Analytical Fresnel laws for curved dielectric interfaces

We shall now see that the backbone structure of these equations is transferred to the curved interfaces. For the convex case, see [6] and the alternative derivation below in section 3, we find the convex Fresnel reflection amplitude r_cx (assuming n > 1 )

$$\begin{eqnarray}&&{r}_{\mathrm{cx}}=\displaystyle \frac{\cos \chi +{\rm{i}}{{ \mathcal F }}_{m}({ka})}{\cos \chi -{\rm{i}}{{ \mathcal F }}_{m}({ka})}.\end{eqnarray} \tag{ 3 }$$

Here, ka is the dimensionless wave number given as product of the wave number k in free space and the local radius of curvature a. The index m is related to the angle of incidence via m = nka sinχ = ka sin η¹ . Furthermore, ${{ \mathcal F }}_{m}$ is given for TM and TE polarization, respectively, as

$$\begin{eqnarray}&&{{ \mathcal F }}_{m}^{\mathrm{TM}}(z)=\displaystyle \frac{{H}_{m-1}^{1}(z)}{n\,{H}_{m}^{1}(z)}-\sin \chi ,\end{eqnarray} \tag{ 4 }$$

$$\begin{eqnarray}&&{{ \mathcal F }}_{m}^{\mathrm{TE}}(z)={n}^{2}{{ \mathcal F }}_{m}^{\mathrm{TM}}(z).\end{eqnarray} \tag{ 5 }$$

The result for the concave reflection amplitude r_cv is the central result of this paper, and has a very similar, noteworthy structure,

$$\begin{eqnarray}&&{r}_{\mathrm{cv}}=\displaystyle \frac{\cos \chi -{\rm{i}}{{ \mathcal F }}_{m}^{* }({ka})}{\cos \chi +{\rm{i}}{{ \mathcal F }}_{m}^{* }({ka})}\end{eqnarray} \tag{ 6 }$$

with the complex conjugation ()* and ${{ \mathcal F }}_{m}$ as given in (4) and (5). It implies that the reflected intensity $R=| {r}^{2}| ={{rr}}^{* }$ is the same at a convex and concave interface, respectively. Note, however, that reversal of the light path and the accompanying change from a concave to a convex interface boundary requires to renormalize the wavenumber by n, and the discussion of convex–concave duality below.

The results are illustrated in figures 2 and 3 for TM and TE polarized light, respectively. The deviation from the planar case (black curve) is clearly visible and characterized by a much later onset of the regime of total internal reflection (i.e. for angles χ larger than the critical angle χ_cr). This effect is more pronounced for smaller wavenumbers ka (higher curvature). The planar case result is approached in the limit ${ka}\to \infty$.

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The reduced total internal reflection at curved boundaries implies a deterioration of the cavity quality (Q) factor and is thus important for many applications. We also point out that the drop of the reflectivity in TE polarization at the Brewster angle is less pronounced at all curved refractive index boundaries and does, in contrast to the planar case, not (quite) reach zero [5, 12].

Having considered the important case of optical microcavities where n > 1, we now generalize the Fresnel equations for curved boundaries to relative refractive indices n < 1 and find

$$\begin{eqnarray}&&{\tilde{r}}_{\mathrm{cx}}=-\displaystyle \frac{\cos \eta +{\rm{i}}{{ \mathcal G }}_{m}^{* }({ka})}{\cos \eta +{\rm{i}}{{ \mathcal G }}_{m}({ka})},\end{eqnarray} \tag{ 7 }$$

$$\begin{eqnarray}&&{\tilde{r}}_{\mathrm{cv}}=-\displaystyle \frac{\cos \eta -{\rm{i}}{{ \mathcal G }}_{m}({ka})}{\cos \eta -{\rm{i}}{{ \mathcal G }}_{m}^{* }({ka})}\end{eqnarray} \tag{ 8 }$$

with

$$\begin{eqnarray}&&{{ \mathcal G }}_{m}^{\mathrm{TM}}(z)=\displaystyle \frac{n\,{H}_{m-1}^{2}(z)}{{H}_{m}^{2}(z)}-\sin \eta ,\end{eqnarray} \tag{ 9 }$$

$$\begin{eqnarray}&&{{ \mathcal G }}_{m}^{\mathrm{TE}}(z)={{ \mathcal G }}_{m}^{\mathrm{TM}}(z)/{n}^{2}.\end{eqnarray} \tag{ 10 }$$

The results are shown in figures 4 and 5. As before, the reflectivity remains finite around the Brewster angle. The most striking difference to the planar case is the rather low reflectivity near grazing incidence at curved interfaces, confirming their larger leakage that we already observed for n > 1.

