Fringe visibility of dual-aperture sampling withpartially coherent illumination

F. T. S. Yu and Y. W. Zhang

The effect on fringe visibility of a dual-aperture imaging system produced by two mutually partially coherentpoint sources is studied. The two mutually partially coherent point sources are generated by an originalextended incoherent source. The problem formulation is developed from the partial coherence theory ofWolf. The results show that the fringe visibility is affected by the spectral bandwidth, source size, samplingaperture size, as well as the defocused distance of the imaging system. These results are quite consistent withthe Thompson's predictions of Young's experiment.

1. Introduction

The earliest investigation of the subject of partialcoherence may be that of Verdet who in 1865 studiedthe region of coherence for light from an extendedsource. Some of the more important developments ofpartial coherence theory are those due to Van Cittert2in 1934 and Zernike3 in 1938. They determined thedegree of coherence for light disturbances at any twopoints on a screen illuminated by an extended lightsource. However, it was Wolf's mutual coherencefunction4 in 1957 that made a broader scope of applica-tions possible for coherence theory. In addition, it wasthe two-beam interferometric technique of Thompsonand Wolf56 that provided a practical measurementtechnique of the degree of coherence.

Recently, Thompson and Sudol7 addressed theproblem of finite-aperture effects in the measurementof the degree of coherence. They presented a 1-Danalysis showing that the fringe visibility of a two-beam interferogram can be predicted by the convolu-tion of two finite apertures under illumination. Theyhave developed simple formulas to gain insight for theapplication of the theory. More recently, Marathayand Pollock8 generalized the study of Thompson andSudol elegantly by utilizing a 2-D approach. Theiranalyses7 8 have shown that the effect of aperture sizebecomes predominant in the neighborhood of the zerosof the coherence or in regions where the coherencefunction is rapidly changing. While Thompson and

Sudol7 and Marathay and Pollock8 intended to pointout that there is no separation of apertures for whichthe fringes disappear completely due to finite size ofthe aperture, we shall analyze the decrease of the fringevisibility due to the spectral bandwidth, source size,sampling aperture size, and defocused distance by fol-lowing a similar approach. 7 8

In this paper we shall study the effects of fringevisibility under partially coherent illumination. Twomutually partially coherent point sources derived byan extended incoherent source are imaged by a dual-aperture imaging system to produce interferencefringes. We shall utilize a partial coherence theory toanalyze a dual-aperture imaging system under partial-ly coherent illumination.

II. Problem Formulation

The layout of the system under consideration isdepicted in Fig. 1. It shows that a diffused surface isilluminated by a collimated partially coherent light atthe object plane of a double-aperture imaging system.The imaging lens of Fig. 1 is assumed very thin so thatthe pair of sampling apertures, which are touching thelens, can be located at either the entrance or exit pupil.

In the problem formulation, we assumed that a lin-ear extended incoherent source is utilized. If the in-tensity distribution of the source is y(O), then, for agiven wavelength X, the mutual intensity function in a1-D form arriving at the object plane Pi would be4

r(Q142;X) = 00 (0) exp[-ik(Q1 - 2)0]d0,

The authors are with Pennsylvania State University, ElectricalEngineering Department, University Park, Pennsylvania 16802.

Received 22 January 1986.0003-6935/86/183191-06$02.00/0.© 1986 Optical Society of America.

(1)

whereAs/2

[(D1 /2)2+ f1

2]1/2 (2)

is the source divergent angle, D is the diameter of the

15 September 1986 / Vol. 25, No. 18 / APPLIED OPTICS 3191

S extended) source

L sampling1' 'aperture

diffusing 'surface I

I(ca;X) = 4

1 L2 JJJJr (4,,2;x)t(,)t*(2)

{ k e (XI - {1 )2 (X2 2)2

x2 + + - L]}

X dldt 2dxldx 2

Fig. 1. Dual-aperture imaging system with partially coherent illu-mination. Li, collimated lens; L2, imaging lens; P,, object plane; P3,

output plane.

collimating lens Li, fi is the focal length of LI, and k =27r/X.

