# Wave Equation for Non-Linear Holographic Materials

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A few years ago I needed to create a tool to compute the incoming angles of the laser beams required to create a specified hologram. The tool was a 3-dimensional generalization of the equations in Kogelnik [1]. At the time I wondered about the effect that material non-linearity might have on the results. Kogelnik assumes that the material has a single sine-wave perturbation of the index of refraction in the material. The higher spatial harmonics will change the relative power of the outgoing diffraction orders, much like the blaze of a grating. Here I work out a generalization of the wave equations, insofar as to include the harmonics.

Light interaction with a hologram is described by the scalar wave equation,

${\nabla }^{2}E+{k}^{2}E=0\text{,}$

(1)

where E (the electric field) and k (related to the dielectric constant and conductivity) are complex functions over space. Since the material making up a hologram is periodic, k can be expanded as a harmonic series

${k}^{2}=\sum _{m}{B}_{m}\left(\stackrel{}{r}\right){e}^{i\stackrel{}{{v}_{m}}\cdot \stackrel{}{r}}\text{.}$

(2)

Bm is a slowly changing function of space. Given that the material is periodic, it is somewhat intuitive that the electric field will be periodic as well, so E can also be expanded as the harmonic series. At this point we shall also assume that the hologram is thick with respect to the wavelength, so that the electric field intensity is changes slowly over space. This gives

$E=\sum _{j}{A}_{j}\left(\stackrel{}{r}\right){e}^{i\stackrel{}{{u}_{j}}\cdot \stackrel{}{r}}\text{.}$

(3)

Inserting both harmonic series into the wave equation gives

$\sum _{j}\left[-{u}_{j}^{2}{A}_{j}+2i\stackrel{}{{u}_{j}}\cdot \stackrel{}{\nabla }{A}_{j}+{\nabla }^{2}{A}_{j}+{A}_{j}\sum _{m}{B}_{m}{e}^{i\stackrel{}{{v}_{m}}\cdot \stackrel{}{r}}\right]{e}^{i\stackrel{}{{u}_{j}}\cdot \stackrel{}{r}}=0\text{.}$

(4)

Next generalize (Aj and uj) to (Ajm and ujm) where m corresponds to the matching Bm.

$\sum _{j,m}\left[-{u}_{\mathit{jm}}^{2}{A}_{\mathit{jm}}+2i{\stackrel{}{u}}_{\mathit{jm}}\cdot \stackrel{}{\nabla }{A}_{\mathit{jm}}+{\nabla }^{2}{A}_{\mathit{jm}}\right]{e}^{i{\stackrel{}{u}}_{\mathit{jm}}\cdot \stackrel{}{r}}+{A}_{\mathit{jm}}{B}_{m}{e}^{i\left({\stackrel{}{u}}_{\mathit{jm}}+{\stackrel{}{v}}_{m}\right)\cdot \stackrel{}{r}}=0\text{.}$

(5)

Now specify j as corresponding to the order of the overtone of v,

${\stackrel{}{u}}_{\mathit{jm}}={\stackrel{}{u}}_{o}+j{\stackrel{}{v}}_{m}\text{.}$

(6)

where uo is the incoming E-wave direction. This gives

$\begin{array}{c}\sum _{j,m}\left[-\left({u}_{o}^{2}+2j{\stackrel{}{u}}_{o}\cdot {\stackrel{}{v}}_{m}+{j}^{2}{v}_{m}^{2}\right){A}_{\mathit{jm}}+2i\left({\stackrel{}{u}}_{o}+j{\stackrel{}{v}}_{m}\right)\cdot \stackrel{}{\nabla }{A}_{\mathit{jm}}+{\nabla }^{2}{A}_{\mathit{jm}}\right]{e}^{i\left({\stackrel{}{u}}_{o}+j{\stackrel{}{v}}_{m}\right)\cdot \stackrel{}{r}}\\ +{A}_{\mathit{jm}}{B}_{m}{e}^{i\left({\stackrel{}{u}}_{o}+\left(j+1\right){\stackrel{}{v}}_{m}\right)\cdot \stackrel{}{r}}=0\text{.}\end{array}$

(7)

Now renumbering the j-1 term results in

$\sum _{j,m}\left[-\left({u}_{o}^{2}+2j{\stackrel{}{u}}_{o}\cdot {\stackrel{}{v}}_{m}+{j}^{2}{v}_{m}^{2}\right){A}_{\mathit{jm}}+2i\left({\stackrel{}{u}}_{o}+j{\stackrel{}{v}}_{m}\right)\cdot \stackrel{}{\nabla }{A}_{\mathit{jm}}+{\nabla }^{2}{A}_{\mathit{jm}}+{B}_{m}{A}_{j-1,m}\right]{e}^{i\left({\stackrel{}{u}}_{o}+j{\stackrel{}{v}}_{m}\right)\cdot \stackrel{}{r}}=0\text{.}$

(8)

Since the various vm terms are most likely harmonically related, sums across equal jvm's are zero. Therefore the expression is zero everywhere if and only if, for every unique C,

$\sum _{j\stackrel{}{{v}_{m}}=\stackrel{}{C}}-\left({u}_{o}^{2}+2j{\stackrel{}{u}}_{o}\cdot {\stackrel{}{v}}_{m}+{j}^{2}{v}_{m}^{2}\right){A}_{\mathit{jm}}+2i\left({\stackrel{}{u}}_{o}+j{\stackrel{}{v}}_{m}\right)\cdot \stackrel{}{\nabla }{A}_{\mathit{jm}}+{\nabla }^{2}{A}_{\mathit{jm}}+{B}_{m}{A}_{j-1,m}=0\text{.}$

(9)

As an example, the j=0 term is

$-{u}_{o}^{2}{A}_{o}+2i{\stackrel{}{u}}_{o}\cdot \stackrel{}{\nabla }{A}_{o}+{\nabla }^{2}{A}_{o}+\sum _{m}{B}_{m}{A}_{-1,m}=0\text{.}$

(10)

The A's are solved for a given set of B's by an eigenvalue problem represented by a matrix which is mostly zeros. Only the diagonal and one off-diagonal are populated.

For a thick hologram, the A's are constant across the material and only change from one side to the other. This results in a second-order differential equation in the material depth. The lowest B-term (after the constant) is the wave programmed in by the original interference setup, while higher terms are due to non-linearity in the material.

## Reference

[1] Kogelnik, Herwig. Coupled Wave Theory for Thick Hologram Gratings. The Bell System Technical Journal. Volume 48, number 9, page 2909-2947. November 1969.

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