JOURNAL OF THE OPTICAL SOCIETY OF AMERICA
VOLUME 65, NUMBER 2
FEBRUARY 1975
Birefringence and dichroism in invertebrate photoreceptors Jacob Israelachvili, Rowland Sammut, and Allan W. Snyder Department of Applied Mathematics and Department of Neurobiology, Institute of Advanced Studies, Australian National University, Canberra 2600, Australia (Received 3 September 1974) Index Headings: Vision; Birefringence.
dipole of moment p=Ab2 ε3 E0/2. The polarizability a of the cylinder in medium 3 is therefore
There a r e two types of photoreceptors: v e r t e b r a t e and i n v e r t e b r a t e . 1 In v e r t e b r a t e r e c e p t o r s , the absorbing photopigments a r e localized in membraneous disks stacked in a p a r a l l e l a r r a y and separated by cytoplasm. In invertebrate r e c e p t o r s , the m e m b r a n e s form into cylindrical, close-packed shells called m i c r o v i l l i , 2 with the cytoplasm concentrated in the microvilli c o r e s . Both types of photoreceptors exhibit birefringence and dichroism, which have an intrinsic component (which depends on the anisotropic orientation of the lipids and photopigments in the membranes) and a form component (due to the over-all anisotropic photoreceptor s t r u c t u r e ) . Expressions for the birefringence and dichroism of i n v e r t e b r a t e r e c e p t o r s a r e derived and briefly discussed.
and if F be the volume fraction occupied by the outer cylinder (of radius b) the bulk polarization i s
Step 2. The field E 0 acting on each cylindrical shell. F o r a s y m m e t r i c (or random) a r r a y of cylinders [Fig. 1(b)], the field E 0 acting on each is given by the average macroscopic field E together with an internal field Ei due to the polarization of distant neighbors. To find Ei we consider a cylindrical region of medium 3 only, from which all cylindrical shells have been removed; the field in this region due to the polarization charges on the boundary wall i s then equal to E i , and is given by 3
An expression for the dielectric constant of a s y m m e t r i c a l (or random) a r r a y of anisotropic cylindrical shells may be derived by a method analogous to that used to derive the Clausius-Mossotti equation. 3 This approach involves two steps: first, we calculate the effective atomic polarizability of a single cylindrical shell placed in a uniform field n o r m a l to the cylinder a x i s . Second, we determine the effective field that acts on each cylindrical shell in the a r r a y , assuming that this field is unaffected by close neighbors.
Equation (5) now becomes
Step 1. The induced polarization of a cylindrical shell of inner and outer radii a and b, in a uniform field E 0 [ F i g . 1 (a)]. Media 1 and 3 of dielectric con stants ε1 and ε3 a r e assumed to be isotropic; medium 2 i s anisotropic, with diagonal dielectric tensor c o m ponents εr2, εθ2, θz2.. In the isotropic media 1 and 3, the electrostatic p o tential Φ obeys Laplace's equation, whereas in medium 2 we have
where D i s the displacement vector. Applying the usual boundary conditions, i . e . , tangential component of E and n o r m a l component of D continuous, we obtain for the potential in medium 3
where
FIG. 1. (a) Single cylindrical shell of inner and outer radii a. and b separating media 1,2, and 3. Media 1 and 3 are iso tropic, with dielectric constants ε1 and ε3, whereas medium 2 is anisotropic, with dielectric constants εr2, εθ2, and εz2. (b) Array of parallel cylindrical shells. F is the volume fraction of media 1 and 2 together; ƒ is the volume fraction of medium 2. In invertebrate photoreceptors, the outer cylinders (micro villi) are close packed (F≃π/2√3≃ 0.91); media 1 and 3 are composed of aqueous cytoplasm, whereas medium 2 is the membrane that contains the absorbing photopigments.