If we assume that the diffused complex light fieldfrom the object plane P1 is t), the mutual intensityfunction reflected from diffused object at P1 would be

rt"Q42;X = r(s,4l2;X)t(t)t*(Q2) (3)

where the superscript asterisk denotes the complexconjugate. Due to the Fresnel diffraction from P1 toP2, the mutual intensity function at the front of thesampling apertures would be

r(xl x2;X) = ' ffrJP(Qi12;X)

X exp -i h [(X1 - 41)2 - (X2 - 42)2] ddt2- (4)

The the mutual intensity function immediately behindthe image lens would be

r'(xlx 2 ;X) = r(XsIx 2;X)T(x,)T*(x 2 ),

where

(exp i-i2 -dX D+d T(x) A ( 2f 2 2 2 2

J, otherwise,

(6)

f is the focal length of the imaging lens L2. Therefore,the output mutual intensity function at P 3 would be

ral,a 2 ;X) = 1 J Jr/(x11x2 ;X)

X exp{-ik [(ca, - x) 2- (a2 - x 2)21} dxdX 2 . (7)

It is apparent that the corresponding intensity distri-bution due to X at P 3 can be written as

We are now in a position to raise a fundamental issue:to what extent would the visibility of the output inter-ference fringes be affected by the degree of coherenceof the illumination? To answer this question we shallutilize the general formulation of Eq. (8) to determinethe effects.

Ill. Effect under Temporally Partially CoherentIllumination

We shall first determine the effect under temporallypartial coherent illumination. To do so, we let thelight source S be a point (i.e., As = 0) but with finitebandwidth (i.e., AX 0). Thus the overall outputimage irradiance at P 3 would be

ro + AX/2> I~~~(aY) = j I(oa;X)dX, (9)

-. -A\/2

where X0 is the center wavelength of the light source,and AX is its spectral bandwidth.

The light field within the sampling aperture withradius d can be kept coherence if It(,-)t*(Q,,q) has ashape and size similar to the Airy spot produced by theaperture. However, in this paper we shall discuss aspecial case where the dual aperture is illuminated bytwo mutually partially coherent point sources.

Let us now take two arbitrary object points at P1with separation equals to the Airy spot projected intothe source plane, i.e.,

> ~~~tQt) = Qt - t + Qt + WI) (10)

where

to Ai .22X (11)

sing = d (12)[12 + D2 )2]1/2

and the magnification of the Airy spot is assumedunity.

We note that 40 could be another value rather thanthe Airy spot in the following analysis, but the Airyspot would be important if an extended object at P isconsidered. By substituting the above equations intoEq. (8), we show that

I(ca;X) = A(a;X)A*(a;X), (13)

where

3192 APPLIED OPTICS / Vol. 25, No. 18 / 15 September 1986

(8)

A(a;A X1L exp{-i2( + -)1

+ + L)] {exp ik( j + ")xi]

+ exp[-ik (•o + a)x 1] }dxj. (14)

Since the output plane P3 is assumed at some dis-tance Az away from the imaging plane, i.e., Az << z, theGaussian lens formula would still be a good approxi-mation for the analysis, i.e.,

1 f

Thus Eq. (14) can be written as

A(a;X) = AI(a;X) + A2 (a,X),

(a) 4 )2 A 1 sin2 7rd(la+L 0 )

71a-2(la + Lto)2 fJA-^A 2 AI

X coS2[ rD(la +L) ]d,XlL jX

4 J0+AX 1 2Frd(la +Lo) lir2 (la - Lt 0 )2 \ A X2 L XIL J

X os2[ rD(la - °L) ]d

43(a) A a 4 )J x 1 sin[ 7rd(la L 0 )]7r(XAan -L 2) A- x2 A AIL

i s47rd(la - Lt0)(15)

+ 1 ,, D +d D d.

