Anomalous dispersion of rhodopsin in rod outer segments of the frog W. S. Jagger* Johnson Foundation, University of Pennsylvania, Philadelphia, Pennsylvania 19104
P. A. Liebman Department of Anatomy, University of Pennsylvania, Philadelphia, Pennsylvania 19104
(Received 14 July 1975) Linear-retardation spectra of single frog-rod outer segments were measured by use of a microspectrophotometer equipped with polarizing optics. The recorded transmitted-flux variation was reduced to a rhodopsin anomalous-dispersion spectrum by use of the Jones calculus. This anomalous dispersion is compared with theoretical predictions.
Retinal-rod outer segments exhibit several crystalline properties. Electron microscopy shows them to be dominated by a regular array of lamellar membranes densely stacked along the rod axis. This well-oriented structure gives rise to the rod form birefringence, whereas the lipid-bilayer character of each membrane of the stack is responsible for the intrinsic birefringence of the outer segments viewed from the side. 1 Furthermore, the photopigment rhodopsin with its dipolar chromophore is embedded within the disk membranes in a fixed orientation so that frog-rod outer segments are dichroic when viewed from the side. 2 Pigment absorption is between four and five times greater for light linearly polarized with its electric vector perpendicular to the outer-segment axis than for light polarized parallel to that axis. A third type of birefringence, in addition to the form and intrinsic birefringence, can be expected to occur because of the anomalous dispersion and dichroism of rhodopsin. Anomalous dispersion, a characteristic change of index of refraction with wavelength in the vicinity of an absorption band, 3 should occur in the region of rhodopsin absorption. Because of the dichroism of rhodopsin, the magnitudes of the two indices of refraction perpendicular n,(X) and parallel n,,(X) to the rod axis will differ, and should give rise to this third type of birefringence (linear retardation) with a spectral distribution similar to that of anomalous dispersion. The very high sensitivity of retardation measurements encouraged us to look for this anomalous dispersion in the linear-retardation spectrum of single rod outer segments. EXPERIMENTAL Frogs (Rana pipiens) were dark adapted at least 2 h at room temperature before decapitation and removal of the eyes. The eyeball was hemisected along the ora serrata and the retina removed with fine forceps. A small piece of retina held in a forceps was gently tapped on a slide that contained a drop of frog Ringer's solution with 12% gelatin, to shake loose outer segments. A cover glass was then placed over the slide and the gelatin allowed a few minutes to set. Under an infra-red microscope, many red rod outer segments were observed lying on their sides, held fast in the gelatin. The procedures described were carried out in dim red
light, to avoid bleaching the rhodopsin. The measuring beam of a recording microspectrophotom56
J. Opt. Soc. Am., Vol. 66, No. 1, January 1976
eter, the details of which are described elsewhere, 2 was then passed through one outer segment perpendicular to its axis, and the reference beam was passed through the immediately adjacent medium. Light incident at the microscope was plane polarized at 450 to the rod axis; an orthogonally oriented polarization analyzer followed the special polarization-optics objective. A sufficiently low numerical aperture was used to eliminate significant distortion of the measurements due to oblique incidence of rays at the outer segment. A Kohler-type retardation plate (compensator) of retardation X/20 at 545 nm was oriented with its slow axis along the rod axis between specimen and analyzer. The retardation of this compensator in fractions of a wavelength is m0 = 1/3. 667 X 105X over the wavelength range of interest. The spectrum from 400 to 700 nm was then scanned with the microspectrophotometer, and the log of the reciprocal specimen transmittance of the outer segment was recorded automatically. The light exposure used for the measurement did not bleach a significant fraction of the rhodopsin, as was determined by repeated scans. When the compensator was rotated 90°, the recorded
curve reversed in sign about its asymptotic value, indicating that the mechanism in the rod outer segment responsible for the curve was predominantly linear retardation. Other mechanisms, such as simple absorption or optical rotation (circular birefringence) would not show such reversal. RESULTS AND DISCUSSION The curves of Fig. 1 can be reduced to n, and n, most conveniently by means of the Jones calculus. 4 By this method, the individual absorbing and refracting elements are represented by matrices that operate upon the incident polarized-light vector. The result of these matrix multiplications (see Appendix) is the transmittance I 1O
a 2 +b 2 - 2ab cos[27r(mr + mc)] 2[1- cos(27rm0 )]
where I is the flux of the light passing the system including the outer segment, 10 is the flux of the light passing the system in the absence of the outer segment (reference beam), a2 and b2 are the transmittances of a rod outer segment for light polarized parallel and perpendicular to the rod axis, respectively, mr is the retardation of the outer segment, and m: is the retardaCopyright © 1976 by the Optical Society of America
56
peak of rhodopsin. The curve exhibits the expected anomalous behavior in the region of the rhodopsin absorption peak. The unusual features of the curve are its relatively rapid return to n, = 0 on either side of the absorption peak, and the displacement of the center zero crossing to about 510 nm, away from the rhodopsin absorption peak at 500 nm.
