## 1. Introduction

Kogelnik's Coupled wave theory [1], published in 1969, has provided an extremely successful approach to understanding diffraction in sinusoidal volume gratings and in providing analytic formulae for the calculation of diffractive efficiency. N-coupled wave theory [2] has extended Kogelnik's approach to provide a useful analytic model of diffraction in spatially multiplexed gratings and in monochromatic holograms.

A more recent and alternative approach to Kogelnik's coupled wave theory, known as the PSM model [3], short for "Parallel Stacked Mirrors", is based on a differential formulation of the process of Fresnel reflection occurring within the grating. This theory has the advantage of providing a particularly useful and more intuitively natural description of diffraction in the reflection volume grating. It also deals with the π-polarisation, which requires significantly greater work under Kogelnik's approach, in a simpler and more natural way.

Although the PSM model is itself a type of coupled-wave theory, it is nevertheless based on an alternative and distinct set of assumptions to standard coupled-wave theory. This in itself is extremely useful as it allows one to look at the problem of diffraction in volume gratings from two relatively separate perspectives. In some cases the PSM assumptions are clearly somewhat superior to Kogelnik's as evidenced by rigorous computational solutions of the Helmholtz equation. But this is not always the case and in various albeit rather extreme cases Kogelnik's theory can provide the superior estimate of diffractive efficiency.

The PSM model naturally treats polychromatic index modulation profiles. This is not to say that Kogelnik's formulism cannot be extended to treat the polychromatic case. Indeed Ning has demonstrated this [4]. But the mathematics and their meaning here is more transparent in the PSM case. Like standard coupled-wave theory, the PSM model can be generalised to an N-coupled wave theory, capable of describing spatially multiplexed gratings and holograms. Again the PSM model provides a simple and trivially transparent generalisation to the polychromatic spatially multiplexed grating allowing a very clear understanding of diffraction in the full-colour reflection volume hologram.

Despite the utility and analytic nature of both Kogelnik's coupled wave theory and the PSM model, a completely accurate description of diffraction in gratings can only be offered by a rigorous solution of the underlying wave equation. Moharam and Gaylord [5] first tackled this problem in 1989 and provided numerical solutions for both transmission and reflection gratings as index modulation increased. Glytis and Gaylord [6] extended this work to cover anistropic media and simple multiplexed gratings.

## 2. Kogelnik's coupled wave theory

Kogelnik's theory [1] assumes that only two plane waves propagate inside and outside a finite thickness grating. The Helmholtz equation is then used to calculate how a specific modulation in the dielectric permittivity intrinsically couples these waves. The approach has its origins in the field of acousto-optics. The first wave is assumed to be the illuminating reference wave and the second wave is the hologram’s response or “signal” wave. The adoption of just two waves is made on the assumption that coupling to higher order modes will be negligible. There is no rigorous mathematical proof for this per se; we therefore look to the results of this two-wave theory to see whether they are sensible and consistent. In addition we shall review a rigorous formulation of the coupled wave equations in section 5 and here we shall see that for the kind of index modulations present in modern holography, the two-wave assumption is pretty good.

### 2.1. Derivation of the coupled wave equations

Assuming a time dependence of

Here

The assumption of small conductivity means that our analysis is restricted to lossless phase holograms with no absorption. The assumption that

#### 2.1.1. One-dimensional grating

A one-dimensional grating extending from

The grating vector

We may write the

Here we have also introduced Kogelnik’s coupling constant

#### 2.1.2. Solution at Bragg resonance

At Bragg resonance the signal and reference wavevectors are related by the condition

The magnitude of both

We now choose a very particular trial solution of the form

The first term represents the illumination or "reference" wave and the second term, the response or "signal" wave. Both are plane waves. Figure 1(a) illustrates how these waves propagate in a reflection grating and Figure 1(b) illustrates the corresponding case of the transmission hologram. Note that the complex functions *R* and *S* are constants. The choice of just using two waves in the calculation - clearly the absolute minimum - and with only a one-dimensional behaviour was inspired by the work of Bhatia and Noble [7] and Phariseau [8] in the field of acousto-optics.

