Chapter 15. Optical Coherence and Lasers
Coherence of light ?
15.3 The Coherence of Light
<Young’s two-slit experiment>
Maxima : Y ( max)  n
Minima : Y ( min )
L
,
n  0, 1, 2, 
d
L
 (n  12 )
,
n  0, 1, 2, 
d
Fringe separation :   Y ( max)  Y ( max)  L d
Nonlinear Optics Lab.
Hanyang Univ.
# Real source of radiation cannot be perfectly monochromatic.
If the source emits radiation with a spread d in wavelength about , the difference in fringe
separation for two wavelengths separated by d should be sufficiently small enough in comparison
with the fringe separation for the wavelength of , in order to get interference pattern ;
d  Ld d d d

  1 : Quasimonochromatic radiation
 L d  
# As we increase d we find the interference pattern becoming less sharp, and beyond a certain value of
d the interference fringes disappear altogether.
# A similar effect is observed when we hold d fixed and bring the slits closer to the observation screen.
As the interference fringes fade out the intensity at any point P on the observation screen becomes
simply the sum of the intensities associated with the two slits individually.
Define, Visibility as V 
I max  I min
I max  I min
: Measure of the ability of light to produce interference fringes ;
Measure of the coherence of light
Nonlinear Optics Lab.
Hanyang Univ.
15.4 The Mutual Coherence Function
Assume,  L  L    L : the fields from the slits will be diffracted enough
a
d
to produce interference fringes on the screen.
and considering a quasimonochromatic field, E (r, t )  ε(r, t )e it
where the complex amplitude ε(r,t )
varies very slowly in time.
Measurable intensity :
I (r, t )  c 0 [Re E (r, t )]2

c 0 2
[ E (r, t )  E *2 (r, t )  2 E * (r, t ) E (r, t )]
4
Re[ E (r, t )]  ( 12 )[ E (r, t )  E * (r, t )]
averaged to 0
 Measurable intensity ;
c
2
 0 [ε 2 (r )e 2i t ε*2 (r )e 2i t 2 ε(r ) ]
4
Nonlinear Optics Lab.
2
c 0
I (r, t )
E (r, t )
2
Hanyang Univ.
<Two-slit interference experiment>
The field at the point P on the observation screen,
E ( P, t )  E1 ( P, t )  E2 ( P, t )
l 
l 


 K1 E  S1 , t  1   K 2 E  S 2 , t  2 
c
c


The intensity at point P, averaged over a few optical periods,
2
c
c
l 
l 


I ( P, t )  0 K1 E  S1 , t  1   0 K 2 E  S 2 , t  2 
2
c
2
c


2
 
l  
l 
 c 0 K1 K 2 Re  E *  S1 , t  1  E  S 2 , t  2 
c 
c 
 
Ensemble average ;
c
l 
2

I ( P,t )  0 K1 E S1 ,t  1 
2
c

2
c
l 
2

 0 K 2 E S2 , t  2 
2
c

2
l  
l 

 c 0 K1K 2 Re E *  S1 , t  1  E S 2 , t  2 
c 
c

Nonlinear Optics Lab.
Hanyang Univ.
Define, mutual coherence function of the fields at r1 , t1 and r2 , t2
(r1 , t1 ; r2 , t 2 )  E * (r1 , t1 ) E (r2 , t 2 )
 I ( P , t )  I1 ( P , t )  I 2 ( P , t )
l
l 

 c 0 K1 K 2 Re  S1 , t  1 ; S 2 , t  2 
c
c

15.6 Fringe Visibility
I ( P)  I1  I 2  2 I1I 2 Re  (S1 , S2 , l c)
where,
General definition :
l l2 l1 ,
 ( S1 , S 2 , l c) 

(c 0 2) K1 K 2 ( S1 , S 2 , l c)
I1 I 2
(c 0 2) K1 K 2 ( S1 , S 2 , l c)
K1

2
I ( S1 )  K 2
2
I (S2 )
 (r1 ,r2 , )
(r1 ,r2 , )
(r1 ,r1 ,0)(r2 ,r2 ,0)
: Complex degree of coherence
(c 0 2)( S1 , S 2 , l c)
I ( S1 ) I ( S 2 )
Nonlinear Optics Lab.
Hanyang Univ.
 i 
Ex) Purely monochromatic field :  (r1 ,r2 , )e
Ex) quasimonochromatic field :  (r1 ,r2 , )  (r1 ,r2 , ) ei
l
2l

I ( P)  I1  I 2  2 I1I 2   S1 , S 2 ,  cos
c


I max  I1  I 2  2 I1I 2 
I min  I1  I 2  2 I1I 2 
I max I min 2 I1 I 2 
 V

: || is a direct measure of the fringe visibility
I max  I min
I1  I 2
Nonlinear Optics Lab.
Hanyang Univ.
15.7 Spatial Coherence of the Light from Ordinary Sources
<Spatial coherence>
: Mutual coherence of the fields at two different points in space at the same time
# A point source of radiation always produces spatially coherent radiation.
Even though the radiated field may vary quite erratically in its amplitude and phase, as a result of
fluctuations in the source, every point on the wavefront has the same variation. Thus the variations
are perfectly correlated across any wavefront, and the emitted field is spatially coherent.
# In the case of an Extended source ?
Consider the case of two independent point sources,
- The source on the axis is perfectly spatial coherent
=> V=1 ; produces interference fringes on the screen
- The second point source is not equidistance from the
two slit => phase difference due to the length
difference, D :
2D 2 ( d )

Nonlinear Optics Lab.


