Simplified polanret system for microscopy

Simplified polanret system for microscopy

Maksymilian Pluta

A variable phase contrast and amplitude contrast device of the polanret type has been developed. It uses only

a single polarizing ring, instead of three zonal polarizers as required in Osterberg's original polanret system,

and is therefore much more easily manufactured, its economy of light is better, and its range of amplitude

contrast is improved. However, it also requires an easily accessible microscope objective exit pupil, where the

polarizing ring must be located to cover the direct image of a conjugate annular diaphragm of a phase contrast

condenser.

1. Introduction

Polarizing and birefringent elements are especiallysuitable for variable phase contrast microscopy. Theyenable the phase and/or the intensity of the direct(undiffracted) light to be continuously varied in rela-tion to the diffracted light. Thus, it is possible toadjust the phase contrast microscope for optimum im-age contrast of microscopic objects whose optical pathdifference 6 or phase shift so = 2r6/X and other opticalproperties differ in a wide range.

Many variable phase contrast systems were devel-oped in the past.1 One of the best known is the polan-ret system invented by Osterberg2 and manufacturedby American Optical Corp.3 Its achromatic versiondevised by Normarski4 and the Nikon interferencephase attachment 5 belong to this category as well. Allthese use three (Osterberg, Nomarski) or two (Nikon)zonal polarizers.

An extremely simple polanret system is now pre-sented. It uses only a single polarizing ring and revealsa number of advantages in comparison with the otherpolanret systems known to date. For the sake of brevi-ty, this simplified system will be referred to by theacronym S-polanret throughout this paper. Similarly,the original polanret system developed by Osterbergwill be denoted by O-polanret.

11. Background

According to Bennett et al.6 it is useful to apply theterms conjugate area and complementary area to the

The author is with Central Optical Laboratory, 18 ul. Kamion-kowska, 03-805 Warsaw, Poland.

Received 23 January 1988.0003-6935/89/081453-14$02.00/0.

) 1989 Optical Society of America.

exit pupil of microscope objectives. Obviously thispupil is coincident with (or near to) the back focalplane of the objective. The conjugate area of thisplane is defined by the image of an opening of theaperture diaphragm located in the front focal plane ofa bright field condenser of the microscope system;whereas the remaining portion of the back focal planeof the objective is called the complementary area.

For phase contrast microscopy the opening of thecondenser diaphragm D (Fig. 1) is typically annular;thus the conjugate area (Aconj) of the objective exitpupil is a ring-shaped zone, which divides the comple-mentary area of this pupil into two portions (Acomp):one is inside and the other is outside the conjugatering-shaped area.

Figure 2 shows the principle of the O-polanret sys-tem. Its basic component is a zonal disk ZD located inthe rear focal plane of the microscope objective Ob.This disk is composed of three zonal polarizers Z1, Z 2,and Z3 made of a polarizing sheet. Polarizer Z coversthe ring-shaped conjugate area of the objective exitpupil. On the other hand, polarizers Z2 and Z3 coverthe complementary area of this pupil. Directions oflight vibration of these polarizers are parallel to eachother, but they are perpendicular to the direction oflight vibration in polarizer Z and form an angle of 450with the slow (or fast) axis Q-Q of the quarterwaveplate Q that precedes the zonal disk ZD. A linearpolarizer P placed before plate Q (or even before thecondenser diaphragm D), and an analyzer A insertedbetween disk ZD and a microscope ocular Oc completethe O-polanret system.

Both the polarizer P and the analyzer A are rotatablearound the objective axis. Rotation of polarizer Pthrough an angle 0 changes the phase difference A

between the conjugate and complementary areas ac-cording to the Senarmont compensator, i.e.,

4' = 20. (1)

15 April 1989 / Vol. 28, No. 8 / APPLIED OPTICS 1453

Aconj Acomp

co I Ac0n1 Acomp n, Oc

Fig. 1. Definition of the conjugate area (ACwnj) and of the comple-mentary area (ACmp) of the focal plane or of the exit pupil of themicroscope objective: D, condenser diaphragm with an annularopening AO; F, front focal point of the microscope condenser Co; II,object plane of the microscope objective Ob; F', back focal point ofthe objective Ob; ni, image plane of the objective Ob; Oc, microscopeocular. The conjugate area Ac,,i,j is simply the image in the objectiveback focal plane of the annular opening AO of diaphragm D, which iscoincident with the front focal plane of condenser Co, whereas thecomplementary area Acomp is the remaining portion of the back focal

plane of objective Ob.

A

(b) A P(b ,

R R

,// \\

I AP pi

A

Fig. 3. Principle of the S-polanret device: D, AO, Co, 1, 0, Ob, P,Q, A, fl', Oc, P-P, Q-Q, and A-A as in Fig. 2; PD, phase amplitudedisk with a single polarizing ring R; C, optical cement which sur-rounds ring R between two glass plates; R-R, direction of light

vibration in ring R.

Tr = A tan 2a,

(a)

(3a)

where A is a constant factor for a given azimuth 0. If 0= m45' (m = 0,I1,±2, .. .), A = 1 and

A p

(b) A 4 ,P

\ I

ZR Z ZR\

/ '

A A

n 0,

.f . ..1

I Z

Fig. 2. Principle of Osterberg's polanret system: D, condenserdiaphragm; AO is its annular opening; Co, condenser; I, object planeof the microscope objective Ob; 0, object under study; P, rotatablepolarizer; Q, quarterwave plate; ZD, zonal polarizing disk; Z1, Z2 , andZ3, zonal polarizers; A, rotatable analyzer; fl', image plane of theobjective Ob; Oc, microscope ocular. P-P, Q-Q, Z 1-ZI, Z2-Z 2 , andA-A indicate, respectively, the directions of light vibration in polar-izer P, quarterwave plate Q, zonal polarizer Z1, zonal polarizers Z2and Z:, and analyzer A. Continuous lines P-P and A-A representthe zero (or initial) positions of the polarizer and analyzer withrespect to the slow axis Q-Q of the quarterwave plate Q, while thebroken lines P-P and A-A indicate any positions of the rotatable

polarizer P and analyzer A.

On the other hand, rotation of analyzer A through anangle a varies the transmittance ratio Tr of these twoareas. This quantity is defined as

-= C/ni (2)T

comp

where Tconj is the intensity transmittance related to theconjugate area, and Trcomp is that related to the comple-mentary area. It can be found [see Eq. (A9) in Appen-dix A and Eq. (17) in Sect. IVE] that

Tr = tan2 a. (3b)

It is self-evident that the conjugate area of the objec-tive exit pupil is primarily filled with the light thatpasses directly through the object under study 0, whilethe complementary area is filled with the light diffract-ed by the object. Consequently, rotating polarizer Pand analyzer A alters both the phase difference A' andthe intensity ratio of the direct relative to the diffract-ed light beams.

111. Principle of the S-Polanret Device

Instead of three zonal polarizers, as required by theO-polanret system (Fig. 2), the S-polanret device con-tains only one polarizing ring R (Fig. 3). This ringcovers the conjugate area of the exit pupil of the micro-scope objective Ob, i.e., the image of the condenserannular opening AO. It is made of a polarizing sheet(Polaroid foil) and cemented between two glass platesby means of a suitably selected optical cement C, whichsatisfies the condition

6CR = (nc - nR)t = m\, (4)

where 5CR is the optical path difference between thelight passing through polarizing ring R and the lightpassing outside this ring through the cement layer C ofrefractive index nc (Fig. 4); t and nR are the thicknessand refractive index of ring R (nR refers, of course, tolight vibrating parallel to the axis of light vibration inthe polarizing sheet of which the ring is made); X is thewavelength of light used (for white light X = 0.55 Mm);and m = 0,±14,2 ....