and
F r o m the functional form of Φ given by Eq. (2) we see that the cylinder in medium 3 i s equivalent to a line 221
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Vol. 65
where α* is the excess polarizability of the cylindrical array in medium 3, and is related to the dielectric con stant ε┴ perpendicular to the cylinders' axes by
stants of the three media are small. It is often conve nient to express a/b in terms of F and the volume frac tion ƒ of medium 2 [see Fig. 1 (Jo)]; thus
Thus, we finally obtain
We shall now calculate the refractive indices (n||, n┴) and absorption coefficients (α || , α┴) parallel and perpen dicular to the cylinders' axes. Media 1 and 3 are a s sumed to be identical and nonabsorbing; thus ε1 =ε3= n12. Medium 2 is assumed anisotropic and absorbing. The dielectric constant ε || may be expressed as 6
Equation (9) reduces to the well-known Rayleigh for mula4 in the limits a = b; and when c = 2, for a = 0, F = 1, ε1 = ε2, and ε2 = ε3. When F=i and a≈ b it reduces, to first order, to a result previously derived 5 for this special case. As regards the dielectric constant ε|| parallel to the cylinders' axes, the continuity of the tangential elec tric field at all boundaries immediately leads to
and similarly for ε┴, εr2, εθ2, and εz2, where it is assumed that the imaginary parts are small, and where λ is the wavelength in vacuum.
Equations (9) and (10) are valid for small F, and for large F when the differences between the dielectric con-
Substituting Eqs. (11) and (12) into Eqs. (9) and (10) and solving for real and imaginary parts, we obtain (for ε1 = ε3)
wherex=n r 2 /n 1 , m = (nθ2/nr2) = 1-(∆/x), ∆ = (nr2-nθ2)/n1, δ = (nr2-nz2)/n1.
In invertebrate photoreceptors [see Fig. 1 (b)] media l a n d 3 are composed of aqueous cytoplasm of refrac tive index n1 ≈ l. 34; medium 2 represents the micro villi membranes (containing the absorbing photopigments) of refractive index n2 ≈ 1. 5. Other typical values are F≈ 0.9, 0.1
VOLUME 65, NUMBER 2
FEBRUARY 1975
Birefringence and dichroism in invertebrate photoreceptors Jacob Israelachvili, Rowland Sammut, and Allan W. Snyder Department of Applied Mathematics and Department of Neurobiology, Institute of Advanced Studies, Australian National University, Canberra 2600, Australia (Received 3 September 1974) Index Headings: Vision; Birefringence.
dipole of moment p=Ab2 ε3 E0/2. The polarizability a of the cylinder in medium 3 is therefore
There a r e two types of photoreceptors: v e r t e b r a t e and i n v e r t e b r a t e . 1 In v e r t e b r a t e r e c e p t o r s , the absorbing photopigments a r e localized in membraneous disks stacked in a p a r a l l e l a r r a y and separated by cytoplasm. In invertebrate r e c e p t o r s , the m e m b r a n e s form into cylindrical, close-packed shells called m i c r o v i l l i , 2 with the cytoplasm concentrated in the microvilli c o r e s . Both types of photoreceptors exhibit birefringence and dichroism, which have an intrinsic component (which depends on the anisotropic orientation of the lipids and photopigments in the membranes) and a form component (due to the over-all anisotropic photoreceptor s t r u c t u r e ) . Expressions for the birefringence and dichroism of i n v e r t e b r a t e r e c e p t o r s a r e derived and briefly discussed.
and if F be the volume fraction occupied by the outer cylinder (of radius b) the bulk polarization i s
Step 2. The field E 0 acting on each cylindrical shell. F o r a s y m m e t r i c (or random) a r r a y of cylinders [Fig. 1(b)], the field E 0 acting on each is given by the average macroscopic field E together with an internal field Ei due to the polarization of distant neighbors. To find Ei we consider a cylindrical region of medium 3 only, from which all cylindrical shells have been removed; the field in this region due to the polarization charges on the boundary wall i s then equal to E i , and is given by 3
An expression for the dielectric constant of a s y m m e t r i c a l (or random) a r r a y of anisotropic cylindrical shells may be derived by a method analogous to that used to derive the Clausius-Mossotti equation. 3 This approach involves two steps: first, we calculate the effective atomic polarizability of a single cylindrical shell placed in a uniform field n o r m a l to the cylinder a x i s . Second, we determine the effective field that acts on each cylindrical shell in the a r r a y , assuming that this field is unaffected by close neighbors.
