Studies on depolarization of light at microscope lens surfaces

199 STUDIES

ON DEPOLARIZATION LENS I.

THE

ORIGIN AT

OF THE

OF LIGHT SURFACES

STRAY LENS

SHINYA Department

of Biology,

AT MICROSCOPE

Princeton

LIGHT

BY

ROT=\TION

SURFACES’

INOUE’ University,

Princeton,

N. J.

Received June 15, 1951

IT is commonly found that only low aperture lenses can be used with the or detecting small retardations. polarization microscope for measuring 115th low aperture, the field of the microscope can be made very dark with crossed nicols, even with a very intense light source. Under this condition a weakly birefringent object will shine brightly against a dark background. If, on the other hand, a wide aperture objective is used and the condenser diaphragm is not stopped down, the field no longer remains dark. In this case a birefringent object does not increase its brightness proportionately \vith the brightness of the field, and so the contrast between the object and its background decreases. Consequently, if the retardation is not large, the birefringent image is obliterated, or nearly so, at higher apertures. For biologists who attempt to study the fine structure of minute ccl1 organelles, A high aperture is necessary for good this is a very distressing limitation. resolution, and \-et at the same time the high sensitivity required for detecting weak birefringence can only be attained at low apertures. InouP and Dan (2, 3) suggested that the stray light dro\vning the birefringent image at high apertures arises by depolarization at lens surfaces. This depolarization gives rise to the so called “polarization cross” (fig. l), obscrvable at the exit pupil of the objective \vhen the ocular is removed, or \vith a Bertrand lens. The cross can be seen xvithout any object on the stage, and becomes clearer the more intense the light source, and the less defective the optical system. 1 Presented to the faculty of Princeton University in partial fulfilment ments for the degree of Doctor of Philosophy. This research was supported of the Eugene Higgins Trust allocated to Princeton University. 2 Present address: Department of Anatomy, School of hledicine, University Seattle 5, \Vash., USr\. 12 - 5-‘3;01

of the requirein part by funds of \\‘ashington,

s. Inoue’ The existence of the polarization cross means that the extinction is nearl! perfect only along the arms of the cross, lvhich are parallel and perpendicular to the axes of the polarizer and the analyzer. Stray light enters into the system from the bright regions between the dark arms. Now, the polarization cross is similar in appearance to the interference figures of monaxial crystals, but is caused in an entirely different manner. Rinne (5) first explained the cross to be caused by rotation of the plane of polarization at lens surfaces. Each of the rays which hits the lenses obliquely is considered to be separated into two vectors at every glass surface, one parallel and the other perpendicular to the plane of incidence. The two vectors are reflected at different rates, and so the two transmitted vectors no longer have the same ratio as the tw-o entrant vectors. The combined transmitted vector, therefore, has a different direction compared to the original entrant vector (fig. 2). This results in a rotation, whose direction in each quadrant of the cross is determinate, since the vector perpendicular to the plane of incidence always suffers the greater reflection. The amount of rotation is greater the greater the angle of incidence, and the greater the angle between the plane of polarization and the plane of incidence. Rinne’s explanation was first confirmed and amplified by Cdsaro (1) and later by Wright (6, 7, 8). Using Fresnel’s formulae both CCsaro and\\‘right calculated the expected rotation for rays passing through tilted glass plates, as well as the amount of rotation at different points of the esit pupil of an idealized objective lens. Their calculations agree well with direct measurements of rotation, but Wright and others have suggested that rotation may not be the only cause of depolarization at high apertures. Therefore, a formula for the total amount of light introduced by rotation alone within given apertures has been developed by the present author to test this suggestion. The new formula will be shown to agree well with the amount of stray light actually measured at various apertures photo-electrically, and so the other possible factors considered by \Vright to cause depolarization must be quite insignificant. The argument is as follows. Between crossed nicols, the amount of light 1, introduced by a transparent plate whose area is A, and which has a rotation of 0 is IA =X0

sin20

. . . . . . . . . . . . . .

[l]

where I, is the intensity (per unit area) of light that passes through the system when the nicols are parallel and when there is no rotation. Knowing the amount of rotation of rays which pass through the different points of the objective exit pupil, it is possible to calculate from equation [l],

Sfudies on the polarizafion

microscope

201

the amount of stray light which would enter into the system by rotation alone within any given aperture. Since the formulae given by CCsaro and Wright are not suited for integration, approximate formulae have been developed as shown belokv. CALCULATION

