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Chapter 4_L6
Ray optics (Geometrical optics)
Wave reflection and transmission
Fabry-Perot interferometer
89
Ray optics 90
Ray optics: light is described by rays that travel in different optical media in accordance
with a set of geometrical rules. Ray optics is concerned with the location and direction of
light rays. The wavelength of light is assumed to be zero (much smaller than the objects).
Paraxial ray approximation: The ray travels close to the optical axis at small inclinations.
Matrix optics: an efficient tool to study the ray’s position and inclination.
The ray at the interface is described by
the position r and the direction as
( , )r 12
1r 2r
z
r
Optical system
Optical axis
Inp
ut
pla
ne
Ou
tpu
t p
lan
e 1 1( , )r 2 2( , )r
If the ray is rotated clockwise to
the optical axis, or else the angle is
negative
0
1 20; 0 sin tan
The optical axis is usually the normal of a surface, if the incident ray is on the optical axis, the output ray will be on it as well .
Marix optics 91
In the paraxial approximation, the relation is generally
2 1 1
2 1 1
r Ar B
Cr D
2 1
2 1
is the ray transfer matrix
r rA B
C D
A BM
C D
Planar interface
n1 n2 1 1 2 2sin sinn n
1 2
1 0
0 /M
n n
Free space propagation
n1 n2 2 1 1 1 2/r r Ln n
1 21 /
0 1
n L nM
n1
L
Snell’s law
Matrix optics 92
Thin lens
1/ 1/ 1/p q f
1 0
1/ 1M
f
p q
Spherical interface
n1 n2
2 1 1
2 2
1 0
1M n n n
n R n
R
Matrix optics 93
For reflection on mirrors, angle is positive for incident ray clockwise,
while for reflected ray anticlockwise
Planar mirror
2 1
1 0
0 1M
/ 2f R
1 0
2 / 1M
R
R
Spherical mirror
R is positive for concave mirror, while negative for
convex mirror
Matrix optics 94
For successive optical elements in a system
2 1
1
2 1
3 2
2
3 2
3 1
2 1
3 1
r rM
r rM
r rM M
M1 M2 MN
2 1...NM M M M
Propagation of a spherical wave 95
The transfer matrix is useful to calculate the propagation (radius) of a spherical wave
2 1
2 1 2 1
1
0 1
( )
z zM
R R z z
2 1
1 0
11
1 1 1
M
f
R R f
11
1
22
2
rR
rR
12
1
AR BR
CR D
Chapter 4_L6
Ray optics (Geometrical optics)
Wave reflection and transmission
Fabry-Perot interferometer
96
Wave reflection and transmission 97
The Electric field E of the optical wave, can be divided into a p-polarized wave Ep (in the plane of incidence) and a s-polarized wave Es (orthogonal to the plane of incidence). Plane of incidence is the plane that formed by the incident wave & the normal of the dielectric interface.
s wave p wave
Wave reflection and transmission 98
Reflection and transmission of electric field (not power)
Fresnel equations
2 1 1 2 1 1
2 1 1 2 2 1 1 2
1 1 2 2 1 2
1 1 2 2 1 1 2 2
cos cos 2 cos
cos cos cos cos
cos cos 2 cos
cos cos cos cos
p p
s s
n n nr t
n n n n
n n nr t
n n n n
For the electric field (p wave):
rp is the reflectivity,
tp is the transmissivity
For the power/intensity
Rp=rp2, Tp=tp
2, Rp+Tp=1
1
2
When 1 0
2 1 1
2 1 2 1
2 p p
n n nr t
n n n n
If rp<0, the reflected field has a 180o phase shift.
Example: Cleaved facet of semiconductor lasers, n1=3.5, n2=1.0 --> R=0.31
99 Wave reflection and transmission
Special angles 100
When the reflectivity Rp is zero, the
incident angle is named Brewster angle.
1 2
1 1 2 2
2 1 1 2
/ 2
sin sin
cos = cos
0
B B
B B
p
n n
n n
r
21
1
tan B
n
n
When n1>n2, it is possible to have total
reflection, that is, the transmission is zero.
