Angular and spectral sensitivity of fly photoreceptors. II. Dependence on facet lens F-number and rhabdomere type in Drosophila

Abstract

A wave-optical model for the integrated facet lens-rhabdomere system of fly eyes is used to calculate the effective light power in the rhabdomeres when the eye is illuminated with a point light source or with an extended source. Two rhabdomere types are considered: the slender rhabdomeres of R7,8 photoreceptors and the wider, but tapering R1–6 rhabdomeres. The angular sensitivities of the two rhabdomere types have been calculated as a function of F-number and wavelength by fitting Gaussian functions to the effective light power. For a given F-number, the angular sensitivity broadens with wavelength for the slender rhabdomeres, but it stays approximately constant for the wider rhabdomeres. The integrated effective light power increases with the rhabdomere diameter, but it is for both rhabdomere types nearly independent of the light wavelength and F-number. The results are used to interpret the small F-number of Drosophila facet lenses. Presumably the small head puts a limit to the size of the facet lens and favors a short focal length.

Appendices

Appendix 1

The thick fly facet lens

The power of a thick lens is given by (Jenkins and White 1976):

$$P_l = P_1 + P_2 + P_3 $$

(A1)

with

$$P_1 = {{n_l - n} \over {r_1 }}\;{\rm ,}\;P_2 = {{n' - n_l } \over {r_2 }}\;{\rm ,}\;{\rm and}\;P_3 = - {t \over {n_l }}P_1 P_2 {\rm }$$

(A2)

P ₁ and P ₂ are the powers of the front and back surface of the lens, respectively; r ₁ and r ₂ are the radii of curvature of front and back surface; t is the lens thickness; n, n _l and n' are the refractive indices of object space, lens, and image space, respectively (Fig. 2).

The primary or object focal length and the secondary or image focal length is given by, respectively:

$$f = {n \over {P_l }}\;{\rm and}\;f' = {{n'} \over {P_l }}$$

(A3)

For a facet lens of Drosophila with r ₁ = 11 μm, r ₂ = −11 μm, t = 8 μm, n = 1.0, n _l = 1.45, and n' = 1.34 follows P ₁ = 0.041 μm^-1, P ₂ = 0.010 μm^-1, P ₃ = −0.002 μm^-1, and P _l = 0.049 μm^-1, or, the lens power is mainly determined by the lens front surface. The object and image focal length then is f = 20.6 μm and f' = 27.5 μm, respectively.

The geometrical optics of an imaging system is fully determined by its 6 cardinal points, i.e., the focal points F and F', the principal points H and H', and the nodal points N and N' (Fig. 3). The corresponding 6 cardinal planes are the planes through the cardinal points perpendicular to the optical axis. Points in one principal plane are imaged in the other principal plane with unit magnification. Rays in object space through the primary focal point F proceed in image space from the primary principal plane parallel to the optical axis, and rays in object space parallel to the optical axis proceed in image space from the secondary principal plane through the secondary focal point F'. Rays in object space through the primary nodal point N proceed in image space through the secondary nodal point N' in the same direction, i.e., the angle with the optical axis, θ, is identical for both rays. The distances of H and H' to the front surface are ntP ₂/n _l P = 1.1 μm and t(1−n'P ₁/n _l P) = 1.8 μm, respectively, and the distances of N and N' to the back surface are f'−t−n(1−tP ₂/n _l )/P = 0.1 μm and n'(1−tP ₁/n _l )−f = 0.8 μm, respectively. The principal points (planes) as well as the nodal points (planes) virtually coincide: their identical distance is 0.7 μm (Figs. 2, 12; for the related case of the blowfly facet lens, see Stavenga 1975; Stavenga et al. 1990).

Fig. 12.

Diagrams for illustrating the different functions of the principal planes, i.e., the planes through the principal points, H and H', and the nodal points, N and N'. F and F' are the focal points. For light focusing, the image focal length, f', is the essential parameter, and for the estimation of the visual axis and visual field, the object focal length, f, must be used

Figure 12 is added to emphasize how the geometrical optics of the fly facet lens determines two crucial aspects of the photoreceptor spatial sensitivity (see Land 1981). Firstly, the channeling of light into the rhabdomere depends on the size of the lens' exit pupil. Its size can be taken to be identical to that of the actual facet lens and its location to be that of the secondary principal plane. The angular aperture on the image side of the cone of light projecting at the rhabdomere thus is given by 2θ _a = 2 tg^-1(D _l /2f'); for a given D _l , the secondary or image focal length, f', is then the essential parameter (see Fig. 12a). Secondly, if the distal entrance of the rhabdomere coincides with the secondary focal plane, the visual axis and the photoreceptor's spatial field are determined by the position of the secondary nodal point. Here the primary or object focal length, f, is the essential parameter (Fig. 12b).

