Lectures on classical optics

Part II

Andreas Hemmerich WS 2018 1

Contents!

•  Laterally confined light beams 3 •  Paraxial wave eqation 12 •  Gauss-Hermite beams 16 •  Kirchhoff integral 21 •  Generalized Kirchhoff integral 23 •  Babinet‘s principle 24 •  Fraunhofer diffraction 32 •  Fresnel diffraction 41 •  ABCD law for Gaussian beams 45 •  Resonators with Gaussian beams 53

2

Light beams

3

Plane waves: parallel infinitely extending wave fronts!Laterally localized beams: must diverge!

Assume laterally localized beam, which does not diverge!

Choose sufficiently small, such that the interference pattern in the intersection region exhibits a periodicity >> beam diameter L. Hence, the intersection region may be chosen to be completely dark, which is incompatible with energy conservation .

!

!S = 0

!

Distance between interference maxima . ! d = !

"Energy conservation requires divergence angle ! such that!

! L< d : " =

#d< $

and hence ! ! / L < "

!

"

4

wx(z) ! wx,0 1+2z

k wx,02

"

#$

%

&'

2

wx,0

tan(!x ) " limz#$

wx(z)z

= 2k wx,0 !x

bx ! k wx,02 confocal parameter

2 wx,0

An important class of confined beams: Gaussian beams

near field far field

! 2 wx,0

= !"wx,0

5

Superposition of plane waves

!(x,y,z) " 1#$%$&

d%d& e'i kz (%,&)z + %x + &y( ) e'

%$%

(

)*

+

,-2

e'

&$&

(

)*

+

,-2

.

paraxial approximation:! kz = k2 ! "2 ! #2 $ k !

12k

"2 + #2( )

=

e!ikz

"#$#%d$d% e!i($x+%y) e

!$#!$

&

'(

)

*+2

e!

%#!%

&

'(

)

*+2

,

1!!"2 =

1!"2 # i z

2k, 1!!$2 =

1!$2 # i z

2k

with!

=

!!"!!#$!"!#

e%ikz e%

x!!x

&

'(

)

*+2

e%

y!!y

&

'(

)

*+2

, !!x , 2!!"

, !!y , 2!!#

!"

6

with abbreviations!

wx,0 !

2"#

, wy,0 !2"$

Rx(z) ! z 1+k wx,0

2

2z

"

#$$

%

&''

2(

)

**

+

,

-- , Ry(z) ! z 1+

k wy,02

2z

"

#$$

%

&''

2(

)

***

+

,

---

! "(x,y,z) #wx,0wy,0

wxwy

e$i kz$1

2%x+%y( ) +k

2x2

Rx

+y2

Ry

&

'((

)

*++

,

-

.

.

/

0

11

e$

x2

wx2+

y2

wy2

&

'

((

)

*

++

Guoy phases Louis Georges Guoy 1854-1926

curvature radii!

beam diameters!

wx(z) ! wx,0 1+2z

k wx,02

"

#$

%

&'

2

, wy(z) ! wy,0 1+2z

k wy,02

"

#$

%

&'

2

!x (z) " arctan2zkwx,0

2

#

$%%

&

'((, !y(z) " arctan

2zkwy,0

2

#

$%%

&

'((

7

Proof

A.

!"2

!!"2= 1# i!"2 z

2k= 1# i

4wx,0

2

z2k

!!"2

!"2 =

1+ i4

wx,02

z2k

1+4

wx,02

z2k

$

%&

'

()

2 =1

1+4

wx,02

z2k

$

%&

'

()

2

1+ i4

wx,02

z2k

1+4

wx,02

z2k

$

%&

'

()

2=

wx,0

wx(z)ei*x (z)

!!"!"

=wx,0

wx(z)e

i12*x (z)

!!"!"

=wx,0

wx(z)ei 12#x (z)

8

!!"2 = !"2

1+ i4

wx,02

z2k

1+4

wx,02

z2k

#

$%

&

'(

2 =4

wx2 +

i2k

zwx,0

2

2kz

#

$%

&

'(

2

+1)

*

++

,

-

.

.

=4

wx2 + i

2kRx

B. e

!x"!x

#

$%&

'(

2

= e!x2 1

wx2+ i

k2Rx

)

*++

,

-..

e!

x"!x

#

$%

&

'(2

= e!"!)x

2

#

$%%

&

'((

2

= e!"!)x

2

#

$%%

&

'((

2

= e!

x2

#

$%&

'(2

4

wx2+i 2k

Rx

*

+,,

-

.//= e

!x2 1

wx2

+ i k2Rx

*

+,,

-

.//

9

Hypersurfaces of constant phase are spherical

C : (x,z){ | z ! (z0 !Rx )( )2

+x2 = Rx2}

Rx(z) = z+ 1z

bx

2

!

"#

$

%&

2'

(

))

*

+

,,, bx - k wx,0

2Relative minimum of

0 = !!z

Rx(z) " zmin =bx

2 , Rx,min(z) = bx

!(x,y,z) " e

#i kz +k2

x2

Rx

+y2

Ry

$

%&&

'

())

*

+

,,

-

.

//, z(x,y) + 1

2x2

Rx(z0)+

y2

Ry(z0)

$

%&&

'

()) = z0 0 z(0,0)

z ! (z0 !Rx )( )2+x2 =Rx

2

" z ! z0( )2+ 2 z ! z0( )Rx +x2 =0

"z0#z! 2 z ! z0( )Rx +x2 =0 " k z+ kx2

2Rx

= k z0

Areas of constant phase:

!"#$%&

$"#$%&

'

($)!*

(+)!+*

(+)+*

,$-.-!

