ON DEPENDENCE OF RESOLVING POWER ON BACKGROU ND INTENSITY. STAGE OF
RESOLUTION AND DETECTING INSTRUMENT
K . C. C H A T U R V E D J A N D M. S. S O D H A KaMLANA^UU,
jot pubUi'Utioi}^ 2, IHr»())
AB STR A CT. In this pafXM* authors ha\ ( ' tho \ srititioti <»I r<"solving |»owor
of'
prism, grating, rf^flecting <H*h('lon and Kahry-INjrol oialorj with haiUgruund inLonsity, stago Oj* resolution desinxi and the detecting instrument, when natural liiu* width is m'gligihle, 'Phe ease when instrumental widtli is negligihle has also heeii discussed.I N t H 0 I) II C’ T 1 () N
D itch bu rn (1930) has jioiiited out th a t the resolving pow er ol an instru- riieiit d ep en d s on the stage o f resolution desired and tlie deteeting instrument.
A g iven eon ib in a tion o f deteetin g instrum ent and the stage o f resolution desired is ch aracterized b y
c,
the ratio o f minimiun to m axim um of t he resultant intensity p attern o f tw o lines fo r optim um resolution. F or exam ple, when the sp cctro- gram is exam ined by a, m ierop h otom eter we have.i) Detection of inhomogeneity in radiation when c — 0.98.
ii) P artial resolu tion (ap p roxim ate measurement o f wavelength seiiara- tion ) when
c -=
0.8.an d iii) C om p lete resolution (m easurem ent of wavelength separation and rela
tiv e intensities) when c — th4.
T h e v a ria tion o f resolving p ow er ot various instrum ents witli c has been stu d ied b y Sharm a and S odha (1954) and M itra (1954), when the background in ten sity is zero. .Sodha (1954)has discussed the variation o f resolving pow er, w ith b a ck g ro u n d in ten sity on R a yleig h s criterion (o = 0.8).
In th is p a p er th e authors h ave investigated th e d ep enden ce o f resolvin g p o w e r o n
Jk
a n d c i.e. d etectin g in stn u n en t, stage o f resolution desire<l and the b a ck g ro u n d in ten sity present. T w o im portan t eases h ave been distinguished,i) when instrum ental width is n egligible and the intensity distribu tion o f a line is g o v e rn e d b y D op p ler effect
ii) w h en D o p p le r w id th is negligible and the intensity d istrib u tion o f a line is g o v e rn e d b y the instrum ent.
643 55
N E G L I G I B L E I N S T R U M E N T A L W I D T H
T h e in ten sity d istrib u tio n o f a sp ectra l line o f w a v e n u m b er d u e to D o p p le r e ff o r t is g iv e n b y
/ ' = l o e x p { — //( v — v„)*}
w h ere / / = / / b e in g th e m a ss o f r a d ia n t a to m s .
T h e in t o n s it y d is t r ib u t io n o f a n o t h e r s p e c tr a l lin e o f w a v e n u m b e r ( v ^ + A v ) a n d s a m e in t o n s it y is
/ " = Jo e x p { - ; ? ( v - ~ V o - A v ) 2 } i f A v is s m a ll {ft s a m e f o r b o t h lin e s).
P u t tin g ^/ft(v~~VQ) “ y a n d \ //? . A v r/, th e r e su lta n t in t e n s it y p a tte r n in th e p re s e n ce o f a b a c k g r o u n d in t e n s it y is g iv e n b y
/ = W „+ /o C -*^ + /o C
N e g le c t in g th e s h r in k a g e e f f e c t , th e in te n s ity m a x im u m (.r — 0 o r a ) a n d in t e n s ity m in im u m ( x — a l 2 ) a re g iv e n b y
544 K . G. Chaturvedi and M . 8 . Sodha
a n d
P u ttin g
_ /r I 2c fn
w e h a v e
T h e r e s o lv in g p o w e r is g iv e n b y
V c . . / I Z
d A H iv a ■ ® ®V2i?!r
w here
a = l
a
(1)
(
2
)(3)
T a b le I , c o m p u te d fro m eq u a tion s (1) and (3), illustrates th e va ria tion o f a w ith
k
a n d c. T h e results h ave been illustrated b y figures 1 and 2,Depe.nde.nce, af Resolving Rower on Background Intensity^ etc. 646
Fig. 1, Variation of a with k and c ( 0 .4 --0 .8 ) when instrumental width is negligible.