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The principle of ray-path reversal as well as the transfer matrix approach outlined below suggest to consider a convex interface together with its concave counterpart as two possible deviations from the planar case for a given n. This implies, however, a renormalization of the reference wavenumber ka in the reversed situation by a factor 1/n. These results are included in figures 2 and 3, where the renormalization increases the effect of curvature (the factor n can be captured in a decrease from a to a/n, or alternatively, in an increase from λ to nλ) and the curves are further away from the ray limit ${ka}\to \infty$. For n < 1, the same reasoning yields the opposite behavior; see figures 4 and 5.

To this end, we point out a symmetry relation between the convex and concave reflectance for a given order of the Hankel function m: $| {r}_{\mathrm{cx}}(n\to {n}_{2}){| }^{2}=| {r}_{\mathrm{cv}}(n\to {n}_{2}){| }^{2}$, i.e. both coincide and deviate in the same manner from the planar case result. Note, however, that r_cx and r_cv differ in a phase such that ${r}_{\mathrm{cx}}(n\to {n}_{2})={r}_{\mathrm{cv}}^{* }(n\to {n}_{2})$. Deviations from this symmetry may occur when light beams that consist of numerous single rays are considered [11].

The transfer matrix relates incoming and outgoing wave amplitudes at (dielectric) interfaces [16]; see figure 6. Here, A₀, B₀ and A₁, B₁ are the amplitudes of the incoming and outgoing waves, respectively, being related by the transfer matrix M as

$$\begin{eqnarray}&&\left(\begin{array}{c}{A}_{1}\\ {B}_{1}\end{array}\right)=M\left(\begin{array}{c}{A}_{0}\\ {B}_{0}\end{array}\right),\end{eqnarray} \tag{ 11 }$$

where

$$\begin{eqnarray}M=\left(\begin{array}{cc}a & b\\ {b}^{* } & {a}^{* }\end{array}\right).\end{eqnarray} \tag{ 12 }$$

Due to the presence of time-reversal symmetry, the 2 × 2 matrix M takes the preceding form, where a and b are complex numbers. The relation to Fresnel coefficients is achieved when writing each outgoing amplitude in terms of a reflected and a transmitted contribution, see figure 6, that are related by

$$\begin{eqnarray}&&{A}_{1}=r^{\prime} {B}_{1}+{{tA}}_{0},\end{eqnarray} \tag{ 13 }$$

$$\begin{eqnarray}&&{B}_{0}=t^{\prime} {B}_{1}+{{rA}}_{0}.\end{eqnarray} \tag{ 14 }$$

Here, r and t (r' and t') are the inner (outer) Fresnel reflection and transmission coefficients, respectively. This yields the following representation of M that holds independent of the curvature of the interface

$$\begin{eqnarray}M=\left(\begin{array}{cc}1/t{{\prime} }^{* } & r^{\prime} /t^{\prime} \\ r{{\prime} }^{* }/t{{\prime} }^{* } & 1/t^{\prime} .\end{array}\right).\end{eqnarray} \tag{ 15 }$$

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In the following, we will use the electromagnetic wave functions ψ₀, ψ₁ on either side of the interface (see figure 6) which are nothing else but the z-component of the electric (magnetic) field E_z (H_z) in the TM (TE) case.

Whereas in the planar case, the ψ's are plane waves and a plane divides space into two half planes [1], a circle/cylinder takes this role in the presence of curvature. Consequently, we use Hankel functions as incoming and outgoing waves ψ because they accommodate the cylindrical symmetry that we assume (locally) for a curved interface.