Appendix 2

Reflectance of the rhabdomere boundary

The reflectance of a boundary between two media with refractive indices n ₁ and n ₂ is given by Fresnel's reflection equations. For polarized light perpendicular and parallel to the plane of incidence the reflectances are given by, respectively:

$$R_s = {{\sin ^2 \left( {\varphi _1 - \varphi _2 } \right)} \over {\sin ^2 \left( {\varphi _1 + \varphi _2 } \right)}}\;{\rm and}\;R_p = {{{\rm tg}^2 \left( {\varphi _1 - \varphi _2 } \right)} \over {{\rm tg}^2 \left( {\varphi _1 + \varphi _2 } \right)}}$$

(A4)

where the angles of incidence and refraction, φ ₁ and φ ₂, are related by Snell's law: n ₁sinφ ₁ = n ₂sinφ ₂. The values of R _s and R _p are virtually identical in the relevant range of small angles. Figure 5a gives the average reflectance, R _r = (R _s + R _p )/2, as a function of the rhabdomere refraction angle, θ _r . The critical angle for light reflection on the boundary of a fly rhabdomere is φ _c  = sin^-1(n ₂/n ₁) = 79.5°, or, its complement θ _c = 10.5° (see Fig. 4a). R _r = 1 when θ _r <θ _c , but R _r rapidly becomes very minor when θ _r exceeds θ _c by more than a few degrees (Fig. 5a).

Figure 5b presents the relation between the facet lens F-number and the rhabdomere refraction angle, using F = 1/(2n'tgθ _a ) (see Eq. 1) and θ _a  = sin⁻¹[(n ₁/n ₂ )sinθ _r ]. Let F _c be the critical value where θ _r = θ _c , then for F ≥ F _c all light rays focused by the facet lens into the rhabdomere will be reflected, but for F <F _c , rays from the lens periphery will leak out of the rhabdomere. The critical F-number, F _c , is used to rescue Land's (1981) expression for an eye's light sensitivity (Eq. 5). The chosen refractive index values yield F _c = 1.98. It could be argued that instead of taking the F-number corresponding to θ _c = 10.5°, where the reflectance R _r = 1, it is better to take as the limiting F-number the value corresponding to the angle where R _r = 0.5: θ _r = 10.7°. Then F _c would be F _c = 1.94 (Fig. 5). For convenience's sake the critical F-number here is approximated to F _c = 2.

Warrant and McIntyre (1993) state that the F-number where all light rays are just captured is about F = 2.8, but this number is estimated too large by a factor n', as the focal distances f and f' have been confused: for focusing light into the rhabdomere, f' and not f is the essential parameter; see Appendix 1.

When a parallel beam from a distant point source enters a Drosophila facet lens with F = 1.25, a marginal ray is given by θ _a = 16.6°. The ray entering a rhabdomere with an angle θ _c = 10.5° makes an angle 10.7° with the visual axis. The ratio between the light power entering from the full angular aperture of the lens and that from a cone where all light rays are totally reflected is (16.6/10.5)² = 2.4. This means that well over half of the light rays are not captured by the rhabdomere. This largely explains the deviation of the sensitivities S ₁ and S ₂ (solid curves) in Fig. 11 from the integral effective power.

Appendix 3

Excitation of waveguide modes in a fly rhabdomere

Consider a facet lens-rhabdomere system illuminated by a distant point source from a directional angle θ (Fig. 3a). A total of 1 W of monochromatic light, wavelength λ, enters the facet lens, and is focused into the rhabdomere. The light power absorbed by the visual pigment then is (Stavenga 2003):

$$P_{abs} \left( \lambda \right) = \sum\limits_p {P_{p,exc} \left( \lambda \right)\left\{ {1 - e^{ - \bar \eta _p (\lambda )\kappa (\lambda )L} } \right\}} $$

(A5)

where the sum is over the different modes; P _p,exc is the power excited into mode p; $\bar \eta _p$ is the effective light fraction of mode p propagating inside the rhabdomere, from which the visual pigment molecules can absorb; κ(λ) is the absorption coefficient of the rhabdomere medium due to the visual pigment; L is the length of the rhabdomere.

Equation A5 is considerably simplified when self-absorption by the visual pigment is minor. Then (Stavenga 2003, Eq. 44):

$$P_{abs} \left( \lambda \right) = \kappa \left( \lambda \right)\;L\;P_{eff} \left( \lambda \right)$$

(A6)

where P _eff is the effective light power from which light can be absorbed (Stavenga 2003, Eq. 45):

$$P_{eff} \left( \lambda \right) = \sum\limits_p {P_{p,exc} \left( \lambda \right)\bar \eta _p \left( \lambda \right)} $$

(A7)

When the rhabdomere entrance coincides with the image focal plane of the facet lens, P _p,exc is found from:

$$P_{p,exc} = {{2n_e } \over {c_p n'J_{l - 1} \left( U \right)J_{l + 1} \left( U \right)}}\left[ {{2 \over C}{W \over V}\int\limits_0^C {J_l \left( {D\Omega } \right)G\left( \Omega \right)} \Omega d\Omega } \right]^2 $$