(+)!+",$*

10

Gaussian beam parameters

2!"(z) # 2

!arctan 2z

b$

%&'

()

w(z)w0

= 1+ 2zb

!

"#$

%&

2

Rmin

R(z)=

4 z b4z2 + b2

�2 �1 1 2

�1.0

�0.5

0.5

1.0

zb

Near field!

beam radius!

Curvature!

Guoy phase!

!

"

11

Gaussian beam: beam radius and curvature radius

w(z)

R(z)

Near field!

beam radius!

curvature radius!

12

Paraxial wave equation

Helmholtz equation ! + k2( ) "(x,y,z) = 0 , "(x,y,z) # u(x,y,z) e$ ik z

!

"2u"x2 +

"2u"y2 +

"2u"z2 #2ik

"u"z

= 0

paraxial approximation:

!2u!z2 ! k

!u!z

,!2u!x2 ,

!2u!y2

!

"2u"x2 +

"2u"y2 # 2ik

"u"z

= 0 paraxial wave equation

Analogous equation in quantum mechanics: Schrödinger equation

!2u!x2 +

!2u!y2 " i!

!u!t

= 0

13

Paraxial spherical waves

Spherical wave

! "

e# ik!r #!r0

!r #!r0

expand

is an exact solution to wave equation

!r !!r0 = z!z0 +

x! x0( )2 + y!y0( )2

2 z!z0( )+ "

!"#$%&

$"#$%&

'$()(!*

'$+()+(!+*

! " e# ik z#z0( ) u(x,y,z) , u(x,y,z) =

1z#z0

e# ik

x# x0( )2+ y#y0( )22 z#z0( )

is an exact solution

of the paraxial wave equation (for arbitrary complex source points x0, y0, z0)

Generalization for different curvature radii in x- and y-directions

u(x,y,z) =1

z!zx z!zy

e! ik

x! x0( )22 z!zx( ) e

! iky!y0( )2

2 z!zy( )

14

Set and choose complex source ccordinates x0 = y0 = 0

zx = zx,0 ! i

k wx,02

2, zy = zy,0 ! i

k wy,02

2

define q-parameters q!" z#z

!= z#z

!,0 + ik w

!,02

2, ! ${x,y}

! u(x,y,z) =1

qx qy

e" ik

x2

2qx e" ik

y2

2qy Astigmatic Gauss mode

1q!

=1

z " z!

=1

R!(z"z

!,0 )" i 2

k w!(z"z

!,0 )2 ="2ik w

!,02

w!,0

w!(z"z

!,0 )ei#! (z"z!,0 )use

1

z ! z"

e! ik

x2

2q" =!2ik w

",02

w",0

w"(z!z

",0 )e

i12#" (z!z",0 )

e!

x2

w" (z!z",0 )2

e! ik

x2

2R" (z!z",0 )

to show that

Résumé: Gauss beams arise as paraxial spherical waves with complex source coordinates

(*)

15

1q!

=1

z"z!,0 + i

k w!,0

2

2

=z"z

!,0 " ik w

!,02

2

z"z!,0( )2

+k w

!,02

2

#

$%%

&

'((

2

=z"z

!,0

z"z!,0( )2

+k w

!,02

2

#

$%%

&

'((

2 "ik w

!,02

2

z"z!,0( )2

+k w

!,02

2

#

$%%

&

'((

2 =1

R!(z"z

!,0 )" i 2

k w!(z"z

!,0 )2

= "ii z"z

!,0( )

z"z!,0( )2

+k w

!,02

2

#

$%%

&

'((

2 +

k w!,0

2

2

z"z!,0( )2

+k w

!,02

2

#

$%%

&

'((

2

)

*

++++++

,

-

.

.

.

.

.

.

= " i 1

z"z!,0( )2

+k w

!,02

2

#

$%%

&

'((

2 ei/! (z"z!,0 )

="2i

k w!,0

2

w!,0

w!(z"z

!,0 )ei/! (z"z!,0 )

(*)

16

! 0 ="2f"x2

# 2ik "f"z

$

%&

'

() =

f(x,z)

p(z)2H xp(z)$

%&

'

()

i

H'' xp(z)$

%&

'

() # 2ik x

p(z)q(z)

#p'(z)$

%&

'

()H'

xp(z)$

%&

'

()#ik

p(z)2

q(z)H xp(z)$

%&

'

() 1+2q(z)q'(z)

A '(q(z))A(q(z))

+ i k x2

q(z)q'(z)#1( )

$

%&

'

()

*

+,,

-

.//

Ansatz

Basis for paraxial wave equation (Cartesian coordinates)

u(x,y,z) ! f(x,z) g(y,z) and f, g satisfy"2 f"x2 # 2ik

"f"z

= 0

$"2u"x2 +

"2u"y2 # 2ik

"u"z

= 0

f(x,z) ! A(q(z)) "H x

p(z)#

$%&

'(" e

) i k x2

2q(z)

In the following we provide a basis system of orthonormal Gauss Hermite modes such that each solution of the paraxial wave equation arises as a superposition of basis members

17

p '(z) = p(z)q(z)

+i

kp(z)

A '(q(z))q'(z) = A(q(z)) i nkp(z)2

!1

2q(z)! i k x2

2q(z)2q'(z)!1( )

"

#$

%

&'

H satifies the Hermitian differential equation Hn '' xp!

