lecture 3b -- diffraction gratings and the plane wave spectrum...kkn exy eek kkn phase matching into...

23
10/2/2019 1 Advanced Computation: Computational Electromagnetics Diffraction Gratings & the Plane Wave Spectrum Outline Fourier Series Diffraction from gratings The plane wave spectrum Plane wave spectrum for crossed gratings Power flow from gratings Slide 2 1 2

Upload: others

Post on 21-Apr-2020

2 views

Category:

Documents


0 download

TRANSCRIPT

Page 1: Lecture 3b -- Diffraction Gratings and the Plane Wave Spectrum...kkn Exy Eek kkn Phase matching into the grating leads to ,inc 2 xx x km k m m ,2,1,0,1,2, Each wave must satisfy the

10/2/2019

1

Advanced Computation:

Computational Electromagnetics

Diffraction Gratings &the Plane Wave Spectrum

Outline

• Fourier Series

• Diffraction from gratings

• The plane wave spectrum

• Plane wave spectrum for crossed gratings

• Power flow from gratings

Slide 2

1

2

Page 2: Lecture 3b -- Diffraction Gratings and the Plane Wave Spectrum...kkn Exy Eek kkn Phase matching into the grating leads to ,inc 2 xx x km k m m ,2,1,0,1,2, Each wave must satisfy the

10/2/2019

2

Slide 3

Fourier Series

Jean Baptiste Joseph Fourier

Born:

Died:

March 21, 1768in Yonne, France.

May 16, 1830in Paris, France.

1D Complex Fourier Series

Slide 4

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

2

22

2

1

x

x

x

x

pxj

pp

pxj

px

f x a e

a f x e dx

Typically, only a finite number of terms are retained in the expansion.

2

x

pxP j

pp P

f x a e

x

3

4

Page 3: Lecture 3b -- Diffraction Gratings and the Plane Wave Spectrum...kkn Exy Eek kkn Phase matching into the grating leads to ,inc 2 xx x km k m m ,2,1,0,1,2, Each wave must satisfy the

10/2/2019

3

2D Complex Fourier Series

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 p qp q A

f x y a e a f x y e dAA

y

x

3D Complex Fourier Series

Slide 6

For 3D periodic functions, the complex Fourier series generalizes to

2 2 2 2 2 2

, , , ,

1, , , ,x y z x y z

px qy ry px qy ryj j

p q r p q rp q r V

f x y z a e a f x y z e dVV

xy

z

5

6

Page 4: Lecture 3b -- Diffraction Gratings and the Plane Wave Spectrum...kkn Exy Eek kkn Phase matching into the grating leads to ,inc 2 xx x km k m m ,2,1,0,1,2, Each wave must satisfy the

10/2/2019

4

Slide 7

Diffraction from Gratings

Fields are Perturbed by Objects

Slide 8

A portion of the wave front is delayed after travelling through the dielectric object.

inck

7

8

Page 5: Lecture 3b -- Diffraction Gratings and the Plane Wave Spectrum...kkn Exy Eek kkn Phase matching into the grating leads to ,inc 2 xx x km k m m ,2,1,0,1,2, Each wave must satisfy the

10/2/2019

5

Fields in Periodic Structures

Slide 9

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

k

1,trnk

2,trnk

3,trnk

4,trnk 5,trnk

x

Diffraction from Gratings

Slide 10

The field is no longer a pure plane wave.  The grating “chops” the wave front and sends the power into multiple discrete directions that are called diffraction orders. 

9

10

Page 6: Lecture 3b -- Diffraction Gratings and the Plane Wave Spectrum...kkn Exy Eek kkn Phase matching into the grating leads to ,inc 2 xx x km k m m ,2,1,0,1,2, Each wave must satisfy the

10/2/2019

6

Grating Lobes

Slide 11

If we were to plot the power exiting a periodic structure as a function of angle, we would get the following.  The power peaks are called grating lobes or sometimes side lobes.  The power minimums are called nulls.