Dd _ 2 2 2 2

X exp[ik ( + .0 xi]dxl}

4 ex ki- +)= r(I2 L )2rX(la + Lt0)

X sin[k(L + - cos

X exp[ik (a _ A°)x1]dx 5}p-k (a2 + '02)]

- ex {-i ( D+ D)

2r (a _ ~0 )

2ir(la -Lt)

X sin[k( - )d] cos[k( - )2 L 1 )(L 1 2

(16)

(17)

With reference to Eqs. (15), (16), and (17), Eq. (13) canbe written in terms of those quantities:

I(a;X) = IAs(a;,)I2 + 1A2 (a;X)1 + Al(a;X)A2(a;)

+ Aj(a;X)A2 (a,X). (18)

The overall output image irradiance at P 3 can bewritten as

I(a) = Il(a) + I2(a) + I3(a), (19)

where

X cost AIL - ] cost AIL I ]dX. (22)[ XIL I XlL j

It is now apparent that the fringe visibility as defined,

VWa A Imax(- -Ija)(23)Imax(a) + Imin(a)

can be written as (see Appendix)

V(a) = {(la - L 0)2 sinc[ 2d(la + ])AXJ

+ (la + Lt 0)2 sinc 2d(la - ])} 2 (12a 2 + L2 t).

(24)

Figure 2 shows plots of fringe visibility as a functionof a for various spectral bandwidths of the light source.From this figure, we see that the visibility decreases asthe spectral bandwidth of the light source increases,which is quite consistent with Thompson's prediction.Figure 3 shows the effect of fringe visibility due tosampling aperture size. Thus we see that the visibilitydecreases as the aperture size increases. Figure 4

,.0

0.8

0o6

0.4

1e600mm L601mm

d 3 mm D=50 mm

,= 5461

AX= a o

0.2 \ \1000 A \ \ \

0.0 1 ~\3000 i A 0.0

0.130 0.154 0.178 0.202 0,26 0.250 a(mm)

Fig. 2. Effect of fringe visibility due spectral bandwidth AX of thelight source under temporally partially coherent illumination.

15 September 1986 / Vol. 25, No. 18 / APPLIED OPTICS 3193

(20)

(21)

whereIk (.2

exp - 2k(Al(a;X) - 2 L

Vca}

1.0

Vca,

1.0

0.8

o.e

0.2

0.0

0.130 0.154 0.178 0.202 0.226 0.250 a(mm)

Fig. 3. Effect of fringe visibility due to sampling aperture size dunder temporally partially coherent illumination.

shows the effect due to deviation of Az (i.e., defocuseddistance). This figure shows that the fringe visibilitydecreases as the defocused distance Az increases.

IV. Effect under Spatially Partially Coherent Illumination

We shall now consider the effect due to spatiallycoherent illumination. In this case, we assume thatthe light source is temporally coherent but with finiteextension (i.e., AX = 0 and As 5d 0). The source irradi-ance may be written as

(0) = {O, 1 s (25l0, otherwise,

where 0o is the source divergent angle.By letting X = o, the quantity of A(a;Xo) can be

derived from Eq. (8), i.e.,

A(a;Xo) = 1 exp(-iko10t)[(Q, - to) + 5(Q, + so)]

Xex { ..Lr(xJt1)2 -2 (a X)211X exp I- 2 I f + L j djdx,

which can be written as

4ex;Ao)= Itk, (I 2 2A(a;X0 ) = 2 x- kLII sin ko a 4o d1

2i0'o(la + L~0) [0 kL 1/)2]

x Cos ko (~+ .~0 - exp[-iko~o0]

4ex i +[ 2 (L )a2

+ 2\ L i k , - 02, \O(la - Lo I (L I r!]

x Cos ko (~ 0 D+~' exp(iko~00), (27)( ) 2](7

where ko = 27r/Xo, and X0 is the wavelength of the lightsource. Thus the overall output image irradiance atP3would be

0.8

0.6

0.4

0.2

1=6oo mm d_3 mm

D=50 mm X.5461 A

AX= 100

m mm

0.0 1 , , , . ,

0.130 0.154 0.178 0.202 0.226 0.250 (MM)

Fig. 4. Effect of fringe visibility due to defocused distance Az undertemporally partially coherent illumination.

I(a)x=JI IA(aX;o)I2dO

= { X.2(la+ Lto)2 sin [ko (L + cos2 [k (a + 'o)D]

X2r2 (a - ~)2 L \L /2 L \L l 2

+ 42(a 2 _ sin2[ko (L _ )-2] cos2[,o (a _o+ Dj

+ 82r(2X _ L2 in k (L + I0)2] cos[k ( I ' ) D

X sin ko (~-!)]cos[ko ( ~~sinc(4-0 )200.[°L )2] [°L 1 2 (Xo |

(28)

As similar to the foregoing temporally partially coher-ent source, the effect of the fringe visibility due to thespatially partially coherent illumination can be evalu-ated from the above equation.

