## Abstract

How good are the Born and Rytov approximations for the case of refraction and reflection at a plane interface? Here, we show the following results: For the reflected field, the Born approximation gives a plane wave at the correct reflection angle but approximates the true reflection coefficient by an expression linear in the scattering strength, which in this case involves both the velocity perturbation across the interface and the cosine of the incident angle. The Rytov approximation, on the other hand, can be interpreted as giving an infinite series of reflected plane waves in which the first term is just the Born approximation to the true reflected wave. Both approximations, however, are uniformly valid for the field above the interface. In contrast, the Born approximation to the transmitted field is not a plane wave and is not uniformly valid since it contains a secular term that grows linearly with distance from the interface. The Rytov approximation to the transmitted field is uniformly valid; in fact, the Rytov approximation gives a transmitted plane wave that satisfies a modified form of Snell’s law. Numerical examples indicate that the Rytov approximation to the transmitted field is surprisingly accurate. For velocity contrasts less than 40% and incident angles less than 30°, the Rytov approximation to the transmitted angle and transmission coefficient is never more than 20% in error.

© 1985 Optical Society of America

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### Equations (41)

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(1)
$${u}_{I}(x,z)=exp(\mathit{\text{ik}}sin{\theta}_{1}x+\mathit{\text{ik}}cos{\theta}_{1}z),$$
(2)
$$n=\frac{{c}_{0}}{{c}_{1}}$$
(3)
$${u}_{R}(x,z)=\mathit{\text{R}}exp(\mathit{\text{ik}}sin{\theta}_{R}x-\mathit{\text{ik}}cos{\theta}_{R}z),$$
(4)
$${u}_{T}(x,z)=Texp(\mathit{\text{ikn}}sin{\theta}_{T}x+\mathit{\text{ikn}}cos{\theta}_{T}z).$$
(5)
$$R=\frac{cos{\theta}_{I}-ncos{\theta}_{T}}{cos{\theta}_{I}+ncos{\theta}_{T}},$$
(6)
$$T=1+R=\frac{2cos{\theta}_{I}}{cos{\theta}_{I}+ncos{\theta}_{T}},$$
(7)
$$nsin{\theta}_{T}=sin{\theta}_{I},$$
(8)
$$ncos{\theta}_{T}={\left(1+\frac{{n}^{2}-1}{{cos}^{2}\hspace{0.17em}{\theta}_{I}}\right)}^{1/2}cos{\theta}_{I}.$$
(9)
$$ncos{\theta}_{T}=cos{\theta}_{I}+\frac{1}{2}\frac{{n}^{2}-1}{cos{\theta}_{I}}+\dots ,$$
(10)
$$\frac{{n}^{2}-1}{{cos}^{2}\hspace{0.17em}{\theta}_{I}}<1.$$
(11)
$${u}_{R}=\frac{1-{(1+\alpha )}^{1/2}}{1+{(1+\alpha )}^{1/2}}exp(\mathit{\text{ik}}sin{\theta}_{I}x-\mathit{\text{ik}}cos{\theta}_{I}z),$$
(12)
$${{u}_{R}}^{(B)}=-\frac{\alpha}{4}exp(\mathit{\text{ik}}sin{\theta}_{I}x-\mathit{\text{ik}}cos{\theta}_{I}z).$$
(13)
$${u}^{(R)}=exp\left[-\frac{\alpha}{4}exp(-2\mathit{\text{ik}}cos{\theta}_{I}z)\right]\times exp(\mathit{\text{ik}}sin{\theta}_{I}x+\mathit{\text{ik}}cos{\theta}_{I}z),$$
(14)
$$u=\left[1+\frac{1-{(1+\alpha )}^{1/2}}{1+{(1+\alpha )}^{1/2}}exp(-2\mathit{\text{ik}}cos{\theta}_{I}z)\right]\times exp(\mathit{\text{ik}}sin{\theta}_{I}x+\mathit{\text{ik}}cos{\theta}_{I}z).$$
(15)