Grating Lobe Level

Main Beam (Zero‐order mode)

Diffraction Configurations

Slide 12

Planar Diffraction from a Ruled Grating

• Diffraction is confined within a plane• Numerically much simpler than other cases

• E and H modes are independent

transmitted

Conical Diffraction from a Ruled Grating

• Diffraction is no longer confined to a plane

• Almost same analytical complexity as crossed grating case, but simpler numerically

• E and H modes are coupled

transmitted

reflected

Conical Diffraction from a Crossed Grating with Planar Incidence

• Diffraction occurs in all directions

• Almost same numerical complexity as next case

• E and H modes are coupled transmitted

reflected

Conical Diffraction from a Crossed Grating

• Diffraction occurs in all directions• Most complicated case numerically

• E and H modes are coupled• Essentially the same as previous case

transmitted

incidentreflected

11

12

Page 7: Lecture 3b -- Diffraction Gratings and the Plane Wave Spectrum...kkn Exy Eek kkn Phase matching into the grating leads to ,inc 2 xx x km k m m ,2,1,0,1,2, Each wave must satisfy the

10/2/2019

7

Grating Equation for Planar Diffraction

Slide 13

The angles of the diffracted modes are related to the wavelength and grating period through the grating equation.

The grating equation only predicts the directionsof the modes, not how much power is in them.

Reflection Region

0trn inc incsin sinm

x

n n m

0ref inc incsin sinm

x

n n m

Transmission Region

ref incn n

trnn

x

Effect of Grating Periodicity

Slide 14

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

13

14

Page 8: Lecture 3b -- Diffraction Gratings and the Plane Wave Spectrum...kkn Exy Eek kkn Phase matching into the grating leads to ,inc 2 xx x km k m m ,2,1,0,1,2, Each wave must satisfy the

10/2/2019

8

Animation of Grating Diffraction (1 of 3)

Slide 15

Animation of Grating Diffraction (2 of 3)

Slide 16

15

16

Page 9: Lecture 3b -- Diffraction Gratings and the Plane Wave Spectrum...kkn Exy Eek kkn Phase matching into the grating leads to ,inc 2 xx x km k m m ,2,1,0,1,2, Each wave must satisfy the

10/2/2019

9

Animation of Grating Diffraction (3 of 3)

Slide 17

Slide 18

The Plane Wave Spectrum

17

18

Page 10: Lecture 3b -- Diffraction Gratings and the Plane Wave Spectrum...kkn Exy Eek kkn Phase matching into the grating leads to ,inc 2 xx x km k m m ,2,1,0,1,2, Each wave must satisfy the

10/2/2019

10

Periodic Functions Can Be Expanded into a Fourier Series

Slide 19

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

Rearrange the Fourier Series (1 of 2)

Slide 20

x A x

A periodic field can be expanded into a Fourier series.

, j rE x y A x e

2

2

2

x

x

yx xz

mxj

j r

m

mxj

j r

m

mxj

j yj x j z

m

S m e e

S m e e

S m e e e e

Here the plane wave term 𝑒 · ⃗ is brought inside of the summation.

19

20

Page 11: Lecture 3b -- Diffraction Gratings and the Plane Wave Spectrum...kkn Exy Eek kkn Phase matching into the grating leads to ,inc 2 xx x km k m m ,2,1,0,1,2, Each wave must satisfy the

10/2/2019

11

Rearrange the Fourier Series (2 of 2)

Slide 21

x A x

x can be combined with the last complex exponential.

2

, yx xz

mxj

j yj x j z

m

E x y S m e e e e

2

xy xz

mj x

j y j z

m

S m e e e

Now let                                and , ,, y m y z m zk k

,

2x m x

x

mk

, jk m r

m

E x y S m e

2ˆ ˆ ˆx x y y z z

x

mk m a a a

The Plane Wave Spectrum

Slide 22

, jk m r

m

E x y S m e

Terms were rearranged to show 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

21

22

Page 12: Lecture 3b -- Diffraction Gratings and the Plane Wave Spectrum...kkn Exy Eek kkn Phase matching into the grating leads to ,inc 2 xx x km k m m ,2,1,0,1,2, Each wave must satisfy the

10/2/2019

12

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

,incinc 0

,inc 0 inc inc

sin

0,

cos

x z

xj k x k z

y

z

k k n

kE 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 z

z x

k m k m k n

k m k n k m

There are two possible solutions here.1. Purely real kz

2. Purely imaginary kz.

Note: kx is always real.

incx

z

Visualizing Phase Matching into the Grating

Slide 24

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

kz is real.  A wave propagates into material 2.

