• No results found

On Dependence of Resolving Power on Background Intensity, Stage of Resolution and Detecting Instrument

N/A
N/A
Protected

Academic year: 2023

Share "On Dependence of Resolving Power on Background Intensity, Stage of Resolution and Detecting Instrument"

Copied!
10
0
0

Loading.... (view fulltext now)

Full text

(1)

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

(2)

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)

(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)

$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 __

3.070 .2 0

4 .0 0 __ __

_

_ 0 .0 0

4 .5 6 0.41 __

6.07 — — - - 0 .5 6

6 .3 0 __ __ __

_

__ 0 .35

0 ,00 0 .0 0 __

1 3 .1] — — — 0 .5 0

10.28 __ __ __ , __ _

_

_ 0 .4 6

24.52 . — — .— 0.4 2

26.81 0.41

36.37 __ __ __ - _ ., - .' 0 .3 5

44.02 — ^— — — __ __ 0 .2 9

49.00

...—

—, — — — — 0 .0 0

(5)

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 y

D ependence o f U eiolving Pow er on Bacicgrotmd in ten sity, etc.

r l+ F

sin^

7^(n^y+n)

_ I + / --^0

w here

F

is th e coefficien t o f fineness and

cr ^ 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 or

n)

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

a

is real, ' ,

(6)

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 i

dA 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*

(7)

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 and

2ft

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 e

r

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

sin

0

A

(8)

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 r

a)

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

2

and

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 r

the-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 )

(9)

Table I I I illustrates the variation o f a with

k

and

c.

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

(10)

TA B LE I I I

Variation o f

a

with

k

and

c

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.

R E F E R E N C E S Ditohbum, 1930,

Proc. Boy. Irish. Acad.,

89, 68.

Sodha Carina, 1964,

Ind. J . Phys.,

88, 437.

Mitra, 1054,

Ind. J . Phys.,

28, 643.

Sodha, 1064,

Ind. J . Phys.,

88, 141.

References

Related documents

As drinking water becomes moderately saline (above 4.11 mS/cm), the marginal effect becomes positive and increasing, with associated high infant mortality and high-level

A comparison of the monsoon activity for two contrasting years, 1908 (which incidentally had only few break days) and 2002 (which incidentally had only few active days) is given

Results of ANOVA indicating significant differences (P&lt;0.05) for Life span and reproductive characteristics of A.franciscana (A.f) and A.parthenogenetica (A.p) from

Attempts to identify poly- observed in all the gels were excluded from the morphic loci from general protein zymograms general pherogram pattern The relative mobility have

The pump power dependence of the emission bands at 550 and 670 nm were investigated on excitation intensity at 980 nm and the experimental data for 550 and 670 nm emission bands

Provided further that in case where continuous supply of service, as defined in sub-section (33) of section 2 of the Central Goods and Services Tax Act, 2017, is provided by

Ditchbum has suggested that a combination of stage of resolution desired and the detecting instrument is characterized by C, the ratio of the minimum to maximum

It can be seen that the background intensity has an important bearing on the choice of an instrument for a particular