Figures 5-7 show the plots of visibility as a functionof distance a for various values of source size As, sam-pling aperture size d, and defocused distance Az, re-spectively. From these figures, we notice that thevisibility decreases as the source size increases, thesampling aperture enlarges, and Az increases. Al-though the variation of the fringe visibility due to anextended source behaves in a similar manner as thebandwidth variation, the effect due to increasingbandwidth (i.e., AX) is more sensitive, as can be seenfrom Figs. 2 and 5. In terms of the sampling aperturesize (i.e., d), we have seen that the effect on fringevisibility is not much different for the two cases. How-ever, fringe visibility is more sensitive to aperture sized than to the separation D. As for the changes ofdefocus distance Az, the effect on the fringe visibilityhas generally behaved similarly for both cases.

Finally we stress that the results that we have ob-tained, using two mutually partially coherent pointsources, are quite consistent with the two-beam inter-ference measurement of Thompson.5 6

3194 APPLIED OPTICS / Vol. 25, No. 18 / 15 September 1986

(25)

s 600 mm L 601 mm

d.3 mm 0=50 mm

X.= 5461 A

"..< As=o.I mm

0.130 0.154 0.178 0.202 0226 0.250 almm)

Fig. 5. Effect of fringe visibility due to source size As under spatial-ly partially coherent illumination.

0.6 U Z

0

o~~~~~~~~o~~s0.2 -- 0

0.0

0.130 0.154 0.178 0.202 0.226 0.250 almm)

Fig. 6. Effect of fringe visibility due to sampling aperture size dunder spatially partially coherent illumination.

Appendix

By completing the integrals of Eqs. (20)-(22), wehave

I (a) = - IL (Ird(la + Lt0) rd(la + L 0)= 7r3(la + Lt 0 )3 d L(X0 - AX) IL(X0 + AX)

1 . 2rd(la + L 0) 1 . F2rd(la + L 0)1+-sinl -sinII2 lIL(X + AX) 2 IL(X0A A)

+ d sin 2rD(la + Lt0 )1 sin2F rd(la + Lt 0)1D+ dl [ IL(X +AX) J IL(X + AX) J

- 2TrD(la + L 0) 1 2rd(la + L)L L(X 0 -AX) 1 [IL(X 0A ) JJd s i 2ir(D - d)(la + L 0)1

2(D- d) |sL IL(X0 -AX) J

sin 27r(D - d)(la + L 0)TILIL(xo -AX) J

d s 7rD(Ia + L 0) S 27rD(a + L 0)12D sl IL(X + AX) J IL(Xo -A) J)

(Al)

0.4

0.2

0.0

IL Ird(la - L40) rd(la - L)12 (a) =r(1a - Lt0)3 d IL(X -AX) IL(X + AX)

1 .[27rd(la - LO) 1 . [2rd(la -LO+ 2 s IL(X+ AX) J 2 LL(XA AX) J+(d ) sin E 2rD(la - Lt0)1 sin2Frd(la -Lto)

(D + d)) IL(X0+ AX) LL(X + AX) J-27D( -l - 1 r7rd(la -Lto)l

- sira 0 sin2I ~)I1L(L-LAX) J L L(X 0 -AAX)

d fi 27r(D - d)(la - L 0)12(D- d) sin IL(xo-AX) J

- sin2r (D - d)la - L~O)[ IL(X0 -A) )

d s. r2rD(la - LO) . F27rD(a - Lt0) \2D L IL(A + AX) LIL( X 0 A) JJ)

(A2)

13 (a) 1 2 d [sin( 27rgd27r2(12a,2 - L2 )d 1 (27r~od -2iraD

27raD ) 1 / 2 rtod 27raD ) A~~-J (~A X A) 1n LJ(A-)

+ d(2rod+ 2lraD

_ + J

2rto 2raD 1 ./2rtod[sink i~ + 27ra sink 7- 0* lrmD) .0 + -AX)

2raD I dL d (AO-+A)2 (27rtod 27rtOD)

