Object-adapted variable-wavelength interferometry. I. Theoretical basis

Vol. 4, No. 11/November 1987/J. Opt. Soc. Am. A 2107

Object-adapted variable-wavelength interferometry.I. Theoretical basis

Maksymilian Pluta

Central Optical Laboratory, ul. Kamionkowska 18, 03-805 Warsaw, Poland

Received December 29, 1986; accepted June 11, 1987

Spectral dispersion of the refractive index and/or that of birefringence of the interference system is utilized for aspecific interferometric method referred to as object-adapted interferometry. The method uses monochromaticlight of continuously variable wavelength (X). In particular, a white-light source and an interference monochroma-tor (wedge interference filter) can be employed. The only parameter that is directly measured is the interfringespacing b, while other quantities are observed, read out from the calibration graph b(X), and derived from quitesimple formulas.

1. INTRODUCTION

Monochromatic light of variable wavelength is commonlyused in conventional interferometry for measuring the spec-tral dispersion of the index of refraction.' Absolute dis-tance measurements by variable-wavelength interferometry(VAWI) were also reported. 2 4 Moreover, variable-wave-length light was used in reflection-contrast microscopy fordetermining the thickness of blood cells.5 .

Quite recently, three methods of VAWI6 9 were developedby the author of this paper. Their specific feature is that theonly parameter that is directly measured is the interfringespacing b; other quantities are observed, read out from thecalibration plot b(X), and derived from quite simple formu-las. These methods permit us now to propose some specificinterferometric procedures, which can be referred to as ob-ject-adapted or adaptive variable-wavelength interferome-try (AVAWI).

2. GENERAL PRINCIPLE OF ADAPTIVEVARIABLE-WAVELENGTH INTERFEROMETRY

Let a fringe interference field be produced by superpositionof two plane wave fronts 2i and Z2 inclined at an angle E toeach other (Fig. 1). The empty interference field, i.e., theinterference pattern that is not perturbed by the objectunder study, is covered by equally spaced straight-linefringes.

Without restricting in any way the generality of the re-sults, we assume that the dark fringes occur for the opticalpath differences A = 0, +X, +2X, . .. and the bright onesoccur for A = +X/2, ±3X/2, +5X/2,.... This is, for instance,the case of the polarization interference system whose polar-izer and analyzer are crossed.

Let 6 be the optical path difference produced by the objectunder study. In general, interferometric techniques permitus to determine 8 from a simple relation,

6 cX (1)

where b is the interfringe spacing of the empty interferencefield, c is the fringe displacement produced by the object,

and X is the wavelength of the light used. Equation (1) canbe formally rewritten as

8 = (ml + q)X = mX, (2)

where m = m1 + q = c/b is the current (generalized) interfer-ence order; m1 is a suitably selected integer number, whichwill be called the initial (or introductory) interference order;and q is the increment (or decrement) of the current inter-ference order m with respect to m1 . From Eqs. (1) and (2) itfollows that

(ml + q)b = m b = c. (3)

The above relations are quite trivial, but they will be usefulfor the following discussion.

A. Transmitted LightLet a transparent object be placed upon the stage of a trans-mitted-light interferometer and transilluminated normallyby a parallel beam of monochromatic light. It is a well-known fact that the interfringe spacing varies with the lightwavelength X. The variation manifests itself as the move-ment of the interference fringes toward or away from thezero-order fringe I0 (Fig. 1) as X decreases or increases. Con-sequently, starting from a long or short wavelength permitsus to select such a particular wavelength X, (Fig. 2a) forwhich the interference fringes I' displaced by the platelikeobject occupy a position coincidents side by side with thelocation of the reference fringes I of the empty interferencefield. Such a coincidence is always possible if the opticalpath difference 6 is not too small, say, not smaller that X. Ingeneral, this situation can be described by

81 = (n,' - n,)t = mll, (4)

where n, and nl' are the refractive indices of the object andits surrounding medium (for X,!), t is the object thickness,and ml is an integer number that was previougly called theinitial interference order.