Fig. 2. Variation o f « with it and c ( 0 .0 ^ 0 .9 8 ) when instrumental width is negligible.
2
$4^. .. K , C, CJ^turvedi and M . B^M odha . .. ,
T A B L E I
Variation o f a with fc -and o when the instrumental width* is negligible ^ •
h 0 - 0 .4 c - 0 .5 c - 0 .6 c - 0 .7 c =: 0 .8 c = 0 .9 c = 0 .9 8
0.00 — — — — 0 .5 3 — —
0.04 — — . — — — — —
0.08 0 .3 8 — 0 .4 5 — — — •—
O.IJ __ 0.41 „
_
- , ■0. J4 — — — 0.48 — — __
0.22 0.35 — — — — •— —
o.:m) __ __ 0.41
_
0 .5 0 0 .56,
0.44 — — - - 0 .4 5 — — __
0.40 — 0.35 — — — —
0.5 0 0 .2 9 _
0.660 0 .0 0 — - — __ ! __ ___
0 .80 — 0 .2 0 *— — — —
0.S2 _
_
0 .35 j_0 .85 — — 0.41 - p__ __ __
1.00 — 0 .0 0 — . r— ^ ; — — —
1.02 — '9.46 '
( •
— __
1.09 — 0.3 2 1
— —. __
1.25 - - — 0i20 ' 1__ \
f 1 f — —
J.43
i
J*. __ ■ 0.3f) -H-*. . . .
J. 50 — — 0 .0 0 — 0 .42 __
1.70 — — 0.32 — 0 .5 0 —
1.00 ___ __ __
_
0 .3 9 __
2.:i8 — 0 .0 0 — — 0 .6 0
2.50 — 0 .3 6 — —
;^.oo ..__ ___ _ 0 .33 0 .4 6
:i.50 — — .— 0.2 9 — __
3.07 — — — 0 .2 0 — —
4 .0 0 __ __
_
_ 0 .0 04 .5 6 — — — 0.41 __
6.07 — — — — - - 0 .5 6
6 .3 0 __ __ __
_
__ 0 .350 ,00 — — — — 0 .0 0 __
1 3 .1] — — — — — — 0 .5 0
10.28 __ __ __ , __ _
_
_ 0 .4 624.52 .— — — — — .— 0.4 2
26.81 — — — — — 0.41
36.37 __ __ __ - _ ., - .' 0 .3 5
44.02 — ^— — — __ __ 0 .2 9
49.00
...—
—, — — — — 0 .0 0
F A B 'R Y -F E () T ' ft T A L () N ’ ’
T h e in te n sity p a tte rn o f a spectral line in the order where
n
is a fVaf- tio 1 an d Uqan in teger is given fo r F a b ry P erot etaloii h yD ependence o f U eiolving Pow er on Bacicgrotmd in ten sity, etc.
r l+ F
sin^7^(n^y+n)
_ I + / --^0w here
F
is th e coefficien t o f fineness andcr ^ n n F ^ '
’'• T h e in te n s ity p a tte rn ’ o f a'ncJther spectral line o f'e q u a l ihlerisity fmi>^irmnn an d sep a ra ted b y a sm all order
An
is given b y ' ' ' 'j" — Jo __ ^0
1 + /^ sin2 77'(i?.„+/7 — A??) I
\ (x ay^
w here
a - n , An
.F \
T h e resu lta n t in ten sity pattern , in the presence o f a ba(’kground intiMisily, w h ich is equ al to k tim es th e intensity m axim um , is given by
I
/ _ , I , J
/ o ^ 1 1 | ‘
N e g le ctin g shrinkage e ffe ct, the m axim um
(x
— 0 orn)
and minimum{sc
a /2 ) o f th e resultard pattern are given b y >r / ' '■ '
14--«^
I <
i.and
^7/1/U — 4_
F o r lim itin g resolu tion , p u ttin g
^
c
we
h a v e(
/■ t.a * (c + c J k -jfc )-« * (5 A :-5 c fc 4 S~ 6 r,).^ 4 {«-+ 2 -cA :-2 < ^ )= - -0
or
, (5ifc— 5 c if c + 8 '- ^ 6 c )+ V ( 5 * —
-^K..