"#$

%& '

2xp

Hn ' xp!

"#$

%& + 2nHn

xp!

"#$

%& = 0

!

Define by the relations q(z), p(z), R(z), w(z)

It follows that

q'(z) = 1

p'(z) = p(z)q(z)

+i

kp(z)

(1)

(2)

(3)

(4)

q(z) = z ! z0 + q0 , q0 = ikw0

2

21

q(z)"

1R(z)

! i 2kw(z)2

p(z) " 12

w(z)

18

It follows with (5), (3) and (2) that

p(z) = 2

ikqq!

q! " q(5) With (4) one gets

A '(q(z))

A(q(z))=

inkp(z)2 !

12q(z)

=n

2q"(z)!

n+12q(z)

and hence

Together

fn(x,z) ! 2"

#

$%&

'(

1/41

2nn!wx,0

qx,0

qx(z)#

$%

&

'(

1/2qx,0qx

)(z)qx,0) qx(z)

#

$%

&

'(

n/2

Hn

2 xwx(z)

#

$%

&

'( * e

+ i kx2

2qx (z)

Normalization

dxdz fn(x,z) fm!(x,z)" = #nm

!nm(x,y,z) = fn(x,z) fm(y,z) e" ikz

d ln(A(q(z))) = n

2q!(z)dz " n+1

2q(z)dz =

n2q!(z)

dq! "n+1

2q(z)dq

= d n

2ln(q!(z)) " n+1

2ln(q!(z))

#

$%

&

'( = d ln q!n/2

q n+1( )/2

)

*++

,

-..

#

$%%

&

'((

/ A = A0q!n/2

q(n+1)/2

19

Various Gauss-modes

qx,0

qx(z)

!

"#

$

%&

1/2qx,0 qx

'(z)

qx,0' qx(z)

!

"#

$

%&

n/2

=w

(,0

w((z)z

(,0 )ei(n+1/2)*( (z)z(,0 )

! fn(x,z) " 2#

$

%&'

()

1/41

2nn!1

wx(z*zx,0)ei(n+1/2)+x (z*zx,0 ) Hn

2 xwx(z)

$

%&

'

() , e

* i k x2

2qx (z)

TEM00! TEM01! TEM02! TEM10! TEM11! TEM12!

zx,0 !zy,0 " astigmatic beam

wx,0 !wy,0 " elliptic beam

20

Diffraction Outline of problem: A volume V2 is bounded by the surfaces A1 and A2. A1 may, for example, represent a screen with some apertures. The electric field, originating from sources inside the volume V1, is known on A1 and on A2. Determine field at point P We apply a scalar theory: only the electric field interacts with the environment in a polarization-independent way. The magnetic field follows the electric field.

•  good approximation for optical frequencies •  not true in the microwave domain

!

"#"$

%&'()*+

,#

,$

21

Derivation of Kirchhoff‘s Integral !

!V

n̂ !P

!+k2( ) "(!x) = 0

!+k2( ) G(!x,!p) = #(

!x$!p)

! "(

!p) = "

##n

G$G ##n

"%

&'(

)*#V+ dA

!!"G#G

!"!

(1)

Proof !(!p) = !"G#G"!( )

V$ dV = !

%%nG#G %

%n!

&

'(

)

*+

%V$ dA

(a) (b)

(a) follows with !+k2( ) "(

!x) = 0, !+k2( ) G(

!x,!p) = #(

!x$!p)

(b) use Gauß-theorem for vector field

22

The Greens-function for empty space: G(!x,!p) ! "

eik!x"!p

4#!x "!p

satifies !+k2( ) G(

!x,!p) = "(

!x#!p)

Proof

!!G = " ik eik

!x"!p

4#!x "!p

!!!x "!p "

eik!x"!p

4#

!!

1!x "!p= " ik eik

!x"!p

4#

!x "!p( )

!x "!p

2"

eik!x"!p

4#

!!

1!x "!p

!!!x "!p =!x "!p

!x "!p

,!!

!x "!p( )

!x "!p

2=

1!x "!p

2,!!

1!x "!p="!!!x "!p

!x "!p

2,!!eik

!x"!p= ikeik

!x"!p !!!x "!p

!

with !

1!x "!p

= " 4# $!x "!p( ), f(

!x)$!x( ) = f(0)$

!x( ) obtain !G = "k2G(

!x,!p) + $

!x "!p( )

(2)

!G =k2eik

!x"!p

4#

!$!x "!p( )!x "!p( )

!x "!p

2" ik eik

!x"!p

4#

!$

!x "!p( )

!x "!p

2" ik eik

!x"!p

4#

!$!x "!p( )!$

1!x "!p"

eik!x"!p

4#!

1!x "!p

=k2eik

!x"!p

4#1!x "!p" ik eik

!x"!p

4#1!x "!p

2+ ik eik

!x"!p

4#1!x "!p

2"

eik!x"!p

4#!

1!x "!p

23

Kirchhoff‘s integral

!(!p) = " 1

4#dA n̂ eikR

RR̂ ik " 1

R$

%&'

()! "!*!

$

%&'

()+V,

!R !!x "!p , R !

!x "!p , R̂ !

!R / R

n̂!!G = "

14#

n̂eik

!x"!p

!x "!p

!x "!p

!x "!p

ik"1!x "!p

$

%&

'

() = "

14#

n̂eikR

RR̂ ik "

1R

$

%&'

()

Proof

(3)

use (1) and (2) ! !(!p) = !

