## ABSTRACT

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.

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.

Particularly simple formulae result for the case of a stack of

*p*plates, optical thickness*λϕ*/2*π*, 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 the number of repeats in the stack is large

*(p*→ ∞), reflexion is complete so long as cos^{2}*ϕ*<*r*^{2}. Outside this range the reflectance is .

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.

## I. INTRODUCTION

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. STATEMENT OF PROBLEM, AND DEFINITIONS

### 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 *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 *j*th plate.

: upward wave, at centre of space below *j*th plate.

*a*_{i}, *a*_{r}, *a*_{t} : defined in § 4.1 and Fig. 2.

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

*n*_{b}, *d*_{b} : each plate in the stack.

*n*_{a}, *d*_{a}: each space between adjacent plates.

*n*_{0}, *d*_{0}; *n*_{z}, *d*_{p}: 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.

*ϕ*_{b} : in each plate of the stack, = *2π d*_{b}*n*_{b}*/λ*.

*ϕ*_{a} : in each space between plates, = *2π d*_{a}*n*_{a}*/λ*.

*ϕ:* equal to *ϕ*_{a} or *ϕ*_{b} when *ϕ*_{a}*= ϕ*_{b}.

*δ:* defined in § 4·1.

*ϕ*_{0}, *ϕ*_{z,}*χ* :defined in § 5.3.

#### Amplitude reflexion (r) and transmission (t) coefficients

*r*_{1}, *t*_{1}, *r*_{2,}*t*_{2}, *r*_{3}, *t*_{3}, *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, r*_{0,}*r*_{z} are real.

*r, t:* at *n*_{b}*/n*_{a} interface.

*r*_{0}: at *n*_{0}*/n*_{a} interface (§ 5·3).

*r*_{z}: at *n*_{a}*/n*_{z} 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.

*r*_{1}, *r*_{2}, *r*_{3}, *t*_{2}, *r*_{3}, *t*_{3} :defined in § 4·1 (see also Fig. 2).

*R*_{z}, *R*_{0z} : defined in § 5·3.

#### Other symbols

*μ, μ*_{1,}*μ*_{2} : ratio of amplitudes in successive layers, defined by equations (6)–(10), (15), (20), (21), (23), (24), (44). (|*μ*_{1}| < |*μ*_{2}| when *μ*’s are real.)

*h, h*_{1,}*h*_{2}: ratio of upgoing to downgoing wave, defined by equations (7) – (9), (11), (12), (16), (17), (22), (26), (45), (46). (|*h*_{1}| < | *h*_{2}| 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.

*v*_{a}, *v*_{b} : 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.

## 3. OUTLINE OF THE METHOD

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

### Step 2

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

### Step 3

### Step 4

### Step 5

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).

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

*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

## 4. DERIVATION OF EQUATIONS

### 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).

*r*_{1}, *t*_{1}, *r*_{2} and *t*_{2} are the reflexion and transmission coefficients for light incident from above, and and are the corresponding quantities for light incident from below.

*ϕ*

_{a}. If

*r*is the reflexion coefficient at an

*n*

_{b}

*/n*

_{a}interface, then by Young’s formula and so that

*t*

_{1}

*t*

_{2}= 1 −

*r*

^{2}. Defining

*ρ*as the reflexion coefficient for one plate flanked by two half-spaces, and

*τ*as the corresponding transmission coefficient, equations (1

*a*) and (2

*a*) become: and

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*).

*ϕ*

_{a}

*= ϕ*

_{b}will be considered (equal optical thicknesses in plates and spaces). Writing

*ϕ = ϕ*

_{a}

*= ϕ*

_{b}, equations (1

*b*) and (2

*b*) become and

### 4.2. The recurrence relations

When this same plate, the *j*th, 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).

These equations are the required recurrence relations.

### 4.3. Solution of the recurrence equations

*μ*is put equal to either

*μ*

_{1}or

*μ*

_{2}, the two solutions of equation (10). The corresponding values of

*h*are obtained by substituting

*μ*=

*μ*

_{1}or

*μ = μ*

_{2}into the following equation: which is obtained by eliminating the

*μh*terms between equations (8) and (9).

*ρ*and

*τ*; they may be rewritten as functions of

*r*and

*ϕ*by substituting from equations (1

*c*) and (2

*c*), giving respectively: and

*h*) should be complex.

^{2}

*ϕ < r*

^{2}, then the

*μ*’s are real and the

*h*’s are complex. Further, since the coefficients of

*h*

^{2}and

*h*

^{0}in equation (17) are equal, always, and when the

*h*’s are complex, |

*h*

_{1}| = |

*h*

_{2}| = 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 .

*ϕ*exactly (or an odd multiple of ), cos

*ϕ*increases in either the positive or negative direction until cos

^{2}

*ϕ*>

*r*

^{2}. Now the

*h*’s are real and the

*μ*’s complex. Since the coefficients of

*μ*

^{2}and

*μ*

^{0}in equation (15) are equal, always, and when the

*μ*’s are complex, |

*μ*

_{1}| = |

*μ*

_{2}| = 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.

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

^{2}

*ϕ*>

*r*

^{2}and the

*μ*’s are complex, more convenient expressions are: or 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 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 upper sign makes |*h*| < 1 so that the (*μ*_{1}*h*_{1}) solution is the one in which the net energy flux is downwards.

