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OLOFBRYNGDAHL AXU STIGLJUNGGREN
"01. 64
A NEW REFRACTIVE ISDEX GRADIEKT RECORDING INTERFEROMETER SUITABLE FOR STUDYING DIFFUSION, ELECTROPHORESIS AND SEDIMEKTATIOK BY OLOFBRYNGDAHL AND STIGLJUNGGREN lnstitute of Optical Research and fhe Department of Physical Chemistry, Royal Institute of Technology, Stockholm 70, Sweden Received February 98, 1960
A new simple and versatile interferometer has been developed in xhich the interference fringes produce a direct plot of the refractive index gradient. The fringe system appears within a sharp image of the cell, each fringe representing the refractive index gradient curve for that part of the cell through which the fringe passes. The distance between the fringes can be varied by varying the parameters of the arrangement. The theory of the optical method is given in detail together with an estimate of possible errors. The records of some diffusion experiments are reproduced.
Introduction There are now a great many methods available for the registration of the concentration of levelled solutions. The preference for any one of these methods often is dictated by habit or other casual circumstances. Since the primary amount of information is unchanged, the methods differ mainly in respect to the form in which the information is conveyed to the observer. This is, indeed, an important item, since the precision of a method is largely determined by the errors introduced in the evaluation of the interferograms. Also, in many instances, it is desirable that the methods should not presuppose any knowledge about the mathematical form of the concentration distribution. If, however, such information is available, it may, of course, be utilized to increase the precision of measurement of a particular property, for example, the diffusion coefficient.' As a property suitable for direct registration, the authors have chosen the refractiT-e index gradieut, since there seems to exist a need for such methods, for example in sedimentation and electrophoresis. Such a method might also be suitable for diffusion measurements. Although there already exist a few such methods, it seems that an alternative method would fill a need. There are a few other desirable features which the author? think should be included. First, the method should give a sharp image of the cell on the photographic plate. This permits impurities and other imperfections in the cell t o be kept under control and allows an accurate determination of the scalc factor. Secondly, the method should tie simple. easy to adjust and insensitive to mechanical \ribrations. I'inally, as has been pointed out elsewhere,2 the interferences should be produced between rays with only a few wave length difference?. Ry a witable combination of birefringent Savart plates, it has proved possible to produce a derivatil-e-recording direct-imaging interferometer which combines all the features mentioned above. Since it is very easy to set up and to adjust, it seems that it could be recommended also for those mho have no previc ,us experience in this field. I. The Optical Method.-The general outline of the method is illustrated by Fig. 1. The entrance slit is illiiniinnted with monochromatic light from a (1) 0.Bryngdahl, A d a Chem. Scond., 11, 1017 (1937). (2) E. Ingelstam, Arkzu Fysak, 9, 197 (1953).
conventional source. From the slit via the cell to the lens Lz, inclusively, the arrangement is the one common to most kinds of interferometers and does not require a separate description. The lest of the arrangement from the polarizer to the image plane 2, inclusively, is essentially new and its theory will be the subject of a following section. The lens Lz produces an image of the cell on the image plane 1. This plane is then, in turn, imaged on the image plane 2 which means that a sharp image of the cell is obtained together with the interference fringes. The core of the method is formed by the two Savart birefringent plates S1 and Sz together with the polarizers P1 a i d Fz. The method works with interference with polarized light. The exact orientation of SI and Szwill Le made clear iii what follows. Figure 2 shows how the Savart p h t e works in parallel light. A plane-polarized wave front entering the plate is split up into two coherent component wave-fronts with oscillation planes described by the unit vectors and ?, respectively, and displaced vertically a distance b with respect to one another and horizontally a dstance hi2 both in the same direction (cf. Fig. 2 ) . The amplitudes of the two component m ave-fronts ale determined by the polarization direction of the incident wave-front according to the rules of ordinary vector decomposition. For example, if the incident wave-front is polarized in such a way that the oscillation plane bisects the t ~ principal o planes of the crystal sub-plates (i.e., 5- or gdirection in Fig. a ) , then the component wavefrolit s will be equally strong. In a previous work' it is shown how a precision interferometer suitable for diff usion studies can be built, using oiily one Sa\ art plate. If, on the other hand, the Savart plate is placed in convergent light, an entering wave-front n ill be split up into two wave fronts with polarization directions perpendicular to one another (in the directions 4 and 7, respectilely) arid niahiiig a small angle with one another, Fig. 3. The optical path difference, A, between the two emerging component wive-fronts in the imagc plane depends on the y-coordinate 1 ia the corresponding iiicident angle i and on the thickness of the Savart plate. For further details we refer to the general treatment in ref. 3 , 4 and 5. (3) 0 Bryngdahl. Technical Report No. TR 16.4.195i.Institute of
r
Optical Research, Royal Institute of Technology, Stockholm 70.