The derivation of Fresnel coefficients at convex interfaces [6] was based on finding the resonances in a disk cavity, and relating their real and imaginary part to the Fresnel reflection coefficient. Here, we use a conceptually different approach that is applicable to concave interfaces as well. To this end, we extend the transfer matrix approach to a Fabry–Perot-type situation with multiple reflections and mimic this behavior by including a perfect mirror, placed at x, r = 0. In the case of a planar interface, the plane-wave ansatz for ψ₀ and ψ₁ reads

$$\begin{eqnarray}&&{\psi }_{0}(x)=\displaystyle \frac{I}{2}\left({{\rm{e}}}^{{\rm{i}}{n}_{x}{k}_{x}x}+{{\rm{e}}}^{-{\rm{i}}{n}_{x}{k}_{x}x}\right)=I\cos ({n}_{x}{k}_{x}x),\end{eqnarray} \tag{ 16a }$$

$$\begin{eqnarray}&&{\psi }_{1}(x)={{\rm{e}}}^{-{\rm{i}}{k}_{x}x}+S{{\rm{e}}}^{{\rm{i}}{k}_{x}x}.\end{eqnarray} \tag{ 16b }$$

The factor I describes (twice) the wave amplitude inside the resonator and S is the transmitted (scattered) amplitude leaving the system. The effective refractive index n_x is given as ${n}_{x}=\tfrac{\tan {\chi }^{{\prime} }}{\tan \chi }$ where χ and χ' are defined in figure 6.

Following the procedure outlined in section 3.1 above, the new transfer matrix M' at the position x = a_p reads

$$\begin{eqnarray}M^{\prime} =\displaystyle \frac{1}{t^{\prime} }\left(\begin{array}{cc}{{\rm{e}}}^{{\rm{i}}({n}_{x}-1){k}_{x}{a}_{p}} & r^{\prime} {{\rm{e}}}^{-{\rm{i}}({n}_{x}+1){k}_{x}{a}_{p}}\\ r^{\prime} {{\rm{e}}}^{{\rm{i}}({n}_{x}+1){k}_{x}{a}_{p}} & {{\rm{e}}}^{-{\rm{i}}({n}_{x}-1){k}_{x}{a}_{p}}\end{array}\right).\end{eqnarray} \tag{ 17 }$$

The analogy to the single-transmission case is established by introducing resonance-dressed Fresnel coefficients ${r}_{r},{r}_{r}^{{\prime} }$ and ${t}_{r},{t}_{r}^{{\prime} }$, characterized by the presence of an additional phase, namely

$$\begin{eqnarray}&&{r}_{r}=r{{\rm{e}}}^{2{\rm{i}}{n}_{x}{k}_{x}{a}_{p}}{r}_{r}^{\prime} =r^{\prime} {{\rm{e}}}^{-2{\rm{i}}{k}_{x}{a}_{p}},\end{eqnarray} \tag{ 18 }$$

$$\begin{eqnarray}&&{t}_{r}=t{{\rm{e}}}^{{\rm{i}}({n}_{x}-1){k}_{x}{a}_{p}}\quad {t}_{r}^{\prime} =t^{\prime} {{\rm{e}}}^{{\rm{i}}({n}_{x}-1){k}_{x}{a}_{p}}.\end{eqnarray} \tag{ 19 }$$

A straightforward explanation of the resonance formation is gained when expressing the amplitude I in terms of a geometric series of resonance-dressed Fresnel coefficients, nicely illustrating the successive reflections of light rays

$$\begin{eqnarray}&&\displaystyle \frac{I}{2}=\displaystyle \frac{{t}_{r}^{{\prime} }}{1-{r}_{r}}={t}_{r}^{{\prime} }+{t}_{r}^{{\prime} }{r}_{r}+{t}_{r}^{{\prime} }{r}_{r}^{2}+{t}_{r}^{{\prime} }{r}_{r}^{3}+\cdots .\end{eqnarray} \tag{ 20 }$$

Note that the wave numbers k at resonance can be obtained by solving equation (1) − r_r = 0 and that $| I/2{| }^{2}$ will oscillate with its maxima reached at resonant wave numbers k.

