2006 가을학기
Laser Optics
(레이저 광학)
담당 교수 : 오 차 환
교 재 : P.W. Miloni, J.H. Eberly, LASERS, John Wiley & Sons, 1991
부교재 : W. Demtroder, Laser Spectroscopy, Springer-Verlag, 1998
F. L. Pedrotti, S.J., L.S. Pedrotti, Introduction to Optics, Prentice-Hall, 1993
Nonlinear Optics Lab.
Hanyang Univ.
Chapter 1. Introduction to Laser Operation
1.1 Introduction
LASER : Light Amplification by the Stimulated Emission of Radiation
1916, A. Einstein : predicted stimulated emission
1954, C. H. Townes et al. : developed a MASER
1958, A. Schawlow, C.H. Townes : adapted the principle of MASER to light
1960, T.H. Maiman : Ruby laser @ 694.3 nm
1961, A. Javan : He-Ne laser @ 1.15 mm, 632.8 nm
…
Nonlinear Optics Lab.
Hanyang Univ.
Einstein’s quantum theory of radiation
E2
(stimulated)
absorption
spontaneous
emission
stimulated
emission
B12N1r
A21N2
B21N2r
E1
[radiative processes]
[light-matter interaction]
* N1, N2 : No. of atoms at E1, E2
* r : photon density
* A21=1/t21 : spontaneous emission rate
* B12, B21 : stimulated absorption/emission coefficients
Nonlinear Optics Lab.
Hanyang Univ.
Spontaneous & Stimulated emissions
Spontaneous emission
Stimulated emission
Phase and propagation
direction of created photon is
random.
Created photon has the same phase,
frequency, polarization, and propagation
direction as the input photon.
Nonlinear Optics Lab.
Hanyang Univ.
Einstein’s A, B coefficients
Rate equation :
dN 2
  N 2 A21  N 2 B21r ( )  N1 B12 r ( )  0 (thermal equilibrium)
dt
N2
 e ( E2  E1 ) / kT  e  h / kT (Boltzman distribution of atoms)
N1
r ( ) 
A21
B12e h / kT
8h 3
1
(Planck’s blackbody radiation law)

3
h / kT
 B21
c
e
1
A21 8h 3
B12  B21,

B21
c3
if N2  N1 (population inversion)
Light amplification ! (Lasing)
Nonlinear Optics Lab.
Hanyang Univ.
Four key elements of a LASER
- Gain medium (Active medium)
- Pumping source
- Cavity (Resonator)
- Output coupler
relaxation
laser
pumping
relaxation
cavity (resonator)
gain medium
Laser light
total
reflector
output
coupler
pumping source
Nonlinear Optics Lab.
Hanyang Univ.
Four key elements of a LASER
1) Pumping source
- Optical : Nd-YAG, Ruby, Dye, Ti:sapphire, …
- Electrical : He-Ne, Ar+, CO2, N2, LD, …
- Chemical : HF, I2, …
2) Active medium
- Gas : He-Ne, Ar+, CO2, N2, …
- Liquid : Dye
- Solid : Nd-YAG, Ruby, Ti:sapphire, LD, …
3) Cavity or Resonator
- Resonator with total reflector : Maximizing the light amplification
- Output coupler : Extracting a laser light
- Resonance condition : ml/2=L (m:integer)
Nonlinear Optics Lab.
Hanyang Univ.
1.2 Lasers and Laser Light (Characteristics of laser light)
Monochromaticity (단색성)
- Linewidth(FWHM) : 7.5 kHz (He-Ne laser)
<< 940 MHz (low pressure Cd lamp)
Coherence (결맞음)
- Definite phase correlation in the radiation field at different locations(spatial)
and different times(temporal)
Directionality (지향성)
- Divergence angle : f1.27l/D < q2.44l/D (diffraction limit angle)
Brightness (높은 휘도)
- Radiance : 106 W/cm2-sr (4mW, He-Ne laser)
<< 250 W/cm2-sr (super-high-pressure Hg lamp)
Focusability (집속도)
- Focusing diameter : d ~ f f
Nonlinear Optics Lab.
Hanyang Univ.
1.5 Einstein theory of light-matter interaction (Laser action)
- Number of photons, q
dq
 anq  bq
dt
- Number of atoms in level 2, n
dn
 anq  fn  P
dt
spontaneous
emission
stimulated loss
emission
pumping
- In steady state : q  n  0
n
b
 nt : threshold number of atoms
a
P f
q  0
b a
fb
 Pt 
 f nt : Minimum(threshold) pumping condition
a
Nonlinear Optics Lab.
Hanyang Univ.
Spatial distribution of laser beam (Gaussian beam)
Maxwell’s curl equations
E
H
H  
,   E  m
t
t
2E
=>  E  m 2  0 : Scalar wave equation
t
2
Put, E ( x, y, z , t )  E0 ( x,y,z)e i t (monochromatic wave)
2