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

kz 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

z

23

24

Page 13: Lecture 3b -- Diffraction Gratings and the Plane Wave Spectrum...kkn Exy Eek kkn Phase matching into the grating leads to ,inc 2 xx x km k m m ,2,1,0,1,2, Each wave must satisfy the

10/2/2019

13

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.

Slide 25

Slide 26

Plane Wave Spectrumfrom Crossed Gratings

25

26

Page 14: Lecture 3b -- Diffraction Gratings and the Plane Wave Spectrum...kkn Exy Eek kkn Phase matching into the grating leads to ,inc 2 xx x km k m m ,2,1,0,1,2, Each wave must satisfy the

10/2/2019

14

Grating Terminology

Slide 27

1D gratingRuled grating

Requires a 2D simulation

2D gratingCrossed grating

Requires a 3D simulation

Diffraction from Crossed Gratings

Slide 28

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

xx

yy

K x

K y

27

28

Page 15: Lecture 3b -- Diffraction Gratings and the Plane Wave Spectrum...kkn Exy Eek kkn Phase matching into the grating leads to ,inc 2 xx x km k m m ,2,1,0,1,2, Each wave must satisfy the

10/2/2019

15

Transverse Wave Vector Expansion (1 of 2)

Slide 29

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.

Transverse Wave Vector Expansion (2 of 2)

Slide 30

The vector expansions can be visualized this way…

xk m yk n

ˆ ˆ,t x yk m n k m x k n y

29

30

Page 16: Lecture 3b -- Diffraction Gratings and the Plane Wave Spectrum...kkn Exy Eek kkn Phase matching into the grating leads to ,inc 2 xx x km k m m ,2,1,0,1,2, Each wave must satisfy the

10/2/2019

16

Longitudinal Wave Vector Expansion (1 of 2)

Slide 31

The longitudinal components of the wave vectors are computed as

2 2 2,ref 0 ref

2 2 2,trn 0 trn

,

,

z x y

z x y

k m n k n k m k m

k m n k n k m k m

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.

,ref ,zk m n ,tk m n

% Compute Longitudinal Wave Vector Componentsk0 = 2*pi/lam;kzinc = k0*nref;kzref = -sqrt((k0*nref)^2 - kx.^2 - ky.^2);kztrn = +sqrt((k0*ntrn)^2 - kx.^2 - ky.^2);

Longitudinal Wave Vector Expansion (2 of 2)

Slide 32

The overall wave vector expansion can be visualized this way…

,ref ,zk m n ,tk m n

,ref ,zk m n ,trn ,zk m n

31

32

Page 17: Lecture 3b -- Diffraction Gratings and the Plane Wave Spectrum...kkn Exy Eek kkn Phase matching into the grating leads to ,inc 2 xx x km k m m ,2,1,0,1,2, Each wave must satisfy the

10/2/2019

17

Slide 33

Power Flow from Gratings

Electromagnetic Power Flow

Slide 34

The instantaneous direction and intensity of power flow at any point is given by the Poyntingvector.

, , ,r t E r t H r t

The RMS power flow is then

*1, Re , ,

2r E r H r

This is typically just written in the  frequency‐domain as

*1Re

2E H

33

34

Page 18: Lecture 3b -- Diffraction Gratings and the Plane Wave Spectrum...kkn Exy Eek kkn Phase matching into the grating leads to ,inc 2 xx x km k m m ,2,1,0,1,2, Each wave must satisfy the

10/2/2019

18

Wave Expressions

Slide 35

To obtain an expression for the RMS Poynting vector, start by write expression for the electric and magnetic fields.

00, ,jk r jk rk E

E r E e H r e

Substituting these into the definition of RMS Poynting vector gives

*

**

** *

0 00 0 *

2* * *0 0 0* *

2

0 2Im

1 1Re Re

2 2

1 1Re Re

2 2

Re2

jk r jk r jk r jk r

j k k rjk r jk r

k r

k E k EE e e E e e

e e eE k E E k

E ke

2 2* *

0 02Im 2Im

* *0 0 r 0 0 r

Re Re2 2

k r k rE Ek ke e

k k

Power Flow Away From Grating

Slide 36

To calculate the power flow away from the grating, it is only the z‐component of the Poynting vector that is of interest. 

