**J -T**

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

**and the Schwarz inequality states that**

*r*** rT ** **^ ** **2***rT * *r***T**

*\*

|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 Yi**2*****

**(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 .**
*J*0
**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.**

**v**1**(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,x**2** ,t) ** **F(Xj,x**2** ,T)**

**y ^ ( T ) = ** *—* ** = ** *—* ** , ** **3.29**

**[r(Xj,Xj,0)T(x2,X**2**,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 -

*X i , X 2 ,*

**r**_{^}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 Y**k — 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 ,x**2

**,t)**

**=**r°° a. . -i27TVT , f ( x i , x 2 , v) e dV ,

*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*

**J**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 j**T|

**< < 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**