Substituting (12) into (3) we obtain

(13) |

Since only two waves are assumed to exist in the solution we must now disregard the third and fourth term of this expression on the pretext that they inherit only negligible energy from the primary modes. Next, second-order derivatives are neglected on the premise that *R* and *S* are slowly varying functions. Then (13) reduces to the two coupled first-order ordinary differential equations:

We can then use (14) and (15) to derive identical uncoupled second order differential equations for

Here the

And if the grating has been written using a reference and object wave of angles of incidence of respectively

#### 2.1.3. Boundary conditions

It has been assumed that the *R* wave is the driving wave and that the *S* wave is the response or "signal" wave. Clearly, and without loss of generality, the input amplitude of the driving wave can be normalised to unity. Then different boundary conditions can be written down for transmission and reflection gratings. For transmission gratings the choice of normalisation means that

With these boundary conditions in hand we can now solve (14) - (15) for the transmission and reflection cases. For the transmission grating we obtain

And for the reflection grating we have

(20) |

These are very simple solutions which paint a rather logical picture. For the transmission case we see that as the reference wave enters the grating it slowly donates power to the signal wave which grows with increasing

In the reflection case the behaviour is rather different. Here, as one might well expect, there is simply a slow transfer of energy from the reference driving wave to the reflected signal wave. If the emulsion is thin then the signal wave is weak and most of the energy escapes as a transmitted R wave. If the emulsion is thick on the other hand then the amplitudes of both waves become exponentially small as

#### 2.1.4. Power balance and diffraction efficiency

Using Poynting’s theorem it can be shown that power flowing along the

Multiplying (21) by respectively

This tells us that at each value of

where

It is now simple to use the forms for *R* and *S* given in (19) and (20) to calculate the expected diffractive efficiencies for the transmission and reflection grating:

Figure 2(a) shows this graphically for

#### 2.1.5. Behaviour away from Bragg resonance

To study the case of a small departure from the Bragg condition Kogelnik continues to use (9) but relaxes the condition that

We then define the “Off-Bragg” or "dephasing" parameter

where

Then, as before, we can solve equations (14) and (26) to arrive at expressions for the diffractive efficiency[1] - . For the transmission grating the result is

whereas for the reflection grating we have

Clearly for

We can understand better the parameter

Then (27) can be written as

This tells us how the parameter

#### 2.1.6. Sensitivity to wavelength and replay angle

We can understand the replay angle and wavelength behaviour of the transmission and reflection gratings by an analysis of equations (29) and (30).

To this end we assume that the illumination wave on playback is of magnitude

We will now adopt a value of

We may then use (34) to show that for the un-slanted transmission grating[2] - ,

and for the corresponding reflection grating,

This shows that a transmission grating is generally more selective in angle than a reflection grating:

## 3. The PSM theory of gratings

The PSM model [3] offers an alternative method to Kogelnik's coupled wave theory for the analysis of diffraction in planar gratings. PSM stands for "Parallel Stacked Mirrors". As might be expected from this name, the theory models a holographic grating as an infinite stack of mirrors, each one parallel to the next. Each mirror is formed by a "jump" or discontinuity in the permittivity profile which constitutes the grating; the process of diffraction is then described entirely by Fresnel reflection. In many ways the PSM model can be thought of as a differential representation of the chain matrix approach of Abeles [9] as described by various authors [10,11] and which was derived from the ideas of Rouard[12]. These ideas, in turn, extend back to Darwin's 1914 work on X-ray diffraction [13]. Early attempts at an analytical formulation of diffraction in the planar grating in terms of Fresnel reflection are also due to Ludman [14] and Heifetz, Shen and Shariar [15].

### 3.1. The simplest model - The unslanted reflection grating at normal incidence

An unslanted holographic grating with the following index profile is assumed

Here,

Now we wish to understand the response of the grating to a plane reference wave of the form

As before we shall assume that the grating is surrounded by a zone of constant index,

(43) |

Here the terms in brackets are just the Fresnel amplitude reflection and transmission coefficients and the exponential is a phase propagator which advances the phase of the *R* and *S* waves as they travel the distance

and consider the limit

These equations are an *exact* representation of Maxwell's equations for an arbitrary index profile,

and the conservation of energy

When

We now make the transformation

where the primed quantities are slowly varying compared to

where the operator

which is just the ratio of the replay wavelength to the recording wavelength. Introducing the pseudo-field,

these equations may now be written in the form of Kogelnik's equations for the normal-incidence sinusoidal grating

where Kogelnik's constant,

For comparison, Kogelnik's coefficients are

By imposing boundary conditions appropriate for the reflection hologram

where d is the grating thickness, equations (53) may be solved analytically. We can then define the diffraction efficiency for both the PSM and Kogelnik models as

Note that we should ensure that

where m is a non-zero integer to prevent a discontinuity in index at

For cases of practical interest for display and optical element holography, substitution of (54) (the PSM coefficients) or (55) (Kogelnik's coefficients) into (57) / (58) yield very similar results. However one should note that the only approximation made in deriving the PSM equations, (53) - (54) and (57) has been that of equation (49). This is an assumption which one would reasonably expect to hold in most gratings of interest. Equation (57) in conjunction with (55) is of course exactly equivalent to (30) for the case of zero grating slant and normal incidence.