When sandwiched between two glass plates, polariz-ing ring R [Fig. 3(a)] and its surrounding medium(optical cement C) constitute a phase amplitude diskPD, which produces the phase difference 4CR = 27rbCR/X between the direct and diffracted light beams. This

1454 APPLIED OPTICS / Vol. 28, No. 8 / 15 April 1989

(a)

n ocAc

- - - - 4 -

I

It t E I1 n

I I

Fig. 4. Illustration of Eq. (4).

quantity is equal to zero if m = 0 in Eq. (4), or it can alsobe equal to ±27r,+47,... if m = 1,2,.... In phasecontrast microscopy, the phase differences 4cR =

±27r,±47r,... are however equivalent to 4cR = 0-When considered separately, disk PD is thereforeequivalent to a conventional phase plate whose phaseshift 4' = 0.

A completely different situation arises when disk PD[Fig. 3(a)], like the zonal disk ZD in the O-polanretsystem [Fig. 2(a)], is placed between two rotatablepolars: polarizer P and analyzer A, and preceded byquarterwave plate Q, whose slow (or fast) axis Q-Q[Fig. 3(b)] makes an angle of 450 with the directionR-R of light vibration being transmitted by polarizingring R. Additionally, we assume that the initial orien-tation P-P and A-A of polarizer P and analyzer A aredefined as follows: P-P is parallel to Q-Q, while A-Ais perpendicular to R-R (P-P and A-A are the direc-tions of light vibration in polarizer P and analyzer A,respectively). The phase difference and also the in-tensity relation between the direct (undiffracted)light, which passes through ring R, and the light dif-fracted by the object under study 0 (this light passesbeyond ring R) can now be changed and optimized ifthe azimuths 0 and a of polarizer P and analyzer A arevaried. Rotating polarizer P changes the phase differ-ence 4' similar to the Senarmont compensator [see Eq.(1)], whereas rotating analyzer A varies the intensityrelation between the direct and diffracted light beams.

IV. Properties and Microscopic Techniques

A. General Remarks

The working principle of the S-polanret device iscompletely defined by the azimuths a and 0 (Fig. 3).For the present, let us assume that the azimuth 0 is,however, equal to zero. Then it can be found [see Eq.(A4) in Appendix A] that the intensity transmittanceratio Tr of the conjugate and complementary areas ofthe exit pupil of the microscope objective Ob is ex-pressed by

rn sina TC cos2(45' + a)

where T is the common intensity transmittance of the

_ _ __ A\_---_ -

\S \~90

0 20 40 60 80 ¶00 120 140 160 180

180 200 220 240 260 280 300 320 340 360225 270 C 0]

Fig. 5. Plots of the transmittance ratio Tr vs the azimuth a. Solidline SP refers to the S-polanret device discussed, whereas broken

line OP relates to the 0-polanret system.

polarizing sheet of which the phase amplitude ring R ismade, and a is the azimuth as shown in Fig. 3(b).

Figure 5 shows the graphic representation of Eq. (5)for T = 0.3, where the graph of Eq. (3b) is also displayedfor comparison purposes. As can readily be seen, anessential difference occurs between both graphs. Forthe O-polanret system, the function Tr(a) is symmetri-cal with respect to the azimuths a = aNI(1) = 900 and a= aNI(2) = 2700, while that for the S-polanret device isasymmetrical with respect to a = aNI(1) = 450 and a =aNI(2) = 2250. This asymmetry, as shown below, is anadvantage.

For the azimuths azNi mentioned above, the trans-mittance ratio Tr is equal to infinity, the diffractedlight is therefore completely extinguished, and thus nodiffraction structure of the object under study is ob-served in the image plane II' [Figs. 2(a) and 3(a)]. Onthe other hand, both graphs attain zero values of Tr forthe same two azimuths a = aDF(1) = 0 and a = aDF(2) =1800. In this instance the direct light is completelyextinguished and a specific situation occurs, which isthe central dark field imagery. There are also fourother specific azimuths a = aBF(1), a = aBF(2) > aBF(1), a

= aBF(3) = 1800 + aBF(1), and a = aBF(4) = 1800 + aBF(2)for which the transmittance ratio Tr = 1. These azi-muths are different for the two polanret systems andthey are responsible for the technique equivalent tobright field microscopy (if the azimuth 0 = 0).

Azimuthal angles a other than aNI, ajDF, and aBF

apply to the variable amplitude contrast technique (if0 = 0) or to the variable phase contrast technique (if 6# 0 or 0 Fd 1800). However, the azimuths a for whichTr > 1 are much less useful in practice than those forwhich Tr < 1. When compared with the O-polanretsystem, the S-polanret device offers us much widerranges of azimuths a for which the transmittance ratioTr is lower than unity. This is an advantageous prop-erty of this new device.

The intensity of the diffracted light produced by themajority of phase or phase amplitude microobjects is


., . . . . I . . . . . . . _

CR

much lower than that of the direct light. The lattermust therefore be reduced, while the former should besustained as high as possible if we want to produce wellcontrasted microscopic images using phase contrasttechniques. Otherwise, if the diffracted light is sup-pressed by optical elements, a high power source oflight is necessary, which is frequently not tolerable inbiological microscopy when living cells and tissues arestudied. We must therefore speak about the economyof light of the polanret phase contrast systems. Toestimate this property of the S-polanret device, wepropose to use the following coefficient:

E = TComp(SP)TComp(OP)

UaDF() | X0 rD F(1) - 1 oL IDE(1) I

L _ ., . ,

0 0.25 0.5 0 0.25 0.5

(a) (b) T

Fig. 6. Particular azimuthal angles a of analyzer A for (a) the S-polanret system (Fig. 3) and (b) the O-polanret system (Fig. 2): aBF,bright field azimuths; DF, dark field azimuths; aNI, azimuths forwhich no microscopic structure of an object under study is observed.The clear areas refer to the ranges of a for which the transmittanceratio Tr < 1, whereas the line areas (less useful than the former) refer

to azimuths a for which T > 1.

E = T /2T3 = 1/2T.

The intensity transmittance X of the typical polariz-ing sheets (H type)7 of stretched and iodine fixed poly-vinyl alcohol is equal to 0.2-0.4. If, for example, =0.25, the economy of light of the S-polanret device istwice as good in comparison with the O-polanret sys-tem when each is used for dark field microscopy withthe same light source and the same microscope illumi-nating system.

C. Bright Field Microscopy (0 = 0; a = BF)

Rotating analyzer A [Fig. 3(a)] from a = 0 to a =3600 permits us to select four specific azimuths aBF

mentioned earlier, for which the transmittance ratio Tr= 1 as bright field microscopy requires. The azimuthsaBF are given [see Eqs. (B2), (B4), and (B5) in Appen-dix B] by

This formula gives four azimuths, BF = BF(1), aBF =

aBF(2), BF = BF(l) + 1800, and BF = BF(2) + 1800, forbright field microscopy.

Equation (8) shows that the bright field azimuthsaBF depend on transmittance of polarizing ring R[Fig. 3(a)]. This dependence is displayed in Fig. 6(a).

Returning to the O-polanret system (Fig. 2), we canfind that the azimuths aBF do not depend on transmit-tance T of the zonal polarizers Z1, Z2, and Z3. If Tr = 1,

360

0.

