real.  A wave propagates into material 2.

ky is imaginary.  The field in material 2 is evanescent.

ky is imaginary.  The field in material 2 is evanescent.

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.

Note:  The “evanescent” fields in material 2 are not completely evanescent.  They have a purely real kx so power flows in the transverse direction.

x

y

Lecture 9 Slide 32

Conclusions About the Plane Wave Spectrum• 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 actually propagating waves. Only these can carry power away from a device. Tunneling is an exception.

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Plane Wave Spectrum from

Crossed Gratings

Grating Terminology

Lecture 9 Slide 34

1D gratingRuled grating

Requires a 2D simulation

2D gratingCrossed grating

Requires a 3D simulation

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Diffraction from Crossed Gratings

Lecture 9 Slide 35

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.

,inc

,inc

..., 2, 1,0,1, 2,...

..., 2, 1,0,1,2,...

x x x

y y y

k m k mK m

k n k nK n

inck

xK

yK

2ˆ

2ˆ

xx

yy

K x

K y

Lecture 8 Slide 36

Transverse Wave Vector Expansion

Crossed gratings diffraction in two dimensions, x and y.

To quantify diffraction for crossed gratings, we must calculate an expansion for both kx and ky.

% TRANSVERSE WAVE VECTOR EXPANSIONM = [-floor(Nx/2):floor(Nx/2)]';N = [-floor(Ny/2):floor(Ny/2)]';kx = kxinc - 2*pi*M/Lx;ky = kyinc - 2*pi*N/Ly;[ky,kx] = meshgrid(ky,kx);

,inc

,inc

2 , , 2, 1,0,1, 2,

2 , , 2, 1,0,1, 2,

ˆ ˆ,

x xx

y yy

t x y

mk m k m

nk n k n

k m n k m x k n y

We will use this code for 2D PWEM, 3D RCWA, 3D FDTD, 3D FDFD, 3D MoL, and more.

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Visualizing the Transverse Wave Vector Expansion

Lecture 9 Slide 37

xk m yk n

tran ˆ ˆ, x yk m n k m x k n y

Longitudinal Wave Vector Expansion

Lecture 9 Slide 38

The longitudinal components of the wave vectors are computed as

ref 2 20 ref

trn 2 20 trn

,

,

z x y

z x y

k m n k n k m k n

k m n k n k m k 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 power.

,zk m n tran ,k m n

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Visualizing the Overall Wave Vector Expansion

Lecture 9 Slide 39

ref ,zk m n tran ,k m n

ref ,k m n trn ,k m n

The Grating Spectrometer

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What is a Grating Spectrometer

Lecture 9 41

Input Light

Separated Colors

Diffraction Grating

Spectral Sensitivity

Lecture 9 42

0avg inc incsin sin

x

n m n m

We start with the grating equation.

We define spectral sensitivity as how much the diffracted angle changes with respect to wavelength                     .

0 avg cosx m

m m

n

0m

This equation tells us how to maximize sensitivity.

1. Diffract into higher order modes ( m).2. Use short period gratings ( x).3. Diffract into large angles ( m)).4. Diffract into air ( navg).

0avg cosx

mm

n m

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Ocean Optics HR4000 Grating Spectrometer

Lecture 9 43

http://www.oceanoptics.com/Products/benchoptions_hr.asp

1 SMA Connector

2 Entrance Slit

3 Optional Filter

4 Collimating Mirror

5 Diffraction Grating

6 Focusing Mirror

7 Collection Lenses

8 Detector Array

9 Optional Filter

10 Optional UV Detector

Fiber Optic Input

Littrow Gratings

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Littrow Configuration

Lecture 9 45

Incident

0+1

In the littrow configuration, the +1‐order reflected mode is parallel to the incident wave vector.  This forms a spectrally selective mirror.

Conditions for the LittrowConfiguration

Lecture 9 46

0incsin sin

x

n m n m

The grating equation is

The littrow configuration occurs when

inc1

0inc2 sin

x

n

The condition for the littrow configuration is found by substituting this into the grating equation.

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Spectral Selectivity

Lecture 9 47

Typically only a cone of angles  reflected from a grating is detected.

We wish to find d/d by differentiating our last equation.

0 2 cosx

dn

d

Typically this is used to calculate the reflected bandwidth.

0 2 cosxn

2

0

2 cosxn ff

c

Linewidth (optics and photonics)

Bandwidth (RF and microwave)

Example (1 of 2)

Lecture 9 48

Design a metallic grating in air that is to be operated in the littrowconfiguration at 10 GHz at an angle of 45.

SolutionRight away, we know that

0

inc

3.00 cm2.12 cm

2 sin 2 1.0 sin 45x n

inc

80

0

1.0

45

3 10 3.00 cm

10 GHz

ms

n

c

f

The grating period is then found to be

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Example (2 of 2)

Lecture 9 49

Solution continuedAssuming a 5 cone of angles is detected upon reflection, the bandwidth is

2

8

2 1.0 2.12 cm 10 GHz cos 455 0.87 GHz

3 10 180ms

f

PatternedFanout Gratings

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Near-Field to Far-Field

, ,0E x y

, ,E x y L

L

FFT

After propagating a long distance, the field within a plane tends toward the Fourier transform of the initial field.

Lecture 9 51

What is a Patterned FanoutGrating?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 52

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Gerchberg-Saxton Algorithm:Initialization

Far‐Field

amplitude phase

Step 1 – Start with desired far‐field image.

Near‐Field

amplitude phase

FFT-1

Step 2 – Calculate near‐field

amplitude phase

Step 3 – Replace amplitude

amplitude phase

FFTStep 4 – Calculate far‐field

Lecture 9 53

Gerchberg-Saxton Algorithm:Iteration

Far‐FieldNear‐Field

amplitude phase

FFT-1

amplitude phase

FFT

amplitude phase

Step 6 – Calculate near‐field

Step 8 – Calculate far‐field

Step 5 – Replace amplitude with desired image.

Step 7 – Replace amplitude

Lecture 9 54

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Gerchberg-Saxton Algorithm:End

Far‐FieldNear‐Field

amplitude phase amplitude phase

FFT

This is what the final image will look like.

This is the phase function of the diffractive optical element.

After several dozen iterations…

Lecture 9 55

The Final Fanout Grating

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.

Phase Fu

nctio

n (rad

)

Lecture 9 56

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Diffractive Optical Elements

What is a Diffractive Optical Element

Lecture 9 58

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).