At Bragg resonance, when

However the PSM model provides a useful insight into what is happening within the grating: multiple reflections of the reference wave simply synthesise the signal wave by classical Fresnel reflection and transmission at each infinitesimal discontinuity. This is a rigorous picture for the normal incidence unslanted reflection grating as equations (45) are an exact representation of Maxwell's equations. The fact that we explicitly need to introduce a "pseudo-field",

### 3.2. The unslanted panchromatic reflection grating at normal incidence

One of the advantages of the PSM model is that it does not limit the grating to a sinusoidal form. This is an advantage over the simplest variants of standard coupled-wave theories including Kogelnik's.

We start by assuming a general index profile

Equations (45) then reduce to the following form

Assuming that the individual gratings have very different spatial frequencies these equations then lead to a simple expression for the diffractive efficiency when the reference wave is in Bragg resonance with one or another of the multiplexed gratings:

In addition, in the region of the j^{th} Bragg resonance, (62) leads to the approximate analytical form

When the spatial frequencies of the different gratings are too close to one another, these relations break down. For many cases of interest however (63) to (65) provide a rather accurate picture of the normal-incidence unslanted polychromatic reflection phase grating. Indeed the following form can often be used to accurately describe an N-chromatic grating at normal incidence:

For example Diehl and George [18] have used a sparse Hill's matrix technique to computationally calculate the diffraction efficiency of a lossless trichromatic phase reflection grating at normal incidence. They used free-space recording wavelengths of 400nm, 500nm and 700nm. The grating thickness was 25 microns and the index parameters were taken as

### 3.3. The unslanted reflection grating at oblique incidence

To treat the case of reference wave incidence at finite angle to the grating planes we must redraw Figure 4(a) using two-dimensional fields, *R* and *S* which we now endow with two indices instead of the previous single index (see Figure 4(b)). We shall make the approximation that the index modulation is small enough such that the rays of both the *R* and *S* waves are not deviated in angle. We shall however retain the proper Fresnel amplitude coefficients.

The Fresnel amplitude coefficients for the σ-polarisation may be written as

(67) |

where r and t pertain respectively to reflection and transmission occurring at the index discontinuity between layers k and k+1. The *R* and *S* waves in the exterior medium of index

where *R* and *S* are functions of x and y. Using the normal rules of Fresnel reflection, the wave-vectors can be written explicitly as

where the angle *R* wave.

We can now use Figure 4 (b) to write down two expressions relating the discrete values of *R* and *S* within the grating. These are an equation for

(70) |

and the corresponding equation for

(71) |

Since we are assuming that

The exponentials are also written using a Taylor expansion. Then using the following additional approximations

and taking the limit *R*and *S* fields

Note the similarity of (76) and (77) to (45). Note also that if we set

#### 3.3.1. Simplification of the PSM equations to ODEs

The PSM equations may be simplified under boundary conditions corresponding to monochromatic illumination of the grating.

Let

Under this transformation equations (76) - (77) yield the following pair of ordinary differential equations

(81) |

Similarly the π-polarisation equations yield

(82) |

Equations (81) and (82) are approximate only because we have assumed an approximate form for the direction vector of the waves within the grating. We may however approach the problem differently and derive exact equations directly from (45). For example, in the case of the σ-polarisation, we use the optical invariant

Then using Snell's law

it is simple to see that (45) reduces to

(86) |

where

then (86) is seen to be an exact solution of the Helmholtz equation. Therefore the solution of (85) and (86) subject to the boundary conditions (56) and

#### 3.3.2. Analytic solutions for sinusoidal gratings

We start by defining an unslanted grating with the following index profile

where we imagine

and using (49), equations (81) reduce to

(90) |

As before we now define the pseudo-field

whereupon equations (90) reduce to the standard form of Kogelnik's equations

The coefficients for the PSM model and for Kogelnik's model are as follows:

(93) |

Equations (92) in conjunction with the boundary conditions (56) then lead, as before to the general analytic expression for the diffractive efficiency of the unslanted reflection grating:

Note that at Bragg resonance both the PSM theory and Kogelnik's theory reduce to the well-known formula

The π-polarisation may be treated in an exactly analogous way, leading to the following pair of ordinary differential equations for *R* and

These are just Kogelnik's equations with a modified

The practical predictions of Kogelnik's model and the PSM model are very close for gratings of interest to display and optical element holography. This is largely due to the effect of Snell's law which acts to steepen the angle of incidence in most situations. But at very high angles of incidence within the grating, larger differences appear.