(6)

where Tcomp(SP) is the intensity transmittance ofthe complementary area of the S-polanret device, andTcomp(OP) is that of the same area of the O-polanretsystem.

The S-polanret device, like the O-polanret system, isprimarily intended for observation of phase microob-jects using the variable phase contrast method. How-ever, both are also suitable for the study of amplitudeand phase amplitude objects in ways equivalent to theabove-mentioned dark field technique, bright field mi-croscopy, and amplitude contrast technique. Thesepotentialities and specific properties of the S-polanretdevice are discussed below.

B. Central Dark Field Technique ( = 0; a = DF = 0 or1800)

As mentioned earlier, the situation equivalent to thecentral dark field technique occurs when the transmit-tance ratio Tr = 0. This means that the image of theobject under study is produced only by the diffractedlight. The intensity of this light is however reduced bythe polars P and A [Fig. 3(a)]. To evaluate this reduc-tion or the economy of light defined by Eq. (6), weassume that the all-polarizing elements (polarizer P,phase amplitude ring R, and analyzer A) are made ofthe same polarizing sheet whose intensity transmit-tance in unpolarized light is T. However, the intensitytransmittance of this sheet is equal to T1 = 2 if theincident light is linearly polarized and its vibrationsare parallel to the axis of light vibration in the polariz-ing sheet.

Since the vibration direction P-P of polarizer P(Figs. 3 and 2) is assumed to be parallel to the quarter-wave plate axis Q-Q (the azimuth 0 = 0) and it forms anangle of 450 with the light vibration axis A-A of analyz-er A (a = aDF = 0 or 1800), the transmittance TcOmp(SP)is given [see Eq. (Al) in Appendix A] by

TC//mp(SP) = 2T 2 COS 45 = T.

On the other hand, the transmittance Tcomp(OP) isexpressed [see Eq. (A6) for a = 0 or 1800] by

TC//mp(0P) = 2T'.

This quantity is smaller than 2; consequently, theeconomy of light [E, see Eq. (6)] offered by the S-polanret device is better by a factor of


(7)

1tanaBF = (8)

as bright field microscopy requires, the following azi-muths aBF result from Eq. (3) for this system: BF(1) =450, aBF(2) = 1350, aBF(3) = 2250, and aBF(4) = 3150.This situation is graphically shown in Fig. 6(b) and isclearly different from that [Fig. 6(a)] offered by the S-polanret device. I

What is now the economy of light (E) of the S-polanret device in comparison with the O-polanretsystem? It can be found [by using Eqs. (Al) and (A6)from Appendix A] that

TCnmp(SP) 2 cos2 (450 + aBF)EP=

Tcompw ) T(9)

where -r is, as before, the common intensity transmit-tance of a polarizing sheet of which all polarizers, P, A,Z1 , Z2, Z3, and R (Figs. 2 and 3), are made, while theazimuths aBF refer to the S-polanret device and aredefined by Eq. (8).

Bright field microscopy requires the transmittanceratio Tr to be equal to unity; this means that Tcomp =TConj. Consequently, for bright field microscopy therelative economy of light can also be expressed as E =-coj(SP)/rcoj(OP). By utilizing Eqs. (A3) and (A8)from Appendix A it can be found that the quantity inquestion is also given by

E = rcoj(SP) = 2 sin2 aBF.TC-..PP)

0 0.1 0.2 0.3 0.4 0.5

Fig. 7. Graphic representation of Eq. (9).

is the phase difference produced by the object; 4 'p andTr(op) are, respectively, the optimum phase differenceand the optimum intensity transmittance ratio of theconjugate and complementary areas of the exit pupil ofthe objective of the phase contrast system.

The phase difference p of pure amplitude objects isequal to zero, and the above equations take the form

Tr(Op) = (1 - r0 )2X (13)

(10) and

For given T and aBF, Eqs. (9) and (10) yield, of course,the same value for E.

Figure 7 shows the variation of E with r and aBF. Ascan be seen, the economy of light E is higher than unityfor aBF(2) and aBF(4), but lower than unity for aBF(1) andaBF(3). The S-polanret device therefore offers us twoversions of the same technique: intensive bright field(IBF) microscopy and damped bright field (DBF) mi-croscopy. These enable the image brightness to bevaried. If, for example, r = 0.3, this variation is as 1 to4. Such a property does not refer to the O-polanretsystem.

D. Amplitude Contrast Microscopy (6 = 0; a is Varied)

If the azimuth 0 = 0 (Fig. 3) and the azimuth a isvaried (but a #F aDF and aBE), a specific imagery withenhanced contrast occurs, which is referred to as am-plitude contrast microscopy. This technique is suit-able for observation of amplitude microobjects whosenatural contrast or artificial contrast obtained bystaining is weak. Its idea follows from both the vectorrepresentation and the algebraic theory of phase con-trast microscopy.1 6

It can be found that the condition for the optimumimage contrast takes the following form:

Tr(op) = 1 -2 g cos'P + Tos, (11)

sin2 q/0 p = sin2 p. (12)Tr(op)

Here ro, is the ratio of the intensity transmittance(To) of the object under study to that (1%) of an equalthickness of the surrounding medium, i.e., ros = ro/Ts; p

V/p = 0 for Tr, < 1, (14a)

orikop = 180° forr., > 1. (14b)

The transmittance ratio -r0 < 1 signifies that the objectis less transparent than its surrounding medium.Conversely, if ro, > 1, the object is more transparentthan its surroundings.

From the above relations it follows that the imagecontrast of amplitude objects can be optimized if 4' = 0(or 0 = 0, see Figs. 2 and 3) and Tr =

Tr(op) is selected

according to Eq. (13). Objects whose relative trans-mittance ratio i-rs changes from unity to zero requiretransmittance 1

rr(op) to be changed from zero to unity;this means that the intensity of the direct light shouldbe decreased. On the other hand, the objects for whichTos > 1 require transmittance ratio lrr(op) to be greaterthan unity; this means that the intensity of diffractedlight should be decreased. The former situation istypical in microscopy, but the latter can occur onlysporadically. The ranges of variation of the azimuth athat give Tr> 1 are therefore less useful in practice thanthose that give Tr < 1. Consequently, microscopy withTr < 1 will be referred to as the useful amplitudecontrast (UAC) technique, and that with rr > 1 will bedesignated as the less useful amplitude contrast(LUAC) technique. Ranges of azimith a for the UACare much larger than those for the LUAC if the S-polanret device is used [Fig. 8(a)], whereas the 0-polanret system offers us two 900 ranges for the UACtechnique [Fig. 8(b)] and also two 900 ranges for theLUAC technique. The S-polanret device appliestherefore better to the majority of amplitude objects.


(a)

Fig. 8. Ranges of azimuthal angles a for the useful (clear areas) andless useful (line areas) amplitude contrast techniques produced by(a) the S-polanret device and (b) the 0-polanret system. B, IBF,and DBF indicate the values of azimuth a for which the bright fieldmicroscopy occurs; DF refers to the central dark field microscopy,

and two no-image positions of the analyzer are marked by NI.

From Eq. (13) it also follows that the transmittanceratio T

r(op) = 1 is an optimum for Tos = 0; the bright fieldtechniques discussed earlier are therefore optimallysuitable for observation of completely opaque objects(the azimuth a = aBEF; see Figs. 6 and 8). If, however,the transmittances of an amplitude object and its sur-rounding medium are alike (o, = 1), the transmittanceratio Tr(op) = 0 appears to be an optimum; this situationcorresponds to the central dark field (DF) technique(the azimuth a = aDF = 0 or 180°).