=> Helmholtz equation :  E0  m E0  0
t 2
Assume, E0   ( x, y, z )e  ikz
2
 2
2 

=>  2  2   2ik   0
y 
z
 x
kr 2
Put,   exp{ i[ p( z ) 
]} , r  ( x 2  y 2 )1/ 2
2q( z )
=>
1
d 1
dp
i

(
)

0
,


q 2 dz q
dz
q
Nonlinear Optics Lab.
Hanyang Univ.
1
d 1

( )  0 =>
2
q
dz q
q  z  q0
 q is must be a complex !
=> put,
q  z  iz 0
=> Assume,
q0 is pure imaginary.
( z0 : real)
At z = z0,
kr 2
 ( z  0)  exp( 
) exp{ ip (0)}
2 z0
 Beam radius at z=0,
2 z0 1/ 2
w0  (
)
: Beam Waist
k
2

w
q at arbitrary z, q  z  i 0
l
z0
1
1
z
1
l

 2
i 2
  i 2 : Complex beam radius
=>
2
2
q z  iz 0 z  z0
R w
z  z0
Nonlinear Optics Lab.
Hanyang Univ.
dp
i

dz
q
2 1/ 2
1
=> ip ( z )  ln[ 1  ( z / z0 ) ]  i tan ( z / z0 )
=> exp[ ip ( z )] 
1
1
exp[
i
tan
( z / z0 )]
2 1/ 2
[1  ( z / z0 ) ]
Nonlinear Optics Lab.
Hanyang Univ.
Wave field


E0 ( x, y, z )  w0
r 2 
kr 2 
1

exp  2   exp  i[kz  tan ( z / z0 ) exp  i

EA
w
(
z
)
w
(
z
)
2
R
(
z
)







2
  lz  2 



z 
2
2
2
   w0 1     : Beam radius
where, w ( z )  w0 1  
2 
z 

  nw0  

  0  


2
  nw 2  2 


z


0
0
   z 1     : Radius of curvature of the wave front
R( z )  z 1  
  lz  
  z  


nw0 2
: Confocal parameter(2z0) or Rayleigh range
z0 
l
Nonlinear Optics Lab.
Hanyang Univ.
Gaussian beam
spread angle :
q / 2  l / nw0
I
2w0
w0
z
z0
Gaussian
profile
z 0
Near field
(~ plane wave)
Far field
(~ spherical wave)
Nonlinear Optics Lab.
Hanyang Univ.
Propagation of Gaussian beam - ABCD law
Matrix method (Ray optics)
yi a i
ao
yo
optical
elements
 yo   A B   yi 
a   C D a 
 i 
 o 
A B
C D  : ray-transfer matrix


Nonlinear Optics Lab.
Hanyang Univ.
Ray-transfer matrices
1) Free space
2) Refracting surface
R
q
r1
z1
n1
n2
q1
r2
d
z2
r2 = r1 + qd
q : constant
(paraxial ray approximation)
 r2   1 d   r1 
  
    
q 2   0 1  q1 
r
q2
s
s’
n1/s + n2/s’ = (n2-n1)/R
r : constant
q2  q1 n1/n2 – (1- n1/n2) (r1/R)
 r2   n 1 n
    2 1

q 2   n2 R
Nonlinear Optics Lab.
0  r 
n1   1 
 q 
n2   1 
…
Hanyang Univ.
Nonlinear Optics Lab.
Hanyang Univ.
Nonlinear Optics Lab.
Hanyang Univ.
ABCD law for Gaussian beam
 yo   A B   yi 
a   C D a 
 i 
 o 
yo  Ayi  Ba i
a o  Cyi  Da i
Ayi  Ba i
Ro 

a o Cyi  Da i
Ayi / a i  B

Cyi / a i  D
yo
Ro (ray optics)  q (Gaussian optics)
optical system
ABCD law for Gaussian beam :
 A B


C D
q1
q2
Aq1  B
q2 
Cq1  D
Nonlinear Optics Lab.
q  z  iz 0
nw0 2
z0 
l
Hanyang Univ.
example) Focusing a Gaussian beam
w01
z1
A

