## Total internal reflection and evanescent gain |

Optics Express, Vol. 19, Issue 22, pp. 21404-21418 (2011)

http://dx.doi.org/10.1364/OE.19.021404

Acrobat PDF (1146 KB)

### Abstract

Total internal reflection occurs for large angles of incidence, when light is incident from a high-refractive-index medium onto a low-index medium. We consider the situation where the low-index medium is active. By invoking causality in its most fundamental form, we argue that evanescent gain may or may not appear, depending on the analytic and global properties of the permittivity function. For conventional, weak gain media, we show that there is an absolute instability associated with infinite transverse dimensions. This instability can be ignored or eliminated in certain cases, for which evanescent gain prevails.

© 2011 OSA

## 1. Introduction

11. A. Siegman, “Fresnel reflection, Lenserf reflection, and evanescent gain,” Opt. Photon. News **21**, 38–45 (2010). [CrossRef]

12. C. J. Koester, “Laser action by enhanced total internal reflection,” IEEE J. Quantum Electron. **2**, 580–584 (1966). [CrossRef]

15. M. P. Silverman and J. R. F. Cybulski, “Investigation of light amplification by enhanced internal reflection. Part II. Experimental determination of the single-pass reflectance of an optically pumped gain region,” J. Opt. Soc. Am. **73**, 1739–1743 (1983). [CrossRef]

11. A. Siegman, “Fresnel reflection, Lenserf reflection, and evanescent gain,” Opt. Photon. News **21**, 38–45 (2010). [CrossRef]

16. B. Nistad and J. Skaar, “Causality and electromagnetic properties of active media,” Phys. Rev. E **78**, 036603 (2008). [CrossRef]

*t*< 0, the solution to Maxwell’s equations for a semi-infinite gain medium equals that of a finite slab for times

*t*less than

*d*/

*c*, where

*d*is the slab thickness and

*c*is the vacuum velocity of light. Hence, understanding semi-infinite media helps explaining transient phenomena.

*iωt*). With respect to Fig. 1 we define the transverse wavenumber (spatial frequency of the source)

*k*. For simplicity we assume both media to be nonmagnetic. Let

_{x}*ɛ*

_{1}and

*ɛ*

_{2}be the relative permittivities of the high-index medium to the left and the low-index medium to the right, respectively. For plane waves, Maxwell’s equations require the longitudinal wavenumbers in the high-index and low-index media to be At some observation frequency

*ω*=

*ω*

_{1}, we assume

*ɛ*

_{2}< 0 and |Im

*ɛ*

_{2}| ≪ 1 (i.e., small gain). The correct sign for the square root in Eq. (1b) is far from obvious: Either Im

*k*

_{2z}> 0 and Re

*k*

_{2z}< 0, or Im

*k*

_{2z}< 0 and Re

*k*

_{2z}> 0, see Fig. 2. None of these solutions are appealing: The first requires the phase velocity and Poynting vector to point

*towards*the boundary. Since there are no sources at

*z*= ∞, one may argue that this scenario cannot be true [11

11. A. Siegman, “Fresnel reflection, Lenserf reflection, and evanescent gain,” Opt. Photon. News **21**, 38–45 (2010). [CrossRef]

9. J. Fan, A. Dogariu, and L. J. Wang, “Amplified total internal reflection,” Opt. Express **11**, 299–308 (2003). [CrossRef] [PubMed]

## 2. Laplace transform frequency-domain analysis

*t*< 0 (see Appendix B), we obtain the causal solution to Maxwell’s equations. The complex frequency-domain fields are usually found from the time-domain fields by a Fourier transform. However, when there is gain in the system, using the Fourier transform can be perilous, since the field may increase with time. At first sight, any instability seems to be convective in our case. This is however not true: A causal excitation involves an infinite band of frequencies. For a single spatial frequency

*k*this means that modes with a wide range of incident angles are involved; in fact even the mode with

_{x}*k*

_{2z}= 0 may be excited. This “side wave” gets amplified and leads to infinite fields at the boundary. This instability is somewhat artificial, since its existence is dependent on infiniteness in the transverse direction; we will argue below how it can be ignored in certain situations. Nevertheless, within a linear medium framework, Fourier transforms do not necessarily exist. Therefore, as in electronics and control engineering, we generalize the analysis by using the Laplace transform, In Eq. (2) a sufficiently large value of Im

*ω*will quench an exponential increase in the time-domain electric field ℰ (

*t*), such that the integral converges. (Note that

*ω*is complex in general, equal to

*is*, where

*s*is the conventional Laplace variable.) The inverse transform is given by The integral is taken along the line

*ω*=

*iγ*, for a sufficiently large, real parameter

*γ*, above all non-analytic points of

*E*(

*ω*) in the complex

*ω*-plane. An important observation is the following: The frequency-domain field

*E*(

*ω*) only has physical meaning through the transforms (2)–(3). Thus, if the field is to be interpreted for all real frequencies, it must be analytic in the upper half-plane Im

*ω*> 0. However, as is shown below, if the non-analytic points are located in the upper half-plane, but close to the real axis and far away from the excitation frequency, we can still attribute a physical interpretation to the frequency-domain expressions.