500 600 WAVELENGTH (NM) v
FIG. 1. Lower trace, absorbance of single rod outer segment placed at 45' between crossed polarizing filters, with X/20 compensator. Upper trace, baseline (no outer segment). Absorbance increases upward; vertical bar indicates absorbance of 0. 044. Two consecutive sweeps are recorded; the duration of each sweep is about 10 s.
tion of the compensator. Because a and b are known from the published absorption spectra, 2 and m, is also known, m, is readily calculated. This yields the birefringence, or the difference of the indices of refraction: birefringence = nil - n =
(pA (Me)
The path length is 6 Am, the diameter of a rod outer segment. Using a dichroic ratio of 4, and considering now only n due to anomalous dispersion, we can assume that for small n, nI/n, = 4, and hence noL= - vr (n, - nj)
Figure 2 presents a plot of n1 due to anomalous dispersion as a function of wavelength, calculated from the data of Fig. 1. The zero level is the asymptotic value reached at wavelengths distant from the absorption
The Kramers-Kronig relation3 ' 5 ' 6 can be used in 'many cases to predict the anomalous-dispersion curve, if the absorption spectrum is known. This relation is based on the mathematical theory of complex variables. For a complex analytical functionf (x)=D (x)+iA (x), of the real variable x, which tends to zero far from the origin, f (
1
Cpf (X)dx J
iTS
X-X
where P denotes the principal value of the integral. Substituting for f (x), we find 1
p
A (x) dx
1
PS
D(x)dx
and A (x)
That is, the real and imaginary parts of a complex analytical function are related to one another by these expressions. The expression for D(x') can be converted to an integral that extends from 0 to 00: D (x')=2 pC xA(x)dx 'IT
J7--X2 0
We can now make the identification between f(x) and the complex index of refraction N1 (Ga)= n, (ac) - iK (w),
10
5 FIG. 2. Solid line, experimentally determined anomalous dispersion of rhodopsin. Dashed line, anomalous dispersion of rhodopsin calculated by use of the Davidov formalism. Dotted line, anomalous dispersion of rhodopsin calculated by use of Kramers-Kronig formaliism.
0
-5
-10
WAVELENGTH (NM) 57
J. Opt. Soc. Am., Vol. 66, No. 1, January 1976
W. S. Jagger and P. A. Liebman
57
where K= acc in 10/2w and I/10= 10-.L-'. pisthepath length, and c is the velocity of light in vacuum. This yields nl(
) = clnlO p iT
J~
al(w) dw -W'
This can be calculated as a sum, =-
cln0
PI
c-uw)Aw 2 - @2
This sum was evaluated from the dichroic absorption spectrum of rhodopsin2 in the range from 300 to 800 nm, by use of a digital computer. The principal value of the sum was obtained by omitting the term for which w= o', which would, of course, make the sum infinite if included. The error at any wavelength was estimated to be less than about 10% of the peak value. The results of this calculation appear in Fig. 2 with the experimental curve of nj(X). Davidov 7 developed another formalism to relate absorption to index of refraction in condensed systems. He considered the interaction of vibrating solvent molecules with an absorbing-solute molecule. The treatment could be expected to apply also to vibration-broadened bands caused by vibration in the absorbing molecule itself. Davidov treated the incoming light beam as a perturbation upon the system that consists of absorber and vibrator, and obtained, for the case of strong interaction between absorber and vibrator, ac= 2K /c lnlO,
where K=A exp(- v 2 ln2)andI/I
-
10-". Here, v= (Q - w)/
/2)2 where a2 is the frequency at maximum Kand
(a -
=
3. 44x 10i 5 /s.