If 61 is several times greater than X,, then decreasing thelight wavelength from Xl to X2, X3, . . . causes the displacedfringes I to be alternately anticoincident (Fig. 2b) and coin-

0740-3232/87/112107-09$02.00 © 1987 Optical Society of America

Maksymilian Pluta

2108 J. Opt. Soc. Am. A/Vol. 4, No. 11/November 1987

a

b '0 -

respect to ml when the light wavelength X decreases from Xito X2, X3, 4 -- -

From Eqs. (4) and (5) it follows that

I I III1Fig. 1. Basic notation for fringe interference produced by twoplane wave fronts inclined at an angle to each other.

a

b

C

1111111111111.I1

ill iIIIIIIIIIIIt11111111

Fig. 2. a, c, Coincident and b, anticoincident configurations ofinterference fringes with decreasing light wavelength. I, Reference(undisplaced) fringes; I', fringes displaced by a platelike strip understudy.

cident (Fig. 2c) with the reference fringes I. These consecu-tive anticoincidences and coincidences may be expressed as

s= (n, - n)t = (ml + q)As, (5)

whore s 2, 3, 4, ... and qs 0.5, 1, 1.5, .. The term ml +q5 is equivalent to the current interference order ms [see Eq.(2)], and q is therefore the increment of this order with

I'

Xsml N - XS

where

n, - n 8,

If the spectral dispersion coefficient

N = 1,

(6)

(7)

(8)

we can speak of the AVAWI that functions in the wavelengthdomain. Interferometry of this kind will therefore be de-noted by AVAWI(X).

Condition (8) permits us to write Eq. (6) as

Xsml = q-1- X"

or

(ml + q,)X, = mll. (10)

When compared with Eq. (2), the above formula shows thatthe optical path difference is a wavelength-independentquantity; i.e., 8 = 81. The latter equality also results fromEqs. (7) and (8).

As was mentioned earlier, the interfringe spacing b (Fig.lb) varies with the wavelength X of monochromatic light.This variation is described by a simple relation,

I

I'b = X

2 tan(e/2)(11)

where is the intersection angle of the interfering wavefronts Z, and 2 as shown in Fig. la. This angle is usuallysmall, and Eq. (11) may be rewritten as b = X/E (here, e isexpressed in radians). Consequently, the wavelengths X,and X occurring in Eq. (6) may be replaced by ble, and b,respectively. Therefore we have

(12)

I'

ml = Hs N 'er teb -b

Here, the coefficient

(13)ES

'els = es

expresses the spectral dispersion of the intersection angle e.If the term

N,,,Els = 1, (14)

then we can also speak of the AVAWI method, but now theobject adaptivity functions in the domain of interfringespacings. Adaptive interferometry of this kind will be de-noted by AVAWI(b).

Condition (14) makes it possible to rewrite Eq. (12) in thefollowing form:

ml s b, - (15)

Maksymilian Pluta

I (9)

I

I

I

I


or

(m + q5)b, = m1bl. (16)

When compared with Eq. (3), the above relation shows thatthe fringe displacement c produced by the object understudy is a wavelength-independent quantity if theAVAWI(b) method can be used. This also means that thezero-order interference fringe in the object image is ideallyachromatic when white light is used.

In general, the coefficient el, is less than or at most equalto unity, whereas Ne,' may be either greater or less thanunity. If, however, the object under study is surrounded byan air medium and its dispersion curve n(X) is normal, thenthe coefficient Nel' = (n, - 1)/(n, - 1) > 1. It is thereforeself-evident that condition (14) is more practical than condi-tion (8). Consequently, the AVAWI(b) method is consid-ered more interesting than the AVAWI(X) method.

The first step in the AVAWI(b) procedure is to measurethe interfringe spacings b, = bl, b2, b3, . . . corresponding toconsecutive fringe coincidences and anticoincidences forparticular wavelengths X = X, X2, X3, ... and respectiveinterference order increments q, = 0, 0.5, 1, .... The initialinterference order ml is then calculated from Eq. (15), andthe mean quantity

sE (ml + q,)b,

(ml + qs)b = =M (16)S

is determined. Here S is the total number of fringe coinci-dences and anticoincidences. If S is high, the anticoincidentconfigurations of interference fringes (Fig. 2b) may be ig-nored, since they cannot be visually fixed as precisely as thecoincident configurations (Figs. 2a and 2c).

The optical path differences for desired light wave-lengths X can then be determined from Eq. (1), in which c isnow replaced by M, while X and b are read out from thecalibration plot b(X).

Finally, the thickness t of the object under study may bedetermined from Eq. (5) or Eq. (4) if the refractive-indexdifference n' - n is known for one wavelength at least;conversely, the difference n' - n (or its spectral dispersion)may be found if the thickness t is known.

