1. A convenient method is presented for calculating the reflectance of a stack of dielectric layers consisting of a series of identical repeats of any particular sequence of layers. The method is closely related to that published by Lord Rayleigh in 1917.

  2. In this method, two quadratic equations are formed from the thicknesses and refractive indices of the layers composing a single repeat unit. The reflectance is obtained by substituting the solutions of these equations into an explicit formula.

  3. Particularly simple formulae result for the case of a stack of p plates, optical thickness λϕ/2π, uniformly spaced in an infinite medium with spaces of the same optical thickness. If r is the amplitude reflexion coefficient at a single interface, the reflectance of the whole stack is as follows :

    • when cos2ϕ < r2,
      formula
      where
      formula
    • when cos2ϕ > r2,
      formula
      where
      formula
    • when the number of repeats in the stack is large (p → ∞), reflexion is complete so long as cos2ϕ < r2. Outside this range the reflectance is .

  4. These results are extended to cover: (a) unequal optical thicknesses in plates and spaces; (b) oblique incidence; (c) layers of materials of other refractive indices above and below the stack itself; and (d) stacks consisting of repeats of more complex units.

Interference between rays of light reflected at successive interfaces in a laminated structure has often been suggested as one of the methods by which colours of animals might be produced (e.g. Biedermann, 1914, p. 1892; Rayleigh, 1919; Onslow, 1921, p. 13; Fox & Vevers, 1960). An even more striking phenomenon is the very high reflectivity, over a substantial part of the visible spectrum, found, for example, in fish scales (Denton & Nicol, 1966), in the argentea of the eye of the scallop, Pecten (Dakin, 1910; Land, 1965) and in tapeta of the eyes of cartilaginous fishes (Denton & Nicol, 1964). In several of these cases it has recently been shown (Land (1966), for the eye of Pecten; Denton & Land (1967), for fish scales) that the reflecting structure consists of a stack of flat transparent crystals, each having an optical thickness close to a quarter of a wavelength and separated from its neighbours by layers of cytoplasm of equal optical thickness. Constructive interference therefore occurs between light reflected at successive interfaces, and a high reflectivity is obtained in the same way as in artificial ‘dielectric reflectors’ made by vacuum deposition of layers of transparent materials of alternately high and low refractive index. Other highly reflecting structures, such as the tapeta behind the retinae of some mammals, also have regularly repeating structural elements at spacings of the right order of magnitude for the same kind of interference to occur, but have not yet been worked out from the optical point of view (e.g. Pedler, 1963; Dartnall et al. 1965).

It might have been thought that the commercial development of multilayer dielectric reflectors and interference filters would have led to the publication of convenient theoretical expressions for dealing with biological systems of this kind. Discussions with Dr M. F. Land during his work on the argentea of the eye of Pecten showed however that this was apparently not the case, and the author derived a number of simple formulae, relevant to reflectors of this kind, which are not given in the accounts of multilayer theory by, for instance, Heavens (1955, 1960), Vašiček (1960), Born & Wolf (1964) or Baumeister (1965). These formulae can presumably be obtained by the powerful matrix method of Abelès (1950), but this method is of little use to those who are not familiar with matrix algebra; another advantage of the method used in the present paper is that it gives greater insight into the processes which give rise to the optical properties of the complete stack of plates than does the method of Abelès. For these reasons it seemed likely that the method, and the formulae obtained by means of it, would be useful to biologists investigating other multilayer reflecting structures.

Most of the steps in this method are the same as were used by Rayleigh (1917); he in turn was adapting for thin layers the method by which Stokes (1862) had treated the case in which the plates are thick enough for interference phenomena to be dis-regarded. These papers seem to have been to a large extent forgotten; of the authors quoted in the preceding paragraph, only Abelès (1950) refers to them.

Rayleigh did not, however, derive all the particular formulae which will be given in the present paper. The author felt it would also be useful to present the whole method afresh in a more elementary style than Rayleigh’s treatment.

In § 2, the problem is stated in its simplest form (optical thickness of spaces equal to that of plates; normal incidence; no reflexion from different materials above or below the stack itself), and symbols are defined. In § 3, the steps of the solution of this problem are outlined and the most important equations are given without proof; these equations and others are derived in § 4. In § 5 the results are extended to some more general cases.

