Abstract

Photodetection by the individual rhabdomeres of the worker-bee photoreceptor (rhabdom) is analyzed by use of electromagnetic theory. The analysis takes full account of the rhabdom’s anisotropic absorption properties. We find, by coupled-mode theory, that only certain modes of a lossless symmetric rhabdom are stable on the lossy rhabdom. Furthermore, the fine structure of the rhabdom (a) enhances the detection of certain modes, whereas it discriminates against others, (b) acts as a polarization detection mechanism, and (c) provides information about the direction of incoming light.

© 1972 Optical Society of America

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References

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  1. E. Snitzer, J. Opt. Soc. Am. 51, 1122 (1961).
    [Crossref]
  2. J. Enoch, J. Opt. Soc. Am. 53, 71 (1963).
    [Crossref]
  3. F. G. Varela and W. Wiitanen, J. Gen. Physiol. 55, 336 (1970).
  4. A. W. Snyder and W. H. Miller, Vision Res. 12, 1389 (1972).
    [Crossref]
  5. A. W. Snyder and P. A. V. Hall, Nature 223, 526 (1969).
    [Crossref] [PubMed]
  6. A. W. Snyder, Z. Vergl. Physiol. 76, 438 (1972).
    [Crossref]
  7. A. W. Snyder, University of London, Ph.D. thesis, 1966.
  8. F. G. Varela and K. R. Porter, J. Ultrastruct. Res. 29, 236 (1969).
    [Crossref] [PubMed]
  9. S. R. Shaw, Vision Res. 9, 999 (1969).
    [Crossref] [PubMed]
  10. S. R. Shaw, Vision Res. 9, 1031 (1969).
    [Crossref] [PubMed]
  11. A. W. Snyder, IEEE Trans. MTT-19, 720 (1971).
  12. A. W. Snyder, IEEE Trans. MTT-17, 1130 (1969).
  13. F. G. Gribakin, Nature 223, 634 (1969).
  14. W. H. Louisell, Coupled Mode and Parametric Electronics (Wiley, New York, 1960).
  15. A. W. Snyder, IEEE Trans. MTT-9, 608 (1970).
    [Crossref]
  16. D. Marcuse and R. M. Derosier, Bell. System Tech. J. 48, 3217 (1969).
  17. K. V. Frisch, The Dance Language and Orientation of Bees (Harvard U. P., Cambridge, Mass., 1967).
    [Crossref]
  18. S. B. Laughlin and G. A. Horridge, Z. Vergl. Physiol. 74, 329 (1971).
  19. A. W. Snyder, IEEE Trans. MTT-17, 1138 (1969).
    [Crossref]
  20. A. W. Snyder, Electron. Letters (London) 7, 105 (1971).
    [Crossref]

1972 (2)

A. W. Snyder and W. H. Miller, Vision Res. 12, 1389 (1972).
[Crossref]

A. W. Snyder, Z. Vergl. Physiol. 76, 438 (1972).
[Crossref]

1971 (3)

A. W. Snyder, IEEE Trans. MTT-19, 720 (1971).

S. B. Laughlin and G. A. Horridge, Z. Vergl. Physiol. 74, 329 (1971).

A. W. Snyder, Electron. Letters (London) 7, 105 (1971).
[Crossref]

1970 (2)

F. G. Varela and W. Wiitanen, J. Gen. Physiol. 55, 336 (1970).

A. W. Snyder, IEEE Trans. MTT-9, 608 (1970).
[Crossref]

1969 (8)

D. Marcuse and R. M. Derosier, Bell. System Tech. J. 48, 3217 (1969).

A. W. Snyder, IEEE Trans. MTT-17, 1130 (1969).

F. G. Gribakin, Nature 223, 634 (1969).

A. W. Snyder, IEEE Trans. MTT-17, 1138 (1969).
[Crossref]

F. G. Varela and K. R. Porter, J. Ultrastruct. Res. 29, 236 (1969).
[Crossref] [PubMed]

S. R. Shaw, Vision Res. 9, 999 (1969).
[Crossref] [PubMed]

S. R. Shaw, Vision Res. 9, 1031 (1969).
[Crossref] [PubMed]

A. W. Snyder and P. A. V. Hall, Nature 223, 526 (1969).
[Crossref] [PubMed]

1963 (1)

1961 (1)

Derosier, R. M.