Let us now find the factor of light economy (E) forthe amplitude contrast techniques. These require twofacts to be taken into account: first, the transmittanceratio 'r is variable with the azimuth a in different waysfor the S-polanret device and for the O-polanret sys-tem [see Eqs. (3) and (5) and Fig. 5]; second, we mustcompare the S-polanret device with the O-polanretsystem at the same level of the transmittance ratio Tr.Consequently, by utilizing Eqs. (Al) and (A6) fromAppendix A it can be found that

E = c (SP) cos'(450 + asp) (15)

T =O'"' ) T COS2a0 p

where asp and ap are the azimuths that produce thesame transmittance ratio Tr for the S-polanret deviceand the 0-polanret system, respectively. These azi-muths can be read from the graphs in Figs. 5 or calcu-lated from Eqs. (3) and (5). The final result is shown

LUJ

3

2

0 60 120 180180 1 240 1 300 360

OrB(3)J Mn

Fig. 9. Relative economy of light (E) of the S-polanret device, as afunction of the azimuth a, arranged for the amplitude contrasttechnique (the azimuths aBF refer to bright field microscopy and a =

0 or 180° to the central dark field microscopy).

in Fig. 9 (for T = 0.3). Curve E(a) shows that in thisinstance the economy of light exceeds unity in a widerange of the azimuths a and only within a narrowinterval (from about a = 10-60°) is lower than unity.The most useful interval of the transmittance ratio Tr isfrom 0.1 to 0.4 for the amplitude contrast technique.Figure 4 shows that this interval is covered by theazimuths a ranging from -100° to 1450, for which theeconomy of light E (Fig. 9) is higher than 3, and evenapproaches E = 4 for a 1350.

E. Variable Phase Contrast Technique ( and a areVaried)

This technique requires the azimuths 0 and a to bevaried (see Fig. 3) and optimized for observation ofdifferent phase objects. Consequently, the intensitytransmittance ratio of the conjugate and complemen-tary areas of the objective exit pupil is no longer ex-pressed by Eq. (5) since this ratio depends not only onthe azimuth a but also on the azimuth 0. By utilizingEqs. (CIO) and (C11) from Appendix C it can be foundthat

= = ~~~~T sin a .(16)

Tr Tr(SP) cos2(45' - 0 + a) + 0.5 sin20 cos(40 - 2a)

A graphic representation of this expression is shown inFig. 10 for selected constant azimuths 0 as a function ofa and in Fig. 11 for selected constant azimuths a as afunction of 0.

The respective graphs concerning the O-polanretsystem are displayed in Figs. 12 and 13 for comparisonpurposes. They result from

tan2aT (OP) =2[cos2(450 - 0) + 0.5 sin20 cos40] (17)

which is a combination of Eqs. (C9) and (C11) takenfrom Appendix C.

As can readily be seen, an essential difference occursbetween the graphs in Figs. 10 and 12. On the otherhand, the curves in Figs. 11 and 13 appear to be of thesame character, especially for azimuths a smaller than200. The curves for 0 = 0 or 180° and for 0 = 90 inFig. 10(a) do not apply to the phase contrast technique,and they are displayed only for comparison purposes.


0 20 40 60 80 100 120 140 160 180

(a) ao]

-180 -90 0

It . . . . . . . . . . . . . .

::: ~~~~~8:±5

0.6 S

0.4

0.2

0 20 40 60 80 100 120 140 160 180

(b) 0C.[0Fig. 10. Graphic representation of Eq. (16) for selected constant

azimuths 0 as a function of a.

90 45 0 -45 -90

Thia zimuth foe segatie -0.8

0.6

0.4-'

0.2 '0

-90 -75 -60 -45 -30 -15 .0 15 30 45 60 75 90

Fig. 11. Graphic representation of Eq. (16) for selected constantazimuths a as a function of 0.

The most important curve is that for = 45'. Itapplies to the pure phase objects which produce smallphase differences s. These objects are mainly used inbiomedical research. It is also important to note thatthe azimuth = 450 or = -45° produces the curveTr(a) with its identical and relatively flat slope. More-over, an advantage of this curve, as well as of othercurves shown in Fig. 10(b), is the fact that the maxi-mum values of Tr do not exceed unity.

-180

4 -

3 -2-

1-

0

0.815 I 75

0.6 30160

0 ;t45;t9o

-301-600.4

-15 ;-75

0.2

0 20 40 60 80 100 120 140 160 180a. [°)

Fig. 12. Graphic representation of Eq. (17) for selected constantazimuths as a function of a.

60 758 [0]

Fig. 13. Graphic representation of Eq. (17) for selected constantazimuths a as a function of 0.

Let us now return to Eqs. (11) and (12). If thetransmittance ratio -r,, of a specimen is equal to unity(pure phase objects), these equations take the form

Tr(op) = 2(1 -cost) = 4 sin 2-2

sin __si_ 0 i--O -

'1 - rt pa 2 sin('p/2) ( 2)

Consequently,

ASP= +90° + 2

(18)

(19)

(20)

If i1Vp (or, generally, 4) is positive, the situation isreferred to as the positive phase contrast (PPC) tech-nique. Conversely, if tp < 0 (or, in general, 4 < 0), wehave the negative phase contrast (NPC) technique.

In the majority of microscopic studies, especially inbiology and biomedicine, the phase difference pc pro-duced by the examined objects is much smaller than90°; thus, Eqs. (18) and (20) take the form:


rr(Op) = (21)

(here p is expressed in radians) and,ap= 90. (22)

Equation (21) shows that the optimum transmittanceratio Tr(op) should be smaller than unity for Isol < 1 rad,while Eq. (22) implies that the azimuth = 45°.Figure 10 (see graphs for 0 = +45° and 6 = -45°) showsthat the S-polanret device is more consistent with theabove requirement than the O-polanret system sincethe former has the transmittance ratio r, < 1 for 100%while the latter (see Fig. 12) only for 50% of the overallrange of the azimuthal angles a.

If the phase difference (p is small, the optimum azi-muth a differs only slightly from 0 or 1800. Varyingthe azimuth changes the image contrast of phasemicroobjects as shown in Table I. When, for example,the image contrast is positive within the azimuthalintervals 0 < 0 < 900 and 180° < < 2700, it is negativewithin the intervals 90° < 0 < 180° and 270° < 0 <3600, and vice versa. The sign of the image contrastalso changes four times (twice from positive to nega-tive, and twice from negative to positive) during therotation of the analyzer through 3600. This phenome-non also occurs in the O-polanret system and is causedby ajump of the phase difference 4 by 180°. Thejumpis sudden and it occurs when the analyzer undergoesthe azimuthal positions a = 900, 1800, 2700, and 3600(or O).

It is worth noting that analyzer A of the S-polanretdevice (Fig. 3) changes not only the transmittance ratioTr, but also slightly shifts the phase of diffracted lightwhen azimuth 0 approaches 450, 1350, 2250 and 3150.