(4)a
— — ^' 2(c+ck—k)
since
ais real, ' ,
T h e resolv in g p o w e r o f th e F a b r y P e r o t e ta lo n is g iv e n b y
^ = ^ =
^ . . n o F i ^ a . n , F idA an a
548 K , C. Ghaturvedi avd M . 8 . 8odha
... (5)
... (
6)
w here
a =
n/a
an d th e va lu e o f
a
is g iv e n b y e q u a tio n (4).T a b le I I illu strates th e v a ria tio n o f a w ith
Ic
an d c. T h e resu lts h a v e been illu strated in figures 3 and 4.Fig. 3. Variation of a with k and c ( 0 .4 —0.8) for F. P. etalon.
Fig. 4. Variation o f « with k and c (0 .9 —0.98) for F. Pi etalon*
T A B L E I I
D&p&nd&nce of Reserving Power on Background Intensity, etc. 649
V a r i a t i o n o f a w i t h k a n d c f o r F . P . e t a l o n .
k c = 0 . 4 0 0,r> c - = 0 . 6 f - - 0 . 7 c = 0 . 8 c = 0 . 9 c = 0 . 9 8
0 . 0 0 . 8 2 0 . 9 6 1.11 1.28 1.4 9 1.7 6 2 . 1 0
0 . 1 0 . 7 4 — — — — — —
0 . 2 0 . 6 6 0 . 8 3 1 .0 0 — - - — —
0 . 4 0 . 4 8 0 . 6 9 __ 1,1 0 1.34 __
0 . 6 — 0 . 5 5 0 . 7 8 — — — —
0 . 6 6 6 0 — — — — —
0 . 8 __
_
0 . 3 8 0 . 9 3 __ — —1 . 0 — . 0 0 .5 5 ,— 1 .1 5 — —
1 .2 — — 0.4 1 0 . 7 6 — • — —
1 . 5 __ __ 0 *,— — — —
1 . 6 — — — 0 . 5 8 — — —
2 . 0 — — — 0 . 3 8 0 .8 5 1.34 —
2 .3 3 3
_ __
__ 0 — — —3 . 0 — — — 0 .5 6 '—. —
4 . 0 — — — — 0 1.01 —
6 , 0
_
___ — 0 .7 2 —8 . 0 — . — — — — 0 .3 8 —
0 . 0 — — — — “ 0
w.o
2 0 . 0 3 0 . 0
— .
__ —
—
—
1.52 1.1 5 0 .8 4
4 0 . 0 — . — — — —• — 0 . 5 4
4 0 . 0 — — '—. — U
G R A T I N G , R E F L E C T I N G E C H E L O N A N D r H I S M T h e in te n sity o f a sp ectral line d iffra cted b y a grating or a reflecting echelon is g iv e n b y
_ 1 sin®
Nfi l \ ~
if® * sin®/?w h ere is th e in ten sity m a x im u m ,
N
is th e n um ber o f lines o f the grating or the n u m b er o f step s in th e_reflectin g ech elon and2ft
th e phase differen ce betw een t w o a d ja c e n t b ea m s.P u ttin g
X
=Nfi,
w e h a v er
sin®a:w h en
p
is sm all.T h e a b o v e ex p ression also represents th e in ten sity d istrib u tion in a
prism
i f
X
sin0
A
560 K . G. Chqturvedi and M . 8. SvcOta;
T h e in ten sity d istrib u tio n o f ianbtlife^ lin e o f th e sam e in te n sity an d an angular sep aration corre sp o n d in g
to Ax — a
is g iv e n b y ,/ " _ sin*
(x—a)
...^0
T h e resu ltan t in te n sity d istrib u tio n o f tlie tw o lines, w hen th e b a ck g ro u n d
in ten sity is ib/g, is g iv en b y ^ *
I j
, sin* X.