2

0 2Im

0 0 r

Re2

k rE ke

k

z

z

x

x

Note: power travelling in the x direction does not transport power into or away from a device that is infinitely periodic along x.  This simplifying statement is not true for devices of finite extent.

2

0 2Im

0 0 r

Re2

zk z zz

E kz e

k

35

36

Page 19: Lecture 3b -- Diffraction Gratings and the Plane Wave Spectrum...kkn Exy Eek kkn Phase matching into the grating leads to ,inc 2 xx x km k m m ,2,1,0,1,2, Each wave must satisfy the

10/2/2019

19

RMS Power of the Diffracted Modes

Slide 37

Recall that the field scattered from a periodic structure can be decomposed into a Fourier series. 

The term 𝑆 𝑚 is the amplitude and polarization of the mth diffracted mode.  Therefore, power flow away from the grating due to the mth diffraction order is

, x zjk m x jk m z

m

E x z S m e e

2 20

2x

x

z x

mk m

k m k n k m

22

0 2Im2Im

0 0 r 0 0 r

Re , Re2 2

zz k m zk z zzz z

S mE k mkz e z m e

k k

Z Directed Power

Slide 38

From the previous equation, the power flow of the incident, reflected, and transmitted waves into the grating are written as

inc

2inc

2Imincinc

0 0 r,inc

Re2

zk z zz

S kz e

k

refm

ref,z m

ref,x m

trnm

trn,z m

trn,x m

inc

incz

incx

ref

2ref

2Imrefref

0 0 r,ref

, Re2

zk m z zz

S m k mz m e

k

trn

2trn

2Imtrntrn

0 0 r,trn

, Re2

zk m z zz

S m k mz m e

k

37

38

Page 20: Lecture 3b -- Diffraction Gratings and the Plane Wave Spectrum...kkn Exy Eek kkn Phase matching into the grating leads to ,inc 2 xx x km k m m ,2,1,0,1,2, Each wave must satisfy the

10/2/2019

20

Definition of Diffraction Efficiency

Slide 39

Diffraction efficiency is defined as the power in a specific diffraction order divided by the applied incident power. 

inc

DE z

z

mm

Despite the title “efficiency,” we don’t always want this quantity to be large.  We often want to adjust how much power gets diffracted into each mode.  So it is neither good nor bad to have high or low diffraction efficiency.  It depends on how the grating is being used.

Putting it All Together

Slide 40

So far, expressions were derived for the incident power and power in the diffraction orders.  Away from the edges of the grating, these are

inc

DE z

z

mm

The diffraction efficiency of the mth diffraction order was also defined as

It is now possible to derive expressions for the diffraction efficiencies of the diffraction orders by combining these expressions.

trn inc

2trntrn

2Imtrn r,trn

DE 2inc incr,incinc

Re

Rez zk m k z zz

z z

S m k mmT m e

kS

Note that: r,inc r,ref

inc ref

2refref

2Imref r,inc

DE 2inc incr,incinc

Re

Rez zk k m z zz

z z

S m k mmR m e

kS

inc

2inc

2Imincinc

0 0 r,inc

Re2

zk z zz

S kz e

k

ref

2ref

2Imrefref

0 0 r,ref

, Re2

zk m z zz

S m k mz m e

k

trn

2trn

2Imtrntrn

0 0 r,trn

, Re2

zk m z zz

S m k mz m e

k

39

40

Page 21: Lecture 3b -- Diffraction Gratings and the Plane Wave Spectrum...kkn Exy Eek kkn Phase matching into the grating leads to ,inc 2 xx x km k m m ,2,1,0,1,2, Each wave must satisfy the

10/2/2019

21

Diffraction Efficiency for Magnetic Fields

Slide 41

Diffraction efficiency equations were just derived from electric field quantities.

inc ref

trn inc

th

2refref

2Imref r,inc

DE 2inc incr,incinc

2trn

2Imtrn

DE 2inc

inc

Electric field amplitude of the diffraction order

Re

Re

Re

z z

z z

k k m z zz

z z

k m k z zz

z

S m m

S m k mmR m e

kS

S m kmT m e

S

trn

r,trn

incr,incRe z

m

k

Sometimes Maxwell’s equations in terms of the magnetic fields.  In this case, the diffraction efficiency equations are

inc ref

trn inc

th

2refref

2Imref r,inc

DE 2inc incr,incinc

2trn

2Imtrn

DE 2inc

inc

Magnetic field amplitude of the diffraction order

Re

Re

Re

z z

z z

k k m z zz

z z

k m k z zz

z

U m m

U m k mmR m e

kU

U m kmT m e

U

trn

r,trn

incr,incRe z

m

k

Reflectance, Transmittance, and Absorptance

Slide 42

The reflectance R is the sum of the diffraction efficiencies of the reflected modes.

