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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
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