Letters to the Editor: Sampling of the incoherent spectrum in two-channel system

(1)

Optica Applicata, Yól. X I I I , No. 3, 1983

Letters to the Editor

Sampling o f the incoherent spectrum in two-channel system

An n a Ma g i e k a

Institute o f Physics, Technical University o f Wroclaw, Wybrzeże Wyspiańskiego 27, 50-370 Wroclaw, Poland.

The information processing scheme of the incoherent optical system is based on the follow ing imaging relation:

I ( x ' , y ' ) = 0 ( x ' , y ' ) ® E ( x ' , y ' ) (1)

where I ( x ' , y'), 0 ( x ’ , y') denote the image and object intensity, respectively, while E ( x ' , y') is the impulse response of the system. In Fourier space the relation (1) has the form

y) = <

5

( | , y)G(£, p) (2)

where I(£,rj), 0(£,rj), and G(£,t]) are Fourier transform of I ( x ’ , y'), 0 ( x ' , y ' ) , E ( x ' , y ' ) , respectively.

From Eq. (2) we see that the object information 0 ( £ , rj) is filtered by the optical transfer function G( £, tj). In papers [1-3] it has been shown that the incoherent spectrum O is attain able through a proper choice of G(£), which should take the form o f a sampling function. The application of the pupil function (Pk) of the slit form the width of which increases prog ressively (Fig. 1) yields Gs (£) in the form of sampling function (Fig. 2).

In the described method the following recurrence formula was needed:

The sampled incoherent spectrum is then

sin(jr £Ax')

0(sA£) = [ f 0(£)Gk (£) 8111 (y æ)

ds_ 2

j 0(£)Gk^ ( £ )

sin (n£ Ax') r ~ sin (n£ Ax') ]

---— --- dtj + J 0(£)Gk- 2( £ ) --- ---d£\

(3)

№

The number of independent samples (N) that can be measured in the incoherent spectrum o f the object is bounded by the finite width o f the photo-diode (Ax') and minimum value o f the increment (A£)mjn, N < l/(A£)min(Ax') [1].

This paper describes the way in which the sampling function Gs (£) in-two-channel system is obtained. For this purpose the pupils functions P(£) (see Table) and the corre sponding autocorrelation were carried out. The results obtained are presented in Fig. 3. As it follows from Figs. 3 f-h the sampling function Gs (£) in two-channel system is obtained by using two slits pupils in phase (channel I) and two slits pupils in antiphase (channel II) - Fig. 4.

(2)

296 A . Mag ie ra

The sampled incoherent spectrum is then equal to OO

O(sAi)

=

J 5(£)&,.(()

— OO

sin (jr£

Ax')

nS

as,

(5)

and does not require the application of the recurrence formula. Numerical examples are presented in Fig. 5. . S-1< Pk s-m l. PIP -J___ -1 Pt?) 1. J Pk-1 -0.9 0.9 \ PR) 1 Pk-2 '0 8 0.8 2 PR) 1 -0.7 07 J Pt?) 1 -0.6 o.6 . 5 GK<5)-z-m GK.,I?)-1.8-|J| Gk.2(5 !-1.6 -|?I I 5 K 2 I ? k 1 8 F ig . 1. T h e p u p il fu n c t io n fo r o b t a in , in g th e s a m p lin g fu n c t io n (?g (£ ) in on e-ch a n n el sy s te m

(3)

The pupil functions used for sampling of the spatial frequence (s<d|) and corresponding transfer functions G({) 1 II (cc/2) 2 — I I (* + 0.5) + I I (* — 0.5) G(i) =

J 2 - I I I

lo

III < 2

III > 2

0 (1 ) =

2 — 3 HI

—

2 + |||

0 I l l < 1 1 < l l l < 2

III > 2

Plxi l l l < 1 . I ll > 1 3 ! ! ( » — 0.5) . X P(x) / 1 — 2111 0< | | | < 0.5 / as+0.75 1t / » — 0.75 \ + n — T T — l 0.5 J1 \ 0.5 / -1 -05 05 1 x « ( I ) 0 III — 1 2 - III VO 0.5 < HI < 1