*k*

_{2z}, it is tempting to start with the response from a slab of finite thickness

*d*, and then take the limit

*d*→ ∞. For finite

*d*the solution to Maxwell’s equations is independent of the sign of

*k*

_{2z}in the slab [17

17. J. Skaar, “Fresnel equations and the refractive index of active media,” Phys. Rev. E **73**, 026605 (2006). [CrossRef]

18. A. Lakhtakia, J. B. Geddes III, and T. G. Mackay, “When does the choice of the refractive index of a linear, homogeneous, isotropic, active, dielectric medium matter?” Opt. Express **15**, 17709–17714 (2007). [CrossRef] [PubMed]

*d*. Thus, for real frequencies, the limit

*d*→ ∞ is not necessarily meaningful [11

**21**, 38–45 (2010). [CrossRef]

17. J. Skaar, “Fresnel equations and the refractive index of active media,” Phys. Rev. E **73**, 026605 (2006). [CrossRef]

*ω*, where the frequency-domain fields exist. There, an exponential increase is quenched by the exponential factor exp(−Im

*ωt*). As a result, we can take the limit

*d*→ ∞ [16

16. B. Nistad and J. Skaar, “Causality and electromagnetic properties of active media,” Phys. Rev. E **78**, 036603 (2008). [CrossRef]

*ρ*and the transmission coefficient

*τ*(including the propagation factor exp(

*ik*

_{2z}

*z*)) become [16

16. B. Nistad and J. Skaar, “Causality and electromagnetic properties of active media,” Phys. Rev. E **78**, 036603 (2008). [CrossRef]

17. J. Skaar, “Fresnel equations and the refractive index of active media,” Phys. Rev. E **73**, 026605 (2006). [CrossRef]

*k*

_{2z}is determined such that

*k*

_{2z}→ +

*ω*/

*c*as Im

*ω*→ ∞, and

*k*

_{2z}is an analytic function of

*ω*. Indeed, even though Eqs. (4) have been derived for large Im

*ω*, we can extend their valid region as follows: The reflected and transmitted frequency-domain fields are given by Eqs. (4) multiplied by the Laplace-transformed incident field. The associated, physical, time-domain fields are obtained by the inverse transform (3). Now, by analytic continuation, we can reduce

*γ*until we reach a non-analytic point of Eqs. (4), without altering ℰ (

*t*). If the expressions (4) are analytic in the entire, upper half-plane, we can set

*γ*= 0 and interpret

*ρ*and

*τ*for real frequencies. On the other hand, if there are non-analytic points in the upper half-plane, the time-domain fields diverge. In that case, real frequencies are not physically meaningful in general.

## 3. Weak gain media

- The permittivity
*ɛ*_{2}(*ω*) obeys the Kramers–Kronig relations. - The gain and dispersion is small, so that the permittivity can be written Here
*ɛ*̄_{2}is required to be a positive constant. In the following we take*ɛ*̄_{2}= 1; the analysis can easily be generalized to the case with another*ɛ*̄_{2}. (In the latter case,*ɛ*̄_{2}is only constant in a wide frequency band including the band where Δ*ɛ*_{2}(*ω*) is nonzero; for very high frequencies it necessarily tends to 1.) - The medium is gainy at the observation frequency
*ω*_{1}and the critical frequency*k*._{x}c

*k*

_{2z}has branch points in the upper half-plane of the complex

*ω*plane. Since

*ɛ*

_{2}(

*ω*) satisfies the Kramers–Kronig relations, it is analytic in the upper half-plane. The maximum modulus principle of complex analysis [19] therefore ensures that property 2 is valid also in the upper half-plane, not only at the real frequency axis. Substituting

*ɛ*

_{2}(

*ω*) = 1 + Δ

*ɛ*

_{2}(

*ω*) into Eq. (7) we find in the upper half-plane, since |Δ

*ɛ*

_{2}(

*ω*)| ≪ 1. Thus, every solution to the dispersion relation in the upper half-plane is located within a distance (Δ

*ɛ*

_{max}/2)

*k*from the critical frequencies ±

_{x}c*k*.