This additional phase shift modifies the phase differ-ence 4' corresponding to azimuthal angles 0, but it canbe balanced by an appropriate overrotation or under-rotation of polarizer P. Such a situation implies thefollowing steps for arranging the correct use of thephase contrast technique: first, polarizer P and ana-lyzer A remain in their initial positions (6 = 0 and a =0); then, the analyzer is rotated by a small angle a toobtain the expected transmittance ratio Tr(op), and thepolarizer is rotated until a satisfactory image contrastis obtained, positive or negative at will; finally, theimage contrast can be optimized by readjusting theanalyzer. At any rate, experience has shown that it is

Table I. Relationship Between Azimuth 0 and the Image Contrast of aPure Phase Microobject that Produces a Very Small Phase Difference p

(Azimuth a Differs Only Slightly from 0 or 1800)

Azimuth 0 (deg) Image contrast

0 or 180 Equal to zero0 < 0 < 45 or 180 < 0 < 225 Increasing

45 or 225 Maximum45 < 0 < 90 or 225 < 0 < 270 Decreasing

90 or 270 Equal to zero90 < 0 < 135 or 270 < 0 < 315 Reversed, increasing

135 or 315 Reversed, maximum135 < 0 < 180 or 315 < 0 < 360 Reversed, decreasing

more important to be able to adjust precisely the trans-mittance ratio Tr than the phase difference 4.

Let us now consider the relative economy of light (E)offered by the S-polanret device arranged for the phasecontrast technique. In this instance we cannot ignorethe quarterwave plate Q (Figs. 2 and 3) since the polar-izer axis P-P remains no longer parallel to the quarter-wave plate axis Q-Q as it was formerly when the darkfield, bright field, and amplitude contrast techniqueswere discussed.

By utilizing Eqs. (C9) and (C10) from Appendix C itcan be found that

E = rcomp(SP)Tcomp(wP)

cos(45' - 0 + asp) + 0.5 sin20 cos(40 - 2aSp)

[cos(45' - 0) + 0.5 sin20 cos40]2T cos2a 0 p(23)

where asp and aop are the analyzer azimuths for whichthe transmittance ratios -rr of the S-polanret deviceand the O-polanret system are the same.

If the phase difference (p produced by the phasemicroobjects under study is small, the optimum azi-muth 0 must be selected close to +45° or - 450, and theabove expression takes a quite simple form:

(24)E= 1 12r cos2 a0 p

where r is, as previously, the common intensity trans-mittance of the polarizing sheet of which polarizingring R (Fig. 3) is made.

For very small s, the optimum azimuth ap thatproduces the optimum transmittance ratio T

r(op) dif-fers only slightly from zero, and E - 1/27-. This resultis the same as for the central dark field technique [seeEq. (7)]. Let r be equal to 0.3. Then E = 1.67 for asp

a aop - 0 and E = 2.68 for the azimuth aop = 380,which produces the same transmittance ratio Tr = 0.6(see Fig. 12 and curve for 0 = ±450) as the azimuth asp= 900 (see Fig. 10 and curve for = d459). On anaverage, the relative economy of light is twice as goodwhen the S-polanret device is used for phase contrastmicroscopy. This is especially important for the studyof living cells and tissues, which generally do not toler-ate high power sources of light.

V. Practical Implementation

Like the O-polanret system3 (Fig. 2), the S-polanretdevice (Fig. 3) requires a microscope with a transferredimage unit that reimages the back focal plane (or theexit pupil) of the middle power and high power objec-tives in an easily accessible intermediate plane wherethe phase amplitude disk (PD) with the polarizing ring(R) must be placed. Otherwise, this device can be usedonly with low power microscope objectives whose backfocal plane is typically outside their lens system. For-tunately, modern inverted biological microscopes andalso some upright universal research microscopes (e.g.,Axiomat available from C. Zeiss Oberkochen, UnivaRfrom C. Reichert Vienna, and Peraval-Interphakofrom C. Zeiss Jena) are constructed so that they pos-


sess a secondary easily accessible exit pupil plane oftheir objectives. Such a two-pupil universal micro-scope (designed several years ago in the Central Opti-cal Laboratory, Warsaw, for the Polish Optical Works,but not yet commercially available) was used for theexperimental verification of the S-polanret device de-veloped.

Obviously low, middle, and high power microscopeobjectives require phase amplitude rings whose diame-ter (dR) and width (WR) are different. A prototypemicroscope with the S-polanret equipment was fittedwith four polarizing phase amplitude disks PD [Fig.3(a)] corresponding to 1OX/0.25, 20X/0.40, 40X/0.65,and 100X/1.25 (oil immersion) objectives of achromat-ic correction. The geometric parameters of phase am-plitude rings R were as follows: dR = 0.5dEp and AR =

0.1AEp. Here dR is the diameter of polarizing ring R,AR is its area (this area is equivalent, with a surplus, tothe conjugate area), dEp is the diameter of the exitpupil, and AEP is its area (it is self-evident that thedifference AEP - AR is equivalent to the complemen-tary area). From the above parameters, which aretypical for fixed phase objectives used in conventionalphase contrast microscopy, it appears that width WR Ofpolarizing ring R should be not larger than 0.40, 0.32,0.26, and 0.21 mm, respectively, for the 10X, 20X, 40Xand 10OX objectives.

A technological problem was encountered in obtain-ing such small polarizing rings with extremely fineborders (edges) and regular (round) contours. After anumber of experiments, the problem was solved satis-factorily using a specific instrument of the punch presstype, which was specially constructed for this purpose.

A typical polarizing sheet of stretched and iodinefixed polyvinyl alcohol was used to cut polarizing rings.Its thickness t was not >0.1 mm and not smaller than0.05 mm, and its refractive index nR for spectral lines F,e, D, and C was equal to -1.526, 1.520,1.517, and 1.512.The degree of polarization of this sheet was not lowerthan 99.98% through the visible spectrum from theblue to the red; it was comparable with that of HN-22type sheets specified by Clarke and Grainger in theirbook.7 The general optical quality of the polarizingsheet used was nearly the same as that of commerciallyavailable polarizing foils of which the polarizers andanalyzers are manufactured for present-day commonpolarizing microscopes.

Polarizing ring R [Fig. 3(a)] is simply cemented be-tween two glass plates, but special attention must bepaid to ensure its central position between the diskglasses; moreover, all air bubbles should be carefullyremoved from the cement layer. No problems arise inthis operation for a person with some experience incementing optical elements. However, an optical ce-ment free from birefringent strains should be used; asemihard cement is therefore recommended. This canbe selected from a manufacturer's list or preparedindividually using some transparent plastics (poly-mers) and solvents which together give a compositionthat does not destroy the polarizing ring and has therefractive index nc satisfying condition (4) for light

Fig. 14. Correct wavefront shear interference image of a phaseamplitude plate (PD, Fig. 3) of the S-polanret device. The interfer-ence image, taken in monochromatic light (green or yellow), showsonly a fragment of the polarizing ring (R, Fig. 3) whose width is, inthis instance, equal to 0.32 mm. Print magnification PM = 5OX.

wavelengths X from the middle region of the visiblespectrum (here the suffix C added to n does not signifythe spectral line C, but it refers to the word cement).It has been experimentally stated that a discrepancyA6CR between an actual situation and theoretical con-dition (4) can be tolerable if A6CR is not higher than±0.12\ in the red and/or blue portions of the visiblespectrum. However, in green and yellow light thiscondition should be fulfilled as accurately as possible.This requirement was verified using a double-refract-ing interference microscope 8 (Biolar PI available fromPolish Optical Works-PZO, Warsaw). A correct inter-ference image of the phase amplitude plate with polar-izing ring R is shown in Fig. 14; no displacement ofinterference fringes is observed in the duplicated(sheared) portions of the image of the polarizing ringtested.