The transmittance T is the sum of the diffraction efficiencies of the transmitted modes.

DEm

R R m

DEm

T T m

The absorptance A is the fraction of power absorbed by the device.

1A R T 0 materials have loss

0 materials have no loss

0 materials have gain

A

A

A

41

42

Page 22: Lecture 3b -- Diffraction Gratings and the Plane Wave Spectrum...kkn Exy Eek kkn Phase matching into the grating leads to ,inc 2 xx x km k m m ,2,1,0,1,2, Each wave must satisfy the

10/2/2019

22

Simplification for TMM

Slide 43

For TMM, the layers are homogeneous so there is no diffraction.  Therefore, only the m = 0 diffraction order exists.

inc ref

trn inc

th

2 refrefr,inc2Im 0ref

DE 2inc incr,incinc

2trn

2Im 0trn

DE 2inc

inc

0 Electric field amplitude of the 0 diffraction order

ˆRe 0000

ˆRe

0 Re00

z z

z z

zk k zz

z z

k k zz

z

S

kSR e

kS

ST e

S

trn

r,trn

incr,inc

0

Re

z

z

k

k

,ref ,incz zk k trn inc

2

ref

DE 2

inc

2trn

2Imtrn r,trn

DE 2 incr,incinc

Re

Rez zk k z z

z

ER R

E

E kT T e

kE

Since there is only one diffraction order,

inc inc

ref ref

trn trn

DE DE

DE DE

0

0

0

0

E S

E S

E S

R R

T T

Diffraction Efficiency Equations in Lossy LHI Materials Right Against Device (i.e. z = 0)

Slide 44

2ref

ref r,inc

DE 2 incr,incinc

2trn

trn r,trn

DE 2 incr,incinc

Re

Re

Re

Re

z

z

z

z

S m k mR m

kS

S m k mT m

kS

Equations for Multiple Diffraction Orders

TMM / Single Diffraction Order2 2

trnref trn r,trn

DE DE2 2 incr,incinc inc

Re

Re

z

z

E E kR R T T

kE E

2ref

ref r,inc

DE 2 incr,incinc

2trn

trn r,trn

DE 2 incr,incinc

Re

Re

Re

Re

z

z

z

z

U m k mR m

kU

U m k mT m

kU

2 2trn

ref trn r,trn

DE DE2 2 incr,incinc inc

Re

Re

z

z

H H kR R T T

kH H

DE DE m m

R R m T T m

43

44

Page 23: Lecture 3b -- Diffraction Gratings and the Plane Wave Spectrum...kkn Exy Eek kkn Phase matching into the grating leads to ,inc 2 xx x km k m m ,2,1,0,1,2, Each wave must satisfy the

10/2/2019

23

Diffraction Efficiency Equations in Lossless LHI Materials Right Against Device (i.e. z = 0)

Slide 45

2ref

ref

DE 2 inc

inc

2trn

trnr,incDE 2 inc

r,trn inc

Re

Re

Re

Re

z

z

z

z

S m k mR m

kS

S m k mT m

kS

Equations for Multiple Diffraction Orders

TMM / Single Diffraction Order2 2

trnref trnr,inc

DE DE2 2 incr,trninc inc

Re

Re

z

z

E E kR R T T

kE E

2ref

ref

DE 2 inc

inc

2trn

trnr,incDE 2 inc

r,trn inc

Re

Re

Re

Re

z

z

z

z

U m k mR m

kU

U m k mT m

kU

2 2trn

ref trnr,incDE DE2 2 inc

r,trninc inc

Re

Re

z

z

H H kR R T T

kH H

DE DE m m

R R m T T m

45