OF THE DIFFERENT

ROTATION POINTS

OF RAYS OF THE

EXIT

PASSING

THROUGH

PUPIL

In the following discussions it will be assumed that “critical” (namely, Kohler’s) illumination is being used. It is also assumed that the field diaphragm (Df) is made so small that all of the rays emerging from the condenser (Lc) converge into a small point at the center of the object plane (PO). Under this condition, the path of light through the microscope can be simply espressed as in fig. 3. It is clear from fig. 3 that, if the nicols are not crossed, a solid cone of light would appear as a bright disc at the exit pupil (D,) of the objective. The radius (r) of this disc has been shown by RIallard (4) and later by Wright (7, p. 148), to be closely proportional to the sine of the half angle 0 of the cone. Since sin 0 is the angular aperture, by studying the optical phenomena at the exit pupil of the objective, it is possible to determine what happens to each of the rays of light passing through the object plane at different angles. RIallard’s relation has been tested in the present optical system by taking two objectives with different apertures and using them alternatively in place of the condenser. The angles of the cones of light which were formed by these lenses were measured carefully and plotted against the diameter of the bright disc at the objective exit pupil. As sholvn in fig. 4, the agreement \vith hIallard’s relation is very close. The hlallard relation can be expressed graphically as in fig. 5. Here, all the beams of light pass through the center of the sphere 0, and change their direction at its surface 0-JIP.4, so that the resultant beam (CC;) is approximately parallel to the axis of the system OJI. If we consider a beam OC lvhich makes an angle 0 with the axis OM and intercepts the sphere at C, and draw a vertical line CI from C to plane OPA, calling the intercept I, then in triangle COZ, < CIO = < R = < MOI

and so

( OCI = < JIOC = 0.

And so 01 = OC sin 0. Therefore, the sphere 0-MPA can be used to represent the exit pupil the objective, while its center 0 would be the center of the object plane

of

202

S. Inoue’

If the sphere Lvere to be observed from an infinite distance in the direction 0111, the radius of a point C on the sphere, or at the csit pupil, would be equal to the projection (OZ) of XC on to plane OPA, and hence equal to OC . sin 0. In this way, rays passing through the object plane at angles betlveen 0 and 0 f dO will emerge at the esit pupil as a ring \vhose radius is r to r i dr,. The amount of rotation for rays emerging at each point along this ring shall first be considered. In fig. 5, let POP’ and d0.4’ include the planes of polarization of the polarizer and the analyzer respectively. The trio simultaneously are at right angles to each other and to OM. -111 of the optical surfaces present in a microscope are spherical and have their centers along OM, and so for a beam of light passing through 0 and emerging at C, the plane of incidence, refraction and reflection at every surface is always MOBC. The angle @ bctwccn OP and MOBC is the azimuth angle for the entrant plane polarized light. Even though the change of the plane of polarization actually occurs at several different surfaces in a microscope, it cm be considered to take place in one step at point C. In figs. 5 and 6, let CD be the vector of the plane polarized light emerging from the polarizer. This is separated into CE and CF at C; CE being perpendicular to the plane of incidence MOBC, and CF being in the plane. Let the reflection for component CE be E’E and for Cl.‘, F’F. Since the reflectivity for both components is dctermincd only by the angle of incidence 0 and is not affected by the azimuth angle p?, for a constant angle of incidence, ck FF’ = k, . CF

EE’ = k,, . CE

\\here Zi, 2 k,. The transmitted

vectors are CE-E’E= (;E’

and so the resulting

vector

cy: CF-F’F=

CF’

CQ is CQ = CE’ + CF’.

The azimuth @’ of this vector is no longer the same as the azimuth di of the entrant vector CD, for the ratio CS/CF is usually not equal to CE’/CF’; in fact this describes the rotation. The angle of rotation (@ -@‘) is and

@ - @’ = QS~CS, QS = QT ’ cos @ QT=QR-TR=k,~CD~sin~-~,;CD~cos~~~tar~~ = CD (k, :.QS

= CD(k,-k,,)

kp) sin @

[email protected]@.

Sfudies on the polarization

microscope

Fig. 2.

Fig. 1.

Df

203

P

DC

Lc

PO Lo

Do

A

Fig. 3. SINtANGLE 1.0 -

OF CONE 2

LIGHT)

RADIUS

OF BRIGHT

DISC

Fig. 4. Fig. 1. Fig. 2. Fig. 3. Fig. 4.

Polarization cross at exit pupil of 4 mm, 0.84 N.A. strain free, coated objective; condenser also strain free lens but not coated. The irregular margin of the upper right quadrant resulted from imperfect blackening of the interior of the lens mount. Mode of rotation at lens surfaces in different quadrants. (Compare with figs. 5 and 6.) OP: plane of polarization, OB: plane of incidence. Trace of light under critical illumination with the field diaphragm (Df) closed down. The radius (r) of the bright disk at the exit pupil is proportional to sin @ (Mallard’s relation). Test for Mallard’s relation.

204

S. Inoue’

Fig. 5.

Fig. 6.

Fig. 5. Fig. 6. Fig. 7.

Fig. 7.

Graphical expression of Mallard’s relation. Vectorial analysis of rotation. (Compare with Figs. 2 and 5.) Appearance of objective exit pupil with the polarizer rotated 0.5” from its crossed position; -the condition otherwise the same as in fig. 1.

Sfudies on the polarizafion ROTATION

N.A.=

0.1

Actually

205

I”)

I

Fig. S.

microscope

0.2

0.84

0.3

0.4

0.5 SINP-COs$’

Rotation measured from photographs (similar to fig. 7) plotted Slope of each curve represents twice the maximum rotation.

against

sin CDcos @.