We adapt the ansatz for the planar cavity (see (16a) and (16b)) to the rotationally invariant case relevant for curved interfaces:

$$\begin{eqnarray}&&{\psi }_{0}(r,\varphi )=\displaystyle \frac{I}{2}\left({H}_{m}^{1}({nkr})+{H}_{m}^{2}({nkr})\right)={{IJ}}_{m}({nkr}){e}^{{im}\varphi },\end{eqnarray} \tag{ 21a }$$

$$\begin{eqnarray}&&{\psi }_{1}(r,\varphi )=({H}_{m}^{2}({kr})+{{SH}}_{m}^{1}({kr})){e}^{{im}\varphi }.\end{eqnarray} \tag{ 21b }$$

Note that we switch from this convex case to the concave situation by exchanging incoming and outgoing Hankel functions (i.e. indices 1 and 2) while properly accounting for the scaling factor n, whereas the distinction between TM and TE polarized light originates in the well-known difference in the transition conditions [1]:

$$\begin{eqnarray}&&\psi ({a}_{p}-0)=\psi ({a}_{p}+0),\end{eqnarray} \tag{ 22a }$$

$$\begin{eqnarray}\psi ^{\prime} ({a}_{p}-0)=\left\{\begin{array}{ll}\psi ^{\prime} ({a}_{p}+0) & \mathrm{TM}\\ {n}^{2}\psi ^{\prime} ({a}_{p}+0) & \mathrm{TE}.\end{array}\right.\end{eqnarray} \tag{ 22b }$$

The transfer matrix takes then the following form

$$\begin{eqnarray}{M}^{({\rm{TM}})}=\displaystyle \frac{1}{D}\left(\begin{array}{cc}{D}_{12} & {D}_{22}\\ -{D}_{11} & -{D}_{21}\end{array}\right),\end{eqnarray} \tag{ 23 }$$

$$\begin{eqnarray}{M}^{({\rm{TE}})}={M}^{({\rm{TM}})}+\,\displaystyle \frac{\sin \chi }{D}\left(n-\displaystyle \frac{1}{n}\right)\left(\begin{array}{cc}-{Q}_{12} & -{Q}_{22}\\ {Q}_{11} & {Q}_{21}\end{array}\right)\end{eqnarray} \tag{ 24 }$$

with (realizing that a_p takes the role of a, and {α, β} ∈{1, 2})

$$\begin{eqnarray}&&{D}_{\alpha \beta }={H}_{m}^{\alpha }({nka}){H}_{m-1}^{\beta }({ka})-\,{{nH}}_{m-1}^{\alpha }({nka}){H}_{m}^{\beta }({ka}),\end{eqnarray} \tag{ 25 }$$

$$\begin{eqnarray}&&D=4/({\rm{i}}\pi {ka}),\end{eqnarray} \tag{ 26 }$$

$$\begin{eqnarray}&&{Q}_{\alpha \beta }={H}_{m}^{\alpha }({nka}){H}_{m}^{\beta }({ka}).\end{eqnarray} \tag{ 27 }$$

By analyzing the resonance-dressed (or multiple) reflection coefficients obtained from (23) and (24), we find the following additional phases in the Fresnel reflection coefficients (see (18) and (19) for the planar case):

$$\begin{eqnarray}&&\mathrm{convex}:\ \ \ \displaystyle \frac{{H}_{m}^{1}({nka})}{{H}_{m}^{2}({nka})},\quad \mathrm{concave}:\ \ \ \displaystyle \frac{{H}_{m}^{2}({ka})}{{H}_{m}^{1}({ka})}.\end{eqnarray} \tag{ 28 }$$

The resulting reflection coefficient at a convex interface reads thus

$$\begin{eqnarray}&&{r}^{({\rm{TM}})}=-\displaystyle \frac{\tfrac{{H}_{m-1}^{1}({ka})}{{H}_{m}^{1}({ka})}-n\tfrac{{H}_{m-1}^{1}({nka})}{{H}_{m}^{1}({nka})}}{\tfrac{{H}_{m-1}^{1}({ka})}{{H}_{m}^{1}({ka})}-n\tfrac{{H}_{m-1}^{2}({nka})}{{H}_{m}^{2}({nka})}}.\end{eqnarray} \tag{ 29 }$$

This coefficient will not oscillate when changing the argument nka. It is therefore suitable to describe open segment boundaries, in consistency with allowing non-integer m = nka sin χ (the integer-m constraint applies formally only in the presence of rotational symmetry).