However, it is important to note that the thickness t orn'(X) - n(X) can also be determined in another way if theAVAWI(b) method works. As can be seen from Eq. (5) andthe relation X, = efb,, the thickness t can be expressed as

t = (ml + q)b 5 Es = M nsS,' S, S, S

(17)

Moreover, Eq. (14) yields e,/(n,' - n) = El/(nl' - nl). Thisrelation means that the term E,/(n' - n) in Eq. (17) is awavelength-independent quantity and can therefore be tak-en as a parameter constant over the wavelength spectrum.On the other hand, if t and e, are known, then the refractive-index difference n,' - n, can be given by

(18)nS' -nS = M st

Moreover, it is worthwhile to note that Eq. (16) permits usto determine additionally a number of interfringe spacings

that cannot be measured. For instance, we are able to findseveral interfringe spacings [and respective light wave-lengths from an extrapolated calibration plot b(X)] in invisi-ble regions of the spectrum when interference patterns andconsecutive fringe coincidences and/or anticoincidences areobserved only in the visible spectral region. Consequently,we can determine optical path differences Is and relatedquantities, e.g., refractive indices, within a larger wavelengthspectrum than may directly be observed.

B. Reflected LightLet an object 0 under study (see Fig. 3) be attached to asubstrate S and placed upon the stage of a reflected-lightinterferometer adjusted to fringe interference in the imageplane. Suppose that the object is a platelike strip of thick-ness (height) t. We also assume, as for transmitted-lightinterferometry discussed above, that a parallel beam ofmonochromatic light strikes the object and its substratenormally. Part of the incident light is reflected back from 0and S, and an optical path difference 8 arises between re-flected rays R and R'. If the object 0 is surrounded by amedium of refractive index n', the above-mentioned opticalpath difference can be expressed as

8 = 2t n' + oi, (19)

where 8 j is an additional optical path difference that is due todifferent phase jumps j and i/.j on reflection for rays R andR'; i.e., bj = 4,X/27r, where 4' is the phase-jump differenceexpressed in radians.

If the light wavelength is continuously varied, the reflect-ed rays R and R' produce interference patterns similar tothose observed in transmitted light (Fig. 2). However, Eqs.(4) and (5) take now the form

81 = 2tnl' + oil = mlX

5 = 2tn,' + j5 = (ml + q5) X

(20)

(21)

From these equations it follows that the initial interferenceorder

P

n'/////// / ////Fig. 3. Light-reflecting object 0 of height t on a light-reflectingsubstrate S. P's, incident beam of parallel rays; R and R', raysreflected from the object and its substrate.

Maksymilian Pluta

'/'/ t//,//c:


ml = qs - + n5 ', - (22)n., X-l X 27r(n 1Xl - X.)

where n, 1' = n'/nl'.Equation (22) shows that it is difficult or even impossible

to obtain object-adapted interferometry in this most-gener-al situation, since the second term of this equation should beequal to zero and the dispersion coefficient n' must beequal to unity. If, however, there is no spectral dispersion ofthe difference of phase jumps, then 4, = As, = 4', and Eq. (22)takes a simpler form:

As + P M = - s X5 27- (23)

The term V//27r is generally much less than unity for manyobjects ( in Fig. 3) and their possible substrates (). Inparticular, 4'/27r = 0 when the object under study and itssubstrate are of the same material or even of different mate-rials, which, however, produce the same phase jumps onreflection (e.g., glass substrate and different dielectric stripson it). In this instance

in, l = l - . (24)

and the AVAWI(X) method works when n' = 1. It istherefore necessary to place the object under study and itsadjacent portion of the substrate in an air medium.

Let X, = b and X = b, be substituted into Eq. (24). Asa result of this substitution, we obtain

ml = qs nfla'dE1 b, -bs' (25)

where e is given by Eq. (13). Here, the product nl,',E5should be equal to unity in order for the AVAWI(b) methodto work. If, however, n 1' = 1 (air medium), then the coeffi-cient el, must also be equal to unity. Consequently, thelatter situation is less suitable than the former for theAVAWI(b) method; the conclusion is quite the reverse ofthat obtained by the AVAWI(X) method.

3. ALTERNATIVE APPROACH TO THEADAPTED VARIABLE-WAVELENGTHINTERFEROMETRY METHODS

The discussion presented above refers to objects that pro-duce optical path differences 8 that are not too small, say,not smaller than 3X. Otherwise, the number of fringe coinci-dences and/or anticoincidences (see Fig. 2) is small or evenequal to zero within the visible spectrum. This limitation isovercome by arranging interference fringe coincidences inanother way. For this operation a gauging graticule, con-sisting of two pointer lines L and L2 (Fig. 4), is used. Thedistance d between these lines is selected to be as long aspossible, say, d = 10bl. One of these lines, L, is permanent-ly brought into coincidence with the center of the zero-orderfringe I0 of the empty interference field, while the consecu-tive high-order fringes, undisplaced I and then displaced I',are brought into coincidence with the other pointer line, L2,when the wavelength of the monochromatic light is varied.