Unfortunately, polarizing sheets available commer-cially are not uniform with reference to their refractiveindex nil, which is denoted by nR in Eq. (4) and referredto as light vibrating parallel to the axis of light vibra-tions transmitted by the polarizing sheet. Moreover,even a single sheet has different nil in its differentfragments; a difference of Anil equal to -0.006 wasobserved. This optical inhomogeneity is primarilycaused by nonuniform stretching of the polyvinyl alco-hol foil. It is therefore necessary to select appropriatefragments of a polarizing sheet for a given opticalcement with which the phase amplitude ring R [Fig.3(a)] is mounted between two glass plates. At anyrate, it is not possible to indicate a concrete polarizingsheet and a concrete optical cement; these must beindividually matched by a manufacturer of phase am-plitude plates PD with a single polarizing ring R asproposed in this paper for polanret phase contrastmicroscopy. In the experiments reported here, differ-ent cementing compositions were used, which satisfiedcondition (4) with m = 0 or m = - 1. No difference wasobserved in the performance of the S-polanret systemwhen the factor m was equal to zero and to minus unity.

A standard phase condenser with four annular dia-phragms D [Fig. 3(a)], arranged in a rotatable disk, was


used in the experiments reported here. Polarizer Pand quarterwave plate Q may, in principle, be locatedanywhere between the phase amplitude plate PD andthe source of light. Analyzer A may also be placedanywhere between plate PD and the eyepiece Oc.Since lenses rotate slightly in the plane of light vibra-tion of linearly polarized waves and produce ellipticalpolarization, polarizer P, quarterwave plate Q andanalyzer A preferably should be located adjacent to thephase amplitude plate PD. The modern two-pupilmicroscope systems are usually suitable for such anadjacent location of all these elements of the S-polan-ret equipment. A family of four phase amplitudeplates (for the four microscope objectives mentionedearlier) was arranged in a rotatable disk. Each platePD was individually mounted in a metallic frame. Po-larizer P, analyzer A, and quarterwave plate Q wereborrowed from a conventional polarizing microscope.Plate Q was a first-order retarder made of quartz crys-tal. Its spectral dispersion of retardance was mean-ingless. In the experiments performed polarizer Pwasplaced before condenser C and analyzer A before theocular Oc. However, the quarterwave plate wasplaced close to the phase amplitude plate PD in thesecondary exit pupil of microscope objectives. Thisposition requires the optical elements of good quality.The quarterwave plate is just among such elements.In contrast, a polarizing sheet has superficial inhomo-geneities, which are, fortunately, canceled by the opti-cal cement with which the polarizing ring is surround-ed between two glass plates perfectly polished.

VI. Results

Satisfactory or even high performance of the S-polanret device was predicted at the moment when theidea of this device was conceived; it is clear that thephase amplitude disk PD [Fig. 3(a)] with a single polar-izing ring R can be made more accurately than theother polarizing phase plates that consist of three (Os-terberg, 2 3Nomarski 4 ) or two (Nikon5 ) zonal polarizers[Z1, Z2, and Z3 or Z2 and Z3 as shown in Fig. 2(a)]. Themultizonal polarizing plates can easily suffer frommore defects and optical artifacts, which disturb theimage of objects under study.

To illustrate the imaging performance of the S-po-lanret device, Figs. 15-18 show photomicrographs tak-en with the two-pupil biological microscope fitted withthe S-polanret equipment developed as described inSec. V.

Figure 15 shows an image variation (from brightfield to positive and negative phase contrast via centraldark field) of a smear of a squashed muscle tissuemounted in Canada balsam between the glass slide andcover slip. In general, the images are quite- consistentwith the theoretical considerations given in Sec. IV.

Figure 16 illustrates the optimal phase contrast im-ages (positive and negative) of a monolayer of dry yeastcells obtained by evaporation from an aqueous suspen-sion of yeast between the glass slide and cover slip.The halo effect, a phenomenon typical of phase con-trast microscopy, is seen in the background as bright

[Fig. 16(a)] or dark [Fig. 16(b)] zones that surround theobject images. The bright halo is attached to the darkimages, while the dark halo accompanies the brightimages of phase objects. The dark halo is, in general,less disturbing than the bright one.

Specimens whose images are shown in Figs. 15 and16 were purely phase objects, which could not be ob-served in a conventional bright field microscope. Incontrast, the specimen whose images are shown in Fig.17 was purely an amplitude object. Its natural con-trast was, however, weak [Fig. 17(a)]. The image con-trast can be significantly improved [Fig. 17(b)] usingthe amplitude contrast technique discussed in Sec.IV.D.

Some experiments were also performed by using aphase amplitude plate PD [see Fig. 3(a)] of whichpolarizing ring R was replaced by a narrow polarizingstripe, and the condenser diaphragm D with annularopening AO was replaced by a slit diaphragm. Such aversion of the S-polanret device appeared to be usefulfor the study of elongated microobjects. An exampleis shown in Fig. 18, where phase contrast images offresh water diatoms are presented. Here we can dis-tinguish elongated individuals with a microstructureconsisting of transverse striae or lines. This structurecan clearly be improved when the elongated individualis oriented at right angles [Fig. 19(b)] to the phaseamplitude stripe PS [Fig. 19(a)]. Conversely, whenthe individual is parallel to the stripe PS [Fig. 19(c)],the structure mentioned above is no longer observed;however, the whole individual is now clearly visible.The specimen of fresh water diatoms was a phasepreparation which could not be observed by means of aconventional bright field microscope or even by usingthe S-polanret device adjusted to the bright field tech-nique discussed in Sec. IV.C.

Some further examples illustrating the performanceof the S-polanret device developed can be found in ashort contribution 8 prepared (simultaneously withthis paper) for biological microscopists.

VII. Conclusions

Like the O-polanret system invented by Oster-berg,23 the S-polanret device has comparable capabili-ty concerning the study of phase, phase amplitude, andamplitude microobjects. However, construction ofthe S-polanret device is much simpler, hence inexpen-sive, and it appears to be better adapted than the 0-polanret system to the study of the majority of objectsoccurring in microscopy practice. Moreover, its econ-omy of light is better and it accepts conventional lightsources used in modern biological microscopes. Thisproperty is especially important when living cells andtissues are studied, which do not tolerate light sourcesof high intensity. No monochromatic light is neces-sary; all photomicrographs shown in Figs. 15-18 weretaken in white light and with typical achromatic micro-scope optics.

Like the O-polanret system and other devices of thistype known to date (Nomarski,4 Nikon5), the S-polan-ret equipment requires microscopes whose objective


a

a

C dFig. 15. Unstained smear of a squashed muscle tissue embedded inCanada balsam between a glass slide and cover slip. The S-polanretdevice was set at (a) bright field, (b) positive phase contrast, (c) darkfield, and (d) negative phase contrast images. A middle power

microscope objective was used; print magnification PM = 160X.

Fig. 16. Monolayer of dry yeast cells obtained by evaporation froman aqueous suspension of yeast between a glass slide and cover slip.The S-polanret device was set at optimum (a) positive and (b)

negative image contrast. Objective 20X/0.40; PM = 270X.

b

Fig. 17. Images of a metallic thin film of 45% intensity transmit-tance with a number of parallel clear slits (this is a pure amplitudespecimen): (a) the S-polanret device was adjusted for bright fieldmicroscopy ( = 0 and Tr = 1); (b) the S-polanret device was adjustedat the optimum level of a pure amplitude contrast image ( = 0, T,. =

0.2). Objective 2.5X/0.06; PM = 35X.

a

b

exit pupils are transferred into an easily accessiblesecondary plane, where the polarizing phase ampli-tude plates must be located. Today this requirementis no serious limitation since present-day microscopes,especially inverted biological microscopes and also up-right universal research microscopes, are constructedso that they possess a secondary (intermediate) andeasily accessible exit pupil plane.