Section 4.3 contains a discussion of the variation of phase and amplitude through the thickness of the stack.

It is assumed throughout that absorption of light is negligible. The results could no doubt be adapted to the case of finite absorption by using complex values for the phase retardations due to the plates and the spaces between them. The method is restricted to stacks consisting of a regularly repeated sequence of layers.

2.1. The problem

Figure 1 illustrates the situation that is dealt with in §§ 3 and 4. Light is incident normally from above on a stack of transparent plates, p in number, uniformly spaced in an infinite medium of different refractive index. The problem is to find the fraction of the incident intensity that is reflected.

The plates are all of equal thickness db, and their refractive index is nb, so that the phase retardation due to traversing a single plate once is given by
formula
where λ is the wavelength of the light in vacuo.
The spaces have thickness da and refractive index na, so the phase retardation due to traversing a single space is
formula

The a’s, denoting the amplitudes (and phases) of the light at different levels within the stack are defined at the broken lines in Fig. 1, i.e. at the centres of the spaces between the plates and at corresponding levels above the top plate and below the bottom plate. This convention is chosen because it makes the structure between any two of the broken lines symmetrical, and therefore simplifies the equations.

2.2. Definitions

Only the symbols which are used throughout the paper are defined here; a number of other symbols are defined where they arise in §§ 4.1, 5.2 and 5.3.

Amplitudes

See Fig. 1. Defined as amplitudes of electric vector, as a complex quantity so as to indicate phase as well as absolute amplitude.

: downward wave, at centre of space below jth plate.

: upward wave, at centre of space below jth plate.

ai, ar, at : defined in § 4.1 and Fig. 2.

Refractive index (n) and thickness (d) of layers

nb, db : each plate in the stack.

na, da: each space between adjacent plates.

n0, d0; nz, dp: defined in § 5.3 and Fig. 4.

Phase retardations for light of wavelength λ

Expressions are for normal incidence; oblique incidence is dealt with in § 5.2.

ϕb : in each plate of the stack, = 2π dbnb.

ϕa : in each space between plates, = 2π dana.

ϕ: equal to ϕa or ϕb when ϕa= ϕb.

δ: defined in § 4·1.

ϕ0, ϕz,χ :defined in § 5.3.

Amplitude reflexion (r) and transmission (t) coefficients

r1, t1, r2,t2, r3, t3, R, T are defined for light incident from above; the same symbols with primes are for light incident from below on the same interfaces or structures. In general, complex, but r, t, r0,rz are real.

r, t: at nb/na interface.

r0: at n0/na interface (§ 5·3).

rz: at na/nz interface (§ 5·3).

ρ, τ: one plate of the stack, with one half-space above and below.

R, T: whole stack, with one half-space above the top plate and one half-space below the bottom plate.

r1, r2, r3, t2, r3, t3 :defined in § 4·1 (see also Fig. 2).

Rz, R0z : defined in § 5·3.

Other symbols

μ, μ1,μ2 : ratio of amplitudes in successive layers, defined by equations (6)(10), (15), (20), (21), (23), (24), (44). (|μ1| < |μ2| when μ’s are real.)

h, h1,h2: ratio of upgoing to downgoing wave, defined by equations (7)(9), (11), (12), (16), (17), (22), (26), (45), (46). (|h1| < | h2| when h’s are real.)

m: = , and equation (32).

α: defined by equations (13) and (14).

θ: defined by equations (25) or (49).

2k : coefficient of h in equation (12), (17) or (46).

2k′ : coefficient of μ in equations (10), (15) or (44).

p : number of plates in the stack.

j: ordinal number of a plate in the stack, from j = 1 for top plate to j = p for bottom plate.

va, vb : defined in § 5·2.

The formulae developed in this paper were used by Land (1966) in connexion with the reflecting layer in the eye of Pecten. The symbols used by him differ from those used in the present paper in the respects shown in Table 1.

The main steps in the procedure are listed here, numbered to agree with the sub-sections of § 4 in which the formulae are derived.

The expressions given in this section are appropriate to the case where the optical thicknesses of plates and spaces are equal, incidence is normal to the interfaces, and there is no reflexion from structures above the topmost plate or below the bottom one. Extensions of these results to more general cases are given in § 5.