The gaussian nature of the main absorption band of rhodopsin is apparent from the fairly good fit possible with this expression on the long-wavelength side of the absorption peak. At shorter wavelengths, additional absorbers account for the poorer fit. These values were then used in Davidov's expression for index of refraction,
100
1i10 n.L(JZ
w11 2
w,/. is the frequency at which K reaches half its maximum value, and p is the path length, 6 ,um. This expression was fitted to the rhodopsin absorption spectrum, a, (Fig. 3), to obtain the values of the parameters,
A=1. 33X10-3 , 62=3.75Xl0 1 5 /s, and
n = 2(1n2)' 1/2Av exp(-v 2 1n2).
The result is plotted in Fig. 2 for comparison with the experimental curve of nl. From Fig. 2, it is clear that the Davidov treatment offers a much better prediction of the anomalous dispersion of rhodopsin than does the Kramers-Kronig relation. The reason for the poor performance of the Kramers-Kronig relation is that it is intended to treat only isolated absorbers. Kramers7 points out that his treatment is not intended for relatively dense systems, in which strong interaction occurs between neighboring atoms. In rhodopsin, this interaction is probably with the vibrational modes of the retinal molecule. Later work8 indicates that failure of the Kramers-Kronig formalism can be considered as a lack of analyticity of the complex index of refraction. Latimer9 also found the Davidov theory useful in describing the anomalous dispersion of various organic liquid systems. For CHCl 3 and CS 2 , he found an agreement between calculated dispersion curves and experimental curves similar to that of rhodopsin. Anomalous dispersion of rhodopsin is also of interest because it may alter the waveguide properties of certain photoreceptors. 11 The outer segment or rhabdomere may be considered to be a cylindrical dielectric waveguide, and its properties largely depend upon its diameter and the indices of refraction inside and outside the receptor. 12 Waveguide properties have a negligible effect upon the axial absorption of frog-rod outer segments because of their relatively large diameter, but for photoreceptors of smaller diameter this is no longer
150r
10
FIG. 3. Solid line, experimental absorption spectrum scl of rhodopsin from Ref. 10. Dashed line, fit of the gaussian-shape curve given by the Davidov theory to the experimental curve.
WAVELENGTH (NM) 58
J. Opt. Soc. Am., Vol. 66, No. 1, January 1976
W. S. Jagger and P. A. Liebman
58
true. A dispersion curve similar to that which we have measured for rhodopsin would be more appropriate for use in these cases than, for example, the lorentzian curve used by Snyder and Richmond. 11 Note added in proof. Stavenga and Van Barneveld [Vision Res. 15, 1091 (1975)] have performed a Kramers-Kronig calculation of the anomalous dispersion of rhodopsin similar to ours. Our calculated KramersKronig peak-to-peak excursion of index of refraction near the rhodopsin absorption peak is about 3. 5 times larger than theirs, however. This discrepancy is apparently due to two factors. First, we used a rhodopsin absorption coefficient about 1. 5 times larger than that which they used. Second, their incorrect use of this coefficient, which is based on decimal logarithms, in their Kramers-Kronig expression, which contains an absorption coefficient based on natural logarithms, accounts for an additional factor of 2. 3. APPENDIX The Jones Calculus The incoming linearly polarized light, with electric vector at O° and of unit irradiance, can be represented by the Jones vector, (1). The irradiance of a Jones vector is given by the sum of the squares of the magnitudes of the elements. The rod outer segment, the compensator, and the analyzer can be represented by the Jones matrices: Dichroism matrix of outer segment with axis at 450, 1 (a+b a-b+ 2\ a-b a+b/ 2
Analyzer matrix, (
;).