""n

G#G""n

!$

%&'

()"V* dA , G(

!x,!p) + #

eik!x#!p

4,!x #!p

24

Generalized Kirchhoff Integral

!+k2( ) "(!x) = 0

!+k2( ) G(!x,!p) = #(

!x$!p)

!k%!&S(!x), k%

!k , 2

!&

1!x $!p

!&S(!x) + 1

!x $!p!S(!x) ' 0

Use Green-function for empty space: G(!x,!p) ! " e

i S(!x)"S(

!p)( )

4#!x "!p

to obtain generalized Kirchhoff‘s inegral

!(!p) = " 1

4#dA n̂ ei S(

!x)"S(

!p)( )

Ri!$S(!x)" R̂

R

%

&''

(

)**!"

!$!

%

&''

(

)**

+V,

!

!V

n̂ !P

!k=!!S(!x)

(5)

(4)

Assume inhomogeneous medium inside the volume V

Note that (*) holds if

(*)

S(!x)!S(

!p) " k

!x !!p ,!k=!#S(!x) = k

!x !!p

!x !!p

25

!!G(!x,!p) = "

!!ei S(

!x)"S(

!p)( )

4#!x "!p

= "ei S(

!x)"S(

!p)( )

4#!x "!pi!!S(!x)" e

i S(!x)"S(

!p)( )

4#

!!

1!x "!p= "ei S(

!x)"S(

!p)( )

4#

!!

1!x "!p+i!!S(!x)

!x "!p

$

%

&&

'

(

))

!G(!x,!p) = "

!#ei S(

!x)"S(

!p)( )

4$!#

1!x "!p+i!#S(!x)

!x "!p

%

&

''

(

)

**"ei S(

!x)"S(

!p)( )

4$!#!#

1!x "!p+i!#S(!x)

!x "!p

%

&

''

(

)

**

= "ei S(

!x)"S(

!p)( )

4$i!#S(!x "!p)!#

1!x "!p+i!#S(!x)

!x "!p

%

&

''

(

)

**"ei S(

!x)"S(

!p)( )

4$!

1!x "!p

+ i!#

!#S(!x)

!x "!p

%

&

''

(

)

**

= "ei S(

!x)"S(

!p)( )

4$i!#S(!x "!p)!#

1!x "!p+i!#S(!x)

!x "!p

%

&

''

(

)

**"ei S(

!x)"S(

!p)( )

4$!

1!x "!p

+ i!S(!x)

!x "!p

+ i!#S(!x)!#

1!x "!p

%

&

''

(

)

**

= "ei S(

!x)"S(

!p)( )

4$i2!#S(!x)!#

1!x "!p+ i!S(

!x)

!x "!p+!

1!x "!p"

!#S(!x)!#S(!x)

!x "!p

+

,

--

.

/

00

use 2!!1R

!!S(!x)+ 1

R"S(!x) # 0 and "

1R=$ 4%&(R)

= !ei S(

!x)!S(

!p)( )

4"#

1!x !!p!

!$S(!x)!$S(!x)

!x !!p

%

&

''

(

)

**= !ei S(

!x)!S(

!p)( )

4"!4"+(

!x !!p)!!$S(!x)!$S(!x)

!x !!p

%

&

''

(

)

**= +(

!x !!p)!G(

!x,!p)!$S(!x)

2%&'

()*

A. Showing that : !+!"S

2#$

%& G(!x,!p) = '(

!x(!p)

Proof

26

B. use (1), (4) , A. with !

!!G(!x,!p) = "

ei S(!x)"S(

!p)( )

4#!x "!p

i!!S(!x) "

!x "!p

!x "!p

$

%&

'

()

!(!p) = "

14#

dA n̂ei S(

!x)"S(

!p)( )

Ri!$S(!x)"

R̂R

%

&'

(

)* ! "

!$!

%

&'

(

)*

+V,to obtain !

Gustav Robert Kirchhoff 1824-1887

27

Assume that

A1 is a screen with apertures B.

A2 is pushed to infinity

In accordance with Sommerfeld‘s

radiation condition no waves come

in from infinity, hence

!"#$%&'

()

(*

+ !P

!R

!R!"

! |A2

=""n

! |A2= 0

Kirchhoff‘s boundary conditions

! |A =

""n

! |A = 0

! |B =

""n

! |BK2 determined by sources

K1

!(!p) = " 1

4#dA n̂ e

ikR

RR̂ ik " 1

R$

%&

'

()!"

!*!

$

%&&

'

())

B+Diffraction integral

Note: K1, K2 are not compatible with Helmholtz equation. Improved solution: choose adequate Green functions for Dirichlet or von Neumann boundary conditions

28

!1(!p) + !2(

!p) = !Source(

!p)

Babinet`s principle

!"#$%&'

(

!"#$%&'

(

!1

!2 !Source

!Source

Jacques Babinet 1794-1872

29

Special case: point source & flat screen

n̂

!P!

P' !R'

!R

!! 'd

!R' !

!x " !P'

!R ! !x " !P

= !1

4"dA eikReikR'

R R'ik ! 1

R#

$%

&

'(n̂R̂ ! ik ! 1

R'#

$%

&

'(n̂R̂ '

#

$%%

&

'((

apertures)

!source(!x) = e

ikR'

R '

!(!P) = " 1

4#dA n̂ e

ikR

R

!RRik " 1

R

$

%&

'

()!source(

!x)"!*!source(

!x)

$

%&&

'

())

apertures+

!