1 < III < 3/2

3/2 < |i|< 2

III > 2

L

ett

er

a

to

t

h

e

E

d

ito

r

2

9

7

(4)

5 - Π æ + 0 . 7 5 Ô~5 G ( f ) = ' l - 2 | f | О Г - i f i 4 I f I — 2 0 0 2 Plxl 6 ( f ) = r 1 — 2 |ξ| 0 I f l - 0 . 8 1 . 8 - I f l ^0 0 < | f | < 0 . 5 0 . 5 1 .8 i.Plx) -Oi -<J3 03 - -Q8 X -0 ( f ) =

(

1 - 2 jf| ° I f l - 0 . 6 1 . 6 - I f l о 0 1 .6 8 П a; + 0 . 8 5 (КЗ Я!- 0 . 8 5 (КЗ Ptx) < ?( f ) = ' 0 . 6 — 2 |f| 0 I f l — 1 .4 2 - I f l kO 0 2 2 9 8 A . M a g i e r a

(5)

O p t ic a A p p li c a t a X I I I /3 n I a:+ 0.65 \ / a; — 0.65 \ A oi3 ) + n ( 0 3 ) PM -o.e -os ' os oa ■ x ' 0.6 — 2 ||| 0 < III < 0 . 3 0 0.3 < III < 1.0 G (l) = { l l l - l - O 1 .0< III < 1.3 1 . 6 - III 1.3 < III < 1.6 .0 III > 1 . 6 1PUI 10 n ( x + 0.45 \ / * — 0.45 \ ( 0 3 ) + n ( 0 3 ) 0 . 6 - 2 III 0 < III < 0 . 3 0 0.3 < III < 0.6 < ? ( { ) = { Ill- 0 . 6 0 6 <||| < 0 . 9 1.2- I I I 0 9 <||| < 1 . 2 .0 III > 1 . 2 Lett er s to th e E d it o r 2 9 9

(6)

16 16 PM

l x -

0.5 \

„ I x -

0.5 \ 1 1 ( 0.2 ) + Π ( 0.2 ) G (l) = < -06 -04 04 06 1 0.6-2111 0 0.6—1^1 II I - 1 .2 0 0< |f|< 0.3 0.3 < III < 0.6 0.6 < III < 0.9 0.9 < |f| < 1.2 III > 1.2 0.4 —2 If 1 0 < |f|< 0.2 0 0.2 < |f| < 0.8 II I - 0 .8 0.8 < |f|< 1 1.2- I I I К i ll< 1.2 0 III > 1.2 3 0 0 Δ . M agieb a

(7)

17 G<l) =

(

0 .4 - 2 HI ° 0 . 8 - III III- 1 . 2 0 0 < |||< 0.2 0.2 < HI < 0.8 0.8 < |i|< 1 1 1-2 pi«) •07 -03 03 07 X G(f> ’’ 0 . 8 - 2 III 0 III- 0 . 6 1 . 4 - III ^0 0 < |||< 0.4 0.4 < |{|< 0.6 0.6 < III < 1 1 < l i K 1.4 III > 1.4 19

-n (.

*'+ 0.5 \ 0.4 / + n

(

0 . 8 - 2 HI ° 0 . 6 - III III- 1 . 4 0 0 < |||< 0.4 0.4 < III < 0.6 0.6 1.4 L et te r» to th e E d it o r 3 0 1

(8)

302 A . Ma g ie b a

θη(ϊ)

-2.0

(9)

Letters to the Editor 303

(10)

804 A . Magieba ,_{■ w} 1 - 0l6 - 0 A P „ l l l - 0 6 -0 .4 0 4 0 6 ? “ 0 « 6 s i Sk : v : : : : : v : v v F ig . 4 . T h e p u p il fu n c t io n s f o r o b ta in in g th e s a m p lin g fu n c t io n Os ( i ) in t w o -ch a n n e l s y s te m

(11)

Zellers to the Editor 305 R eferen ces [ i ] Gt j i l l a m e B., Du v e r n o y J., Opt. Commun. 36 (1981), 4. £2] Gö r l i t z D., La n z l F., Opt. Commun. 20 (1977), 11. £3] Lo i i m a n A. W ., Rh o d e s W . T., Appl. Opt. 17 (1978), 7. Received February 28, 1983