The equations are numbered according to their sequence in § 4.

Step 1

Obtain expressions for the reflexion and transmission coefficients of a single plate, taking account of multiple reflexions within the plate. They are respectively:
formula
and
formula

Step 2

Derive a pair of recurrence relations connecting the amplitudes of the upward-and downward-propagating waves at successive layers in the stack.

They are
formula
and
formula

Step 3

Show that these equations are solved by
formula
and
formula
where
(a) μ1 and μ2 are the solutions of the equation
formula
(b) h1 and h2 are obtained by putting μ = μ1 or μ = μ2 in the equation
formula
and (c) α is a constant whose value depends on the boundary conditions.

Step 4

Obtain α by considering the boundary condition at the bottom of the stack. The result is
formula
where
formula

Step 5

The amplitude reflexion coefficient is now obtained from equation (14). It is
formula

This is a complex quantity, indicating the phase as well as the absolute amplitude of the reflected light.

Step 6

Take the square of the modulus of R to obtain the reflectance of the stack (fraction of incident intensity that is reflected).

When the optical thickness of each plate (and each space) is close enough to a quarter wavelength so that cos2ϕ < r2, then m is real and
formula

This approaches unity (complete reflexion) as the number of plates is increased.

When the optical thicknesses deviate from a quarter wavelength far enough so that cos2ϕ > r2, then m and the μ’s are complex, and
formula
where
formula
The right-hand side of equation (39) is an oscillatory function of θ, which in turn varies with ϕ and therefore with wavelength. When the number of plates p is large, it is convenient to average this function over one cycle of its oscillation; the result is
formula
When cos2ϕ = r2, equations (35) and (39) reduce to
formula

4.1. Reflexion by a single plate

Suppose that two partially reflecting surfaces are separated by a layer of material which introduces a phase delay of δ (Fig. 2).

r1, t1, r2 and t2 are the reflexion and transmission coefficients for light incident from above, and and are the corresponding quantities for light incident from below.

Adding the amplitudes of the successive reflected beams (expressed as complex numbers so as to indicate phase), we obtain
formula
The second and later terms between the curly brackets form a geometrical progression which can be summed to give
formula
Now ar/ai is the combined reflexion coefficient r3, so
formula
In the same way
formula
or
formula
When the two reflectors are simple interfaces between media of different refractive index, the r’s and t’s are connected by
formula
(e.g. Born & Wolf (1964), p. 324), so that
formula
and
formula
These well-known results give the phases of the reflected and transmitted beams as they just leave the plate, relative to the phase of the incident beam as it just reaches the plate. To apply them to the amplitudes in our stack, we note that both for and for (see Fig. 1) relative to , there are two half-spaces to be traversed in addition to the single plate, giving an extra phase delay of ϕa. If r is the reflexion coefficient at an nb/na interface, then by Young’s formula
formula
and
formula
so that t1t2 = 1 − r2. Defining ρ as the reflexion coefficient for one plate flanked by two half-spaces, and τ as the corresponding transmission coefficient, equations (1a) and (2a) become:
formula
and
formula

Since the system now under consideration (one plate flanked by two half-spaces) is symmetrical about the centre of the plate, these expressions for ρ and τ are appropriate for light incident either from above or from below (i.e. ρ= ρ and τ′ = τ).

In the rest of § 4 only the case ϕa= ϕb will be considered (equal optical thicknesses in plates and spaces). Writing ϕ = ϕa= ϕb, equations (1b) and (2b) become
formula
and
formula

4.2. The recurrence relations

When this same plate, the jth, is in place in the stack, light is incident on it from below as well as from above because of reflexion from lower layers in the stack (Fig. 3).

The amplitude of light leaving it in the upward direction is the resultant of a component from reflected by the plate and a component from transmitted by the plate, or
formula
Similarly, for the downward wave leaving the plate,
formula

These equations are the required recurrence relations.