A. Transmitted LightInitially the empty interference field is processed (pointed).Starting from long wavelengths permits us to select a clearlyvisible red first wavelength X, for which one of the high-order dark fringes I of the empty interference field becomescoincident with the pointer line L2 (Fig. 4a). The distance dis now covered by ml interfringe spacings b; i.e., d = mbl.Here, ml is referred to as the initial interference order for theempty interference field at the pointer line L2, where theoptical path difference Al between the wave fronts 11 and 12(see Fig. la) is given by

Al = de = mlXl. (26)

Let the light wavelength then be decreased from X, to X2, X3,X4, ... for which the pointer line L2 becomes consecutivelycoincident with bright (Fig. 4b) and dark (Fig. 4c) fringes Iwhose interference orders are higher by q = 0.5, 1, 1.5, 2, . . .with respect to m1 . In the analogy of Eq. (26), these consec-

a X IfJJJI

o10

L1 L2

Fig. 4. a, c, High-order dark and b, bright interference fringes I ofthe empty interference field are brought into coincidence with thepointer line L by continuously decreasing the light wavelength,while the zero-order dark fringe Io. is permanently coincident withthe pointer line L.

Maksymilian Pluta


10 ~~d

lo10 L2

big~~~~~~~~~~~~~~~~~~~~~~i

lo0 L, L2

Fig. 5. a, c, High-order dark and b, bright interference fringes I'displaced by an object under study are brought into consecutivecoincidence with the pointer line L2 by continuously decreasing thelight wavelength.

utive coincidences may be expressed by the optical pathdifferences A, at the pointer line L2, i.e.,

A = de, = (ml + qs)X,

where s = 2, 3, 4, ....From Eqs. (26) and (27) it follows that

Xs

(27)

ml = qs A, psi kZZ)

where e 8 = s/el. In order for the AVAWI(X) method towork, the dispersion coefficient E81 must be equal to unity.This requirement is, of course, troublesome.

Quite another situation occurs when we use the interfringespacings, which permit us to write the above equation as

b-s (29)Mli = Vs

where e = E/es. Note that the coefficient e81 = /e18 ; thusEslel = 1, and Eq. (29) takes the form of Eq. (15). Wetherefore have the situation that applies directly to theAVAWI(b) method.

The angle e8 that occurs in Eq. (27) is equal to X/b 8.Consequently (ml + q,)b8 = d. It is, however, recommendedthat, instead of taking d, we take the mean quantity(m1 + qS)b, = msb, = E, defined as

S

Z (ml + q8 )b8

E s= 1

where S is the overall number of consecutive coincidences ofthe dark (Figs. 4a and 4c) and the bright (Fig. 4b) interfer-ence fringes with the pointer line L2 -

Let us now consider the interference fringes I' (Fig. 5)displaced by a platelike transparent object transilluminatednormally by a parallel beam of monochromatic light. Thepointer line L1 is still coincident, as shown before (Fig. 4),with the center of the zero-order fringe of the empty inter-ference field. On the other hand, the pointer line L2 servesnow for pointing the fringe I' displaced by the object understudy. Starting from long wavelengths permits us to fix ared wavelength X1' for which one of the high-order darkfringes I' becomes coincident with the pointer line L2 (Fig.5a). At this line, the optical path difference between inter-fering wave fronts in now given by

A1 = A1 + 61 = del + (nl' - nl)t = ml'X, (31)

where 61 is the optical path difference produced by the objectunder study, and A1 = d El' is the optical path difference thatis due to the empty interference field (at the line L2!). It isself-evident that A1 is now different from the A1 in Eq. (26).However, m1 ' may be equal to m1 if 51 is small (smaller thanX).

When the light wavelength is decreased from X1' to theshort-wavelength region of the spectrum, we can select par-ticular wavelengths X2 , X3 , X4 , ... for which consecutivebright (Fig. 5b) and dark (Fig. 5c) fringes become coincidentwith the pointer line L2 . These coincidences may be ex-pressed as

A'= A + as = de, + (ne' - n 8)t = (ml' + q)X.', (32)

where s = 2, 3, 4, ... and q8 = 0.5, 1, 1.5, .From Eqs. (31) and (32) it follows that

ml' = qs N -XS/ 'Xd'-X''NS Usl 1 -Xs(33)

where Nsl' is defined by Eq. (7). In order for the AVAWI(X)method to work it is now necessary not only that N 1' = 1 butalso that the second term of the above equation be equal tozero. This requirement is impossible to satisfy in practice.