(O
1-
The analyzer passes light with electric vector at 90°. The Jones vector of the emerging light is then found by performing the matrix multiplication
(emerging (vector
x
(
compensatory =(analyzerA retardation / matrix / matrix
outer-segment /outer-segment retardation dichroism matrix /\matrix
veco (
The relative order of the two outer-segment matrices is immaterial. The numerator and denominator of the ratio I/Io are obtained by performing the multiplication for the system with and without the rod, respectively.
*Present address: Department of Physiology, Justus Liebig University, Giessen, West Germany. 'W. J. Schmidt, Kolloid-Z 85, 137 (1938). 2 p. A. Liebman, Ann. N. Y. Acad. Sci. 157, 250 (1969). 3 R. W. Ditchburn, Light (Interscience, New York, 1963). 4 W. A. Shurcliff, PolarizedLight(Harvard, Cambridge, 5
Mass., 1966).
E. Corinaldesi, Nuovo Cimento Suppl. XIV, 369 (1959).
6
M. H. A. Kramers, Atti Congr. Int. Fisica, Como 2, 545
2
where a and b are the transmittances 1/10 for light with electric vector parallel to, and perpendicular to the rod axis, respectively. Linear-retarder matrices (outer segment and compensator),
59
where m is the retardation, expressed as a fraction of a wavelength. The slow axes of the retarders lie along the outer-segment axis.
1ml
M20
m, = 2 (1 + e-i2rm) A
VM2
Mly )
m
= - 2(1 -e-i2Tm),
J. Opt. Soc. Am., Vol. 66, No. 1, January 1976
(1927). A. S. Davidov, Izv. Akad. Nauk SSSR, Ser. Fiz. 17, No. 5,
7
523 (1953). 8J. M. Jauch and F. Rohrlich, The Theory of Photons and Electrons (Addison-Wesley, Reading, Mass., 1955). 9 P. Latimer, J. Opt. Soc. Am. 51, 116 (1961). lOP. A. Liebman and G. Entine, Vision Res. 8, 761 (1968). "A. W. Snyder and P. Richmond, J. Opt. Soc. Am. 62, 1278 (1972). 12 J. M. Enoch, J. Opt. Soc. Am. 50, 1025 (1960).
W. S. Jagger and P. A. Liebman
59
P. A. Liebman Department of Anatomy, University of Pennsylvania, Philadelphia, Pennsylvania 19104
(Received 14 July 1975) Linear-retardation spectra of single frog-rod outer segments were measured by use of a microspectrophotometer equipped with polarizing optics. The recorded transmitted-flux variation was reduced to a rhodopsin anomalous-dispersion spectrum by use of the Jones calculus. This anomalous dispersion is compared with theoretical predictions.
Retinal-rod outer segments exhibit several crystalline properties. Electron microscopy shows them to be dominated by a regular array of lamellar membranes densely stacked along the rod axis. This well-oriented structure gives rise to the rod form birefringence, whereas the lipid-bilayer character of each membrane of the stack is responsible for the intrinsic birefringence of the outer segments viewed from the side. 1 Furthermore, the photopigment rhodopsin with its dipolar chromophore is embedded within the disk membranes in a fixed orientation so that frog-rod outer segments are dichroic when viewed from the side. 2 Pigment absorption is between four and five times greater for light linearly polarized with its electric vector perpendicular to the outer-segment axis than for light polarized parallel to that axis. A third type of birefringence, in addition to the form and intrinsic birefringence, can be expected to occur because of the anomalous dispersion and dichroism of rhodopsin. Anomalous dispersion, a characteristic change of index of refraction with wavelength in the vicinity of an absorption band, 3 should occur in the region of rhodopsin absorption. Because of the dichroism of rhodopsin, the magnitudes of the two indices of refraction perpendicular n,(X) and parallel n,,(X) to the rod axis will differ, and should give rise to this third type of birefringence (linear retardation) with a spectral distribution similar to that of anomalous dispersion. The very high sensitivity of retardation measurements encouraged us to look for this anomalous dispersion in the linear-retardation spectrum of single rod outer segments. EXPERIMENTAL Frogs (Rana pipiens) were dark adapted at least 2 h at room temperature before decapitation and removal of the eyes. The eyeball was hemisected along the ora serrata and the retina removed with fine forceps. A small piece of retina held in a forceps was gently tapped on a slide that contained a drop of frog Ringer's solution with 12% gelatin, to shake loose outer segments. A cover glass was then placed over the slide and the gelatin allowed a few minutes to set. Under an infra-red microscope, many red rod outer segments were observed lying on their sides, held fast in the gelatin. The procedures described were carried out in dim red
light, to avoid bleaching the rhodopsin. The measuring beam of a recording microspectrophotom56
J. Opt. Soc. Am., Vol. 66, No. 1, January 1976
