Appendix 1: intensity distribution of waveguide modes
Light propagates in a cylindrical optical waveguide in modes, where the light power exists partly inside and partly outside the cylinder. Each mode has a specific mode propagation constant, β:
$$ \beta = n_{e} k = {\sqrt {n^{2}_{1} k^{2} - {\left( {\frac{U} {b}} \right)}^{2} } } = {\sqrt {n^{2}_{2} k^{2} + {\left( {\frac{W} {b}} \right)}^{2} } } $$
(A1)
where b is the rhabdomere radius; n
e
is the effective refractive index of the waveguide for the propagated mode, and n
1 and n
2 are the refractive indices of the media inside and outside the waveguide; k is the wave number, where k = 2π/λ, with λ the light wavelength; U and W are the roots of the characteristic waveguide equation (Snyder 1969):
$$ U\frac{{J_{{l + 1}} (U)}} {{J_{l} (U)}} = W\frac{{K_{{l + 1}} (W)}} {{K_{l} (W)}} $$
(A2)
where J
l
and K
l
are (modified) Bessel functions. U and W together determine the waveguide number V (with Eq. A1):
$$ V = {\sqrt {U^{2} + W^{2} } } = kb{\sqrt {(n^{2}_{1} - n^{2}_{2} } } = \frac{{\pi D_{r} }} {\lambda }{\sqrt {(n^{2}_{1} - n^{2}_{2} } } $$
(A3)
where D
r
= 2b is the diameter of the rhabdomere. The number of allowed modes depends on the cut-off V-number, V
co
, which is 2.405, 3.832, 3.847 and 5.136 for modes p = 1, 2, 3 and 4, respectively (for mode nomenclature and the value of parameter l for mode p, see Table 2 of Stavenga 2003a).
When unpolarized, monochromatic incident light excites a total power of P
p
= 1 W in mode p, the average intensity of this mode is given by:
$$ \bar{I}^{*}_{p} (R) = {\left| {M^{*}_{p} (R)} \right|}^{2} H(U,W)/\pi b^{2} $$
(A4a)
with
$$ \begin{array}{*{20}c} {{M^{*}_{p} (R) = J_{l} (UR)/J_{l} (U),}} & {{R \leqslant 1}} \\ \end{array} $$
(A4b)
$$ \begin{array}{*{20}c} {{M^{*}_{p} (R) = K_{l} (WR)/K_{l} (W),}} & {{R \geqslant 1}} \\ \end{array} $$
(A4c)
and
$$ H(U,W) = - \frac{{W^{2} }} {{V^{2} }}\frac{{J^{2}_{1} (U)}} {{J_{{l - 1}} (U)J_{{l + 1}} (U)}} $$
(A4d)
where R = r/b is the radial distance from the waveguide axis, r, normalized to its radius value. The mode intensity at the rhabdomere border (r = b, or R = 1) is:
$$ B_{p} = \bar{I}_{p} (b) = \bar{I}^{*}_{p} (1) = H(U,W)/\pi b^{2} $$
(A5)
because \( M^{*}_{p} (1) = 1 \). The physical quantities are adorned with an asterisk when the expressions are given as a function of the normalized spatial coordinate, R. They are without an asterisk when the spatial coordinate is r; e.g., \( \bar{I}^{*}_{p} (R) \equiv \bar{I}_{p} (r) \) and \( M^{*}_{p} (R) \equiv M_{p} (r) \). Considering Eq. A4a, it may be useful to note that in case the total power of 1 W were confined within the rhabdomere boundary and equally distributed there, the intensity would be 1/πb
2; πb
2 is the cross-sectional area of the rhabdomere.