1i"

QR R'

dA eik R+R'( )

apertures# , Q $

12

n̂ R̂% R̂'( ) = 12

cos(&)+ cos(& ')( )

R, R' > d , R, R' >> !

30

Generalized case: point source & flat screen around

n̂

!P!

P' !R'

!R

!! 'd

!R' !

!x " !P'

!R ! !x " !P

= !14"

dA #source(!x) e

i S(!x)!S(

!p)( )

Rn̂ i!$S(!x)! R̂

R

%

&'

(

)*! ik ! 1

R'

%

&'

(

)*n̂R̂ '

%

&''

(

)**

apertures+

!source(!x) = e

ikR'

R '

!(!P) = " 1

4#dA n̂ e

i S(!x)"S(

!p)( )

Ri!$S(!x)" R̂

R

%

&'

(

)*!source "

!$!source

%

&''

(

)**

apertures+

!

1i"

QR

dA#source(!x) ei S(

!x)$S(

!p)( )

apertures% , Q &

12

n̂ 1k

!'S(0)$ R̂'

(

)*+

,-=

12

cos(.)+ cos(. ')( )

R, R' > d , R, R' >> !

S(!x)

!x=0

31

Expansion of R and R‘

RP=

!x ! !P

P = 1 ! P̂

!P

!x + 1

21

P2

!x2 ! P̂

!x( )

2"

#$

%

&' + 1

2P̂!x

P3

!x2 ! P̂

!x( )

2"

#$

%

&' + "

! dP

! dP"

#$

%

&'

2

! dP"

#$

%

&'

3

Fresnel limit

RP! 1 " P̂

!P

!x Fraunhofer limit

P̂!x ! 0 n̂

!P!

P' !R'

!R

!! 'd

(x1,x2)!plane

(x1,x2,x3) = 0

R, R' >> d , R, R' >> !

RP

! 1 + 12P2

!x2 " P̂

!x( )

2#

$%

&

'(

32

Fraunhofer diffraction

!(!P) = k

2" iQ

R R' dA eik R+R'( )

apertures# , Q $ 1

2n̂ R̂ % R̂ '( )

= k2" i

Q R R'

dA &(x1,x2) eik R+R'( )

%'

'

#%'

'

# ( k Q2" i

eik P+P'( ) P P'

dA &(x1,x2) e%ik P̂+P̂ '( )!x

%'

'

#%'

'

#

= k Q2" i

eik P+P'( ) P P'

dx1 dx2 &(x1,x2) e%ik

P1P+

P1'P'

)

*++

,

-..x1 +

P2P+

P2 'P'

)

*++

,

-..x2

/

011

2

344

%'

'

#%'

'

#

RP

! 1 " P̂!P

!x

eik R+R'( ) ! eik P+P'( ) " eik P̂+P̂ '( ) !x n̂

!P!

P' !R'

!R

!! 'd

(x1,x2)!plane

(x1,x2,x3) = 0

R, R' >> d , R, R' >> !

!(!x) transmission function describes the apertures

Joseph von Fraunhofer 1787-1826

33

Fraunhofer diffraction

!(!P) =

kn̂ R̂ " R̂ '( ) 4# i

eik P+P'( ) P P'

dx1 dx2 $(x1,x2) e"ik

P1P+

P1'P'

%

&''

(

)**x1 +

P2P+

P2 'P'

%

&''

(

)**x2

+

,--

.

/00

"1

1

2"1

1

2

! "(!P) # Fourier $ transform of %(

!x)

Experimental realisation:

!

!

"

"

R, R' >> d , R, R' >> !

= kn̂ R̂ ! R̂ '( )

4" ieik P+P'( )

P P' dx1 dx2 #(

!x) e

!ik sin($1)+sin($1')( )x1 + sin($2 )+sin($2 ')( )x2%&

'(

!)

)

*!)

)

*

P!

P= sin("

!) ,

P!'

P'= sin("

!')

34

Fraunhofer diffraction for rectangular aperture

!(!P) =

kn̂ R̂ " R̂ '( ) 4# i

eik P+P'( ) P P'

dx1 dx2 $(!x) e

"ik sin(%1)+sin(%1')( )x1 + sin(%2 )+sin(%2 ')( )x2&'

()

"a2

a2

*"a1

a1

*

Intensity:

= n̂ R̂ ! R̂ '( )eik P+P'( )

" i P P'

sin k sin(#1)+ sin(#1 ')( )a1$%

&'

k sin(#1)+ sin(#1 ')( ) sin k sin(#2)+ sin(#2 ')( )a2$

%&'

k sin(#2)+ sin(#2 ')( )

!1 ' = !2 ' = 0 " I = I0sin2 ksin(!1) a1

#$ %&k2 sin2(!1)

sin2 ksin(!2) a2

#$ %&k2 sin2(!2)

minima ! sin("1) a1 = n #2

or sin("2) a2 = n #2

2a1

2a2

35

Fraunhofer diffraction for circular aperture

!(!P) " r dr d# '

0

2$

%0

a

% e&ikr sin(') cos(#&# ') = r dr d# '0

2$

%0

a

% e&ikr sin(') cos(# ')

= r dr J0(&kr sin(')) 0

a

% = 2$k2 sin2(')

dx x J0(x) 0

kasin(')

%

=2$

k2 sin2(') dx d

dxx J1(x) ( )

0

kasin(')

% = 2$aksin(')

J1(kasin('))

Intensity:

2a

ddx

x J1(x) ( ) = x J0(x) , J0(x) = 12!

d" eixcos(") 0

2!

