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Hello all,

Apologies in advance for the text-wall; this is a rather involved question.

I am trying to compute the effective transmission coefficient for a medium of non-uniform refractive index. For simplicity I am assuming the slab has thickness ##d##, that ##n(0)=1##, and that ##n(d)=n## (where ##n## is a constant).

I am familiar with the normal incidence reflection and transmission coefficients, ##R=(n_1-n_2)/(n_1+n_2)##, for light incident from a region of index ##n_1## upon a region of index ##n_2##. I also know how to derive these coefficients from the continuity and smoothness conditions of the electric and magnetic fields at the boundary.

One can equally well consider the problem of a wave on a string of nonuniform density. My question is largely of a mathematical nature.

At first, there does not appear to be a clear problem with applying these same conditions to "infinitesimal discontinuities," i.e. ##n(x)## to the left of the boundary and ##n(x)+dn(x)=n(x+dx)## to the right of the boundary. Now our reflection coefficient ##dR(x)## is infinitesimal, and the transmission coefficient ##T(x)=1+dR(x)##.

I'm aware that the principles leading to the derivation of the wave equation do make assumptions about the uniformity of the medium - this is most clearly illustrated by the wave-on-a-string example. Assuming a non-uniform string breaks the derivation. However, the fact of the matter is an answer exists, and my current approach (of the three I've tried) seems to be the most sensible.

I treat the medium like an infinite sequence of media with "infinitesimal discontinuities" that have infinitesimal reflection coefficients, and transmission coefficients which differ from unity by an infinitesimal.

Much like integrals are the limits of sums of terms whose values tend to zero as the number of terms tends to infinity, what I have for the effective transmission coefficient (assuming no back-and-forth reflections) is a product of factors whose values tend to unity as the number of factors tends to infinity. (This is akin to an exponential of an integral of logarithms.)

This leads to an effective transmission coefficient of ##1/\sqrt{n}## for the non-uniform medium, times ##2n/(n+1)## for the final transmission.

My question is about corrections to this. If the incident wave is reflected back at ##x_1##, then reflected forward at ##x_2##, there will be a doubly-infinitesimal contribution. We can choose any pair of coordinates such that ##0<x_1<d## and ##0<x_2<x_1##, meaning we will have to integrate over a two-dimensional region. This gives a finite and non-infinitesimal correction to the above.

Hence my current predicament. Here are the approaches I've attempted, in decreasing order of preference:

1. Using a "hopscotch"-style boundary conditions approach, I can generalize the relations ##E_t(d+x)=TE_i(d+(n_t/n_i)x)## and ##E_r(d-x)=RE_i(d+x)##. This gives me some nasty recursion relations in the general case, that simplify in the event of infinitesimal layers. However, they simplify to the same expression used to get the na\"ive estimate above, without considering multiple reflections. I would like to use this method, if possible, but I would greatly appreciate some help with figuring out how to get the NLO contribution.

2. Computing the multiple integrals analytically is impossible except for a very specific parametrization of the refractive index, ##n(x)=(1-x/L)^{-1}## for some choice of length scale ##L##. Even for this choice of refractive index, this method requires integration by parts and leads to a dynamic mess at every order, and while I most trust the results it gives me, I am least confident in my ability to generalize to the correction for ##2n## reflections.

3. Using the BCs that are used to derive the reflection and transmission coefficients in the first place, I get equations that don't even agree with my verified results for a uniform slab of thickness ##d## (for which ##T_\mathrm{eff}=T_i^*T_i(1-2R_0^2\cos(nkd)+R_0^4)^{-\frac{1}{2}}##, where ##R_0## is the reflection coefficient at the interface, ##n## the refractive index of the slab, and ##k## the wavenumber of the incident light). Moreover, when I attempt to use this method to compute infinitely many contributions, all I get is a telescoping product, the result of which is ##T=1##. This method would appear to require the least work, but seeing as it does not reproduce either transmission coefficient above, I am reticent to trust it.

I don't see another way forward at this point. Could anybody please look over the work I have done? I really just want to know: Whence comes the NLO contribution?

Thanks,

QM

Apologies in advance for the text-wall; this is a rather involved question.