#### 3.3.3. Multi-colour gratings

A multi-colour unslanted reflection grating can be modelled in the following way

In this case the PSM σ-polarisation equations yield

(100) |

Once again, if we assume that the individual gratings have very different spatial frequencies, then these equations lead to a simple expression for the diffractive efficiency when the reference wave is in Bragg resonance with one or another of the multiplexed gratings:

The corresponding result for the π-polarisation is

In the region of the j^{th} Bragg resonance, (100) leads to the approximate analytical form[4] -

Again, as long as there is sufficient difference in the spatial frequencies of each grating we can add each response to give an convenient analytical expression for the total diffraction efficiency:

In cases where the individual gratings are too close to one another in wavelength or where small amplitude interaction effects between gratings are to be described, equations (100) must be solved numerically.

### 3.4. The slanted reflection grating at oblique incidence

We may use the PSM equations for the unslanted grating to derive corresponding equations for the general slanted grating. To do this we define rotated Cartesian coordinates

In the un-primed frame we have

whereas in the primed frame we have

Derivatives in the primed system are related to those in the un-primed system by Leibnitz's chain rule

The PSM equations for the σ-polarisation may therefore be written as

and

Note that we have kept the un-primed frame on the RHS on purpose as in this system the index profile is one dimensional and so much easier to evaluate.

#### 3.4.1. Analytic solutions for sinusoidal gratings

To study the single colour grating we use the unslanted index profile (88) in the un-primed frame leading to the following profile in the primed frame

Letting

Equations (110) and (111) then become

(114) |

Next we make the transformation

whereupon once again the PSM equations reduce to a simple pair of ordinary differential equations of the form of Kogelnik's equations, (92) with coefficients

For comparison, Kogelnik's coefficients are

With the usual reflective boundary conditions

Substitution of either (117) or (118) into (119) gives the required expression for the diffractive efficiency in either the Kogelnik or PSM model. When

which is identical to Kogelnik's solution. Note that the behaviour of the π-polarisation is simply described by making the transformation (98) in all formulae of interest. The PSM model for the slanted grating under either the σ or π polarisations gives expressions very similar to Kogelnik's theory. For most gratings of practical interest to display and optical element holography, the two theories produce predictions which are extremely close.

#### 3.4.2. Polychromatic gratings

As before the formulae (102) - (105) with coefficients (117) give useful expressions for the diffractive efficiency of the general polychromatic slanted reflection grating at oblique incidence.

### 3.5. Slanted transmission gratings at oblique incidence

The PSM model can be applied to transmission gratings by simply using the appropriate boundary conditions to solve the PSM equations in a rotated frame. We use the transmission boundary conditions

to solve equations (92) with coefficients (117) which at Bragg resonance result in the standard formula given by Kogelnik's theory.

## 4. Theory of the spatially-multiplexed reflection grating

Both Kogelnik's Coupled wave model and the PSM model can be extended to model diffraction from spatially multiplexed gratings of the form [19]

In PSM this is done by considering the Fresnel reflections from N grating planes, each having a slant

where for the σ-polarisation

and where

Here the

At Bragg resonance

and (125) then gives the following expression for the diffractive efficiency of the ^{th} grating:

The total diffraction efficiency of the entire multiplexed grating is likewise found by summing the diffractive response from each grating:

Here ^{th} signal wave. These results are identical to the expressions obtained from an extension of Kogelnik's theory - the N-coupled wave theory of Solymar and Cooke [2]. At Bragg resonance the N-PSM model of the multiplexed grating therefore gives an identical description to the corresponding N-coupled wave theory just as the simple PSM theory gives an identical description at Bragg resonance to Kogelnik's theory. Away from Bragg resonance however, the predictions of the two models will be somewhat different.