eter, the details of which are described elsewhere, 2 was then passed through one outer segment perpendicular to its axis, and the reference beam was passed through the immediately adjacent medium. Light incident at the microscope was plane polarized at 450 to the rod axis; an orthogonally oriented polarization analyzer followed the special polarization-optics objective. A sufficiently low numerical aperture was used to eliminate significant distortion of the measurements due to oblique incidence of rays at the outer segment. A Kohler-type retardation plate (compensator) of retardation X/20 at 545 nm was oriented with its slow axis along the rod axis between specimen and analyzer. The retardation of this compensator in fractions of a wavelength is m0 = 1/3. 667 X 105X over the wavelength range of interest. The spectrum from 400 to 700 nm was then scanned with the microspectrophotometer, and the log of the reciprocal specimen transmittance of the outer segment was recorded automatically. The light exposure used for the measurement did not bleach a significant fraction of the rhodopsin, as was determined by repeated scans. When the compensator was rotated 90°, the recorded
curve reversed in sign about its asymptotic value, indicating that the mechanism in the rod outer segment responsible for the curve was predominantly linear retardation. Other mechanisms, such as simple absorption or optical rotation (circular birefringence) would not show such reversal. RESULTS AND DISCUSSION The curves of Fig. 1 can be reduced to n, and n, most conveniently by means of the Jones calculus. 4 By this method, the individual absorbing and refracting elements are represented by matrices that operate upon the incident polarized-light vector. The result of these matrix multiplications (see Appendix) is the transmittance I 1O
a 2 +b 2 - 2ab cos[27r(mr + mc)] 2[1- cos(27rm0 )]
where I is the flux of the light passing the system including the outer segment, 10 is the flux of the light passing the system in the absence of the outer segment (reference beam), a2 and b2 are the transmittances of a rod outer segment for light polarized parallel and perpendicular to the rod axis, respectively, mr is the retardation of the outer segment, and m: is the retardaCopyright © 1976 by the Optical Society of America
56
peak of rhodopsin. The curve exhibits the expected anomalous behavior in the region of the rhodopsin absorption peak. The unusual features of the curve are its relatively rapid return to n, = 0 on either side of the absorption peak, and the displacement of the center zero crossing to about 510 nm, away from the rhodopsin absorption peak at 500 nm.
500 600 WAVELENGTH (NM) v
FIG. 1. Lower trace, absorbance of single rod outer segment placed at 45' between crossed polarizing filters, with X/20 compensator. Upper trace, baseline (no outer segment). Absorbance increases upward; vertical bar indicates absorbance of 0. 044. Two consecutive sweeps are recorded; the duration of each sweep is about 10 s.
tion of the compensator. Because a and b are known from the published absorption spectra, 2 and m, is also known, m, is readily calculated. This yields the birefringence, or the difference of the indices of refraction: birefringence = nil - n =
(pA (Me)
The path length is 6 Am, the diameter of a rod outer segment. Using a dichroic ratio of 4, and considering now only n due to anomalous dispersion, we can assume that for small n, nI/n, = 4, and hence noL= - vr (n, - nj)
Figure 2 presents a plot of n1 due to anomalous dispersion as a function of wavelength, calculated from the data of Fig. 1. The zero level is the asymptotic value reached at wavelengths distant from the absorption
The Kramers-Kronig relation3 ' 5 ' 6 can be used in 'many cases to predict the anomalous-dispersion curve, if the absorption spectrum is known. This relation is based on the mathematical theory of complex variables. For a complex analytical functionf (x)=D (x)+iA (x), of the real variable x, which tends to zero far from the origin, f (
1
Cpf (X)dx J
iTS
X-X
where P denotes the principal value of the integral. Substituting for f (x), we find 1
p
A (x) dx
1
PS
D(x)dx
and A (x)
That is, the real and imaginary parts of a complex analytical function are related to one another by these expressions. The expression for D(x') can be converted to an integral that extends from 0 to 00: D (x')=2 pC xA(x)dx 'IT
J7--X2 0
We can now make the identification between f(x) and the complex index of refraction N1 (Ga)= n, (ac) - iK (w),
10
5 FIG. 2. Solid line, experimentally determined anomalous dispersion of rhodopsin. Dashed line, anomalous dispersion of rhodopsin calculated by use of the Davidov formalism. Dotted line, anomalous dispersion of rhodopsin calculated by use of Kramers-Kronig formaliism.