sin*(x—a)
/ ; = * ’ + - y - +
N eglectin g shrinkage effect th e in te n sity m a x im u m
{x = 0
o ra)
a n d m inim u m (a? = a /2 ) o f th e resu ltan t p attern are g iv en b y= l + i f c + sin*
a a
2and
2 s in * (a /2 )
h
(«/2 )*F o r lim itin g resolu tion , p u ttin g
^♦ntn _= C
^max
w e h a v e
^ ^ 8 s iii* (a /2 )
\
1 ^ ) I m - c )
a * ( l - c )T h e resolv in g p o w e r o f th e
grating
o rthe-reflecting echelon
is g iv en b y:N u = a :'Nn
dX Afi a
f f- > <
a n d th e resolvin g p o w e r o f th e
prism
is g iv en b y-r (7)
... (8)
dX
where
a
=nja.
the value of a being given by equation <7).
• ,^.v {9)
... ( 10 )
Table I I I illustrates the variation o f a with
k
andc.
The results have been iljjustrated in figures .5 and 6. _Dependence of Mesolving Power on Background Intensity, etc. 561
Fig. r>. Varifttion of a with,./* and c 0.8) for grating, reflecting echelon and pfisin.
F ig. 6. Variation o f a with k and c ( 0 . 9 —0.9 8 ) for grating, reHecting ocholon and prism
TA B LE I I I
Variation o f
a
withk
andc
for grating, reflecting echelon and prism.552 K . C. Chaturvedi and M . 8. Sodha
h c = 0 .4 c = 0 .5 c = 0 .6 c = 0 .7 c = 0 .8 c = 0 .9 c = 0 .98
0 .00 ____ ____ ____ ____ 1.00 ____ 1.115
0.01 — 0.83 — — — — —
0 .00 — — — — — 1.05 —
0.17 — ____ ____ _ 0.91 ____ ____ ____
0.26 0.71 — 0.83 — — 1.04 —
0.61 0.63 0.71 — — — — —
0.63 0.55 _____ ____ ____ ____ ____ ____
0.67 O.iiO — — — — ____ —
0.60 — — — 0.83 — — —
0.76 — — — — 0.91 ____ 1.11
0.82 — 0.6 3 — — — ____ —
0.89 — — 0.71 — — 1.00 —
0.96 ____ 0.65 ____ ____ ____ ____ ____
1.00 — 0.5 0 — ____ ____ ____ ____
1.28 — — 0.63 — — — —
1.46 ____ ____ 0,55 ____ ____ ____
1.60 — — 0 .50 ____ — ____ —
1.65 — — — 0.71 0.83 —
2.06 — — . _____ 0.63 ____ ____
2.28 — — — 0.55 — .—
2.33 — — — 0 .50 — — —
2.83 — ____ — ____ 0.71 ____
3.60 — — — — 0.63 — —
3.92 — — — — 0.55 — —
4.00 ____ ____ ____ ____ 0.50 ____ ____
4.13 — — — — — 0.83 —
4.49 — — — — 1.05
6.72 — — — ____ ____ 0.71 —
6.8 6 — — — — — — 1.02
8.23 — — — — — 0.63 —
8.47 ____ ____ — _____ ____ ____ 1.00
8.86 — — — — . — . 0.55 —
9.00 — — — — — 0 .50 —
16.69 ____ ____ ____ - . ____ ____ 0.91
24.76 ____ ____ — ____ ____ 0.83
37.79 — — — — — — 0.71
46.28 - — — 0.63
48.34 — ____ ____ — — 0.55
49.00 — — — — — — 0 .50
A C K N O W L E D G M E N T S
T h e a u th ors are g ra tefu l t o D r. D . S. K o t h a r i, S cie n tific A d v ise r t o th e Ministry o f Defence, for permission to publish this paper.