The fraction of the total light power propagated in mode p inside the rhabdomere follows from spatial integration of Eq. A4 using Eqs. A4b and A4c:
$$ \eta _{p} = (W/V)^{2} + H(U,W) $$
(A6a)
and the fraction of the total light power propagated in mode p outside the rhabdomere is:
$$ \zeta _{p} = 1 - \eta _{p} = (U/V)^{2} - H(U,W). $$
(A6b)
The light intensity outside the rhabdomere (r ≥ b), given by Eq. A4a together with Eqs. A4c and A4d, can be approximated with an exponential function:
$$ \ifmmode\expandafter\bar\else\expandafter\=\fi{I}_{p} (r) = B_{p} e^{{ - \frac{{r - b}} {{\rho _{p} }}}} $$
(A7)
where r - b is the radial distance to the rhabdomere border and ρ
p
is the space constant of mode p. The fraction of the total light power propagated in mode p outside the rhabdomere is then simply derived by spatial integration of Eq. A7, yielding:
$$ \zeta _{p} = 2\pi B_{p} \rho _{p} (\rho _{p} + b). $$
(A8)
The right-hand term of Eq. A8 should formally contain a factor \( P^{{ - 1}}_{p} \), because ζ
p
is dimensionless, the dimension of ρ
p
and b is μm, and that of B
p
is W μm-2, but the factor is omitted here because P
p
= 1 W. Of course, the shape of the intensity distribution in a mode is independent of the total propagated light power, and when P
p
≠ 1 W, the intensity B
p
is proportionally modified.
Numerical results of Eqs. A6b and A8 are given in Fig. 5c. It appears that the calculation of Eq. A8 with the values of ρ
p
derived from the exponential fit, Eq. A7, yields ζ
p
values which deviate from the exact values (calculated from Eq. A6b) at wavelengths near cut-off (Fig. 3b). A more correct space constant is obtained by using the exact ζ
p
-value and then solving ρ
p
from Eq. A8:
$$ \rho _{p} = {\left[ {{\sqrt {\frac{{b^{2} }} {4} + \frac{{\zeta _{p} }} {{2\pi B_{p} }}} } - \frac{b} {2}} \right]}. $$
(A9)
ρ
p
values calculated with this relation are presented in Fig. 3b (bold curves).
Appendix 2: light absorption by visual pigment and pupil
On its way along a fly rhabdomere, some of the light power is absorbed by the visual pigment inside the rhabdomere and/or by the screening pigment of the pigment granules in the soma, depending on the light wavelength. When the extinction is sufficiently small, so that the intensity distribution in the modes is not strongly affected, Lambert-Beer’s law can be applied. If the medium of the rhabdomere interior with the visual pigment is homogeneous and has an absorption coefficient κ
v
, and the exterior medium with the screening pigment is equally homogeneous with absorption coefficient κ
s
, it is straightforward to prove that the effective absorption coefficient of the rhabdomere is given by (see Snyder 1975):
$$ \kappa _{p} = \kappa _{{p,v}} + \kappa _{{p,s}} $$
(A10a)
where
$$ \kappa _{{p,v}} = \eta _{p} \kappa _{v} $$
(A10b)
is the effective absorption coefficient for mode p due to the visual pigment, and
$$ \kappa _{{p,s}} = \zeta _{p} \kappa _{s} $$
(A10c)
is the effective absorption coefficient due to the pupillary screening pigment.
The assumption of a homogeneous distribution of the screening pigment granules, underlying Eq. A10c, will in general not hold for the photoreceptor soma. When the density of the granules is non-homogeneous and a light power P
p
= P
p
(z) flows in mode p through a plane perpendicular to the rhabdomere axis with longitudinal coordinate z, then P
p
changes in a layer dz due to absorption by the pupil (with \( \bar{I}_{p} (r) \equiv \bar{I}^{*}_{p} (R) \)):
$$ dP_{p} = - dz{\int\limits_b^\infty {} }{\int\limits_0^{2\pi } {\kappa _{s} (r)\bar{I}_{p} (r)rdrd\varphi = - dz2\pi } }{\int\limits_b^\infty {\kappa _{s} (r)\bar{I}_{p} (r)rdr} }. $$
(A11)
Here the pupil is taken to be circular symmetric around the rhabdomere waveguide. This may be well-approximated by the fused rhabdoms of bees and butterflies, but it is less realistic for fly eyes. When the cross-section of the soma of the fly photoreceptor under consideration spans an angle Δφ
c
, i.e. occupies a fraction ψ = Δφ
c
/2π of the total 2π, the right-hand term of Eq. A11 has to be corrected by a factor ψ.