D. Marcuse and R. M. Derosier, Bell. System Tech. J. 48, 3217 (1969).

Enoch, J.

Frisch, K. V.

K. V. Frisch, The Dance Language and Orientation of Bees (Harvard U. P., Cambridge, Mass., 1967).
[Crossref]

Gribakin, F. G.

F. G. Gribakin, Nature 223, 634 (1969).

Hall, P. A. V.

A. W. Snyder and P. A. V. Hall, Nature 223, 526 (1969).
[Crossref] [PubMed]

Horridge, G. A.

S. B. Laughlin and G. A. Horridge, Z. Vergl. Physiol. 74, 329 (1971).

Laughlin, S. B.

S. B. Laughlin and G. A. Horridge, Z. Vergl. Physiol. 74, 329 (1971).

Louisell, W. H.

W. H. Louisell, Coupled Mode and Parametric Electronics (Wiley, New York, 1960).

Marcuse, D.

D. Marcuse and R. M. Derosier, Bell. System Tech. J. 48, 3217 (1969).

Miller, W. H.

A. W. Snyder and W. H. Miller, Vision Res. 12, 1389 (1972).
[Crossref]

Porter, K. R.

F. G. Varela and K. R. Porter, J. Ultrastruct. Res. 29, 236 (1969).
[Crossref] [PubMed]

Shaw, S. R.

S. R. Shaw, Vision Res. 9, 999 (1969).
[Crossref] [PubMed]

S. R. Shaw, Vision Res. 9, 1031 (1969).
[Crossref] [PubMed]

Snitzer, E.

Snyder, A. W.

A. W. Snyder and W. H. Miller, Vision Res. 12, 1389 (1972).
[Crossref]

A. W. Snyder, Z. Vergl. Physiol. 76, 438 (1972).
[Crossref]

A. W. Snyder, IEEE Trans. MTT-19, 720 (1971).

A. W. Snyder, Electron. Letters (London) 7, 105 (1971).
[Crossref]

A. W. Snyder, IEEE Trans. MTT-9, 608 (1970).
[Crossref]

A. W. Snyder, IEEE Trans. MTT-17, 1130 (1969).

A. W. Snyder and P. A. V. Hall, Nature 223, 526 (1969).
[Crossref] [PubMed]

A. W. Snyder, IEEE Trans. MTT-17, 1138 (1969).
[Crossref]

A. W. Snyder, University of London, Ph.D. thesis, 1966.

Varela, F. G.

F. G. Varela and W. Wiitanen, J. Gen. Physiol. 55, 336 (1970).

F. G. Varela and K. R. Porter, J. Ultrastruct. Res. 29, 236 (1969).
[Crossref] [PubMed]

Wiitanen, W.

F. G. Varela and W. Wiitanen, J. Gen. Physiol. 55, 336 (1970).

Bell. System Tech. J. (1)

D. Marcuse and R. M. Derosier, Bell. System Tech. J. 48, 3217 (1969).

Electron. Letters (London) (1)

A. W. Snyder, Electron. Letters (London) 7, 105 (1971).
[Crossref]

IEEE Trans. (4)

A. W. Snyder, IEEE Trans. MTT-17, 1138 (1969).
[Crossref]

A. W. Snyder, IEEE Trans. MTT-9, 608 (1970).
[Crossref]

A. W. Snyder, IEEE Trans. MTT-19, 720 (1971).

A. W. Snyder, IEEE Trans. MTT-17, 1130 (1969).

J. Gen. Physiol. (1)

F. G. Varela and W. Wiitanen, J. Gen. Physiol. 55, 336 (1970).

J. Opt. Soc. Am. (2)

J. Ultrastruct. Res. (1)

F. G. Varela and K. R. Porter, J. Ultrastruct. Res. 29, 236 (1969).
[Crossref] [PubMed]

Nature (2)

A. W. Snyder and P. A. V. Hall, Nature 223, 526 (1969).
[Crossref] [PubMed]

F. G. Gribakin, Nature 223, 634 (1969).

Vision Res. (3)