However, a quite different situation occurs if we use theinterfringe spacings. These permit us to rewrite Eq. (33) inthe form

ml' = q N b' + d N1'fls' - 1 1N .,E 'bl9 / / sl ils 1l- s (34)

where El,' is equal to E1'/E8'. As can readily be seen, this

(30)

Maksymilian Pluta


formula takes the form of Eq. (15), and thus the AVAWI(b)method works, when the term Nslt'l,' = 1 [cf. Eq. (14)]. Ifthis requirement is met, we have

m1 = qs b,' -b 5 (35)

or

(m' + qs)b,' = m5 'b'. (36)

It is recommended that we take the mean quantity (m' +q)b'= ins'b,,' = M' defined as

E (ml' + q5)bs'M = S , (37)

S

where S is the overall number of consecutive coincidences ofthe dark (Figs. 5a and 5c) and the bright (Fig. 5b) interfer-ence fringes I' with the pointer line L2. The two AVAWI(b)methods (compare Fig. 2 with Figs. 4 and 5) are thereforeequivalent, but the latter is suitable for studying objects thatproduce small optical path differences.

From Eq. (32) it follows that

6 = (ml' + q,)X5' - As = min'Xs' - AS, (38)

where m' = ml' + qs is the current interference order of thedisplaced fringes I' at the pointer line L2. The term ms'X5'may be expressed as

M's = ms'bs' = M' _, (39)~~~ ~bs' bsAs in Eq. (27), the optical path difference As that occurs inEq. (38) may be expressed as

but the term E,'/(n,' - n) is a wavelength-independentquantity if the AVAWI(b) works. Note that the above ex-pression is similar to Eq. (17).

B. Reflected LightThe above discussion can readily be extended into reflected-light interferometry. For the empty interference field wehave, of course, the same relations as for transmitted light[see Eqs. (26)-(30)]. If the object 0 under study (see Fig. 3)is surrounded by an immersion medium of refractive indexn' and the phase-jump difference on reflection () is equal tozero, we may write the following equations:

Al' = 2tnl' + A, = 2tnl' + de'= ml'X,',

A,,' = 2tn,' + A, = 2tn,' + de' = (ml' + qs)XS',

(44)

(45)

which express the total optical path differences in the objectimage at the pointer line L2. By combining these equationswe obtain

ml=U tX I A, X d n I'Al'-X 'ns, 1 Si5 1 - X

(46)

or

in,'=q5 bs' nl,'Els' -1

nsl Els'b' - bs' ns'Els'b' -bs(47)

The AVAWI(X) method applies if n5,' = 1 and E' - ES' = 0.This requirement is rather troublesome. On the other hand,the AVAWI(b) method can be used if n 1'el, = 1. When,however, an air medium surrounds the object, the coefficientn,,' may be assumed to be equal to unity, and the samerequirement e' - Es' = 0 applies to the two methods.

As = (ml' + qS k ps)Xs' = (ml' + qs + ps)bsd bs

(40)

where ps is an additional interference order increment (ordecrement) for the empty interference field, which resultssimply from a difference between Xs and Xs'. However, theterm (mi' + q, + p,)b 5' is the same constant quantity E thatwas defined earlier by Eq. (30). Consequently, A, = EX,'/b,'.Substituting this equation and Eq. (39) into Eq. (38) yields

8s= (M'-E) 5' - (M'- E)-. (41)If the object thickness t is known, then the refractive-

index difference n,' - n may be determined from the rela-tion 8s = (n,' - n) t, or, conversely, the thickness t can befound if In,' - n is known.

On the other hand, Eq. (32) permits us to express theobject thickness t as

(m' + qs)Xs' - des'

ns-n ''