Equation (5) now becomes
Step 1. The induced polarization of a cylindrical shell of inner and outer radii a and b, in a uniform field E 0 [ F i g . 1 (a)]. Media 1 and 3 of dielectric con stants ε1 and ε3 a r e assumed to be isotropic; medium 2 i s anisotropic, with diagonal dielectric tensor c o m ponents εr2, εθ2, θz2.. In the isotropic media 1 and 3, the electrostatic p o tential Φ obeys Laplace's equation, whereas in medium 2 we have
where D i s the displacement vector. Applying the usual boundary conditions, i . e . , tangential component of E and n o r m a l component of D continuous, we obtain for the potential in medium 3
where
FIG. 1. (a) Single cylindrical shell of inner and outer radii a. and b separating media 1,2, and 3. Media 1 and 3 are iso tropic, with dielectric constants ε1 and ε3, whereas medium 2 is anisotropic, with dielectric constants εr2, εθ2, and εz2. (b) Array of parallel cylindrical shells. F is the volume fraction of media 1 and 2 together; ƒ is the volume fraction of medium 2. In invertebrate photoreceptors, the outer cylinders (micro villi) are close packed (F≃π/2√3≃ 0.91); media 1 and 3 are composed of aqueous cytoplasm, whereas medium 2 is the membrane that contains the absorbing photopigments.
and
F r o m the functional form of Φ given by Eq. (2) we see that the cylinder in medium 3 i s equivalent to a line 221
222
L E T T E R S TO THE E D I T O R
Vol. 65
where α* is the excess polarizability of the cylindrical array in medium 3, and is related to the dielectric con stant ε┴ perpendicular to the cylinders' axes by
stants of the three media are small. It is often conve nient to express a/b in terms of F and the volume frac tion ƒ of medium 2 [see Fig. 1 (Jo)]; thus
Thus, we finally obtain
We shall now calculate the refractive indices (n||, n┴) and absorption coefficients (α || , α┴) parallel and perpen dicular to the cylinders' axes. Media 1 and 3 are a s sumed to be identical and nonabsorbing; thus ε1 =ε3= n12. Medium 2 is assumed anisotropic and absorbing. The dielectric constant ε || may be expressed as 6
Equation (9) reduces to the well-known Rayleigh for mula4 in the limits a = b; and when c = 2, for a = 0, F = 1, ε1 = ε2, and ε2 = ε3. When F=i and a≈ b it reduces, to first order, to a result previously derived 5 for this special case. As regards the dielectric constant ε|| parallel to the cylinders' axes, the continuity of the tangential elec tric field at all boundaries immediately leads to
and similarly for ε┴, εr2, εθ2, and εz2, where it is assumed that the imaginary parts are small, and where λ is the wavelength in vacuum.
Equations (9) and (10) are valid for small F, and for large F when the differences between the dielectric con-
Substituting Eqs. (11) and (12) into Eqs. (9) and (10) and solving for real and imaginary parts, we obtain (for ε1 = ε3)
wherex=n r 2 /n 1 , m = (nθ2/nr2) = 1-(∆/x), ∆ = (nr2-nθ2)/n1, δ = (nr2-nz2)/n1.
In invertebrate photoreceptors [see Fig. 1 (b)] media l a n d 3 are composed of aqueous cytoplasm of refrac tive index n1 ≈ l. 34; medium 2 represents the micro villi membranes (containing the absorbing photopigments) of refractive index n2 ≈ 1. 5. Other typical values are F≈ 0.9, 0.1
