( x 1 ,t)V (x2 ,t) dt

In document Light coherence properties in optical fibres and visual receptors (Page 34-39)

J -T

IV(x: ,t)V (x2 ,t) I dt 3.13

and the Schwarz inequality states that

r rT ^ 2 rT rT

\

|v(xlft)V (x 2 , t) I dt[ < IV (x2 , t) I 2 dt |v(x2 ,t)|2 dt ,

\J __rp J ' _ r n

3.14

so that in terms of the mutual intensity function

i . e .

I r I < [F (Xi ,x1) F (x2 ,X 2 ) ]

I Yl 2 I ^ 1 •

3.15

Wolf used his generalised Huygens' principle to analyse a simple interference experiment, where the screen A (in fig. 2.3) has

small openings at Pi and P 2 . The intensity distribution on a plane

through P can then be expressed, after some algebra, as

I (x) = I1 + I 2 + 2/lj I2 j Yj 2 I cos (ar9 Y j 2 b (r ^ ~ r2 )) , 3.16

where I is the intensity at P(x) due to the opening at P^ alone

(s = 1,2) .

This generalised interference law for partially coherent light is very similar to the expression used by Hopkins to define his "phase- coherence factor" in [13], and corresponds to Zernike's somewhat

differently derived equation in [12] used to define his degree of coherence.

It is in this situation that it becomes obvious that the degree of coherence is an observable, i.e. measurable, quantity, as has

been pointed out before. The actual measurement of I(x), when I x , I2 ,

r 1 , r2 and k are known, yields both |y1 2 1 an<3 ar9 Yi2 ' anc* i-n particular,

2.3 23 1/ I - I . max m m

T

+ 1

max min + LlP ' Y l2' '

Uz iJ

3.17

while the position of the central fringe, relative to the two openings in the screen A, gives the relative value of arg Yi2*

(b) Polychromatic

All the previous analysis, involving the definition of the mutual intensity function in (3.10), is built around a statistically stationary, quasimonochromatic optical field represented by the truncated real function V(x,t). In order to extend his analysis to polychromatic fields, Wolf introduced a second more general definition

in terms of the complex representation of the real fields. Before introducing this definition, which takes into account any path length differences involved in a typical situation, it is necessary to consider this complex representation of V(x,t).

Born and Wolf [1] and Beran and Parrent [2] devote a consider­ able amount of time to discussing the analytic signal, or complex half­ range function, representation of the real optical fields, as introduced into coherence theory by Wolf in [20]. Essentially, a real, optical disturbance is assumed to be expressible by the Fourier integral

v r (t)

,oo

J — oo

a t

V (v) e- 2ti ivt dv , 3.18

where the superscript r is used to indicate a real quantity.

This may be rewritten as a sum of two integrals, i.e.

v r (t) /srv r (V) e-i2TTVt dv + v r /\ r(V) e-i27TVt dV

/\ 23 *

and then noting that [V (V)] = V (-V), changing the order of integration and setting V = -V in the first part,

v r (t) (V) e' i2TTVt 3.20 where R indicates the real part. If v r (v) = a(v) e ^ ^ , where both a(v) and (j) (v) are real functions, then

v r (t) *00 2a (V) cos [-27TVt + (J) (V) ] dV . J0 3.21 IT

A second function is now introduced, derived from V (t) by changing the phase of each frequency by tt/ 2, i.e.

v1(t)

*00

2a(V) sin [-27TVt + <j) (V) ] dV ,

J o

where the superscript i indicates an imaginary quantity.

3.22

The complex function representation of V r (t) may then be defined by

V(t) = V r (t) + iV1 (t) , 3.23 and then

V(t)

where

2a(V) e -i27TVt + i(j)(V) ndv V(v) e-i2TTVt dv o

3.24

V(V) = 2vr (v) .

This complex representation is common in communications theory where V(z) is known as the analytic signal representation, where z is a complex variable. V(z) is analytic in the lower half-plane of z. It is also known as the complex half-range function, as all the information about the function is contained in the positive frequency components.

The Fourier inversion theorem then gives

V(V) V(t) e i2WVt dt V > 0

v < o

3.25

r i

2 .3 25

This is of relevance to the definition of the coherence function that is

introduced below, as Beran and Parrent [2] discuss in detail. They show

r i

that as V (t) and V' (t) are then orthogonal,

-TO .TO

V (x1 ,t)V (x2 ,t) dt = 2 V r (xj,t)Vr (x2 ,t) dt . 3.26

^ — OO ' — T O

The general coherence function introduced by Wolf to include polychromatic fields is defined in terms of the analytic signal

representation of the real field disturbances, and takes the form of the complex cross-correlation function, defined by

r(xj,x2 ,T ) lim

T -> OO _1_

2T V ( X j ,t + t)V (x2 ,t) dt , 3.27

where the disturbance at P x (Xj) is considered at a time T later than

that at P 2 (x2) . It is a property of the analytic signal that r(xlfx 2 ,T)

so defined is also an analytic signal, so that

r (Xi,x2 ,T) = 2Tr (x1 ,x2 ,T) , 3.28

I T 1C

where T is defined in terms of V (t) as above, in section (a).