By substituting Xs' = e 5'b,' and using Eqs. (30) and (37), weobtain

t = (M'-E) j , (43)n - n

4. DOUBLE-REFRACTING ADAPTEDVARIABLE-WAVELENGTH INTERFEROMETRYSYSTEM

In Sections 2 and 3 we dealt with general aspects of theAVAWI methods; this section deals with a concrete interfer-ometric system, the double-refracting wave-front shear in-terferometer, which is especially suitable for the AVAWI(b)technique. This system (Fig. 6) is quite simple; it consists ofa polarizer (P), an analyzer (A), and a Wollaston prism (W)made of quartz crystal. The polarizer and the analyzer arecrossed, and their directions of light vibration form an angleof 450 with the apex edge of the Wollaston prism. Thiswave-front-shearing device is inserted into an optical systemthat consists of a condenser (C), an objective (Ob), and anocular (Oc). The condenser is preceded by a slit diaphragm(D) that is located at its front focal point. The slit () of thisdiaphragm is parallel to the wedge edges of the Wollastonprism. Immediately before the diaphragm, a wedge inter-ference filter (IF) is located, which extracts monochromaticlight from a source of white light. An object 0 to be exam-ined is placed in the object plane r of the objective (Ob),while the objective image plane r' is occupied by a graticule(G) with two pointer lines L and L In fact, this graticule isassociated with the ocular Oc, whose front focal plane iscoincident with the image plane r'. Additionally, the grati-cule incorporates a central pointer line L.

Maksymilian Pluta


I Oc

I L

1-I'~~~~~~~~~~~~~~~~~~~

I Li2

A

w ICW s<- P o -N

the object is isotropic, then its two images are equivalent toeach other, and we can apply to each of them the theoreticaldiscussion given in Sections 2 and 3.

From Eqs. (11) and (48) it follows that the wavelength Xcan be expressed as

X = be = b2D tan a.

Consequently, Eqs. (4) and (5) may be rewritten as

61 = (nl'- nl)t = mlbl2Dtan a

and

= (n,'- n,)t = (ml + q5)b52Dtan a.

The intial interference order m1 is now found to be

bs,

in1 = q5 N'D b -b'

where

D, (ne - no),D, (ne - no),

(49)

(50)

(51)

(52)

(53)

I Ob

0

C

, S- ~ ~~~~I IF

S

L == P

Fig. 6. Optical system of a double-refracting interferometric sys-tem. P polarizer; IF, wedge interference filter (interference mono-chromator of continuous wavelength); D, slit diaphragm; S, slit; C,condenser; 7r, object plane; 0, object under study; Ob, objective; W,symmetrical Wollaston prism; MS, micrometer screw; A, analyzer;7r', image plane; G, graticule with three pointer lines (L, L1 , L2 ); Oc,ocular.

If the condenser slit is sufficiently narrow, a fringe inter-ference field is produced by the Wollaston prism in theimage plane -r'. The fringe field may be considered a resultof superposition of two wave fronts 31 and 22 (Fig. 1) in-clined to each other at an angle E. This angle is very smalland is given by

E = 2(ne-no) tan a = 2D tan a, (48)

where D = ne - nO is the birefringence of the crystal of whichthe Wollaston prism is made and a is the apex angle of thisprism. If a symmetrical Wollaston prism is used, as shown inFig. 6, the empty interference field in the image plane r'consists of equally spaced straight-line fringes (this does notapply to a typical Wollaston prism, which produces an inter-ference pattern with slightly variable interfringe spacingacross the image plane' 0).

This interference system permits observation of an objectunder study as two completely or partially split images. If

The AVAWI(b) method works if the term

N,,tDl = 1; (54)

then Eq. (52) takes the form of Eq. (15).In general, the coefficient Dl, is smaller than unity, where-

as N 1' may be either greater or less than unity. If, however,the object under study is surrounded by an air medium andits dispersion curve n(X) is normal, the coefficient Ns,' takesthe form Nel' = (1 - n)/(1 - nl) and becomes greater thanunity. It is therefore self-evident that condition (54) is morepractical than N 1' = 1, as the AVAWI(X) method requires[see Eqs. (8) and (9)].

It will be interesting to analyze condition (54) and findsome hypothetical materials of different refractive index nD

that can be applied to the double-refracting AVAWI(b) sys-tem fitted with the Wollaston prism (Fig. 6) made of differ-ent birefringent crystals. In order to do this analysis, weassume that nl' = 1, n,' = 1 (air medium), n1 = nD, and D, =DD. As a result of this assumption, we can rewrite Eq. (54)in the following form:

ni= 1+ D (nD 1),DD

(55)

where n5 can now be replaced by either nF or nc and DS can bereplaced by either DF or DC, respectively (here, C, D, and Fdenote the spectral lines of wavelengths Xc = 656.3 nm, XD =589.3 nm, and XF = 486.1 nm). Consequently, we can findthat the Abbe number V = (nD - 1)/(nF - nc) of the materi-als in question is given by

DF-DC(56)

where DD, DF, and DC are the birefringences of the crystals ofwhich the Wollaston prism is made. Similarly, the meandispersion of the materials in question is given by

nD - 1nF C '-U V (57)ti~tiC

--

Maksymilian Pluta


For Wollaston prisms made of quartz, calcite, ammoniumdihydrogen phosphate, and potassium dihydrogen phos-phate crystals, the coefficients Vw are 33.61, 23.69, 18.12,and 15.47, respectively.

