2/23/2016
ECE 5322 21st Century Electromagnetics
Instructor:
Office:
Phone:
E‐Mail:
Dr. Raymond C. Rumpf
A‐337
(915) 747‐6958
rcrumpf@utep.edu
Lecture #9
Diffraction Gratings
Lecture 9
1
Lecture Outline
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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
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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
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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
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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
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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  kinc  K r A  j  kinc  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.
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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 
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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
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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
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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).
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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    
m0
navg  x

sin   m   
m0
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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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 
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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
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Patterned
Fanout Gratings
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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 
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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.
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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.
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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
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Step 5 – Replace amplitude with desired image.
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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…
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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.
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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).
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