We proceed by simplifying the expressions using the large-argument approximation

$$\begin{eqnarray}&&\displaystyle \frac{{H}_{m-1}^{1}({nka})}{{H}_{m}^{1}({nka})}\approx {{\rm{e}}}^{-{\rm{i}}\left(\chi -\tfrac{\pi }{2}\right)}=\sin \chi +\mathrm{icos}\chi ,\end{eqnarray} \tag{ 30 }$$

where χ = arcsin (m/nka) as before. This relation can be derived using a somewhat less-common approximation for the Hankel function [17] that holds when m and the argument nka of the Hankel function are of similar order as in the present case, namely

$$\begin{eqnarray}&&{H}_{m}^{1}(z)=\displaystyle \frac{\sqrt{2/\pi }}{\sqrt{z\cos \chi }}{{\rm{e}}}^{{\rm{i}}{\varphi }_{m}(z)}\left(1-\displaystyle \frac{{b}_{1}(z)}{z\cos \chi }+O({z}^{-2})\right),\end{eqnarray} \tag{ 31 }$$

where

$$\begin{eqnarray}&&{\varphi }_{m}(z)=z\cos \chi -m\left(\displaystyle \frac{\pi }{2}-\chi \right)-\displaystyle \frac{\pi }{4}\end{eqnarray} \tag{ 32 }$$

and

$$\begin{eqnarray}&&{b}_{1}(z)=\displaystyle \frac{1}{8}+\displaystyle \frac{5}{24}{\tan }^{2}\chi .\end{eqnarray} \tag{ 33 }$$

Equation (30) follows in the limit of large m and nka, and eventually we find indeed the results stated in (3) and (7).

In analogy, we find for the concave case

$$\begin{eqnarray}&&{r}^{{\prime} ({\rm{TM}})}=-\displaystyle \frac{\tfrac{{H}_{m-1}^{-}({ka})}{{H}_{m}^{-}({ka})}-n\tfrac{{H}_{m-1}^{-}({nka})}{{H}_{m}^{-}({nka})}}{\tfrac{{H}_{m-1}^{+}({ka})}{{H}_{m}^{+}({ka})}-n\tfrac{{H}_{m-1}^{-}({nka})}{{H}_{m}^{-}({nka})}}.\end{eqnarray} \tag{ 34 }$$

We proceed by simplifying the expressions using the previously introduced approximation for the argument ka being in the order of m, and η = arcsin (m/ka),

$$\begin{eqnarray}&&\displaystyle \frac{{H}_{m-1}^{1}({ka})}{{H}_{m}^{1}({ka})}\approx {{\rm{e}}}^{-{\rm{i}}\left(\eta -\tfrac{\pi }{2}\right)}=\sin \eta +\mathrm{icos}\,\eta ,\end{eqnarray} \tag{ 34a }$$

$$\begin{eqnarray}&&\displaystyle \frac{{H}_{m-1}^{2}({ka})}{{H}_{m}^{2}({ka})}\approx {{\rm{e}}}^{+{\rm{i}}\left(\eta -\tfrac{\pi }{2}\right)}=\sin \eta -\mathrm{icos}\,\eta .\end{eqnarray} \tag{ 34b }$$

From this, (6) and (8) follow. For the derivation of the TE-case results in section 2, we proceed as before but use the appropriate transfer matrix (24).

To summarize, we have completed the picture of Fresnel coefficients at generic curved interfaces by deriving a formulae for the concave case in addition to the previously known convex case coefficients, and by illustrating the power of the transfer matrix approach to this problem.