( I L J

[(2d I L) (O + AX) ( - LD)1 + d - .i2irtod

L + J

2rtod 2rtODX -sin + 2

L (AX)

d r [sin( 27rad 2raD(27rad _ LraD L

1 . (2irad 2iraD 1 1(OX + AX) 1 L I / (x-AX)I

15 September 1986 / Vol. 25, No. 18 / APPLIED OPTICS 3195

( d2irad+ 2raD\

L I)

Vca)

1.0

0.8

[sin(27rad + 2iraD ) A sin( 2 rad. 2aL (A + A X) L

+2raD I I (A) TxvA)I

0.6

0.4

0.2d . 2sin(2rad 21rtoDX 1

I L)

- (27rad - 27rIoD ) (X0A- d- sin Lrad )7D ( h 2lrad + 22rtoD)- I L J(27 d + 2ir(2r)d(X AX

r ,i 2ad 27rt0P\ 11 L )(A + A)

0.0

1 600 mm d=3 mm

Do 50 mm X5461 A

mz= mm

0.130 0.154 0.178 0.202 0.226 0.250 a(mm)

Fig.7. Effect of fringe visibility due to defocused distance Az underspatially partially coherent illumination.

- sin2 7rad + 2rOD) 1 A

-sn I +L ) o-AA, X) .f

(A3)

If we assume that D >> d, the quantity of I3(a) is muchsmaller than Il(a) and I2(a). Thus I3(a) can generallybe neglected, and I,(a) and I2(a) can be further simpli-fied as

Thus the corresponding fringe visibility can be writtenas

V(a) = (la - L 0)2 sinc E l+L 0)A

+ (la + Lt 0)2 sin [2d(la + LO)AX}/2(12a2 + L2 2).

(A7)

L fird(la + Lt0) rd(la + L 0)Ir

2(Ia +L)

3 d | lL(o AX) IL(X0 + AX)

1 2rd(la + L 0) 1 i 27rd(la + Lt 0)

2 IL(Xo+ AX) - 2 [ IL(X -AX) J(A4)

We acknowledge the support of the U.S. Air ForceOffice of Scientific Research AFOSR grant 83-0140.

I2(a) = IL { rd(la - Lt0) ird(la - L 0)r 3(1a -L )3 d lIL(X0 - AX) IL(X0 + AX)

+1 sing 2rd(la - L 0) 1 sin 2rd(la - Lo)12 IL(Xo + AX) J 2 L 1L(X0 + AX) IJ

(A5)

By expanding 1/(Xo + AX) and 1/(Xo - AX) in binominalseries and neglecting the higher-order terms of (AX)2,the output image irradiance can be written as

I(a) I(a) + I2(a) = 2(Ia + L )2Xg

X {1 +sinc 2d(la + ) A Cos 27rd(la + Lto)]}

smc \L2 gcs L,\ j

+ 72(aAX {1 + sinc 2d(la - L 0)AAILX~ J

27rd(la - L ) Icos LO

References1. E. Verdet, Interferences en General, Vol. 5/6, Lecons d'Opt.

Phys. (L'Imprimerie Imperiale, Paris, 1869), 1, p. 106.2. P. H. Van Cittert, "Die Wahrschcinlicke Schwingungs verteilung

in einer von einer lichtquelle direkt Oden Mittels einer linse,"Physica 1, 201 (1934).

3. F. Zernike, "The Concept of Degrees of Coherence and Its Appli-cation to Optical Problems," Physica 5, 785 (1938).

4. M. Born and E. Wolf, Principles of Optics (Pergamon, New York,1964), pp. 499-508.

5. B. J. Thompson and E. Wolf, "Two-Beam Interference withPartially Coherent Light," J. Opt. Soc. Am. 47, 895 (1957).

6. B. J. Thompson, "Illustration of the Phase Change in Two-BeamInterference with Partially Coherent Light," J. Opt. Soc. Am. 48,95 (1958).

7. B. J. Thompson and R. Sudol, "Finite Aperture Effects in theMeasurement of the Degree of Coherence," J. Opt. Soc. Am. A 1,598 (1984).

8. A. S. Marathay and D. B. Pollock, "Young's Interference Fringeswith Finite-Size Sampling Apertures," J. Opt. Soc. Am. A 1, 1057(1984).

3196 APPLIED OPTICS / Vol. 25, No. 18 / 15 September 1986