In general, phase contrast microscopy suffers fromthe halo and shading-off effects.' These are, however,in the essential nature of the phase contrast methodand can never be removed. Nevertheless, the S-polan-ret device (like the 0-polanret system) enables theseundesirable effects to be controlled and reduced tosome extent to optimize the quality and fidelity ofphase contrast imaging.

I wish to thank the reviewers for their constructivecomments, which permitted me to expand Secs. IV.Eand V of this paper.

Fig. 18. Fresh water diatoms mounted in Canada balsam between aglass slide and cover slip. The S-polanret device with a slit condens-er diaphragm and a stripe polarizing phase amplitude plate [see Fig.

19(a)] was used. Objective OX/0.25; PM = 15OX.

I also thank E. Sobolewska for her help in makingthe photocopies of the illustrations.

Appendix A: Derivation of the Function r,(a)

S-polanret device. We assume that azimuth = 0[see Fig. 3(b)], while azimuth a is varied. Let T be thecommon transmittance in unpolarized light of a polar-izing sheet of which polarizer P, phase amplitude ringR, and analyzer A are made [see Fig. 3(a)]. Ring R andanalyzer A are, however, struck by light linearly polar-


Ps s'

(IA lulJJIIIii[1111 IIllh 1(a) (c)

Fig. 19. Illustration of the situation shown in photomicrographs ofFig. 18: (a) orientation of the polarizing phase amplitude stripe PSthat covers image S' of the condenser slit; (b) and (c) perpendicularand parallel orientations with respect to PS of an elongated individ-

ual with transverse line structure.

ized; their transmittance is therefore equal to 2r whenthe vibrations of the incident light are parallel to theR-R and A-A axes of these polarizing elements.

The complementary area of the S-polanret device iscovered by two polarizers, P and A, whose directionsP-P and A-A of light vibration form an angle of 450 +a with each other. The transmittance Tmp of thisarea is therefore given (according to the Malus law) by

TComp = 2r2 COS2(45 + a). (Al)

On the other hand, the conjugate area is covered bythree polarizers. The first two, P and R, give thetransmittance of this area expressed by

Tro,,j = 2Tr cos45 - 12, (A2)

and the resultant transmittance -rconj, when analyzer Ais added to P and R, is given by

lcnjl T0onj21 COS2(90, + a) = 2T sin2a. (A3)

From Eqs. (Al) and (A3) we obtain

= Iconj T sin a (A4)1 comp COS 2 (450 + a)

0-polanret system. Both the complementary andconjugate areas of this system are covered by threepolarizers (see Fig. 2). The complementary areatransmittance produced by polarizers P, Z2, and Z3 isexpressed by

Tcomp = 2 cos2450 = T. (A)

When analyzer A is added, the resultant transmittanceof this area is given by

Tcomp = Tcomp2T cos a = 27 cosa. (A6)

Similarly, the conjugate area transmittance producedby polarizers P and Z, is expressed by

rco,,j = 2T cos2450 = T. (A7)

This is modified by analyzer A, and the resultanttransmittance of this area is given by

rconj = conj2rcos(90 + a) 2T sin ca.

By combining Eqs. (A6) and (A8), we obtain

(A)

A

Fig. 20. Illustration of Eq. (B2).

1COfli 2Tr = = tan a.T

comp

(A9)

Appendix B: Derivation of Eq. (8)

Trigonometric approach. To find the angles aBFfor bright field microscopy, we can assume that theamplitude transmittances of the conjugate and com-plementary areas are represented by two vectors, V1and V2 (Fig. 20), and that the length of vector V2 isequal to unity (V 21 = 1) before analyzer A (see Fig. 3)and its direction is parallel to P-P and Q-Q [see Fig.3(b)]. On the other hand, vector V1 is equal to V2before polarizing ring R, but after this ring it becomesparallel to R-R and its length is reduced to IV1 ex-pressed by

V1 I = V21X-I cose = 2r COSE,

where r is, as before, the common intensity transmit-tance of a polarizing sheet of which polarizer P, phaseamplitude ring R, and analyzer A are made. Conse-quently, 4/S is the amplitude transmittance of ring Rfor the light linearly polarized by polarizer P and inci-dent normally on this ring. The angle e between vec-tors V1 and V2 (Fig. 20) is equal to 450, thus IV,1 = 4;F.

Starting from its zero position, analyzer A (Fig. 3)can be oriented so that the normal projections V'1 andV'2 of vectors V, and V2 on the analyzer axis A-A (Fig.20) have the same lengths, i.e., IV'= IV21. It is self-evident that for this situation the transmittance ratioTr = 1.

From the geometry of Fig. 20 it follows that

Iv;I = 1v11 cos(90' - aBF) = a sinaBF,

Iv1 = Iv21 cos(450 + aBF) = cos(45' + aBF)-

Consequently, we obtain the following equation:

(B1)cos(45 + aBF) = a,/ sinaBF,

from which it follows that


A

l

A

A

A

Fig. 21. Illustration of Eq. (4).

1tanaBF = (B2)

1 + 2r

This formula gives two azimuths: aBF = aBF(1) andaBF = aBF(3) = 1800 + aBF(1) for the microscopic tech-nique equivalent to bright field microscopy.

Lengths V'1 and IV21 of vectors V1 and V2 are alsoequal to each other when vectors V1 and V2 are normal-ly projected on the analyzer axis A-A oriented asshown in Fig. 21. From the geometry of this figure itfollows that

IV'11 = V11 cos(90' - aBF) = a/r sinaBF,

Iv2l = IV21 cos(180' - aBF - 450) = -cos(45' + aBF).

Therefore we have the following equation:

cos(450 + aBF) =-a sinaBF, (B3)

which yields

tanaBF (B4)1 - 2r

This formula gives two further azimuths, aBF(2) andaBF(4) = 1800 + aBF(2), for bright field microscopy.Note that aBF(2) > aBF(1)-

Derivation based on the Malus law. We assumethat T

comp = T conj as required for bright field microsco-py. Thus, from Eq. (A4) we obtain

cos2(45' + a) = r sin2a,

or

cos(450 + a) = ±f sina.

This equation can be rewritten as

cos45' cosa - sin45' sina = 4r sina,

orcosa - sina = 2-i- sina,

from which it follows that

A

0 B X

Fig. 22. Illustration of the derivation of Eq. (C7).

tana = tanaBF IBF Zr (B5)

Appendix C: Derivation of the Intensity Transmittance ofTwo Polars Separated by a Birefringent Retarder

Let a ray of unpolarized light of intensity Io normallystrike the linear polarizer followed by a birefringentretarder and the analyzer. The intensity transmit-tance of the polars is -. The axes (fast and slow) of theretarder are coincident with the rectangular coordi-nate axes x and y (Fig. 22), the origin of which isdenoted by 0. Directions OP and OA of light vibra-tion in the polarizer and analyzer form angles 0 and 6 -f with the x axis ( is the angle between OP and OA).The amplitude component OE of light vibration trans-mitted by the polarizer is equal to duo. The projec-tions of the amplitude OE on the x and y axes are givenby OB = ,j/o cos6 and OC = JIo sin6. The analyzertransmits only the OD and OF portions of light vibra-tion parallel to OA. These portions are the projectionsof amplitudes OB and OC on the analyzer axis OA.The lengths or amplitudes of these projections areexpressed by

OD = OB27 cos(O - O) = TXo cos0 cos(O -),

OF = OC Z2 sin(O - f) = T2o sin0 sin(O - A).