### 4.4. Boundary conditions at bottom of the stack

*p*th, 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 whence or where

### 4.5. Amplitude reflexion coefficient of whole stack

*R*, we have, from equation (14):

### 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 (cos^{2} ϕ < r^{2})

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

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 μ_{1} and μ_{2} are complex (cos^{2}ϕ > r^{2})

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 *pθ* = 0 and *r*^{2}/cos^{2}*ϕ* when sin *pθ* = ±1.

*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

*pθ*corresponds to less than in

*ϕ*.

#### (c) Limiting case when cos^{2} ϕ = r^{2}

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.

*r*

^{2}− cos

^{2}

*ϕ*) is small, the right-hand side of equation (32) may be expanded by the binomial theorem, giving as the first two terms

## 5. EXTENSION TO MORE GENERAL CASES

### 5.1. Optical thicknesses of plates and spaces not equal (ϕ_{a} ≠ ϕ_{b})

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

*ϕ*

_{a}=

*ϕ*

_{b}. The equations equivalent to equations (15), (16) and (17) can be obtained by substituting from equations (1

*b*) and (2

*b*) into equations (10), (11) and (12), obtaining and

*μ*is real,

*h*is complex, and vice versa, as is the case when

*ϕ*

_{a}=

*ϕ*

_{b}, but this can be seen easily from equation (45). Writing 2

*k*′ for the coefficient of

*μ*in equation (44),

*k*

^{′2}

*>*1 is the condition both that

*μ*is real and that

*h*is complex.

*k*′

*>*+1 or

*k*′

*<*− 1. The former reduces to which becomes the same as equation (18) when

*ϕ*

_{a}=

*ϕ*

_{b}. The other range for real

*μ, k*′

*<*− 1, reduces to which is now a finite range of (

*ϕ*

_{a}+

*ϕ*

_{b}) centred around any even multiple of

*π*. When

*ϕ*

_{a}=

*ϕ*

_{b}, so this range is reduced to zero extent in the case treated in §§ 3 and 4.

*θ*(equation (25)) is modified to 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, *μ*_{1} is the solution of equation (44) whose absolute value is less than unity. The correct sign for *h*_{1} is obtained by substituting *μ*_{1} into equation (45).

(*b*) When the *μ*’s are complex, *μ*_{1} is the solution of equation (46) whose absolute value is less than unity. The correct sign for *μ*_{1} is obtained by substituting *h*_{1} into equation (45) and rearranging to obtain *μ*.

Calculation of the reflectance of the whole stack is a little more laborious than when *ϕ*_{a} = *ϕ*_{b}, 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 2*μ* in equation (44), must be evaluated, and from it, *μ*_{1} and *μ*_{2}. *m*^{2} is then obtained as (*μ*_{1}*/μ*_{2})^{p}. *k*, the coefficient of 2*h* in equation (46), is evaluated, and *k* and *m*^{2} 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, ϕ*_{a} and *ϕ*_{b}. Equations (1*b*) and (2*b*) will then give the new values for *ρ* and *τ*, and the rest of the treatment is unchanged.

*v*

_{a}in the spaces (refractive index

*n*

_{a}) and

*v*

_{b}in the plates (index

*n*

_{b}).

*r*differs according to the direction of polarization of the incident light, and is given by Fresnel’s equations: when the electric vector is in the plane of incidence, and 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

*r*

_{1}and

*r*

_{⊥}). The phase retardations for a single passage through each plate and space are now and respectively (cf. Born & Wolf, 1964, p. 282).

Note that if *ϕ*_{a} = *ϕ*_{b} 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 *n*_{5} and may also be covered with a material of refractive index *n*_{0}. *r*_{z} and *r*_{0} are the amplitude reflexion coefficients for *n*_{a}*/n*_{a} and *n*_{0}*/n*_{a} 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).

*R*

_{5}due to the stack and the

*n*

_{a}

*/n*

_{5}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

*r*

_{1}=

*r*

_{1}

^{′}=

*R*and

*r*

_{2}=

*r*

_{z}in equation (1), where is the extra phase lag due to material of refractive index

*n*

_{a}below the lower broken line.

*R*and

*T*are always different in phase; if

*T*= |

*T*| exp (−

*iχ*) then

*R*= ±

*i*|

*R*| exp (−

*iχ*) and

*T*

^{2}−

*R*

^{2}exp (− 2

*iχ*) since |

*T*|

^{2}+ |

*R*|

^{2}= 1 for the conservation of energy. Hence, equation (54) becomes

*n*

_{0}

*/n*

_{a}interface at the top is also present, then equation (1) can be applied again; this time

*r*

_{t}

*= r*

_{0},

*r*

_{1}

^{′}

*= — r*

_{0}and

*t*

_{1}

*t*

_{1}

^{′}−

*r*

_{1}

*r*

_{1}

^{′}= 1, while

*r*

_{2}is

*R*

_{s}. Hence, the combined reflexion coefficient

*R*

_{0-s}of the whole system is where

If these results are used when the incident light is oblique, it is important that the same sign convention should be used throughout for *r*_{1} (see equation (50)).

*R*_{s}, and *R*_{0s}, 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 *R*_{z} or *R*_{0z} 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 *n*_{b} and one space of refractive index *n*_{a}. 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.

## ACKNOWLEDGEMENT

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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