#use

Px

P= sin(!)cos(") ,

Py

P= sin(!)sin(") , Px ' =Py ' = 0

!P

!P'

I = I0 2J1(kasin(!))

kasin(!)

"

#$

%

&'

2

36

Diffraction at a transmission grating

Maxima: a!" integer

! " FT[#](k$) , # = f % g , $ & (sin(')+ sin(' '))

! " FT(f # g) = FT(f) $FT(f) " sinc % b&'

(

)*

+

,- e

i n 2%a&'

n=0

N.1

/

! sinc " b#$

%

&'

(

)*

sin N " a#$

%

&'

(

)*

sin " a#$

%

&'

(

)*

!

"f(x) ! 1 if x " [#b

2,b2

]

0 otherwise

$

%&

'&

, g(x) ! ((x #na)n=0

N#1

)

!

!2

a!= 50 , b

!= 8 , N = 10

�0.03 �0.02 �0.01 0.00 0.01 0.02 0.03envelope

�0.3 �0.2 �0.1 0.0 0.1 0.2 0.3

37

Diffraction at a reflective surface

u'+ v ' = x + w '( ) sin(! ')

v ' = f(x)cos(! ')

, w ' = f(x) tan(! ')

! u' = x sin(" ') + f(x) tan(" ')sin(" ') # 1cos(" ')

$

%&

'

() = x sin(" ') # f(x)cos(" ')

u = x sin(!) + f(x) cos(!)

(1)

(2)

(3)

(1),(3)

! eik u'"u( ) = eik x #"f (x) $%& '( , # ) sin(* ')" sin(*) , $ ) cos(* ')+ cos(*)

! " # dx eik x $%f (x) &'( )* + # FT[e% i k f (x) & ]

k$|

(4)

(2),(4) (5)

!

! "

#

$%&'

&()&

38

Diffraction at a reflective grating

f(x+na)= f(x)Assume periodic function

! " dx eik x #$f (x) %&' ()

0

Na

* = dx eik x#$f (x) %&' ()

na

n+1( )a

*n=0

N$1

+

= dx ' eik x '+na( )!"f (x '+na) #$% &'

0

a

(n=0

N"1

) = eikna!

n=0

N"1

)*

+,-

./0 dx ' eik x '!"f (x ') #$% &'

0

a

(

x = x '+nasubstitue

!

"#!$

%

= ei(N!1)"

a#$

sin N " a#$

%

&'

(

)*

sin " a#$

%

&'

(

)*

+ dx eik x$!f (x ) ,%& ()

0

a

- envelope

! " eiN# a

$%e& ikf0'

sin N # a$%

(

)*

+

,-

#$%

Trivial case: flat surface

f(x)= f0 ! dx eik x"#f (x ) $%& '(

0

a

) = e# ikf0$ei*

a+"

sin(*a+")

*+"

,

specular reflection ! ' = !!

39

Special case: blazed grating

f(x)

[0,a]| = tan(!)x

dx eik x!"f (x ) #$% &'

0

a

( = dx eik !"tan())#( )x

0

a

( = ei*+!"tan())#( )a

sin*+! " tan())#( )a,

-./

01

*+! " tan())#( )

! " ei(N#1)$

a%&

sin N $ a%&

'

()

*

+,

sin $ a%&

'

()

*

+,

- dx eik x&#f (x ) .'( *+

0

a

/

= ei! a"

(#N$tan(%)&)sin N ! a

"#

'

()

*

+,

sin ! a"#

'

()

*

+,

-

sin !a"# $ tan(%)&( ).

/01

23

!"# $ tan(%)&( ) envelope

!

"#!$

%

40

Very useful example: blazed grating

! " ei# a$

(%N&tan(')()sin N # a

$%

)

*+

,

-.

sin # a$%

)

*+

,

-.

/

sin #a$% & tan(')(( )0

123

45

#$% & tan(')(( )

Maxima: m =!

a"

integer # $ = $m($')

By chosing the appropriate blaze angle one may position the maximum of the envelope to a higher order maximum :

envelope

m ! 0!

�0.4 �0.2 0.0 0.2 0.4 �0.4 �0.2 0.0 0.2 0.4�0.4 �0.2 0.0 0.2 0.4

m = 0, !' = 0.2" m = 1 m = 2

! / " ! / " ! / "

41

! = arctan "

#

$

%&

'

() = arctan sin(* ')+sin(*)

cos(* ')+cos(*)

$

%&

'

() =

12* '+*( ) with *=*m

sin(! ')" sin(!) = 2cos( 12 ! '+!#$ %&)sin( 1

2 ! '"!#$ %&)

cos(! ')+ cos(!) = 2cos( 12 ! '+!#$ %&)cos( 1

2 ! '"!#$ %&)

!

"

!2 = "+ #1+ $!2 = " '+ #2 % $

! "#" '+ $1 # $2 + 2% = 0

(*)

! "1 = "2(*)

42

Fresnel Diffraction Augustin Jean Fresnel 1788 - 1827

P̂!x ! 0, P̂ '

!x ! 0

!"#

!P'

!P

RP

! 1 + 12P2

!x2 " P̂

!x( )

2#

$%

&

'( ! 1 +

!x2

2P2

Q ! 12

n̂ R̂ " R̂ '( ) # 1 , eik R+R'( ) # eik P+P'( ) ei $%

1P+

1P'

&

'(

)

*+ !x2

!(!P) " k

2# iQ

R R' dA eik R+R'( )

apertures$ " 1

i%eik P+P'( )

P P' dx1 dx2 e

i #%

1P+

1P'

&

'(

)

*+ x1

2+x22( )

apertures$

!" # 2

$1P+

1P'

%

&'

(

)* x"

+ ,(!P) - 1

2ieik P+P'( )

P+P' d!1 d!2 e

i .2

!12+!2

2( )apertures/

Fresnel approximations

Fresnel integral: (Cornu spiral) F(!) " dz e

i #2

z2

0

!