S. R. Shaw, Vision Res. 9, 999 (1969).
[Crossref] [PubMed]

S. R. Shaw, Vision Res. 9, 1031 (1969).
[Crossref] [PubMed]

A. W. Snyder and W. H. Miller, Vision Res. 12, 1389 (1972).
[Crossref]

Z. Vergl. Physiol. (2)

A. W. Snyder, Z. Vergl. Physiol. 76, 438 (1972).
[Crossref]

S. B. Laughlin and G. A. Horridge, Z. Vergl. Physiol. 74, 329 (1971).

Other (3)

K. V. Frisch, The Dance Language and Orientation of Bees (Harvard U. P., Cambridge, Mass., 1967).
[Crossref]

W. H. Louisell, Coupled Mode and Parametric Electronics (Wiley, New York, 1960).

A. W. Snyder, University of London, Ph.D. thesis, 1966.

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Figures (8)

Fig. 1
Fig. 1

Schematic representation of the cross section of a worker-bee rhabdom, based on electron microscopy of Varela and Porter (Ref. 8). Each quadrant is made up of two rhabdomeres, which are numbered 1 to 8. The photopigment is held within the parallel dark lines known as the microvilli. Each rhabdomere is an outgrowth of a retinular cell, which senses the absorbed light.

Fig. 2
Fig. 2

Schematic representation of the electric vectors for the x-polarized l, m modes (e lm x ) that can exist on the lossless bee rhabdom taken in order of increasing-cutoff dimensionless frequency V = V c , where V is defined by Eq. (17). Replacing the x vector by a y vector gives the y-polarized mode set (e lm y ).

Fig. 3
Fig. 3

The fraction of modal light within the rhabdom vs dimensionless frequency V defined by Eq. (17). If n1=1.347 and n 2 = 1.339, there are no other modes on the bee rhabdom.

Fig. 4
Fig. 4

Angles used to specify the incidence of light on the bee rhabdom. The direction of the plane-wave propagation is given by k ˆ . The x and y axes are aligned with the microvilli as in Fig. 1.

Fig. 5
Fig. 5

Graphs of Q lm ( i ) (ϕ0, ψ) = power absorbed in rhabdomere i for the l = 0, m = 1, and l = m = 1 mode family excited with polarization cosϕ0 x ˆ + sinϕ0 y ˆ , initial power unity, and azimuthal angle ψ. The shaded circles indicate the lossless mode excited at the start (z = 0) of the rhabdom, oriented as in Fig. 1. The results are for d = 0.2, αη11L = 3.15. —– Q11(1), - - - Q11(2), —– Q01(1).

Fig. 6
Fig. 6

Continuation of Fig. 5 (see Fig. 5 caption).

Fig. 7
Fig. 7

l = 0, m = 1 and l = 1, m = 1 modes excited by the plane wave of Eq. (29) with ψ = ϕ0. (a) illustrates the lossless mode excited, whereas (b) illustrates the combination of stable modes excited. As stated in earlier sections, the l = 0 lossless modes are stable, so they need not be split into x- and y-polarized modes.

Fig. 8
Fig. 8

The ratio of the modal power loss of an unstable mode, assumed stable, to the correct answer for the l = m = 1 modes with ψ = ϕ0. The solid curve is for d = 0.2, the broken curve for d = 0.4. Both curves have α(λ)η11 (λ)L = 3.15. This figure provides data on the severity of unstable mode coupling.

Tables (1)

Tables Icon

Table I Absorbed power for individual modes of the bee rhabdom assuming α=1.8 × 10−2/μm, L=350 μm, and V=6.16. Each mode is excited with unit power.

Equations (74)