R
Hanyang Univ.
=> The interference pattern associated with the second source is shifted a distance on the screen :
 L L

2 d
R
Therefore, the interference pattern associated with two sources can be observed, if
L
R

L
d


 d  R : Airy disk size region
  (r1 ,r2 ,0) 
2 J1 ( x)
x
where, x
2
2 d
r1 r2 
R
R
ex) x=1 => 2J1(x)/x ~ 0.88
2d
1 R  | |  88%
: if x 
1 or d 
R
2 
# Spatial coherent region (diameter) of ordinary source with radius of  :
1 R
R
d coh 
 0.16
2 

Nonlinear Optics Lab.
Hanyang Univ.
<Method for obtaining spatially coherent beam of large area from an ordinary source>
<Lens-and-pinhole arrangement>
#  a, R f
 d coh 0.16
f
a
Ex) Sunlight,  ~ 550 nm, r ~ 6.96 x 1010 cm, R ~ 1.5 x 1013 cm
(0.16)(5500108 cm)(1.51013 cm)
d coh 
6.961010 cm
0.02mm
# Measurement of the angular diameter of stars :  

R
d when the interference
fringes disappear.
 d coh  0.16
Nonlinear Optics Lab.


Hanyang Univ.
15.8 Spatial Coherence of Laser Radiation
# What is unique about lasers is that they can combine spatial coherence with high intensity.
# However, a laser operating on more than one transverse mode does not have perfect spatial coherence.
<= The different modes are being excited by quite different groups of active atoms, and are therefore
associated with completely independent sources.
single mode
He-Ne laser
double mode
multimode
thermal source (calculation)
Nonlinear Optics Lab.
Hanyang Univ.
<Speckle effect>
Random bright/dark spots
The bright and dark areas are associated respectively
with constructive and destructive interference of light
from the various surface scattering elements.
Because the surface has more or less random
irregularities, the speckle pattern itself appears random
and irregular.
If the radiation incident on the scattering surface were
not spatially coherent, the uncorrelated fluctuations in
the field at nearby points on the surface would wash out
the interference pattern.
Nonlinear Optics Lab.
Hanyang Univ.
15.10 Coherence and the Michelson Interferometer
<Temporal coherence>
: Mutual coherence of the fields at the same point in space but at two different time
<Michelson interferometer>
The total field at P at time t is E ( P, t )  12 E  R, t  l1   12 E  R, t  l2 

c

c
where, l1  L  2d1  l
l2  L  2d 2  l
Nonlinear Optics Lab.
Hanyang Univ.
Measured intensity at P ;
c 0
2
E ( P, t )
2
2
2
c 0  1 
l1 
l2 
l1  
l2  

*
1
1

 E  R, t    4 E  R, t    2 Re E  R, t   E  R, t  
2  4 
c
c
c 
c 


I ( P, t ) 
For stationary fields, every term in the equation is independent of t, and mutual
coherence function depends only on the time difference,
d  d2
 l2   l1  l1  l2
2 1

t    t   
c
c
 c  c

I ( P) 
1
4

1
2
 I ( R)
 I ( R)  c 0 Re ( R, R, )
I ( R) 1  Re  ( R, R, )
where, the complex degree of coherence,  ( R,R, )
(c 0 /2)( R,R, )
I ( R) I ( R)
Nonlinear Optics Lab.

(c 0 /2)( R,R, )
I ( R)
Hanyang Univ.
Ex) Purely monochromatic field :
(15.6.4) =>
 e i t
I ( P) 
1
2
I ( R) (1  cos  )
 I ( R) cos 2 12 


 I ( R) cos 2  (d1  d 2 ) 
c

 2

 I ( R) cos 2 
(d1  d 2 ) 
 

Ex) For quasimonochromatic field :
I ( P) 
1
2

 2

I ( R) 1   ( R, R, ) cos
(d1  d 2 ) 
 


Visibility : V 
I ( P)
I ( P)
max
max
 I ( P)
 I ( P)
min
  ( R, R,  )
min
Nonlinear Optics Lab.
Hanyang Univ.
15.11 Temporal Coherence
Suppose we have radiation of spectral width d incident upon a Michelson interferometer.
=> Interference pattern is smeared out if |d1-d2| is large enough that the largest
wavelength +(1/2)d corresponds to an intensity maximum whereas the smallest
wavelength (1/2)d corresponds to an intensity minimum :
 

c
c




d 1
d1  d 2  n(  12 d)
2 d 2d 2d
d 
d1  d 2 2 

22 
d  d2
1
d1  d 2  (n  12 )(  12 d)


 1

or d1 d 2 c 
c
2d
2
d
 1

1
1
 d1 d 2  1  1 
: The time difference at which the


d


d
2
2
 2

interference pattern to be smeared out.
Nonlinear Optics Lab.
Hanyang Univ.
<Conventional definition>
1
 coh 
: coherence time ( c coh : coherence length)
2d
Ex) Non-laser source, d 100 MHz
 coh 
1
 1.6nsec
(2 )(108 sec 1 )
c coh  48cm
Ex) Laser source, d 300 Hz
lcoh c coh ~160 km
# A laser perating on a single transverse mode will be have perfect spatial coherence,
whereas its temporal coherence will be determined by the bandwidth of the output
radiation.
# A laser operating om more than one longitudinal mode will have a much smaller
coherence length than in the single-mode case.
Ex) Two longitudinal modes,  c/2 L  d c/2 L   coh ~ L/c, lcoh c coh ~ L/
Nonlinear Optics Lab.
Hanyang Univ.