N-PSM can be extended to the polychromatic case in which case (130) generalises to

In the limit that

and where

## 5. Rigorous coupled wave theory

Moharam and Gaylord [5] first showed how coupled wave theory could be formulated without approximation. This led to a computational algorithm which could be used to solve the wave equation exactly. Although earlier approaches such as the Modal method [20] were also rigorous they involved the solution of a trancendental equation for which a general unique algorithm could not be defined. This contrasted to the simple Eigen formulation of Maraham and Gaylord. Here we provide a derivation of rigorous coupled wave theory for the more complicated spatially multiplexed case. For brevity we shall limit discussions to the σ-polarisation for which the Helmholtz equation may be written

where

defines the multiplexed grating[5] - . We consider the case of illumination of the grating by a wave of the form

In both the front region (

(139) |

This expression may be substituted into (135) and(136). On taking the Fourier transform and applying orthogonality we then arrive at the following rigorous coupled wave equations:

(140) |

Note that for the case of the simple sinusoidal grating, the transformation

reduces (140) to the more usual form

### 5.1. Boundary conditions

In the zones in front of and behind the grating where

These equations define which

where the square roots are real for un-damped propagation[6] - . Accordingly we may deduce that the form of the front solution comprising the illumination wave and any reflected modes must be of the form

(145) |

Likewise the rear solution comprising all transmitted modes must be of the form

By demanding continuity of the tangential electric field and the tangential magnetic field at the boundaries

And at the rear surface they take the form

The modes available for external (undamped) propagation are calculated using the condition

Moharam and Gaylord [5] solved the single grating equations (142) using a state-space formulation in which solutions are obtainable through the eigenvalues and eigenvectors of an easily defined coefficient matrix. But one can also solve the more general equations (140), subject to the boundary conditions (147) and (148), using simple Runge-Kutta integration. This is a practical method as long as the number of component gratings within the multiplexed grating is relatively small. Diffraction efficiencies of the various modes are defined as

where the fields in this equation are defined either at the front boundary in the case of reflected modes or at the rear boundary in the case of transmitted modes. Note that we are treating the lossless case here and so the sum of all transmitted and reflected efficiencies totals to unity[7] - .

## 6. Comparison of PSM and Kogelnik's theory with rigorous CW theory

Equations (140), subject to the boundary conditions (147) and (148) can be solved using either Runge-Kutta integration or through the eigen-method referred to above. This permits the rigorous calculation of the diffraction efficiencies of all modes which are produced by a general grating. Fig.5 shows an example for a simple reflection grating and a simple transmission grating at Bragg Resonance[8] - . In the case of the reflection grating a very high index modulation has been assumed. Nevertheless the PSM/Kogelnik estimation is still only 20% out and it is clear that most of the dynamics of the grating is associated with the +1 reflected mode as both PSM and Kogelnik's coupled wave theories assume. In the case of the transmission hologram, a relatively high index modulation is assumed and also a large incidence angle with respect to the grating planes. Here we see again only a small departure from the PSM/Kogelnik estimation but also the presence of the +2 mode.

Fig.6 compares the N-PSM theory with the rigorous equations (140) for the case of a reflection duplex grating formed by the sequential recording of two simple reflection gratings of different slant. The first plot shows the on-Bragg behaviour at high index modulation where evidently good agreement is seen between the two "+1" rigorous modes and the two signal waves in N-PSM despite many other waves being present at much smaller amplitude. The second plot shows the off-Bragg behaviour of the duplex grating at a typical index modulation where excellent agreement is seen between N-PSM and the rigorous calculation.

In general, for the type of index modulations encountered typically in display and optical element holography, the Kogelnik and PSM theories produce fairly accurate estimations of diffractive efficiencies. For multiplexed gratings and for holograms, N-PSM and N-Coupled wave theory similarly produce usefully accurate estimations.

## 7. Discussion

In this chapter we have presented two analytic methods to describe diffraction in loss-free volume holographic gratings. These are Kogelnik's model and the PSM model. We have shown briefly how the PSM model can be extended to describe spatially multiplexed gratings and holograms. At Bragg resonance the N-PSM model is in exact agreement here with the extension of Kogelnik's model which is known as N-coupled wave theory. Away from Bragg resonance Kogelnik's model and the PSM model give slightly different predictions. But the differences are rather small. This is the same situation when one compares the N-PSM theory with N-coupled wave theory.

We have briefly discussed rigorous coupled wave analysis. Here we have seen that even at high values of index modulation diffraction in the simple reflection grating is controlled predominantly by the "+1" mode. In the simple transmission grating, higher order modes such as the "+2" can become significant if index modulation and incidence angle with respect to the grating planes is high. However the overall conclusion is that for index modulations characteristic of modern display and optical element holography both Kogelnik's coupled wave theory and the PSM model provide a rather good description of diffraction in the volume grating. And this is particularly so in the reflection case where Snell's law conspires to reduce the incidence angles and where RCW analysis shows that the dynamics are controlled really by the "+1" mode alone. RCW analysis also shows that this picture extends to the case of the multiplexed grating - with the implication that it should also apply to holograms.

Finally we should point out that all the theories presented here can be extended to cover more complex cases such as the presence of loss and anisotropy.