0
-5
-10
WAVELENGTH (NM) 57
J. Opt. Soc. Am., Vol. 66, No. 1, January 1976
W. S. Jagger and P. A. Liebman
57
where K= acc in 10/2w and I/10= 10-.L-'. pisthepath length, and c is the velocity of light in vacuum. This yields nl(
) = clnlO p iT
J~
al(w) dw -W'
This can be calculated as a sum, =-
cln0
PI
c-uw)Aw 2 - @2
This sum was evaluated from the dichroic absorption spectrum of rhodopsin2 in the range from 300 to 800 nm, by use of a digital computer. The principal value of the sum was obtained by omitting the term for which w= o', which would, of course, make the sum infinite if included. The error at any wavelength was estimated to be less than about 10% of the peak value. The results of this calculation appear in Fig. 2 with the experimental curve of nj(X). Davidov 7 developed another formalism to relate absorption to index of refraction in condensed systems. He considered the interaction of vibrating solvent molecules with an absorbing-solute molecule. The treatment could be expected to apply also to vibration-broadened bands caused by vibration in the absorbing molecule itself. Davidov treated the incoming light beam as a perturbation upon the system that consists of absorber and vibrator, and obtained, for the case of strong interaction between absorber and vibrator, ac= 2K /c lnlO,
where K=A exp(- v 2 ln2)andI/I
-
10-". Here, v= (Q - w)/
/2)2 where a2 is the frequency at maximum Kand
(a -
=
3. 44x 10i 5 /s.
The gaussian nature of the main absorption band of rhodopsin is apparent from the fairly good fit possible with this expression on the long-wavelength side of the absorption peak. At shorter wavelengths, additional absorbers account for the poorer fit. These values were then used in Davidov's expression for index of refraction,
100
1i10 n.L(JZ
w11 2
w,/. is the frequency at which K reaches half its maximum value, and p is the path length, 6 ,um. This expression was fitted to the rhodopsin absorption spectrum, a, (Fig. 3), to obtain the values of the parameters,
A=1. 33X10-3 , 62=3.75Xl0 1 5 /s, and
n = 2(1n2)' 1/2Av exp(-v 2 1n2).
The result is plotted in Fig. 2 for comparison with the experimental curve of nl. From Fig. 2, it is clear that the Davidov treatment offers a much better prediction of the anomalous dispersion of rhodopsin than does the Kramers-Kronig relation. The reason for the poor performance of the Kramers-Kronig relation is that it is intended to treat only isolated absorbers. Kramers7 points out that his treatment is not intended for relatively dense systems, in which strong interaction occurs between neighboring atoms. In rhodopsin, this interaction is probably with the vibrational modes of the retinal molecule. Later work8 indicates that failure of the Kramers-Kronig formalism can be considered as a lack of analyticity of the complex index of refraction. Latimer9 also found the Davidov theory useful in describing the anomalous dispersion of various organic liquid systems. For CHCl 3 and CS 2 , he found an agreement between calculated dispersion curves and experimental curves similar to that of rhodopsin. Anomalous dispersion of rhodopsin is also of interest because it may alter the waveguide properties of certain photoreceptors. 11 The outer segment or rhabdomere may be considered to be a cylindrical dielectric waveguide, and its properties largely depend upon its diameter and the indices of refraction inside and outside the receptor. 12 Waveguide properties have a negligible effect upon the axial absorption of frog-rod outer segments because of their relatively large diameter, but for photoreceptors of smaller diameter this is no longer
150r
10
FIG. 3. Solid line, experimental absorption spectrum scl of rhodopsin from Ref. 10. Dashed line, fit of the gaussian-shape curve given by the Davidov theory to the experimental curve.