I am trying to compute the effective transmission coefficient for a medium of non-uniform refractive index. For simplicity I am assuming the slab has thickness ##d##, that ##n(0)=1##, and that ##n(d)=n## (where ##n## is a constant).

I am familiar with the normal incidence reflection and transmission coefficients, ##R=(n_1-n_2)/(n_1+n_2)##, for light incident from a region of index ##n_1## upon a region of index ##n_2##. I also know how to derive these coefficients from the continuity and smoothness conditions of the electric and magnetic fields at the boundary.

One can equally well consider the problem of a wave on a string of nonuniform density. My question is largely of a mathematical nature.

At first, there does not appear to be a clear problem with applying these same conditions to "infinitesimal discontinuities," i.e. ##n(x)## to the left of the boundary and ##n(x)+dn(x)=n(x+dx)## to the right of the boundary. Now our reflection coefficient ##dR(x)## is infinitesimal, and the transmission coefficient ##T(x)=1+dR(x)##.

I'm aware that the principles leading to the derivation of the wave equation do make assumptions about the uniformity of the medium - this is most clearly illustrated by the wave-on-a-string example. Assuming a non-uniform string breaks the derivation. However, the fact of the matter is an answer exists, and my current approach (of the three I've tried) seems to be the most sensible.

I treat the medium like an infinite sequence of media with "infinitesimal discontinuities" that have infinitesimal reflection coefficients, and transmission coefficients which differ from unity by an infinitesimal.

Much like integrals are the limits of sums of terms whose values tend to zero as the number of terms tends to infinity, what I have for the effective transmission coefficient (assuming no back-and-forth reflections) is a product of factors whose values tend to unity as the number of factors tends to infinity. (This is akin to an exponential of an integral of logarithms.)

This leads to an effective transmission coefficient of ##1/\sqrt{n}## for the non-uniform medium, times ##2n/(n+1)## for the final transmission.

My question is about corrections to this. If the incident wave is reflected back at ##x_1##, then reflected forward at ##x_2##, there will be a doubly-infinitesimal contribution. We can choose any pair of coordinates such that ##0<x_1<d## and ##0<x_2<x_1##, meaning we will have to integrate over a two-dimensional region. This gives a finite and non-infinitesimal correction to the above.

Hence my current predicament. Here are the approaches I've attempted, in decreasing order of preference:

1. Using a "hopscotch"-style boundary conditions approach, I can generalize the relations ##E_t(d+x)=TE_i(d+(n_t/n_i)x)## and ##E_r(d-x)=RE_i(d+x)##. This gives me some nasty recursion relations in the general case, that simplify in the event of infinitesimal layers. However, they simplify to the same expression used to get the na\"ive estimate above, without considering multiple reflections. I would like to use this method, if possible, but I would greatly appreciate some help with figuring out how to get the NLO contribution.

2. Computing the multiple integrals analytically is impossible except for a very specific parametrization of the refractive index, ##n(x)=(1-x/L)^{-1}## for some choice of length scale ##L##. Even for this choice of refractive index, this method requires integration by parts and leads to a dynamic mess at every order, and while I most trust the results it gives me, I am least confident in my ability to generalize to the correction for ##2n## reflections.

3. Using the BCs that are used to derive the reflection and transmission coefficients in the first place, I get equations that don't even agree with my verified results for a uniform slab of thickness ##d## (for which ##T_\mathrm{eff}=T_i^*T_i(1-2R_0^2\cos(nkd)+R_0^4)^{-\frac{1}{2}}##, where ##R_0## is the reflection coefficient at the interface, ##n## the refractive index of the slab, and ##k## the wavenumber of the incident light). Moreover, when I attempt to use this method to compute infinitely many contributions, all I get is a telescoping product, the result of which is ##T=1##. This method would appear to require the least work, but seeing as it does not reproduce either transmission coefficient above, I am reticent to trust it.

I don't see another way forward at this point. Could anybody please look over the work I have done? I really just want to know: Whence comes the NLO contribution?

Thanks,

QM

**[Post edited by the Mentors at the request of the OP]**
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