29 .
d i s c us s i on on t he use o f gas laser s and c onvent ional s our c e s in i n t er f�r ometr y , and is appr opriat e in t he c ont e xt of t he pr e se nt w or k .
2 . BAS IC COHEREN CE TH EOR Y
Br i e fly , c oher en c e may b e c onsider e d by using the opt i c a l ar rangement d&pi c t e d in figur e 2 - 1 . I f t h e s our c e i s i nfini t e sima l ly sma l l and the li ght emi t t e d i s p er fe c t ly monochr omat ic , t h e n t he e l ec tr i c f i e l d ve c t or s ( E1 ( t ) and E2 ( t ) ) at t h e two aper t ur e s o f t h e Rayle i gh mask i n t h e pup i l o f l e ns L2 will b e i n c ompl e t e agr e ement , since they b e l ong t o t h e same wave fr ont at any instant of t i me. H e nc e , t h e c ompl ex amplitude o f
t he e l e c tr i c fie ld v e c t or ER ( t ) a t a p oi nt
PC § ' )
i n t h e f o ca l p lane o f a p er fe c t l ens L2 w i l l b e t he r e sult of the super p osi t i on o f t h0 tw o i nt er fe ring wav e s.i . e . , ER ( t ) = E 1 ( t ) + E2 ( t )
= H1 exp
,
i( tllt+k..6 1 + <i,)
+ A 2 exp i ( ult +kj.)2 +�
)Wher e ' A ' and 'iJ 1 ar e t he r e sp e c t ive amp l i t ud e and path l engths in the i nt er fer om e t er , an d
�
i s t he c ommon phase d i f f er en c e at t ime t = 0 • H er e it i s a s sume d t hat E1 ( t ) and E2 ( t ) have a c ommon p lane of p o lar i zat i on.The obser va b l e int e n s i t y at P i s I =
i nt ens i t y ( c feq ( 1 -3 ) ) gi ven by
i . e . , t he t i me aver a g e d ( +T 1 I = 2T
\
ER ( t ) . ER ( t ) * d t wher e T > > T = 0 H e nc e ) -T 2n {)) i n whi ch .e = ...62--fi1 t he p er i o d of osc i l lat i o n . t h e d i f fer e n c e o f t he t w o i n t er fer ing waves at P •( 2-1 )
( 2- 2 )
3 0 .
Obvi ously maxima o f int e ns i t y , I max -- ( A
1
+ A 2 ) 2 ' oc c ur sat a l l point s P for whi c h l i s an i nt e gr a l multiple o f t h e wave length ( £ = NA ) or
k£ = 2nN ( 2- 4 )
wher e a s minima l = (N +·�) A .
2
I . = ( A 1 - A ) are obser v e d whe re ver
m�n 2
N ow int ermedi a t e poi nt s are c har a c t er i z e d by t h e i r order numb er (m) , whi ch has an i mp o r t ant r ole i n thi s e l ement ary d i s c u s si on of c oherenc e .
m =
f
=�!
= N + c ( 0 � c� 1 )
wher e N and c are the i n t e gral and frac t i ona l order numb e r respe c t i ve ly .
The o f the patt ern i s given by
V = I I max - I . + I . m�n
max m�n
=
and obvi ously i f t h e t w o int e r fering wave s have e qual amp l i tude ( A1 = A2 ) t h e n maxi mum visibi l i t y is a c hi e ve d throughout the f i e l d o f view , i . e . , V = 1 . 0 •
( 2- 5 )
( 2- 6 )
Thi s i d e a l c a s e i s never achi e v e d i n prac t i c e and although sharp fring e s o f reduced vi s i b i l i t y are f ound
for small path d i f f e r e nc e s l , and fringe s fade out as £ is inc rea s e d f or obs e r vati on point s far t h e r away fr om
t he c e ntral fringe. The reas ons for t hi s ar e e s sentially t w o f old in a real s ourc e , s i n c e (a ) any r e al s ourc e ha s finit e ext ens i on , and (b) a lways e mit s l i ght o f fini t e spe c t ral width . Thus t h e assumpt i ons made i n t he analysis o f e quat i ons ( 2- 1 ) t hr ou gh t o ( 2 - 6 ) above are i nvali d . N ow f or any real s ource t he r e e x i s t s unc e rt ai n t y in t he order number
'I
6 m = 2
whi c h i s a s s o c i at e d w i t h e very p oi nt of ob s e r vat i on ( p ) • Equat i on ( 2 -7 ) i s u s e ful s i n c e it shows t hat t he
( 2 -7 )
"bt"ISION OF \1/A\fttA(oNT (AP&fTuet�
U��K)
FI GURE 2- 1 . DI AGRAM FO R BAS I C COHERENCE TH EO R Y .
FIGURE 2-2 . DI AGRAM FOR TH E DER I V A TI ON OF TH E S P ATI AL
COHERENCE CONDI TI ON .
3 2 .
i n path di f fe r e n c e ( 6 £ ) c aused by t h e f i ni t e ext ens i on of t he sour c e , and t h e l a c k o f monochromat i c i t y given by the uncertainty in t he wavenumber (6k ). Obviously , i f l t m l � � ' no i nt e r f e r e n c e fringes will b e observed
and the t w o l i ght b eams are said t o be mut ually n oncoherent. For the limi t i n g c a s e \ t m ' = J we obtain fr om e quat i ons
( 2 - 5 ) and ( 2 - 7 ) , t hat
+
- n ( 2 - 8 )
whi c h d e fines t he l i mi t o f c oherence.
Nat ura lly , i f / llm l �
�
i . e . , i s smallc ompared with the half width of the fringe s , then t h e r e w i l l be n o Appre c i ab l e r e duc t i on i n the vi s i b i l i t y o f t he fringes from the c oher enc e point o f view , and this d e f i n e s t h e d e s i r e d c oherenc e c on di t i on.
i . e . ,
I f e quat i on ( 2 - 9 ) i s sat i s f i e d t h e n the t w o l i ght be ams are said t o be mut ually c oher9nt .
I n t hi s analys i s w e w i l l c onside r the two limi t i ng cases s eparat e ly where 6 k = 0 and .6£ = 0
respe c t i vely. Thi s i s e xperiment ally and t h e or e t i c al ly ( 2-9 )
c on v e ni ent , and allows a meaningful c ompari s on of t he t w o
i nt e r ferome t er sour c e s und er d i s cussi on t o b e made. ( a ) .:J P,:..TIAL
I n i t i ally , c on s ider p e r f e c t ly mon ochr omat i c l i ght the order numb er unc ertainty t he ext ens i on of the s ourc e ,
an (
( 6 m and
ext ended s our c e e m i t t ing
r 0) . I n t hi s 0 k = c a s e , + k.6 £ ) i s caused by = - t he sour c e i s sai d t o have per f e c t t e mp oral c oherenc e but li mit e d spa t ial c o h e r e nc e . The c ondi t i on for spat i a l c oher e n c e was
d e r i ved by Verdet3 0 in 1 8 69 , and may be obtained r ea d i ly from f i gure ( 2 - 2 ) . H e r e w1 and w2 repre s ent t w o wave front s e mana t i n g a t a small angle a t o each other
from a point sour c e at Q • C onside r t h e t w o beams t o i nt erfere at a d i stant point P and let £ be t h e p a t h d i fferenc e a t P owing t o t h e fini t e s o�r c e p o i nt Q • N ow at t he ot her e d ge of t h e s our ce re mot e f r om Q
33 .