$ �1.0 �0.5 0.5 1.0

�1.0

�0.5

0.5

1.0

Re F(!)"# $%

Im F(!)"# $%

�1.0 �0.5 0.5 1.0

�1.0

�0.5

0.5

1.0

43

Diffraction at an edge

!(!P,") # 1

2ieik P+P'( )

P+P' F(") $ F($%)( ) F(%) $ F($%)( )

= 1+ i( )2i

eik P+P'( )

P+P' F(") +

1+ i( )2

&

'

((

)

*

++ , " , 2

-1P+

1P'

&

'(

)

*+ x

Re F(x)!" #$

Im F(x)!" #$

!"#

!

!(!P,") = 1

2 P+P'( ) F(")#F(#$)

Re F(x)!" #$

!(!P,")

2

!�4 �2 0 2 4

44

Fresnel zone plate Consider Fresnel diffraction at a circular aperture

!"#

$

%

&

rn ! n "1P+

1P'

, n = 0,1,2,... #

rn

! " 1

i#eik P+P'( )

P P' dx1dx2 e

i $#

1P+

1P'

%

&'

(

)* x1

2+x22( )

apertures+ = 2$

i#eik P+P'( )

P P' dr r ,(r) e

i $#

1P+

1P'

%

&'

(

)* r2

0

rmax

+

Fresnel! int egrand ei "#

1P+

1P'

$

%&

'

() rn

2

= (!1)n

The region between and is called the n-th Fresnel zone. Each zone rn rn!1

encloses the same area !

Covering the even or odd zones yields Fresnel‘s zone plate: on the optical axis locations arise with constructive or destructive interference. !

A = ! rn2 " rn"1

2 ( ) = !#1P+

1P'

45

Fresnel zone plate

Consider Fresnel zone plate with rn = n ! f

Hence, constructive inerference arises if odd or even zones are covered and if

1P+

1P'

!

"#

$

%& f

! " dr r ei #$

1P+

1P'

%

&'

(

)* r2

0

rmax

+ = dr r ei #$

1P+

1P'

%

&'

(

)* r2

rn

rn+1

+n=0

nmax

, = 12

dz ei #$

1P+

1P'

%

&'

(

)* z

rn2

rn+12

+n=0

nmax

,

" ei #$

1P+

1P'

%

&'

(

)* rn+1

2

- ei #$

1P+

1P'

%

&'

(

)* rn

2%

&

''

(

)

**

n=0

nmax

, = ei# 1

P+

1P'

%

&'

(

)* f n+1( )

- ei # 1

P+

1P'

%

&'

(

)* f n%

&

''

(

)

**

n=0

nmax

,

= ei# 1

P+

1P'

%

&'

(

)* f

- 1%

&

''

(

)

** e

i # 1P+

1P'

%

&'

(

)* f n

n=0

nmax

,

is odd.

The first focus arises, if 1P+

1P'

= 1f

46

Paraxial optics with Gaussian beams (ABCD law)

is incident upon an optical system described by a transfer matrix M = A B

C D!

"#

$

%&

q!(z) = z"z

!,0 + ik w

!,02

2, ! #{x,y}

u(x,y,z) =1

qx(z)qy(z)e! ik

x2

2qx (z) e! ik

y2

2qy (z)

!"

"

!"!#$% !#$%

& &

!

Assume that a Gaussian beam

Statement: the transmitted beam is a Gaussian beam with

q!'(z) = z " zM ' + q

!,M ' ,q!,M 'n'

=C +

q!,M

nD

A +q!,M

nB

, q!,M # q

!(zM)

47

Proof

zx,M zx,M '

Consider conjugate points P, P‘. According to Fermat‘s principle, each path connecting P and P‘ has the same optical path length:

S1 = kn d + !S " k n'd'

S2 = kn d + x2

2d

!

"#

$

%& + S(x ',zM ')' S(x,zM) ' kn' d' + x '2

2d'

!

"#

$

%&

S1 = S2 ! S(x ',zM ') " S(x,zM) = #S + kn' x '2

2d'"kn x2

2d

! !

"

#$% &

'()

#$ % &"

' *)+

48

n'! ' = An! + Bx

d' = x '

! '=

x 'n'An!+ Bx

=x 'n'C

An!C+ AD"1( )x=

x 'n'CA x '" x

x ' = Cn! + Dx (1)

(2)

d = x

!=

xCnx ' "Dx

(1)

AD !BC = 1 (3)

(2) (3) (1)

S(x ',zM ') ! S(x,zM) = "S + kn' x '2

2d'!kn x2

2d

= "S +k

2Cx ' A x '! x( ) ! k

2Cx x ' !Dx( ) = "S +

k2C

Ax '2+ Dx2 ! 2xx '( )

(4)

(5)

(4,5)

Determination of S(x ',zM ') ! S(x,zM)

(6)

u'(x ',zM ') !

i"L

#

$%&

'(

1/2

dx u(x,zM)e) i S(zM '))S(zM)( )