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E t ( x , y , z ) = p a p ( z ) e p ( x , y ) ,
( 1 μ ) 1 2 A e p · e q d A = δ p q ,
P ¯ ( z ) = p a p ( z ) 2 ,
P ( L ) = P ¯ ( 0 ) - P ¯ ( L ) .
d a p d z + i β p a p = - q a q C p q ,
C p q = ω 2 A ɛ ¯ · e p · e q d A ,
ɛ ¯ = 1 ( λ ) { x ( ϕ ) x ˆ x ˆ + y ( ϕ ) y ˆ y ˆ } ,
x ( ϕ ) = 1 ,             y ( ϕ ) = d
y ( ϕ ) = 1 ,             x ( ϕ ) = d
P p q | C p q γ b | 2 ,
γ b = ( γ d 2 - C p q 2 ) 1 2 ,
2 γ d = β p - β q + i ( C q q - C p p ) .
a p ( z ) a p ( 0 ) exp ( - ( i β p + C p p ) z ) .
C p p ( λ ) = ( 1 4 ) η 0 m α ( λ ) ( 1 + d ) ,             l = 0
C p p ( λ ) = ( 1 4 ) η l m α ( λ ) ( 1 + d ± 2 l π ( 1 - d ) sin l π 2 ) ,             l 1
α ( λ ) = ω ( μ 1 ) 1 2 1 ( λ ) .
η l m = ( 1 μ ) 1 2 A e l m 2 d A .
V = ( 2 π ρ λ ) ( n 1 2 - n 2 2 ) 1 2 ,
P p ( L ) = 1 - exp ( - 2 C p p L ) .
P ( i ) = ω 0 L d z A i ɛ ¯ E t · E t * d A ,
P ( i ) = p b p 2 ( C p p ( i ) C p p ) [ 1 - exp ( - 2 C p p L ) ] + 4 Re p , q p < q P p q ( i ) ,
P p q ( i ) = b p b q * C p q ( i ) ( C p p + C q q - i ( β p - β q ) ( C p p + C q q ) 2 + ( β p - β q ) 2 ) × { 1 - exp ( - [ i ( β p - β q ) + C p p + C q q ] L ) } ,
C p q ( i ) = ω 2 A i ɛ ¯ · e p · e q d A .
β p - β q ( C p p + C q q )
P p q ( i ) b p b q * ( C p q ( i ) C p p + C q q ) × [ 1 - exp ( - ( C p p + C q q ) L ) ] .
C p q ( 1 ) = v p 4 l π α η l m sin 2 l π 4 ,             l 1
C p q ( 2 ) = ( - 1 ) l - 1 v p 4 l π α η l m sin 2 l π 4 ,             l 1 ,
C p p ( 1 ) = v p 16 α ( λ ) η 0 m ,             l = 0
= v p 16 α ( λ ) η l m ( 1 ± 2 π l sin l π 2 ) ,             l 1
P p ( i ) = b p 2 ( C p p ( i ) C p p ) [ 1 - exp ( - 2 C p p L ) ] .
b p = ( 1 μ ) 1 2 A E inc · e p d A .
E ˆ inc = cos ϕ 0 x ˆ + sin ϕ 0 y ˆ
E inc = E ˆ inc e - i k · r .
b p = cos ϕ 0 B 0 m , x ˆ mode = sin ϕ 0 B 0 m , y ˆ mode
b p = cos ϕ 0 cos ( l ψ ) B l m , x ˆ even mode = cos ϕ 0 sin ( l ψ ) B l m , x ˆ odd mode = sin ϕ 0 cos ( l ψ ) B l m , y ˆ even mode = sin ϕ 0 sin ( l ψ ) B l m , y ˆ odd mode
P ( i ) = B 01 2 Q 01 ( i ) + B 11 2 Q 11 ( i ) .
Q 01 ( i ) = Q 01 ( i ) ( ϕ 0 ) = 1 4 ( cos 2 ϕ 0 + d sin 2 ϕ 0 ) 1 + d × ( 1 - e - 1 2 ( 1 + d ) α ( λ ) η 01 ( λ ) L ) ,
Q 01 ( 7 ) ( ϕ 0 ) = Q 01 ( 2 ) ( ϕ 0 ) = Q 01 ( 1 ) ( ϕ 0 ± π / 2 ) .