4.3. Solution of the recurrence equations

Try whether equations (4) and (5) are solved by equations of the form:
formula
and
formula
Insert these expressions into 4 and 5 and divide through by , obtaining:
formula
and
formula
Eliminating h between these two equations gives the following quadratic for μ :
formula
Equations (4) and (5) are therefore solved by equations (6) and (7), provided that μ is put equal to either μ1 or μ2, the two solutions of equation (10). The corresponding values of h are obtained by substituting μ = μ1 or μ = μ2 into the following equation:
formula
which is obtained by eliminating the μh terms between equations (8) and (9).
Also, μ can be eliminated between equations (8) and (9) to obtain the following quadratic for h:
formula
Equations (6) and (7) are therefore solutions of equations (4) and (5) provided that either
formula
or
formula
Since equations (4) and (5) are linear, any constant multiple of a solution, or any sum of solutions, is also a solution, and a general solution is therefore:
formula
and
formula
Equations (10) to (12) give the coefficients in terms of ρ and τ; they may be rewritten as functions of r and ϕ by substituting from equations (1c) and (2c), giving respectively:
formula
formula
and
formula
The condition for real roots in equation (15) is that
formula
this reduces to
formula
which is also the condition that the roots of equation (17) (for h) should be complex.
Hence, whenever the optical thickness of each plate and each space is close enough to a quarter wavelength (or to any odd number of quarter wavelengths) so that cos2ϕ < r2, then the μ’s are real and the h’s are complex. Further, since the coefficients of h2 and h0 in equation (17) are equal,
formula
always, and when the h’s are complex, |h1| = |h2| = 1. Thus the upward wave has the same amplitude as the downward one that it accompanies, but it is shifted in phase. The amplitudes vary exponentially with distance through the stack; in an infinite stack, only the solution with |μ| < 1 can exist and its amplitude decays away toward zero, all the energy being reflected. The solution with |μ| > 1 exists only in so far as energy is reflected at the bottom of the stack. Since μ is real, the phase of each wave is the same or 180° out at every layer in the stack; this means that the velocity of the wave is being pulled by the periodic structure so that an integral number of half-waves corresponds exactly to one repeat, even when ϕ is not exactly .
If the wavelength is changed further from the ideal value which makes ϕ exactly (or an odd multiple of ), cos ϕ increases in either the positive or negative direction until cos2ϕ > r2. Now the h’s are real and the μ’s complex. Since the coefficients of μ2 and μ0 in equation (15) are equal,
formula
always, and when the μ’s are complex, |μ1| = |μ2| = 1. Thus, each wave travels through the stack with unchanging amplitude but with its phase being shifted by an equal amount for each layer that is passed. The accompanying wave in the opposite direction has a different amplitude (|h| ≠ 1) so that there is a finite transfer of energy through the stack, and the reflexion coefficient of the whole stack is less than unity however many plates it contains.
When cos2ϕ < r2 and the μ’s are real, the convenient expressions for μ and h (upper signs for μ1 and h1) are
formula
or
formula
and
formula

With this convention for the signs, |μ1| < 1 irrespective of the signs of r and sin ϕ, so that the (μ1, h1) solution is always the one with amplitude decreasing in the down-ward direction through the stack.

When cos2ϕ > r2 and the μ’s are complex, more convenient expressions are:
formula
or
formula
where θ is real; this form is permissible because |μ| = 1 as pointed out earlier. Equating real parts of equations (23) and (24) shows that θ is given by
formula
to give the correct sign to the imaginary part in equation (23), the solution to be chosen is the one which makes lie in the same quadrant as ϕ.
The corresponding equation for h is
formula

The upper sign makes |h| < 1 so that the (μ1h1) solution is the one in which the net energy flux is downwards.

4.4. Boundary conditions at bottom of the stack

Equations (13) and (14) contain an arbitrary constant a. The appropriate value for this constant can be obtained by considering the boundary condition at the bottom of the stack. represents the amplitude of the upward wave below the pth, or lowest, plate. In the case we are now considering there is no reflecting surface below this level, so this wave is non-existent and . Substituting this in equation (14) gives
formula
whence
formula
or
formula
where
formula

4.5. Amplitude reflexion coefficient of whole stack

represents the upward (reflected) wave above the first or topmost plate of the stack, while is the incident amplitude. Denoting the reflexion coefficient of the whole stack by R, we have, from equation (14):
formula
Substituting for α from equation (27) gives
formula
Now h1h2 = 1 (equation (19)), so
formula
It is convenient to rewrite this as
formula
since and are simple expressions obtained from equations (12), (17), (22) or (26).
In the same way, the transmission coefficient of the whole stack is
formula

4.6. Intensity reflexion coefficient of whole stack

This quantity (reflectance, the fraction of the incident light intensity that is reflected) is the square of the modulus of R. It is convenient to use different procedures for obtaining the modulus when the μ’s are real and when they are complex.