)*

*

+ =i"L

#

$%&

'(

1/2

dx u(x,zM)e) i ,S +

k2C

Ax '2+Dx2)2xx '( )-

./

0

12

)*

*

+

49

(6)

Assume that at the -distribution of the incident field is zM !

u(x,zM) = u0 e!ikn x2

2q , q = q",M = q

"(zM) , L # zM '! zM( ) , u0 # 1

2$

%

&'

(

)*

1/4kw0

q

%

&'

(

)*

1/2

Using Kirchhoff‘s generalized integral (Page 30 with R = L, Q = 1)

=

i!L

"

#$

%

&'

1/2

u0 e(i)S e(i k

2CA x '2

dx e( ik n

2q+

D2C

"

#$

%

&'

*

+,

-

./ x2

(0

0

1 e(2 i(kx '

2C

"

#$

%

&'x

=i!L

"

#$

%

&'

1/2

u0 e(i)S 2*C

ik nq

C+D"

#$

%

&'

e(i n'kx '2

21

n'CA( n

qC+D

"

#$

%

&'(1"

#

$$

%

&

''

+

,

--

.

/

00

use

dxe!ax2

e!bx

!"

"

# =$a

eb2

a

50

= e!i"S u0 Cnq

C+D#

$%

&

'(L

e!ikn' x '2

2q' , 1q'

= 1n'C

A !nq

C+D#

$%

&

'(

!1#

$

%%

&

'

((

n'q'

= 1C

A !1

nq

C+D

"

#

$$$$

%

&

''''

= 1C

nq

AC+ AD!1

nq

C+D

"

#

$$$$

%

&

''''

=

nq

A +B

nq

C+D

"

#

$$$$

%

&

''''

! q'n'

=

nq

C+D

nq

A +B =

C+qn

D

A +qn

B

51

Recall geometrical optics expression for conjugate points

!M =

A +RnB B

C+RnD!R'

n'A +R

nB

"

#$

%

&' D!R'

n'B

"

#

$$$$$

%

&

'''''

with M ( A BC D

"

#$$

%

&''

! !

""

# $#%

0 = C+Rn

D!R'n'

A +Rn

B"

#$

%

&' ( R'

n' =

C+Rn

D

A +Rn

B

Paraxiale Wellenoptik Paraxiale geom. Optik q ! R

52

Theorem: q1 = fM1

(q0) , q2 = fM2(q1) ! q2 = fM2! M1

(q0)

Re[ !q] = Re[q]+ q

2B

1+ 2Re[q] B + q2B2

Example 1: lens with focal length f

!

M ! 1 B

0 1"

#$%

&' , B = (

1f

Im[ !q] = Im[q]

1+ 2Re[q] B + q2B2

q ! q(0) = "z0 + q0, q0 = i

#w02

$

!q ! !q(0) = " !z0 + !q0, !q0 = i

# !w02

$

!

!"# !z0, !w0 z0,w0

53

Example 1: lens with focal length f

! !q0

f =

q0

f

1+ z0

f

!

"#

$

%&

2

+q0

2

f2

z0 = !f " !q0 q0 = f 2 , !z0 = f

!z0

f = 1 !

1+ z0

f

1+ z0

f

"

#$

%

&'

2

+q0

2

f2

!q0 / f

!z0 / f

z0 / f

z0 / f

f / !q0

f / 2 !q0

1+ f / !q0

1! f / !q0

1

1 1+ q0 / f1! q0 / f

1 1+ q0 / f1! q0 / f

Special case:!

54

Gaussian beams in optical resonators

M ! A BC D

"

#$$

%

&'' consider a resonator with unit cell

!q = fM(q) = q ! q2 + q A "DB

#

$%

&

'( " C

B = q

! q = D " A

2B ± (A +D)2 " 4

4B2 = D " A2B

± i 4 " (A +D)2

4B2

criterion for stability: 12

Trace(M) = 12

A +D < 1 same condition as in geometrical optics:

55

Example: resonator with two identical mirrors

! "!

#M = 1 0d 1

!

"##

$

%&&

1 '2R

0 1

!

"

###

$

%

&&& =

1 '2R

d 1' 2dR

!

"

####

$

%

&&&&

unit cell

! q = d2

± i 12

(2R "d) d

plane resonator waist position = d2

, confocalparameter = !

confocal resonator waist position = d2

, confocalparameter = d

concentric resonator waist position = d2

, confocalparameter = 0

56

Resonance condition

kd ! (n+m+1) "(d2

)!"(! d2

)#

$%

&

'( = ) * , ) integer

w0 ! wx,0 = wy,0

!(z) " arctan 2zb

#

$%

&

'( , b =

2)w02

* = (2R +d) d

! kd " (n+m+1) 2 arctg d(2R "d) d

#

$%%

&

'(( = ) * , ) integer

! kd " (n+m+1) arccos 1" dR

#

$%

&

'( = ) * , ) integer

! ""FSR

= # $ 1%

(n+m+1) ar cos 1$ dR

&

'(

)

*+ , # integer

2 arctg x( ) = arctg 2x1! x2

"

#$

%

&' , arctg y( ) = arccos 1

1+ y2

"

#

$$

%

&

''

use

57

nearly plane resonator

n+m = 0

R >> d

!

n+m = 0!+1

! ""FSR

= # $ 12

(n+m+1) , # integer

!FSR

confocal resonator R = d

n+m = even!

n+m = odd!

n+m = even!+1