12 1
2
21
1
sin sin 1
The critical angle: sin =c
n
n
n
n
1
2
Brewster polarizer
Optical fiber
Chapter 4_L6
Ray optics (Geometrical optics)
Wave reflection and transmission
Fabry-Perot interferometer
101
Fabry-Perot interferometer/etalon 102
A F-P eltalon is usually made of two parallel highly
reflecting mirrors. The power reflectivity of each mirror is R,
and the transmissivity is T. (Note: R+T=1)
At point b, the transmitted electric field is
Ein
0 expcos
in
lE E T jk
At point c, the transmitted electric field is
'
1
3exp
cosin
lE E RT jk
At the wave front, the phase difference between wave E1 and wave E0 is
0 0
0 0
0 0
2
cos
2 tan sin
sin sin
2 cos
lk k l
l l
n n
kl
1 0
0
0 1
exp
exp
...
1exp
cos 1 exp
m
m
out m
in
E E R j
E E R jm
E E E E
lE T jk
R j
nwave number k=2 /
Fabry-Perot interferometer 103
Then, the total power transmission TFP is
2
2
2
2
/
1 exp
1 2 cos
FP out in
FP
T E E
T
R j
TT
R R
2 coskl
The maxima occur at , therefore,
the corresponding frequencies are
2m
2 cosm
cv m
nl
The free spectral range (FSR) is
2 cosFSR
cv
nl
The maximum transmission is:
,max 1FPT
The minimum transmission is 2
,min 2(1 )FP
TT
R
Fabry-Perot interferometer 104
2
21 2 cosFP
TT
R R
The width of the transmission peak
at half maximum:
2 2
2 2 2
1 2cos
2
1 21
2 2
R T
R
R T
R
T
R
2 coskl
2 cosFWHM
c Tv
nl R
The finesse describes how sharp the transmission peak with respective to the FSR.
2 cosFSR
cv
nl
1
FSR
FWHM
v RF
v R
Example 4.3
F-P interferometer as a spectrometer 105
is the resolution
is the working range
FWHM x y FSR
FWHM
FSR
v v v v
v
v
The spectrometer is used to measure the optical spectrum, i.e., power versus
frequency/ wavelength
Chapter 4_L7 106
Diffraction (wave optics)
Wave equation 107
( , , ) ( , , )exp( )
( ) ( )exp( )
E x y z u x y z jkz
E t u t j t
Wave optics is a branch of optics which studies interference, diffraction, polarization, and other phenomena for which the ray approximation of geometric optics is not valid. The scalar electric field is described by the wave equation.
Slowly varying amplitude approximation,
where u(t) is the slowly varying electric field (envelope, vary little on a wavelength scale).
( , , ) ( , , )exp ( )
( ) ( )exp ( )
u x y z A x y z j z
u t A t j t
2 2
2
( , , ) 0
Under paraxial approximation
( , , )( , , ) 2
k E x y z
u x y zu x y z jk
z
Wave equation
We discuss the electric field distribution after transmission with diffraction.
Hugents-Fresnel-Kirchhoff diffraction theory 108
Huygens-Fresnel principle: Each point at the wavefront becomes a source for the secondary spherical wave. At any subsequent time, the wavefront can be determined by the sum of these secondary waves.
Fresnel-Kirchhoff integral equation: It is the mathematical description of the Huygens-Fresnel principle.
1 1 1 1 1
exp( )( , , ) ( , , ) cos
j jkrE x y z E x y z dx dy
r
cos is the obliquity factor ( ) proposed by Fresnel 倾斜因子
1 1 1 1 1( , , ) is the electric field in the elemental areaE x y z dx dy
exp( ) stands for spherical wave
jkr
r
indicates for a /2 phase shiftj
109 Hugents-Fresnel-Kirchhoff diffraction theory
Fresnel approximation under paraxial wave approximation,
1
2 2 2
1 1 1
2 2
1 1
1 2
1
2 2
1 1
1 2
1
cos 1
But for the phase term:
1
11
2
r z z
r x x y y z z
x x y yz z
z z
x x y yz z
z z
1 1 1 1 1
exp( )( , , ) ( , , ) cos
j jkrE x y z E x y z dx dy
r
1
1
2 2
1 1
1 1 1 1 1
1
exp ( )( , , )
( )
( , , )exp2( )
j jk z zE x y z
z z
x x y yE x y z jk dx dy
z z
1
2 2
1 1
1 1 1 1 1
1
( , , )( )
( , , )exp2( )
ju x y z
z z
x x y yu x y z jk dx dy
z z
When the wave passes through an optical system with a ABCD matrix,
2 2 2 2
1 1 1 1
1 1 1 1 1
2 2( , , ) ( , , )exp
2
A x y D x y x x y yju x y z u x y z jk dx dy
B B
Chapter 4_L8 110
Gaussian beams
Gaussian beam 111
Gaussian beam is a beam of monochromatic electromagnetic radiation whose transverse magnetic and electric field amplitude profiles are given by the Gaussian function; this also implies a Gaussian intensity profile. It is one important solution of the Fresnel-Kirchhoff integral equation.