(a) Case when µs are real (cos2 ϕ < r2)

From equation (21) it follows that
formula

This quantity is real, and lies between 0 and 1.

(h1 + h2) is real and h1h2) is imaginary, so from equation (30)
formula
since h1h2 = 1 (equation (19)), so
formula
where 2k is the coefficient of h in equations (12) or (17), so
formula
In equation (17), k = cos ϕ/r, so in this case
formula

With an infinite number of plates m approaches zero and equation (35) shows that the reflectance then approaches unity, i.e. all the incident light is reflected.

(b) Case where μ1 and μ2 are complex (cos2ϕ > r2)

Using for μ the expression given in equation (24), we obtain
formula
(θ being defined by equation (25)), and equation (30) becomes
formula
(h1h2) is now real, so
formula
As before, , and , so
formula
From equation (17), k = cos ϕ/r, so in this case
formula

As the wavelength is changed, ϕ, and therefore also θ, changes, and this function fluctuates, the fluctuations becoming closer as the number of plates, or their optical thickness, is increased. This behaviour of multilayer dielectric reflectors is well known (e.g. Vašiček, 1960). The reflectance varies between zero when sin = 0 and r2/cos2ϕ when sin = ±1.

For a thick stack of plates it is useful to calculate the mean reflectance over one cycle of this fluctuation. Equation (39) may be rewritten.
formula
whence
formula
if ϕ can be considered as constant within the limits of integration. From equation (25) it can be seen that the range of ± π in 2 corresponds to less than in ϕ.

(c) Limiting case when cos2 ϕ = r2

This may be approached from either equation (35) or equation (39); the result is the same in both cases but the approach from equation (35) is the simpler.

Since (r2 − cos2ϕ) is small, the right-hand side of equation (32) may be expanded by the binomial theorem, giving as the first two terms
formula
This can be taken as 1 except in the term (1 − m2), which becomes
formula
Substituting these results in equation (35) gives
formula
Since r2 = cos2ϕ, this may be written as either
formula
or
formula

5.1. Optical thicknesses of plates and spaces not equal (ϕa ≠ ϕb)

The appropriate expressions for ρ and τ have already been derived (equations (1b) and (2b)).

Equations (4)(14) inclusive are valid whether or not ϕa = ϕb. The equations equivalent to equations (15), (16) and (17) can be obtained by substituting from equations (1b) and (2b) into equations (10), (11) and (12), obtaining
formula
formula
and
formula
It is laborious to show directly from equations (44) and (46) that whenever μ is real, h is complex, and vice versa, as is the case when ϕa = ϕb, but this can be seen easily from equation (45). Writing 2k′ for the coefficient of μ in equation (44),
formula
Substituting this into equation (45) gives
formula
so that k′2> 1 is the condition both that μ is real and that h is complex.
This condition is satisfied by either k> +1 or k< − 1. The former reduces to
formula
which becomes the same as equation (18) when ϕa = ϕb. The other range for real μ, k< − 1, reduces to
formula
which is now a finite range of (ϕa+ϕb) centred around any even multiple of π. When ϕa = ϕb, so this range is reduced to zero extent in the case treated in §§ 3 and 4.
Equations (19) and (19’) are still valid. The equivalents of equations (20)(23) and (26) are too cumbersome to be useful. Equation (24) is still valid provided that the definition of θ (equation (25)) is modified to
formula
the solution to be chosen is the one which makes lie in the same quadrant as .

Equations (27)(31), (33), (34), (36)(38) are still valid provided that 2k is taken as the coefficient of h in equation (46) instead of in equation (17), and θ is defined by equation (49) instead of equation (25). The signs to be chosen in evaluating the μ’s and h’s should be selected by the following criteria:

(a) When the μ’s are real, μ1 is the solution of equation (44) whose absolute value is less than unity. The correct sign for h1 is obtained by substituting μ1 into equation (45).