(Cl)

(C2)

These expressions can be considered the amplitudes oftwo coherent waves which interfere with each otherand produce the resultant intensity I defined by thewell-known formula

I = I + I2 + 2 cosp, (C3)

where p is the phase difference of these waves. In thisinstance, the phase difference is produced by the bire-fringent retarder, which separates the polarizer andanalyzer, while intensities I, and I2 are given by thesquared Eqs. (Cl) and (C2), i.e.,

I = 2I COS2

0 CoS2

(0 - ),

I = 2T21o sin2 0 sin2 (0 - ).

(C4)

(CM)


ID

By substituting these expressions into Eq. (C3) andapplying some trigonometric identities we obtain

I = 22ECos2# -sin20 sin2(0 - /3) sin2-] * (C6)

10 L 2J

It is self-evident that the ratio I/1o expresses the resul-tant transmittance Tres of the system that consists oftwo polars separated by a birefringent retarder. Forthe quarterwave retarder (Q, see Figs. 2 and 3) thephase difference p is equal to 900, and Eq. (C6) takesthe form

rres = I/b0 = 2r2 [cos2 /3 - /2 sin20 sin2(0 - /3)]. (C7)

This equation applies directly to the intensity trans-mittance Tcomp(SP) of the complementary area of theobjective exit pupil of the S-polanret device (see Fig.3). Whereas the respective transmittance Tcomp(OP)of the 0-polanret system (Fig. 2) is given by

,rcomp(OP) = 2T2 [COS2 0, - 1/2 sin2(0 - /')]2 COS2a, (C8)

where f' is the angle between the P-P axis of polarizerP and the Z2-Z2 axis of the zonal polarizers Z2 and Z3.From the geometry of Figs. 2(b) and 3(b) it follows that' = 450 - and = 450 - + a. Consequently, Eq.(C8) takes the form

Tcomp(OP) = 2T2 [COS2 (45' - 0) + /2 sin20 cos4O]2T COS2a. (C9)

Similarly, -rcomp(SP) can be expressed as

Tcomp(SP) = 2 2[COS

2 (45' - 0 + a) + 1/2 sin20 cos(40 - 2a)]. (C10)

It is worth noting that Eqs. (C9) and (C10) are equiv-alent, respectively, to Eqs. (A6) and (Al) if azimuth 0 =0.

In contrast, the intensity transmittances rconj(SP)and rconj(OP) of the conjugate areas of the S-polanret

device and of the 0-polanret system are identical andcan be expressed by a formula similar to Eq. (C8) inwhich, however, 3' denotes the angle between the P-Paxis of polarizer P and the Z1-Z 1 axis of the zonalpolarizer Z1 (Fig. 2) or the R-R axis of polarizing ring R(Fig. 3). This angle is now equal to 45° + 0. More-over, the geometry of Figs. 2(b) and 3(b) shows that theangle a in Eq. (C8) should be replaced by 900 + a.Consequently, the intensity transmittances in ques-tion can be expressed as

rTon(SP) = 'ron(OP)

= 2r 2[CoS2(450 + 0) + 1/2 sin20]2r sin2 a. (Cli)

It can readily be seen that this expression takes theform of Eqs. (A3) and (A8) if azimuth 0 = 0.

References1. M. Pluta, "Non-Standard Methods of Phase Contrast Micro-

scopy," Adv. Opt. Electron Microsc. 6, 49 (1975).2. H. Osterberg, "The Polanret Microscope," J. Opt. Soc. Am. 37,

726 (1947).3. 0. W. Richards, "The Polanret Variable Densiphase Micro-

scope," J. Microsc. 98, 67 (1973).4. G. Nomarski, "A Variable Achromatic Phase Contrast Micro-

scope," J. Opt. Soc. Am. 58,1568 (1968).5. Nikon's Interference Phase Attachment, Technical Description

64.11.B.E. (Nippon Kogaku K.K., Japan, 1964).6. A. H. Bennett, H. Jupnik, H. Osterberg, and 0. W. Richards,

Phase Microscopy (Wiley, New York, 1951).7. D. Clarke and J. F. Grainger, Polarized Light and Optical Mea-

surement (Pergamon, Oxford, 1971).8. M. Pluta, "A Double Refracting Interference Microscope with

Continuously Variable Amount and Direction of WavefrontShear," Opt. Acta 18, 661 (1971).

9. M. Pluta, "Single-Ring Polanret Phase-Contrast System," J.Microsc. 148, 11 (1987).

Books continued from page 1444In 1917 Raman became Palit Professor of Physics at the Universi-

ty of Calcutta, and in 1919 Honorary Secretary of the Association.During this period in February 1928 he made a major discovery inthe scattering of light by molecules, now known as Raman scattering.With ordinary or Rayleigh scattering there is no change in thefrequency of light. In Raman scattering there is a change in fre-quency and random change in phase of a photon traveling through atransparent medium due to the change in vibrational or rotationalenergy of the scattering molecule transferred to the photon. Thebook fully discusses the history of this type of scattering, the physicsinvolved, and the persons in the research and their relationships,including the work of Mandelshtam and Landsberg. Raman wasknighted by the British Government in India in 1929 and receivedthe Nobel Prize in December 1930.

Raman made a great physical investigation of diamonds startingabout 1930 and lasting for many years. He began building up apersonal collection out of his own money (including Nobel Prizemoney), and by 1944 he had a collection of 310 diamonds.

After a controversy with Meghad Saha lasting some years, Ramanin April 1933 joined the Institute of Science in Bangalore, where helater became head of the Raman Research Institute, organized in1948. His last papers were on vision and the perception of color. Hedied 21 Nov. 1970. His scientific papers, published in six volumes,number 495 papers grouped under scattering of light, acoustics,optics, crystal physics, and perception of color.

He contributed some papers to the Journal of the Optical Societyof America. His paper in Vol. 1 of Applied Optics on the spectro-

scopic behavior of rock salt and his volume on Lectures in PhysicalOptics, Vol. 1 (1926) [reviewed in J. Opt. Soc. Am. 50,124 (1960)] arenot listed with his publications. He was elected an honorary mem-ber of the Optical Society of America in 1941.

While working on diffraction some years ago this reviewer noticedRaman and Krishnan's work in the Proc. Phys. Soc. 38,350 (1926) onthe diffraction of light by a sphere, which had some interestingresults in the shadow along the optical axis. This led to somecorrespondence with Raman who generously sent all his publicationson the subject.

Raman was involved in many controversies. This was partly dueto the remoteness of India, the difficulty of financial support, butalso to his attempts to develop and maintain scientific excellenceunder the conditions in India at that time. His personal ego contrib-uted to many controversies as his biographer remarks: "He wasknown for his colorful discussion of his rivals and their theories."He attempted to get Max Born to join him in Bangalore in 1935 butfinances were too limited. He later had a controversy with Born onthe interpretation of second-order Raman scattering, which is de-scribed in detail. His biographer notes that in the end Born wassubstantially correct.

This is a readable book on science in India and the life of C. V.Raman, giving details of his scientific contributions and controver-sies in which he was involved, interwoven with many anecdotes andpersonal sidelights.

FRANKLIN S. HARRIS, JR.

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Simplified polanret system for microscopy