$$\begin{array}{ll}{u}^{(R)}\hfill & =[1-\frac{\alpha}{4}exp(-2\mathit{\text{ik}}cos{\theta}_{I}z)+\frac{{\alpha}^{2}}{32}\hfill \\ \hfill & \times exp(-4\mathit{\text{ik}}cos{\theta}_{I}z)+\dots ]{u}_{I}\hfill \\ \hfill & ={u}_{I}-\frac{\alpha}{4}exp(\mathit{\text{ik}}sin{\theta}_{I}x-\mathit{\text{ik}}cos{\theta}_{I}z)+\frac{{\alpha}^{2}}{32}\hfill \\ \hfill & \times exp(\mathit{\text{ik}}sin{\theta}_{I}x-3\mathit{\text{ik}}cos{\theta}_{I}z)+\dots ,\hfill \end{array}$$
(16)
$${u}_{T}=\frac{2}{1+{(1+\alpha )}^{1/2}}exp[\mathit{\text{ik}}sin{\theta}_{I}x+\mathit{\text{ik}}{(1+\alpha )}^{1/2}cos{\theta}_{I}z].$$
(17)
$${{u}_{T}}^{(B)}=\left[1-\frac{\alpha}{4}(1-2\mathit{\text{ik}}cos{\theta}_{I}z)\right]\times exp(\mathit{\text{ik}}sin{\theta}_{I}x+\mathit{\text{ik}}cos{\theta}_{I}z).$$
(18)
$${{u}_{T}}^{(R)}={e}^{-\alpha /4}exp[\mathit{\text{ik}}sin{\theta}_{I}x+\mathit{\text{ik}}(1+\alpha /2)cos{\theta}_{I}z],$$
(19)
$$k{({n}^{2}+{\alpha}^{2}{cos}^{2}\hspace{0.17em}{\theta}_{I}/4)}^{1/2}.$$
(20)
$${n}^{(R)}={({n}^{2}+{\alpha}^{2}{cos}^{2}\hspace{0.17em}{\theta}_{I}/4)}^{1/2}$$
(21)
$$k{(1+\alpha )}^{1/2}cos{\theta}_{I}z-k\left(1+\frac{\alpha}{2}\right)cos{\theta}_{I}z=-k\frac{{\alpha}^{2}}{8}cos{\theta}_{I}z+\dots $$
(22)
$$|{(1+\mathrm{\Delta})}^{-2}-1|<|{(1-\mathrm{\Delta})}^{-2}-1|.$$
(23)
$$u(x,z)=f(z)exp(\mathit{\text{ik}}sin{\theta}_{I}x),$$
(24)
$${\nabla}^{2}u+{k}^{2}{n}^{2}(x)=0,$$
(25)
$$\frac{{\text{d}}^{2}f}{\text{d}{z}^{2}}+{{k}_{z}}^{2}[1+V(z)]f=0.$$
(26)
$${k}_{z}=kcos{\theta}_{I}$$
(27)
$$V(z)=\{\begin{array}{ll}0,\hfill & z<0\hfill \\ \frac{{n}^{2}-1}{{cos}^{2}\hspace{0.17em}{\theta}_{I}}\equiv \alpha ,\hfill & z>0\hfill \end{array},$$
(28)
$${f}^{(B)}=\text{\u2211}_{m=0}^{\infty}{\u220a}^{m}{f}_{m},$$
(29)
$${f}^{(R)}=exp\left(\text{\u2211}_{m=0}^{\infty}{\u220a}^{m}{\psi}_{m}\right).$$
(30)
$${{f}_{1}}^{(B)}={f}_{0}+\u220a{f}_{1}$$
(31)
$${{f}_{1}}^{(R)}=exp({\psi}_{0}+\u220a{\psi}_{1}).$$
(32)
$$\begin{array}{cc}{{f}_{1}}^{(R)}& ={f}_{0}exp(\u220a{f}_{1}/{f}_{0})\hfill \\ & ={f}_{0}exp[{{f}_{s}}^{(B)}/{f}_{0}],\end{array}$$
(33)
$${f}_{0}=exp(\mathit{\text{ik}}cos{\theta}_{I}z)={e}^{i{k}_{z}z}$$
(34)
$${{f}_{s}}^{(B)}(z)=\frac{i{k}_{z}\alpha}{2}{\mathit{\int}}_{0}^{\infty}\text{d}{z}^{\prime}exp(i{k}_{z}|z-{z}^{\prime}|){e}^{i{k}_{z}{z}^{\prime}}.$$
(35)
$$\begin{array}{cc}{{f}_{s}}^{(B)}(z<0)& =\frac{i{k}_{z}\alpha}{2}{e}^{-i{k}_{z}z}{\mathit{\int}}_{0}^{\infty}\text{d}{z}^{\prime}{e}^{2i{k}_{z}{z}^{\prime}}.\\ & =-\frac{\alpha}{4}{e}^{-i{k}_{z}z},\end{array}$$
(36)
$$\begin{array}{ll}{\mathit{\int}}_{0}^{\infty}\text{d}{z}^{\prime}{e}^{2i{k}_{z}{z}^{\prime}}\hfill & \equiv \underset{\delta \to {0}^{+}}{lim}{\mathit{\int}}_{0}^{\infty}\text{d}{z}^{\prime}exp[2i({k}_{z}+i\delta ){z}^{\prime}]\hfill \\ \hfill & =-\frac{1}{2i{k}_{z}}.\hfill \end{array}$$
(37)
$${{f}_{s}}^{(B)}(z>0)=-\frac{\alpha}{4}(1-2i{k}_{z}z){e}^{i{k}_{z}z}.$$
(38)
$${{u}_{R}}^{(B)}=-\frac{\alpha}{4}exp(\mathit{\text{ik}}sin{\theta}_{I}x+\mathit{\text{ik}}cos{\theta}_{I}z).$$
(39)
$${{u}_{T}}^{(B)}=\left[1-\frac{\alpha}{4}(1-2\mathit{\text{ik}}cos{\theta}_{I}z)\right]exp(\mathit{\text{ik}}sin{\theta}_{I}x+\mathit{\text{ik}}cos{\theta}_{I}).$$
(40)
$${u}^{(R)}=exp\left[-\frac{\alpha}{4}exp(-2\mathit{\text{ik}}cos{\theta}_{I}z)\right]\times exp(\mathit{\text{ik}}sin{\theta}_{I}x+\mathit{\text{ik}}cos{\theta}_{I}z),$$
(41)
$${{u}_{T}}^{(R)}={e}^{-\alpha /4}exp(\mathit{\text{ik}}sin{\theta}_{I}x+\mathit{\text{ik}}(1+\alpha /2)cos{\theta}_{I}z].$$