Illumination of a dark-adapted fly eye triggers the migration of pigment granules in the photoreceptor from a remote area towards the rhabdomere. A useful case to consider then is that where the pupillary granules are uniformly dispersed outside a cylinder with radius s = b + h, so that an annulus with width h is left empty (Fig. 4, inset). Then κ
s
(r) = κ
s
for r >s, and κ
s
(r) = 0 for r <s, so that Eq. A11 yields:
$$ dP_{p} = - \kappa _{s} \varepsilon _{p} (s)P_{p} dz $$
(A12)
where
$$ \varepsilon _{p} (s) = \frac{{2\pi }} {{P_{p} }}{\int\limits_s^\infty {\bar{I}_{p} (r)rdr} }. $$
(A13)
Here \( \varepsilon _{p} (s) = \varepsilon ^{*}_{p} (S) \), with S = s/b, can be derived from waveguide theory via spatial integration, with Eqs. A4a, A4c and A4d, yielding:
$$ \varepsilon ^{*}_{p} (S) = S^{2} \frac{{K_{{l - 1}} (WS)K_{{l + 1}} (WS) - K^{2}_{l} (WS)}} {{K^{2}_{l} (W)}}H(U,W). $$
(A14)
A more accessible expression is obtained with the exponential approximation of the distribution of light intensity outside the rhabdomere, Eq. A7, yielding
$$ \varepsilon _{p} (s) = 2\pi B_{p} \rho _{p} (\rho _{p} + s)e^{{ - \frac{h} {{\rho _{p} }}}} . $$
(A15)
The much friendlier Eq. A15 is the preferred choice for gaining insight into the interaction of the pupil and the rhabdomere boundary wave, since calculations show that the values obtained from Eqs. A14 and A15 closely correspond for all modes p when using the corrected ρ
p
values (Eq. A9, Fig. 3b); only slight deviations occur near the cut-off wavelengths for h ≈ 1 μm. Fig. 4 presents Eq. A15 for the case of D
r
= 1.6 μm. When the pigment granules are homogeneously dispersed throughout the soma up to the rhabdomere, so that s = b or h = 0, Eq. A15 becomes equivalent to Eq. A8, because ε
p
(b) = ζ
p
, where ζ
p
, the fraction of the mode light power propagated outside the rhabdomere, is given by Eq. A6b.
Appendix 3: light sensitivity
The light sensitivity of a photoreceptor is defined as the fraction of the incident light power absorbed by the visual pigment. A general, simple and/or analytic expression for the light power absorbed by the visual pigment cannot be given, because the granules are distributed over a limited longitudinal distance in the photoreceptor soma, and this distribution will be quite inhomogeneous. Electrophysiological experiments indicate however that the pupil is effectively concentrated distally in the fly photoreceptors (Roebroek and Stavenga 1990b), with the result that the fly pupil is in fact comparable to the case of the pupil in the human eye. The pupil action then may be approximated by that of a set of transmission filters T
p
, for each mode p, given by:
$$ T_{p} = e^{{ - \kappa _{{p,s}} \delta _{s} }} = e^{{ - K_{{p,s}} }} $$
(A16)
with δ
s
the longitudinal distance over which the pupil extends, and where the absorption coefficient κ
p,s
increases with light adaptation proportional to the increase in the number of pigment granules in the boundary wave area.
Let us first consider the case of a fly eye illuminated with a monochromatic point source delivering a total power P(λ) = 1 W at wavelength λ through a facet lens. The light flux excites a power P
p,exc
in mode p in the rhabdomere, which is a function of the angle of incidence and wavelength and is calculated by a convolution of the facet lens diffraction pattern and the rhabdomere waveguide modes (Pask and Barrell 1980a, 1980b; van Hateren 1989; Stavenga 2003a, 2003b).