Let us now consider a platelike birefringent object placedin the object plane r of the double-refracting interferencesystem shown in Fig. 6. Part of the image plane r', wherethe split images of the plate overlap, contains information onthe optical path difference produced by birefringence B =ny - nta of the plate under study (ny and nta are the refrac-tive indices of the plate for the slow and fast components oflight wave). This optical path difference is defined as 6 = Btand observed as a displacement of interference fringes whenthe fast or slow axis of the birefringent plate forms an angleof 45° with the directions of light vibration in the polarizer(P) and the analyzer (A) (Fig. 6). By analogy with Eqs. (50)and (51), we may now write the following equations:

81 = Blt = mlbl2Dltan a, (58)

a = Bt = (ml + q,)b,2Dstan a, (59)

from which it follows that

b-bB,,,Dlsbl - b

(60)

where B8l = B 8/B 1.The coefficients B8l and Dl, express the spectral disper-

sion of birefringence; however, B 8l refers to the object understudy, whereas Dl, applies to the Wollaston prism of thedouble-refracting interferometer (Fig. 6). If the dispersioncurves B(X) and D(X) satisfy the equation

BslDI = 1, (61)

then the AVAWI(b) method can be used. In this instanceEq. (60) takes the form of Eq. (15), and we have now a new,simple, and practical technique for accurate interferencemetrology of birefringent objects.

First, the thickness t of birefringent plates can be deter-mined precisely by measuring only the interfringe spacings.From Eq. (59) it follows that

D,t = (ml + q)b,, B 2 tan az. (62)SB

If condition (61) is true, as the double-refracting AVAWI(b)method requires, then the term DS/BS is a wavelength-inde-pendent quantity. In particular, B8 may be equal to D,;consequently, Eq. (62) reduces simply to

t = (ml + q8)b,2 tan a = 2M tan a. (63)

The apex angle a of the Wollaston prism is a constant pa-rameter that is valid in perpetuity and can be determinedextremely accurately. There is no problem in securing Aa =

t2 arcsec and Ab, = ±0.01 gm; thus the thickness t may bedetermined with an accuracy of AŽt = 0.OOOlt or At/t =0.01% (if t is not smaller than several micrometers, say, 10/,m). If the thickness t is accurately known, the birefrin-gences B8 of the plate may then be determined preciselyfrom the relation B 8 = 88/t = M X8/t b, for the correspondingwavelengths X, read out from the calibration plot b(X).

The calibration plot b(X) can be performed accurately

since highly monochromatic light (He-Ne, Ar, and He-Cdlasers) may be used for this operation. Once it is carefullyplotted, graph b(X) is valid in perpetuity for the double-refracting interference system shown in Fig. 6. For quartzWollaston prisms, this graph is almost linear over the visiblespectrum and quite consistent with Eq. (49); thus it can bereadily extrapolated toward UV and/or IR regions.

The measurement of the interfringe spacing b is per-formed with the transverse movement of the Wollastonprism (Fig. 6). This movement (marked by the arrow in Fig.6) is accompanied by an equivalent movement of the inter-ference fringes observed in the image plane 7r', and the fringecenters are precisely guided onto the pointer line L. Themeasurement of multiple interfringe spacings, e.g., the dis-tance = 40b or even = 100b, is recommended becausehighly accurate values for b can be obtained in this fashion.

The above discussion refers to the interferometric proce-dure shown in Fig. 2. If, however, the optical path differ-ence 6 is small, say, smaller than 3X, the procedure shown inFigs. 4 and S is recommended. In this instance, Eq. (60)takes the form

b- BlDlS -1

s Bsl~lsbl-be BslDlbl - bs (64)

The AVAWI(b) method can be used when B,1D = 1. Con-sequently, this requirement is the same as that for the inter-ferometric procedure shown in Fig. 2.