WAVELENGTH (NM) 58
J. Opt. Soc. Am., Vol. 66, No. 1, January 1976
W. S. Jagger and P. A. Liebman
58
true. A dispersion curve similar to that which we have measured for rhodopsin would be more appropriate for use in these cases than, for example, the lorentzian curve used by Snyder and Richmond. 11 Note added in proof. Stavenga and Van Barneveld [Vision Res. 15, 1091 (1975)] have performed a Kramers-Kronig calculation of the anomalous dispersion of rhodopsin similar to ours. Our calculated KramersKronig peak-to-peak excursion of index of refraction near the rhodopsin absorption peak is about 3. 5 times larger than theirs, however. This discrepancy is apparently due to two factors. First, we used a rhodopsin absorption coefficient about 1. 5 times larger than that which they used. Second, their incorrect use of this coefficient, which is based on decimal logarithms, in their Kramers-Kronig expression, which contains an absorption coefficient based on natural logarithms, accounts for an additional factor of 2. 3. APPENDIX The Jones Calculus The incoming linearly polarized light, with electric vector at O° and of unit irradiance, can be represented by the Jones vector, (1). The irradiance of a Jones vector is given by the sum of the squares of the magnitudes of the elements. The rod outer segment, the compensator, and the analyzer can be represented by the Jones matrices: Dichroism matrix of outer segment with axis at 450, 1 (a+b a-b+ 2\ a-b a+b/ 2
Analyzer matrix, (
;).
(O
1-
The analyzer passes light with electric vector at 90°. The Jones vector of the emerging light is then found by performing the matrix multiplication
(emerging (vector
x
(
compensatory =(analyzerA retardation / matrix / matrix
outer-segment /outer-segment retardation dichroism matrix /\matrix
veco (
The relative order of the two outer-segment matrices is immaterial. The numerator and denominator of the ratio I/Io are obtained by performing the multiplication for the system with and without the rod, respectively.
*Present address: Department of Physiology, Justus Liebig University, Giessen, West Germany. 'W. J. Schmidt, Kolloid-Z 85, 137 (1938). 2 p. A. Liebman, Ann. N. Y. Acad. Sci. 157, 250 (1969). 3 R. W. Ditchburn, Light (Interscience, New York, 1963). 4 W. A. Shurcliff, PolarizedLight(Harvard, Cambridge, 5
Mass., 1966).
E. Corinaldesi, Nuovo Cimento Suppl. XIV, 369 (1959).
6
M. H. A. Kramers, Atti Congr. Int. Fisica, Como 2, 545
2
where a and b are the transmittances 1/10 for light with electric vector parallel to, and perpendicular to the rod axis, respectively. Linear-retarder matrices (outer segment and compensator),
59
where m is the retardation, expressed as a fraction of a wavelength. The slow axes of the retarders lie along the outer-segment axis.
1ml
M20
m, = 2 (1 + e-i2rm) A
VM2
Mly )
m
= - 2(1 -e-i2Tm),
J. Opt. Soc. Am., Vol. 66, No. 1, January 1976
(1927). A. S. Davidov, Izv. Akad. Nauk SSSR, Ser. Fiz. 17, No. 5,
7
523 (1953). 8J. M. Jauch and F. Rohrlich, The Theory of Photons and Electrons (Addison-Wesley, Reading, Mass., 1955). 9 P. Latimer, J. Opt. Soc. Am. 51, 116 (1961). lOP. A. Liebman and G. Entine, Vision Res. 8, 761 (1968). "A. W. Snyder and P. Richmond, J. Opt. Soc. Am. 62, 1278 (1972). 12 J. M. Enoch, J. Opt. Soc. Am. 50, 1025 (1960).
W. S. Jagger and P. A. Liebman
59