The excited light power is first filtered by the pupil mechanism and then propagates in the rhabdomere until it is eventually absorbed by the visual pigment. The total power absorbed, summed over all modes, is:
$$ P_{{abs}} (\lambda ) = {\sum\limits_p {P_{{p,abs}} (\lambda ) = } }{\sum\limits_p {P_{{p,exc}} (\lambda )T_{p} (\lambda ){\left\{ {1 - e^{{ - \bar{\eta }_{p} (\lambda )\kappa _{v} (\lambda )L}} } \right\}}} } $$
(A17)
thus yielding the light sensitivity of the photoreceptor; η̄
p
is the averaged light fraction of mode p propagating inside the rhabdomere (Stavenga 2003b). The angular and spectral sensitivity of the photoreceptor are derived by normalization to the angular or spectral peak values, respectively.
Concerning the angular sensitivity, in Eq. A17 only P
p,exc
depends on the incident angle of the light source. When no more than one mode contributes to the light sensitivity, the angular sensitivity at a certain wavelength is independent of the degree of light absorption by the pupillary and visual pigments, due to normalization. This does not hold however when two or more modes are present.
Equation A17 is considerably simplified when κ
v
(λ)L is sufficiently small. Then (Eq. 44 of Stavenga 2003a):
$$ P_{{abs}} (\lambda ) = \kappa _{v} (\lambda )L{\sum\limits_p {P_{{p,eff}} (\lambda ) = \kappa _{v} (\lambda )LP_{{eff}} (\lambda )} } $$
(A18)
where P
p,eff
is the effective mode power from which light can be absorbed by the visual pigment (Eq. 45 of Stavenga 2003b):
$$ P_{{p,eff}} (\lambda ) = P_{{p,exc}} (\lambda )T_{p} (\lambda )\bar{\eta }_{p} (\lambda ). $$
(A19)
In this case the angular sensitivity at a given wavelength is independent of visual pigment absorption and can be obtained directly from P
eff
, again due to normalization. Figure 10 presents the effective light power for both modes present at 355 nm in the 1.6 μm rhabdomere, as well as their sum.
When the pupil absorption is negligible, T
p
= 1 for all modes p. This case is illustrated for an axial point source in Fig. 3 and for a uniform light source in Fig. 4. When the illumination comes from a uniform light source, P
p,exc
in Eq. A17 and A19 is obtained by integrating the excited mode power over the angle of incidence (Eq. A14 of Stavenga 2003b).
Figure 8 shows the spectral changes in effective power due to pupil closure for 6 degrees of pupil closure, indicated by the pupil distance h = s - b, and where the exponent of the transmittance (Eq. A16) for mode p is:
$$ K_{{p,s}} (\lambda ,h) = m_{s} \alpha _{s} (\lambda )\varepsilon _{p} (\lambda ,h) $$
(A20)
where m
s
is a factor proportional to the concentration of the pigment, α
s
(λ) is the normalized absorption coefficient of the pupillary screening pigment (given in Fig. 7), and ε
p
(λ,s) is the mode power fraction outside the cylinder with radius s (Eq. A15). In the calculations for Fig. 8 a value m
s
= 50 was chosen so that the maximally closed pupil yielded a peak absorbance of ca. 3 log units (Roebroek and Stavenga 1990a). If the action of the pupil is approximated by that of a filter controlling the light flux distally of the visual pigment, the pupil transmittance is:
$$ T(\lambda ,h) = \frac{{P_{{abs}} (\lambda ,h)}} {{P_{{abs}} (\lambda ,\infty )}} $$
(A21)
where P
abs
(λ,h) is given by Eq. A17, with T
p
(λ) = T
p
(λ,h), and h = ∞ represents the case where the pupil is negligible. The absorbance of the pupil then follows from:
$$ A(\lambda ,h) = - ^{{10}} \log T(\lambda ,h). $$
(A22)