There are in practice birefringent objects, e.g., X, X/2, or /4 plates or other phase retarders, that do not permit us toobserve simultaneously the reference (undisplaced) inter-ference fringes and the fringes displaced because of the ob-ject birefringence. If, in this instance, the optical path dif-ference 6 is small, say, smaller than 3, the procedure shownin Figs. 4 and 5 may be used. Otherwise, if > 3X, thenneither the procedure shown in Figs. 4 and 5 nor that shownin Fig. 2 can be used effectively. This limitation is over-come, however, by using a technique based on the samegeneral principle as that shown in Figs. 4 and 5 but with thedistance d between the pointer lines L, and L2 reduced to thezero value; consequently, a single pointer line L (Fig. 6) isused. Initially, the zero-order interference fringe of theempty interference field is brought into coincidence withthis line; the light wavelength is then varied, high-orderfringes displaced by the object under study are consecutivelybrought into coincidence with the pointer line L, and respec-tive interfringe spacings b are measured. Since the dis-tance d is now equal to zero, Eq. (64) takes the form of Eq.(60).

Many experiments have been performed with a double-refracting interferometer similar to that shown in Fig. 6.Results of these experiments are fully consistent with thetheoretical discussion presented above; they will be de-scribed in separate papers.

5. CONCLUSIONS

The AVAWI method presented in this paper cannot be con-sidered a universal method; it offers us, rather, special butsimple and accurate interferometric tecnliques suitable fora number of selected objects.

Maksymilian Pluta


In comparison with common interferometric techniques,in which monochromatic light of constant wavelength isused, the AVAWI method is simpler and more accurate forestablishing the integral number of interfringe spacings bywhich the object under study displaces interference fringes.

It has been shown that the AVAWI(b) method is morepractical than the AVAWI(X) method. The former is espe-cially well accepted by double-refracting interference sys-tems that use Wollaston prisms. The Wollaston prisms canbe made of different common crystals, whose spectral dis-persion of birefringence appears to be adapted to, similar to,or even the same as the spectral dispersion of the refractiveindex and/or of the birefringence of many objects and mate-rials to be examined or tested in transmitted light.

A defect of the AVAWI methods is that the calculationsleading to the final interferometric results are time consum-ing and require a personal computer. However, the calcula-tion procedure is quite simple; moreover, the techniquespresented here are suitable for fully automatic photoelec-tronic operation. It is self-evident that two pointer lines L,and L2 (see Figs. 4 and 5) or a single pointer line L (see Fig. 6)can be considered slit apertures of photoelectronic detec-tors. Similarly, the procedure shown in Fig. 2 is highlysuitable for automatic photoelectronic processing.

Finally, it is worthwhile noting that the AVAWI(b) and/orAVAWI(X) methods can be useful for IR and/or UV interfer-ometry for which photoelectronic recording of the interfer-ence fringe coincidences or anticoincidences and photoelec-

tronic measurement of the interfringe spacings are neces-sary.

REFERENCES

1. K. Kerl, "Continuous wavelength interferometry. An effectivemethod for measuring the dispersion of the complex index ofrefraction," Opt. Acta 26, 1209-1224 (1979).

2. F. Bien, M. Camac, H. J. Caulfield, and S. Ezekiel, "Absolutedistance measurements by variable wavelength interferome-try," Appl. Opt. 20, 400-403 (1981).

3. C. W. Gillard, N. Buholz, and D. W. Rider, "Absolute distanceinterferometry," Opt. Eng. 20, 129-134 (1981).

4. V. P. Burpashov, "Application of group parameters to an inter-ferometric system for absolute distance measurements by usinga light source with continuously variable wavelength," Avtome-triia 6, 91-94 (1985) (in Russian).

5. J. Piper and F. Pera, "Reconstruction des Oberflichenreliefsvon Erythrocyten mit Hilfe der Leitz-Reflexionskontrast-Ein-richtung," Leitz-Mitt. Wiss. Tech. 7, 230-234 (1980).

6. M. Pluta, "Variable wavelength interferometry. I. Fringe fieldmethod for transmitted light," Opt. Appl. 15, 375-393 (1985).

7. M. Pluta, "Variable wavelength interferometry. II. Uniform-field method for transmitted light," Opt. Appl. 16, 141-157(1986).

8. M. Pluta, "Variable wavelength interferometry. III. Reflected-light techniques," Opt. Appl. 16, 159-174 (1986).

9. M. Pluta, "Variable wavelength interferometry. IV. Alterna-tive approach to the fringe-field method," Opt. Appl. 16, 301-323.

10. M. Pluta, "On double-refracting microinterferometers whichsuffer from a variable interfringe spacing," J. Microsc. (Oxford)146, 41-54 (1987).

Maksymilian Pluta

Documents

Object-adapted variable-wavelength interferometry. I. Theoretical basis