(A8)

The five linearly polarized modes (p = 1–5) that are bound at wavelength 300 nm by a rhabdomere with diameter 2.0 μm are called LP01, LP11, LP21, LP02 and LP31 (see Table 2 of Stavenga 2003). For each mode, the value of l is the first of the number pair. When l = 0, c _p = 1, otherwise c _p = 2. U and W are roots of the waveguide characteristic equation:

$$U{{J_{l + 1} \left( U \right)} \over {J_l \left( U \right)}} = W{{K_{l + 1} \left( W \right)} \over {K_l \left( W \right)}}$$

(A9)

J _l and K _l are (modified) Bessel functions; n' is the refractive index of the facet lens image space (Fig. 2), and n _e is the effective refractive index of the rhabdomere. n _e and n' have very approximately the same value, or n _e /n' = 1. Furthermore, D = (2FD _l /D _r )tgθ; and C is an integration constant:

$$C = {{\pi D_r } \over {2\lambda F}}$$

(A10)

$$G\left( \Omega \right) = {{V^2 } \over {\left( {\Omega ^2 - U^2 } \right)\left( {\Omega ^2 + W^2 } \right)}}\left[ {\Omega J_l \left( U \right)J_{l + 1} \left( \Omega \right) - UJ_l \left( \Omega \right)J_{l + 1} \left( U \right)} \right]\;\left( {\Omega \; = \;{\rm U}} \right)$$

(A11a)

or

$$G\left( \Omega \right) = {1 \over 2}\left[ {J_l^2 \left( U \right) - J_{l - 1} \left( U \right)J_{l + 1} \left( U \right)} \right]\;\left( {\Omega \; = \;{\rm U}} \right)$$

(A11b)

Finally, $\bar \eta _p$, the effective light fraction of mode p propagating inside the rhabdomere from which the visual pigment molecules can absorb is given by:

$$\bar \eta _p = {1 \over L}\int\limits_0^L {\eta _p \left( {\lambda ,z} \right)dz} $$

(A12)

where z is the longitudinal coordinate of the waveguide; η _p , the local light fraction inside the waveguide, is related to the waveguide number V and the corresponding U and W for mode p by:

$$\eta _p = {{W^2 } \over {V^2 }}\left( {1 - {{J_l^2 \left( U \right)} \over {J_{l - 1} \left( U \right)J_{l + 1} \left( U \right)}}} \right)$$

(A13)

When the rhabdomere has a constant diameter, which is the case for the rhabdomeres of the central photoreceptors R7 and R8, $\bar \eta _p = \eta _p$. The rhabdomeres of the peripheral photoreceptors, R1–6, taper. The tapering is approximately parabolic in the housefly Musca (Boschek 1971): D _r = D _r (0)[1-(z/L)²/2], i.e., with a distal diameter D _r (0) = 2.0 μm, the proximal diameter, for z = L, is 1.0 μm. This parabolic tapering is assumed in Fig. 6b.

Appendix 4

Integral effective power due to an extended light source

When the facet lens-rhabdomere system is illuminated by a uniform light source with irradiance Q(λ), the total light power excited in mode p is (e.g., Snyder 1979):

$$P_{p,{\mathop{\rm int}} } \left( \lambda \right) = 2\pi A_l Q\left( \lambda \right)\int\limits_0^{{\pi \over 2}} {\theta E_{p,exc} \left( {\theta ,\lambda } \right)} {\rm d}\theta $$

(A14)

where A _l = πD _l ²/4 is the facet lens area, and E _p,exc (θ,λ) is the fraction of the light incident at the facet lens surface which is excited into mode p by light coming from an angle θ. E _p,exc (θ,λ) equals the dimensionless P _p,exc (θ,λ), as the latter is the excited light power when 1 W of monochromatic light enters the facet lens (see Appendix 3).

The integral effective light power then is, assuming negligible self-absorption (see Eq. A6 of Appendix 3):

$$P_{{\mathop{\rm int}} } \left( \lambda \right) = \sum\limits_p {P_{p,{\mathop{\rm int}} } \left( \lambda \right)\bar \eta _p \left( \lambda \right)} $$

(A15)

This quantity is independent of the facet lens diameter for a given F-number, because an increase in the facet lens diameter is fully compensated by the concomitant narrowing of the angular sensitivity function. In the calculations of the integral effective light power of Fig. 9, Q is taken as Q = 1 W sr⁻¹ μm⁻² (μm because the dimension of the facet lens is μm).

The spectral sensitivity of a photoreceptor for an extended light source is calculated by normalization of the total light absorption by the visual pigment, which is obtained by multiplying P _int with the absorption coefficient κ(λ) and length L (see Eq. A5 of Appendix 3).