C
w02 ?
q1
z2 ?
B   1 z 2  1
0  1 z1 
  



D   0 1   1 / f 1  0 1 
1  z 2 / f z1  z 2  z1 z 2 / f 

 
0
1  z1 / f


 q2 
(1  z 2 / f )q1  ( z1  z 2  z1 z 2 / f )
 q1 / f  (1  z1 / f )
Nonlinear Optics Lab.
Hanyang Univ.
2
1
1  z1 
1  w01 



1



2
2 
2 

f 
f  l 
w02
w01 
2
f 2 ( z1  f )
z2  f 
( f )
2
2
2
( z1  f )  (w01 / l )
- If a strong positive lens is used ; w01  w02
=> w02 
=> w02 
fl
 fq 1
w01
2l f N
2 fl

, f N  f / d : f-number
 (2w01 )

; The smaller the f# fo the lens, the smaller the beam waist at the focused spot.
2
- If w01 / l  ( z1  f )
2
=> z2  f
Note) To satisfy this condition, the beam is expanded before being focused.
Nonlinear Optics Lab.
Hanyang Univ.
Chapter 2. Classical Dispersion Theory
2.1 Introduction
Maxwell’s equations :   D  0 ,   B  0 ,   E  - B ,   H  D
t
t
B  μ 0 H (for nonmagnetic media)
D   0E  P
Wave equations :
1  2E
1  2P
 E- 2 2 
c t
ε 0 c 2 t 2
2
(2.1.13)
Nonlinear Optics Lab.
Hanyang Univ.
2.2 The Electron Oscillator Model
Equation of motion for the electron :
d 2 re
me 2  eE(re , t )  Fen (ren )
dt
Electric-dipole approximation :
d 2x
m 2  eE(R , t )  Fen ( x )
dt
where, x : relative coordinate of the e-n pair
R : center-of-mass coordinate of the e-n pair
m : reduced mass
Electron oscillator model (Lorentz model) <refer p.30-31>
d 2x
m 2  eE(R , t )  k s x
dt

P  Np  Nex
Nonlinear Optics Lab.
Hanyang Univ.
2.3 Refractive Index and Polarizability
 d2
e
2
d 2x


m 2  eE(R , t )  k s x   2  0 x  E(R , t )
m
dt
 dt

Consider a monochromatic plane wave, E( z, t )  ε̂ E 0 cos(t  kz)

 eE 0 / m 
 cos(t  kz)
x  ε̂  2
2 
 0   
Dipole moment : p  ex  a E
e2 / m
where, polarizability : a ( ) 
02   2
Polarization :
 Ne 2 / m 
E cos(t  kz)
P  Np  ˆ 2
2  0
 0   
Nonlinear Optics Lab.
Hanyang Univ.
From (2.1.13),
 2 2 
 2 Na ( )
 -k  2 ˆ E 0 cos(t  kz)   2
ˆ E 0 cos(t  kz)
c 
c
0


Na ( )   2 2
  2 n ( ) : dispersion relation in a medium
k  2 1 
c 
0  c
2
2 
1/ 2

Na ( ) 

n( )  1 
0 

: refractive index of medium
For a medium with the z electrons in an atom :
 eE 0 / m 
 cos(t  kz) ,
x i  ε̂  2
2 
 i   
z
p   ex i

1/ 2
 Na ( ) 

n( )  1 
0 


N
 1 
 0
2
i 1
Nonlinear Optics Lab.
1/ 2
e /m 


2
2 
i 1  i   
z
Hanyang Univ.
(2.3.22a)
Electric susceptibility (macroscopic parameter),

:
P   0  E    Na ( ) /  0
n( )  [1   ( )]1/ 2
Ne 2
 ( ) 
m 0
z
1

2
2
i 1  i  
Nonlinear Optics Lab.
Hanyang Univ.
2.4 The Cauchy Formula
From (2.3.22),
Ne2
2
n (l )  1  2
4  0 mc2
li 2l2

2
2
i 1 l  li
z
If li << l2 ,
2
Ne2
n (l )  1 
4 2 0 mc2
2
 li 2 
li 1  2 

l 
i 1

z
2
If we suppose further that | n 2 (l )  1 |  1 (as in like a gas medium)
Ne 2
n (l )  1  2
8  0 mc 2
 li 2 
B



: Cauchy formula
l
1


1

A
1



i 
2 
2 
l 
 l 
i 1

z
2
z
where, A 
2
Ne
8 2 0 mc 2
z
l
i 1
i
2
, B
l
i 1
z
l
i 1
4
i
2
i
Nonlinear Optics Lab.
Hanyang Univ.