For a spherical wave, the E-field at the place with the wave’s radius R is
2 2
( , , ) exp /
1exp
2
E x y z jkR R
x yjk z
z R
R
z (0,0,0)
The Gaussian beam solution 2 2
2
( , , ) exp2
1 1
x yE x y z jk z
q
with jq R W
(x,y,z)
2 2 2 2
2( , , ) exp exp
2
x y x yE x y z jk z
R W
Gaussian beam 112
2 2 2 2
2( , , ) exp exp
2
x y x yE x y z jk z
R W
2 2
0 2exp
x yu
W
2
2
( )( ) exp
x bf x a
c
Note: Gaussian function form
The amplitude term
For the amplitude u0, the maximum is at (x,y)=(0,0) The width W is determined by u0=e-1, that is, the intensity reduces to e-2.
2 2
0 exp2
p
x yu jk z
R
The phase term is identical to the
spherical wave, that is, the Gaussian
beam has a spherical wavefront
Propagation of Gaussian beam 113
Assume the beam parameter at (x1,y1,z1) is q1, when the beam is propagated to (x,y,z) through an optical system with ABCD matrix:
2 2
1
1
1
1( , , ) xp
/ 2
with
x yu x y z e jk
A B q q
Aq Bq
Cq D
Assume at z1=0, R is infinite large, then
2
1 0
1 jq W
In the free space,
2
1
1 1 1= j
q q z R W
2
2 2
0 2
0
22
0
1
1
zW W
W
WR z
z
2 2 2 2
0
2
1
2
0
( , , ) exp exp exp2
tan
W x y x yu x y z j jk
W W R
zwith
W
114 Propagation of Gaussian beam
The Rayleigh range is defined as
2
0 /Rz W
Then, the solutions become
2
2 2
0
2
1
1
1
tan
R
R
R
zW W
z
zR z
z
z
z
2 2
0
2
2 2
( , , ) exp
exp2
exp
W x yu x y z
W W
x yjk
R
j
Amplitude factor
Trans. phase
Long. phase
When z=zR,
0
min
2
2
/ 4
R
W W
R z
0
/d W zW
The term W0/W ensures the beam power is independent of z.
Gaussian beam and the ABCD law 115
Through a thin lens, just before and just after the lens
1
2 1
2 1
2 1
2 1
1 /
/
1 1 1
1 1 1
C D q
q A B q
q f q
W W
R R f
Through a thin lens, input waist just before the lens, where is the output waist
02 02
1 1
2 2
2
1
2
02 01 1
01
1 /
1/ 1
1/ /
1/ /
1 /
1 /
R
R
m
R
R
z f zM
f
q j z
q j z
fz f
f z
f fW W f z
W
Examples 4.5 & 4.6
High-order modes 116
For the free space propagation, the eigensolutions of the electric field are the product of Hermite polynomial with a Gaussian function:
2 2
1 1 1 11 1 1
1 1 1
2 2( , , ) exp
2lm l m
x y x yu x y z H H jk
W W q
When at the waist 2
1 0 /q j W
2 2
2
0
2 2
2 2( , , ) exp
exp (1 )2
lm l m
W x y x yu x y z H H
W W W W
x yjk j l m
R
2 2
( ) 1m
m X X
m m
dH X e e
dX
Hermite polynomial
0
1
2
2
3
3
( ) 1
( ) 2
( ) 4 2
( ) 8 12
...
H X
H X X
H X X
H X X X
Homework 117
Page 159: 4.1 4.2 4.3 4.7 4.9 4.12
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