( ψ 11 ) 1 2 e 11 = 2 f 11 ( x ˆ cos ϕ 0 + y ˆ sin ϕ 0 ) cos ( ϕ - ϕ 0 ) ,
P ˜ p ( i ) = B 11 2 K p p ( i ) K p p [ 1 - exp ( - 2 K p p L ) ] ,
K p p ( i ) = α η 11 16 [ 1 + 2 π ( cos 2 ϕ 0 ± sin 2 ϕ 0 ) ] × ( cos 2 ϕ 0 + d sin 2 ϕ 0 ) ,
K p p ( 2 ) ( ϕ 0 ) = K p p ( 1 ) ( π 2 - ϕ 0 ) ,
K p p ( 7 ) ( ϕ 0 ) = K p p ( 1 ) ( ϕ 0 - π 2 ) ,
K p p ( 8 ) ( ϕ 0 ) = K p p ( 1 ) ( - ϕ 0 ) ,
K p p = α η 11 4 [ 1 + d + 2 π ( 1 - d ) cos 2 2 ϕ 0 ] .
E ( x , y , z ) = P α p e p ( x , y ) e i β p z ,
[ β ( HE l m ) - β EH l - 2 , m ) ] L π ,             l 2
ρ [ β ( HE l m ) - β ( EH l - 2 , m ) ] [ 1 - ( n 2 n 1 ) 2 ] 3 2 ,
( L ρ ) [ 1 - ( n 2 n 1 ) 2 ] 3 2 π .
[ 1 - ( n 2 n 1 ) 2 ] 10 - 2             and             ( L ρ ) 10 2 .
( ψ l m ) 1 2 e x ( R , ϕ ) = x ˆ γ l f l m ( R ) cos l ϕ ,
( ψ l m ) 1 2 e x ( R , ϕ ) = x ˆ γ l f l m ( R ) sin l ϕ ,
( ψ l m ) 1 2 e y ( R , ϕ ) = y ˆ γ l f l m ( R ) cos l ϕ ,
( ψ l m ) 1 2 e y ( R , ϕ ) = y ˆ γ l f l m ( R ) sin l ϕ ,
f l m ( R ) = J l ( U l m R ) J l ( U l m ) ,             R 1
= K l ( W l m R ) K l ( W l m ) ,             R 1 ;
V 2 = U l m 2 + W l m 2 ,
U l m J l + 1 ( U l m ) J l ( U l m ) = W l m K l + 1 ( W l m ) K l ( W l m ) .
U l m ( V ) ~ U l m ( ) exp ( - 1 / V ) ,
ψ l m = π ρ 2 ( 1 μ ) 1 2 ( V U l m ) 2 K l - 1 ( W l m ) K l + 1 ( W l m ) K l 2 ( W l m ) ,
( ρ β l m ) 2 = V 2 δ ( 1 - δ U l m 2 V 2 ) ,
Q 11 ( 1 ) ( ϕ 0 , ψ ) = ( 1 / C 11 ) ( cos 2 ϕ 0 cos 2 ψ C ¯ 11 ( 1 ) + sin 2 ϕ 0 sin 2 ψ d C ¯ 11 ( 2 ) ) ( 1 - e - 2 C 11 L ) + ( 1 / C 22 ) ( cos 2 ϕ 0 sin 2 ψ C ¯ 11 ( 2 ) ) + sin 2 ϕ 0 cos 2 ψ d C 11 ( 1 ) ) ( 1 - e - 2 C 22 L ) + ( C 12 C 11 + C 22 ) sin ( 2 ψ ) ( cos 2 ϕ 0 + d sin 2 ϕ 0 ) × ( 1 - e - ( C 11 + C 22 ) L ) ,
Q 11 ( 2 ) ( ϕ 0 , ψ ) = Q 11 ( 1 ) ( π / 2 - ϕ 0 , π / 2 - ψ ) ,
Q 11 ( 7 ) ( ϕ 0 , ψ ) = Q 11 ( 1 ) ( π / 2 - ϕ 0 , π / 2 + ψ ) ,
Q 11 ( 8 ) ( ϕ 0 , ψ ) = Q 11 ( 1 ) ( ϕ 0 , - ψ ) ,
C 11 = g { 1 + d + 2 π ( 1 - d ) } ,
C 22 = g { 1 + d - 2 π ( 1 - d ) } ,
C ¯ 11 ( 1 ) = ( g / 4 ) ( 1 + 2 / π ) ,
C ¯ 11 ( 2 ) = ( g / 4 ) ( 1 - 2 / π ) ,
C ¯ 12 = g / π ,
g = 1 4 α ( λ ) η 11 ( λ ) .
Q 11 ( 1 ) ( ϕ 0 , ψ ) = 1 4 cos 2 ϕ 0 ( 1 + 2 π sin ( 2 ψ ) - sin 2 ψ e - 2 C 22 L ) .
i = 1 8 Q 11 ( i ) ( ϕ 0 , ψ ) = 1             for             L ,
i = 1 , 2 , 7 , 8 Q 11 ( i ) ( ϕ 0 , ψ ) = 1 2             for             L .