(b) When the μ’s are complex, μ1 is the solution of equation (46) whose absolute value is less than unity. The correct sign for μ1 is obtained by substituting h1 into equation (45) and rearranging to obtain μ.

Calculation of the reflectance of the whole stack is a little more laborious than when ϕa = ϕb, but it can be carried out by means of equations (34) or (38). When μ is real (|k′| > 1), equation (34) is used. k′, the coefficient of 2μ in equation (44), must be evaluated, and from it, μ1 and μ2. m2 is then obtained as (μ12)p. k, the coefficient of 2h in equation (46), is evaluated, and k and m2 are substituted into equation (34), giving the required result. When μ is complex, θ is evaluated from equation (49), and k and θ are substituted into equation (38).

5.2. Oblique incidence

When the incident light is not directed along the normal to the reflecting surfaces in the stack, different expressions must be used for r, ϕa and ϕb. Equations (1b) and (2b) will then give the new values for ρ and τ, and the rest of the treatment is unchanged.

Let the angle between the normal and the direction of propagation be va in the spaces (refractive index na) and vb in the plates (index nb). r differs according to the direction of polarization of the incident light, and is given by Fresnel’s equations:
formula
when the electric vector is in the plane of incidence, and
formula
when it is perpendicular to the plane of incidence (see, for example, Born & Wolf (1964) p. 40, and footnote on p. 41 for the difference between the signs of r1 and r). The phase retardations for a single passage through each plate and space are now
formula
and
formula
respectively (cf. Born & Wolf, 1964, p. 282).

Note that if ϕa = ϕb for normal incidence, this will no longer be true for obliquely incident light.

5.3. Additional reflecting surfaces present

The situation to be considered is shown in Fig. 4, where the stack is deposited on a material of refractive index n5 and may also be covered with a material of refractive index n0. rz and r0 are the amplitude reflexion coefficients for na/na and n0/na interfaces respectively.

The whole stack, from the upper to the lower broken line in Fig. 4, can be regarded as a single partially reflecting surface with reflexion and transmission coefficients R and T defined by equations (30) and (31).

The combined reflexion coefficient R5 due to the stack and the na/n5 interface at the bottom can be obtained by means of equation (1). Since the stack (including the half-spaces at both ends) is symmetrical, R and T are the same for light incident in either direction, i.e. R= R and T= T. Hence, putting r1 = r1 = R and r2 = rz in equation (1),
formula
where
formula
is the extra phase lag due to material of refractive index na below the lower broken line.
It can be seen from equations (30) and (31) that R and T are always different in phase; if T = |T| exp (− ) then R = ±i|R| exp (− ) and T2R2 exp (− 2) since |T|2 + |R|2 = 1 for the conservation of energy. Hence, equation (54) becomes
formula
If the n0/na interface at the top is also present, then equation (1) can be applied again; this time rt= r0, r1= — r0 and t1t1r1r1 = 1, while r2 is Rs. Hence, the combined reflexion coefficient R0-s of the whole system is
formula
where
formula

If these results are used when the incident light is oblique, it is important that the same sign convention should be used throughout for r1 (see equation (50)).

Rs, and R0s, like R, are complex quantities indicating the phase as well as the absolute amplitude of the reflected light. To obtain the corresponding reflectances, it is necessary to evaluate Rz or R0z as a complex quantity and take the sum of the squares of its real and imaginary parts.

5.4. More complex sequences

It has so far been assumed that each repeat in the stack consists of one plate of refractive index nb and one space of refractive index na. If the repeating unit is more complex, the reflexion and transmission coefficients ρ and τ for a single unit can be obtained by repeated application of equations (1) and (2); this is the procedure described for example by Vašiček (1960), and used in § 5·3. μ, h, k and k′ are then found by substituting this ρ and τ into equations (10), (11) and (12), and from them, m and θ. The reflectance of the stack is then found from equation (34) or (38) according as the μ’s are real or complex.

I wish to acknowledge the helpful suggestions of Professor W. A. H. Rushton, who read the typescript, and of Professor O. S. Heavens.

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