In a complete parallel fashion to that outlined in section (a),

the mutual coherence function may be normalised to give a complex degree

of coherence, by setting

r(Xj,x2 ,t) F(Xj,x2 ,T)

y ^ ( T ) = = , 3.29

[r(Xj,Xj,0)T(x2,X2,0)] 2 [I(xj)I(x2 )] 2

although now I (x^) is defined in terms of the analytic signal at P_^ and so, as shown above in (3.26),

I ( x J = ( V t x ^ x ^ O J V (x^,x^,0) ) = 2ir (xi) . 3.30

Again, it may be shown that |Yi2 (t)| ^ 1, and the generalised

u s i n g t h e s e d e f i n i t i o n s i n v o l v i n g T a n d t h e a n a l y t i c s i g n a l r e p r e s e n t a t i o n . T h u s , f o r t h e H u y g e n s ' p r i n c i p l e , I (x) [ I (Xj ) I ( x 2 ) ] AJ A r i r 2 r 0 - r X i , X 2 , ^1 ^ 2 d x i d x 2 3 . 3 1 w h e r e t h e A ^ ' s r e p r e s e n t some mean v a l u e s o f t h e i n c l i n a t i o n f a c t o r s . I n f a c t Y may b e r e p l a c e d b y t h e r e a l p a r t a s a l l t h e o t h e r q u a n t i t i e s a r e r e a l . S i m i l a r l y , t h e i n t e r f e r e n c e l a w b e c o m e s I (x) I i + I 2 + 2 [ I j l 2 ] 2 Yk — 1' -_2 3 . 3 2 T h a t t h e p r e v i o u s l y o b t a i n e d f o r m s o f t h e s e l a w s d o a c t u a l l y r e p r e s e n t t h e a p p r o p r i a t e q u a s i m o n o c h r o m a t i c a p p r o x i m a t i o n s may b e s e e n b y e x a m i n i n g t h e f o r m o f r ( x 1 , x 2 , T) f o r q u a s i m o n o c h r o m a t i c l i g h t . P u t t i n g H x i ,x2,t) = r°° a. . -i27TVT , f ( x i , x 2 , v) e dV , J o 3 . 3 3 a n d t h e n r e w r i t i n g t h i s a s r ( X j , x 2 , ! ) e-2iTiVT f ( X x , X2 ,V) e i2TT(V-V)T d v , 3 . 3 4

t h e q u a s i m o n o c h r o m a t i c a p p r o x i m a t i o n may now b e made t h a t Av << V . Th e n c o n s i d e r i n g o n l y s m a l l v a l u e s o f T, s u c h t h a t A v jT| < < 1 / g i v e s r ( x x , x 2 , T) e-2TTiVT ' roo 0 ? ( x x , x 2 , V) dv -2TTiVT = e

r (Xj

x 2 , 0 ) . 3 . 3 5 3 . 3 6 N o t i n g t h a t s u b s t i t u t i n g a p p r o p r i a t e l y i n t o ( 3 . 2 8 ) a n d ( 3 . 2 9 ) a b o v e a n d t a k i n g t h e n e c e s s a r y r e a l p a r t s , t h e r e q u i r e d q u a s i m o n o c h r o m a t i c f o r m s a r e r e t r i e v e d [20] .

2.3 27

While the propagation of the mutual coherence function may be described in terms of the Huygens-Fresnel integral, this is necessarily an approximate description, with a rather restricted range of validity

(see e.g. [1], Chap. VIII). A complete and rigorous law for the

propagation of r(x1 ,x2 ,l) was obtained by Wolf in the following manner.

The real field disturbance is assumed to obey the scalar wave equation, i.e.

V 2 Vr (t) JL 32v r (t)

=2 9t2

3.37

hence the analytic signal associated with the field also obeys such an equation. Consequently each Fourier frequency component of the analytic signal obeys the equation

V 2 v(x1#v)

r2TfV 2

V ( X j ,V) , 3.38

where

V

2 is the Laplacian operator with respect to the position vector

In document Light coherence properties in optical fibres and visual receptors (Page 34-39)