Sept,., 1960
REFRACTIVE INDEX GRADIEXT RECORDING INTERFEROMETER
LIGHT SOURCE
\
Fig. 1.-Optical A = 6 8in i cos $
arrangement.
no 1 - -__--___ ___ sin i sin il.) ne dno2 ne*
+
(1)
where i is the angle between the crystal-surface normal and the entering ray, $ its azimuthal angle and no and ne the principal refractive indices. The first Savart plate, S1, introduces a vertical displacement between the two wave-fronts polarized in the e and q-directions. The second crystal plate, Sz, in convergent light, is turned through YO” with respect to SI, which is irrelevant in this connection. The wave-fronts entering Sz are oscillating in the 5- and q-directions and are equally strong owing to the setting of the polarizer P1. The principal planes of Sz also lie in the E- and q-directions and thus each entering wave-front emerges without intensity reduction, polarized in the same direction as before Sz. The second Savart plate only introduces a small angle between the wave-fronts coming from SI. The interference fringes become visible in the image plane 2 by means of the analyzer Pz. PZ ordinarily is used perpendicularly to PI and the resulting wave-fronts interfere destructively and constructively according as the path difference between the two wave-fronts is an int,egral number of wave lengths or a half-integral number of wave lengths, respectively. The interferogram gives us directly the derivative curve as will be clear from the theoretical section. 11. Apparatus 1. The Lens System.-The lens system is shown schematically in Fig. 1. E is a horizontal slit illuminated by monochromatic light. It is located in the focal plane of the first lens in the system 4, so that parallel light traverses the cell. The system LI is composed of two equal achromatic lenses corrected for spherical aberration, each having a focal length of 100 cm. By means of the lens LP,f = 12 em., an image of the cell plane is obtained in the image plane 1. A strict collimation, necessary in the wave-fronts traversing the crystal plate, is accomplished by the ad.us.L ment of Lz. An optical reduction is introduced by in order to keep down the dimensions of the Savart plates. By means of the lens L1, an ordinary microscope objective with a magnification factor = 5, the image plane 1 is transformed to the image plane 2. 2. The Savart Plates.-The first crystal plate, SI, which is traversed by strictly parallel light, introduces a displace-
Rf. Born, “Optik,” Springer Verlag. Berlin, 1933, p. 253. ( 5 ) hi. Francon, J . O p t . Sac. Amer., 47, 528 (1957).
(4)
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For notation see text.
ment between thc wave-fronts in the r-direction equal to b,. If the Savart plate is adjusted so a b to be perpendicular to the optical axis, so that no phase shift is introduced between the two wave-fronts, then b is determined by the equation
-
- ne2 no2 b = e 4 2 ___ ne2 no2
+
where e is the thiclrness of each part of the double plate and
n, and no are the principal refractive indices. The quartz plates (SI and &), which are used in our experimental arrangement, have e = 10 mm. and b = 84.2 p.
The Savart plates were purchased from Bernhard Halle Nachfolger, Hubertusstrasse 11, Berlin-Steglitz. 3. The Monochromator.-The horizontal entrance slit is illuminated by a 100 w. fine-capillary, low pressure mercury lamp oyith a filter that only transmits the green Hg line (5461 A.). The width of the entrance slit is carefully adjusted as a compromise between light intensity and coherence. I n our apparatus, the width of the entrance slit is 75 p. 4. The Camera and Photography.-In the experiments with liquid gradients, where a quick registration of the beginning of the run is desired (an exposure every tenth second), a Robot camera which automatically feeds the film was used. For all photographic recordings, Ilford HPS film was used with Ilford Microphen fine grain developer. HPS film was used to attain a minimum exposure time (about 2 see.). 5 . Support.-All of the optical components were mounted on a 3 m. steel beam resting on sheets of sponge rubber on concrete tubes standing on the floor. In this way, vibrations from the floor were effectively damped out.
III. Theory.-In the light entering the cell, we may fasten our attention exclusively on the horizontal oscillation component. This is possible, since we later on introduce a polarizer PI in the light path. The oscillation plane of the polarizer can be set horizontally or vertically as desired. I n this way, the two sheared wave fronts ([ and 7) will have the same amplitude after the Savart plate S1. MTecan therefore describe the horizontal component of the wave entering the cell by a transversal electric field strength amplitude vector referred to the base system of vectors t; and 7 and to the cell plane U =
(f
A + 7) esp(ikz) 4 2
(3)
where k = 2 ir/X and A is the scalar amplitude. If, now, we denote the refractive index of the solution in the cell by n ( x ) and the cell thickness by a,
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Yol. 64.