SD < < CS and so ~-~,=CD(k,,-l;,,).siil~.cos~

CD

~~~ = f(@).sin

@.cos 0 . . .

[2]

(Q-Q’) is maximum at @ = x/1 and sin @ cos @ = 0.5. Therefore, f (0) is twice the maximum rotation at any given angle of incidence. The actual amount of rotation can be measured by rotating the polarizer or the analyzer slightly off from the crossed position. Then the polarization cross opens up into two dark hgperbolae (fig. 7) which represent the lines of equal rotation. In fig. 8, the rotation along certain apertures measured from photographs of the dark hyprrbolae are plotted against sin @. cos @. Clearly within the range of the measurements, the approximate formula very closely describes the relation between rotation and azimuth angles at different angles of incidence. The maximum rotation at different apertures can be detrrmined from figs, for the slope of each line represents f (0) or twice the maximum rotation for rach of the corresponding apertures. The log of f (0) determined in this manner has been plotted against 0, and the relation proves to be pcrfectl? linrar (fig. 9). This corresponds to the straight line portion of a sigmoid curve described by CEsaro’s and \Vright’s formulae.

s. Inoue’

206 The straight

line can be expressed

as,

The value O.-N \\-a~ cletcrniined graphically from several sets of data and from Ckaro’s ancl\\‘right’s formulae. Regardless of the mode of calculation, this ant1 only this universal value \\‘as obtained. Transferring the base of the logarithm to e,

f(O) = f(o)e”-‘~‘“~‘” Equation THE

. . . . . . . . . . . . . [3]

[3] is the e?cpresGon for n1asimum TOTAL

FLUX

OF LIGHT

WITHIN

rotation

at tlif’ferent apertures.

INTRODUCED

A GIVEN

BY

ROTATION

APERTURE

From equations [2] and [3], the rotation R (r, di) for a heam which any pOiilt (r, CD) at the esit pupil of the objective is,

f (0) sin @. cos = f (0)eS..L./o.l”sin

R (r, @) =

@ @

. cOs

@.

ITsing AIallard’s relation and by choosing a proper of the exit pupil of the objective, namely r = S.d.,

R (r, @)= f( O>er!0.10sin From equation

[l],

the amount

unit

for the radius

@ . cos @.

of light introduced

hy this rotation,

L r, @ = ;iI, sin2 CR(r, @)I = -II, sin” [f(0)ePi”~lY and since the rotation

is smaller

I ‘r, a, = From

sin @ . cos @}

than 10 degrees sin2 0 = 02, and

Al. 1,. f’(o) . ~zriO.lnsit12@. ~052@,.

fig. 10, ;l=rtl@.

dr

passes

is,

r

Studies on the polarixtion

207

microscope

LOG (ROTATION1

0.2

0.6

0.4 Fig.

0.0

N. A.

0

9.

r Fig. 10.

dr

N. A

Fig. 11 Log (Rlaaimum rotation) plotted against numerical aperture. \Iaximum rotation was determined graphically from figures similar to fig. 8. Fig. 10. Coastructiou for iutegratiou of amouut of stray light iutroducctl at different apertures. Fig. 11. Curve giving the calculated amount of light (L) eutering the system withiu any given aperture. Circles represent actual photoelectric measurements of stray light. Departure of mea\urcmeuts from calculated values at low aperture can he accounted for; it arises owing to the constant aperture of the objective diaphragms. Fig. 9.

S. Inout!

208

Since

Equation [A] gives the total flux of light introduced given apcrturc r. Taking the log, we obtain 1ogL = log 19 ~-Io~f’(0)n1W3 14

+ log(eri.u”“(rl.095-

by rotation

1) + l}

mAthin a

_, . . . .

I

The first term on the right is a specific constant for the system, and the second term esprcssrs the change of log 15 \vith r, the aperture. In fig. 11, the line log L (calculated from the equation above) is plotted against r, together \vith the individual results of direct photo-electric nieasurerncnts. The vcr! good agreement shmvs quite definitely that stray light at lvider apertures is caused by the rotation of the plane of polarization. The figure also sho\vs wry clearly that a small increase in the aprturc causes a wr\- great increase in the amount of stray light. \\‘hcrc the aperture is not wry small, a tenfold incrcmc of stray light is brought nhorlt by un trdditional 0.2 numerictri nperture. This relation then acltyuatcly csplains how a great amount of stray light is introduced Lvhen the numerical aperture of the system is increased in polarization microscopy ~vhcrc the elements are of the usual construction and finish. REFERENCES 1. CBsmo,

G., Bull.

;ictrd. Roy. .IfPd. IMg., 459 (1906). 19, 111 (1949) (in .Japanesr).

2. hour& s., and DAN, I<., Krcguku, 3. -J. Jforphol., 89, (1951). 4.

iuALLARD,

E.,

hi!.

.$Ci.

.!Iin.

h.,

5,

77

(1882).

5. Rrss~, F., Zenfr. .Ilinernl. Geol., 88 (1900). 6. \VRIGHT, 1:. E., Am. J. Sci., 31, 157 (1911).

i. --,

The hIethods

8. --

rx., 1911. J. Opftcul Sot.

of Petrographic dm.,

7, ii9

Research.

(1323).

Carnegie

Inst. of \Vashington,

\Vashington.