Transcript Interaction of radiation with atoms and ions

Interaction of radiation with atoms and ions (I)
Absorption- Stimulated emission
2
2

2
| a2 (t ) |2 
I
|

|
 (  0 ) t
21
2
3n 0 ch
E2
E2  E1  h 0
W12 =W21
E1
 
E  E0 sin( t ) h  
 (  0 )  g (  0 )
More definitions
W
F
 ( N1  N2 )

Spontaneous emission
Cross section
Absorption
coefficient
 ( N2  N1 )
16 3 3n
2
A
|

|
21
3 0 c 3h
Gain
Interaction of radiation with atoms and ions (II)
Line broadening
Homogeneous
(Lorentz)
Inhomogeneous
(Gauss)
 0 
1
Collisions
Spont. emission
c
 0 
1
Phonons
2sp
Doppler
 0  2 0
Local field
Saturation
2
2
1
1
High
absorption

Reduced
absorption
0
1 I / Is
2kT ln( 2)
Mc 2
Ray and wave propagation through optical media
Matrix- formulation of Geometric Optics
 r2   A B  r1 
   
 
 2   C D 1 
lens
 1

 1/ f
Gaussian Beam
w
~ 
Em (r )  A 0 H  ( w(2zx) ) H m ( w(2zy) )e
w( z )
1/ 2
 z 
w( z )  w0 1  2 
 zR 
2
ABCD law
2
R
z
R( z )  z 
z
0

1 

Material with
index n
x2  y2
w( z ) 2
e ikz e
ik
e i (   m1) ( z )
1  z 
 ( z )  tan  
 zR 
1
1
i


q( z ) R( z ) w( z ) 2
Aq1  B
q2 
Cq1  D
x2  y 2
2R( z )
1 C  D / q1

q2 A  B / q1
1 L / n


0
1


w02
zR 

Stable resonators
Stability condition
Two-mirror resonator
General case
Round trip matrix
A B


C
D


Single passage
matrix
 A1

 C1
 mn 
B1 

D1 
A D
1 
1
2
L
gi  1 
Ri
Gaussian beam
solution
B1 D1
qi 
A1C1
R( z1 )  R1
1/ 4

g2
 L  

w1  
 
    g1 (1  g1 g 2 ) 
1/ 2
1/ 4
 L   g 2 g1 (1  g1 g 2 ) 
w0  

2
    g1  g 2  2 g1 g 2  
1/ 2
Frequencies  mn 

c  1   m
1
n

cos
( A1 D1 )

2L 


z1 
 Lg 2 (1  g1 )
g1  g 2  2 g1 g 2
c  1   m

1
n

cos
(

g
g
)
1 2 
2 L 


  g2
  B1D1
Spectral width
   ln R1 R2 (1  Ti ) 2 
1
2
0  g1 g 2  1
c 
L
c
 c 
1
2c
Continuous Wave Laser Behavior
Rate equations
B
dN
N
 R p  BN 
dt

cw-laser
Put 
 2c
2 Le
Va Le
Le
c
h
Va
Ab
N ,
Threshold

Nc 


Rcp 

Stationary
solution
N0  Nc
0  Va c ( R p  Rcp )

Ab I s 2  Pp
  1
Put 
2  Pth 
c 

d

 BNVa 
dt
c
N (0),  (0)  N (t ),  (t )
c
Nc
dP
 s  ut
dPp
Rp
Rcp
Transient Laser Behavior
Q-switching:
High losses
Mode-locking
Switching of losses
Pulse (ns)
Frequency domain: Modes “in phase”
N   R p
High population inversion
Time domain: Pulse train, periodicity: 2 L
 p  L  1 (0.44) Gaussian
c
 c
Phase velocity
Group velocity
v ph  
 n

d




  ' ( L )
d
Group delay
vg
d  
d
vg 
d  L
d d
d 2

d
d 2
d d
 d 
 L
d
L
Group delay
dispersion
 " ( L )
  L
Properties of Laser beams
Spatial coherence
Temporal coherence
 (r , r , )  E (r , t   ) E (r , t ) 
(1)
*
Measurement: Michelson
interferometer
Temporal coherence:
monochromaticity
 co
1


Laser: good temporal
coherence if monomode
P
Brightness: B 
A
For a Gaussian beam
(high brightness)
B
(1) (r1 , r2 ,0)  E (r1 , t ) E * (r2 , t ) 
Measurement: Young’s
double slit interferometer
Spatial coherence:
directionality


d
Laser: good spatial coherence
if one transverse mode
L
Thermal light: d coh  0.32
d
4P
2
Laser and thermal light have different
statistical properties
different high-order coherence