OLOF BRYNGUilHL AXI) STIG IiJUNGGREX
the component of U in the direction f i h displac.cd downward and laterally by an amount bl 2 , whereay the component in the direction q is displaced upward and laterally t o the same side by the same amount. The refractive index being a fuiictioii of 2, the sideways displacement is of no concern. After SI we have therefore the following amplitude vector, also referred t o the first image plane U
=
'? Fig. 2.-The Savart plate: the splitting-up of an entering ray into two rays is shown. The arrows a t the rays indicate polarization directions.
+
A rd2
6 5 exp {ik[w(z b 1 / 2 )
+ q -r ~Ap 2exp { zk[w(z- b1/2) + zJ - x : J ] )
4 ..A
then the optical path of a ray passing through the cell will be W ( z )=
(4)
a&)
independent of y. The amplitude vector of the light leaving the cell referred t o the cell plane will therefnre be
where xIl is an arbitrary reference plane in front of the cell. Equation 5 is true apart from certain very small corrections, the theory of which has been studied by Suensson.6-8 Among these corrections, we note, for example, the Wieiier skewness which can be eliminated by focusing on a plane one third of the cell width from the front surface of the cell. These corrections are as a rule very sniall and usually are neglected. Sext, the light passes through the last lens of the system TJ1 and L2, the purpose of which is to effect a vale reduction. Denoting the reduction factw by r , we have to introduce a new function w(7-z) = W ( Z )
U
=
(t +a)
9
-
+
-cup (zk[w(~) 7-11 I r
(7)
S e n constants tl, z2, 23. . . ., are introduced after each transformation In passing the Sarart, plate, 16) H S ~ e n s s o n O . p i i c a Acta, 1, 2 5 (1951) 17) H Sxensson zbid 3 , 164 ;19it3) (8) R I'orshetg and H S ~ e n s s o n ,? h i d 2 , 90 ( 1 9 j I )
+ + x/21 I + 7-3
+ 7 r m-4d 2 esp (ik[V(z- mb1/2) + z3 On passing the second Savart plate, there is introduced firstly a vertical displacement, which ib of no concern, and secondly an optical path displacement of different sign for the polarization directions 4' and q and of magnitude a/2
1 bz sin i cos +(1 - c sin i sin +) 2
=-
(11)
where i is the angle between the surface normal and the entering ray, its azimuthal and
+
If we now introduce the coordinates in the iniage plane (denoting by h the distance between the focal plane of Ls and the final image plane 2 ) L = h y =h
tg i sin tg i COS
+ +
we call rewrite (11) in the following way (13)
The expression for the emergent wave, referred to the second image plane, then c m be written as U
=
A= exp { i k [ V ( s+ mb1/2) + A,@ --t .4 eup ( ~ k [ V ( . c- m b l / 2 ) zq + x ' / 2 ] I + rmq2 rm42
r) -
(6)
and can the71 write down the amplitude vector of the wave entering the first Savart plate S1 in the following way as referred to the first image plane
+
A - esp (ik[V(z mb1/2)
\
Fig. 3.-(Ltft) the refractive index as a function of z for the concentration gradient. A doubling of the wave front results from the first Savart plate. (Middle) introduction of a small angle between the wave-fronts due to the second Savart plate. (Right) the appearance of destructive interference in the image plane.
(9)
W(L)
Referred to this plane, the light enteriiig the second Savart plate Sp is described by the vector
Y
\
(8)
where me have introduced the quantity x, thc path difference between the wave-fronts, related to a possible tilting angle of the Savart plate. If we wish t o refer instead to the second image plane, we have to introduce the magnification factor m and a new function ?'(mz) =
tX
+ a + x/211 +
+
a/2 24 - X ' P I 1 (14) here, for simplicity, we have neglected explicitly t o write out the obvious transition 2 -+ x b2J2 resulting from the vertical displacement. hfter passing the second polarizer, we can write the resulting amplitude as
T\
Ti
+
=
U(E A/2
'4 + 7)/,'2 = 2 r m [exp ( ? k [ V ( i + ?nb1/2) + $+ x'/Z]} + eup Izk[T7(r V ~ b 1 / 2 )A/L + - x ' / 2 ] I ] (15) -
-
Zg
2,
Sept., 1960
KTLFRACTIVE
INDEX GRADIENT Ih2ORUIlvC; ISTERFEROMETBR
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Vsiiig tlie Lagraiige mean value theorem we have T7(x
+ mb1/2) - V ( x - mb1/2) = mblV’(x -
mb1/2
where 0 V(x