_{x}c*k*in more detail. If there were two solutions

_{x}c*ω*and

_{a}*ω*to the dispersion relation, then Eq. (8) would predict that By property 4 this is impossible unless

_{b}*ω*=

_{b}*ω*. Thus there is a unique solution to Eq. (7) in the first quadrant, located in the vicinity of

_{a}*k*: In addition there is a mirrored solution in the second quadrant, located at

_{x}c*ω*= −

*k*–

_{x}c*δ*′ +

*iδ*. Note that

*k*

_{2z}and the Fresnel coefficients (4), these solutions appear as branch points. Hence, when evaluating the physical time-domain fields by the inverse Laplace transform, we must integrate above the associated branch cuts, from −∞ +

*iγ*to +∞ +

*iγ*, see Fig. 3. By path deformation this path is the same as the path from −∞ to ∞ plus the paths around the branch cuts in the upper half-plane (Fig. 3). Thus we may use the inverse Fourier transform to determine the time-domain fields, but only if we add the integrals around the branch cuts. Due to the exponential factor exp(−

*iωt*), the integrals around the branch cuts diverge and dominate after some time.

*u*(

*t*)cos(

*ω*

_{1}

*t*), is

*ω*≠ 0. One of these frequencies is the branch-point frequency for which

*k*

_{2z}= 0, that is,

*ω*≈

*k*+

_{x}c*iδ*. This frequency is complex; the imaginary part

*δ*means that the associated eigenmode is a growing wave with envelope exp(

*δt*). Physically, a wave with

*k*

_{2z}= 0 propagates along the boundary. Because the medium is gainy, this side wave picks up gain on its way. Consider a fixed observation point, e.g. the point

*z*= 0

^{+}and

*x*= 0. Since the medium and the excitation are unbounded in the transverse

*x*-direction, there are side waves that start arbitrarily far away from the observation point. Thus the field at the observation point diverges. As the field in medium 2 becomes infinite, the field in medium 1 is infinite as well. Since the field at a fixed point in space diverges and the instability is not a result of amplified, multiple reflections, the instability for the system in Fig. 1 can be classified as an absolute instability [17

**73**, 026605 (2006). [CrossRef]

20. P. A. Sturrock, “Kinematics of growing waves,” Phys. Rev. **112**, 1488–1503 (1958). [CrossRef]

*x*-direction, leading to an infinite spectrum of

*k*modes (see Appendix A and Ref. [8

_{x}8. A. A. Kolokolov, “Fresnel formulas and the principle of causality,” Phys. Usp. **42**, 931–940 (1999). [CrossRef]

*ω*

_{1}is sufficiently remote from the branch points, the side wave with

*k*

_{2z}= 0 is only excited very weakly, and can be neglected up to a certain time. The condition that the excitation frequency is remote from

*k*means that the incident angle is not close to the critical angle. This condition is imperative in order to distinguish between the reflected wave, with an angle of reflection equal to the angle of incidence, and the wave associated with the growing side wave, with “reflection” (or propagation) angle equal to the critical angle.

_{x}c*u*(

*t*)exp(

*ik*–

_{x}x*iω*

_{1}

*t*), with Laplace transform exp(

*ik*)/(

_{x}x*iω*

_{1}–

*iω*), is given by at

*z*= 0. The integral (12) can be evaluated by a generalized version of the residue theorem, in which we find the contour integral around all poles and branch cuts of the integrand in half-plane Im

*ω*<

*γ*. Provided

*ω*

_{1}is sufficiently remote from any resonances of the two media, the transients due to all poles and branch cuts for Im

*ω*< 0 can be ignored. Alternatively, for times larger than the maximum inverse bandwidth Γ

^{−1}of the resonances, the transients will have died out. Then the reflected field for

*x*= 0 is given by where the wavenumbers

*k*

_{1z}and

*k*

_{2z}have been evaluated at the frequency

*ω*

_{1}. The term ℰ

_{bc}(0,

*t*) is the integral (12) around the two branch cuts above

*ω*= ±

*k*. This integral is bounded by Here, the constant depends on the specifics of the active medium (see Appendix C). In other words, for Γ

_{x}c^{−1}≲

*t*≲

*δ*

^{−1}and provided

*ω*

_{1}is not too close to

*k*, we can ignore ℰ

_{x}c_{bc}(0,

*t*). Then the reflected field is well described by the first term in Eq. (13).

*k*

_{2z}must be determined such that

*k*

_{2z}is analytic everywhere, except at the two branch cuts in the upper half-plane. Since

*k*

_{2z}→ +

*ω*/

*c*as

*ω*→ +∞, we can determine the sign by decreasing

*ω*from +∞ to

*ω*

_{1}, ensuring that

*k*

_{2z}is continuous everywhere except at

*ω*=

*k*where it changes sign. From Fig. 4 we find that Im

_{x}c*k*

_{2z}> 0 at the observation frequency

*ω*

_{1}. Hence, for weak conventional gain media, provided the “reflected” field from the side wave can be ignored, evanescent gain is possible. This result is consistent with [7,8

8. A. A. Kolokolov, “Fresnel formulas and the principle of causality,” Phys. Usp. **42**, 931–940 (1999). [CrossRef]

10. K. J. Willis, J. B. Schneider, and S. C. Hagness, “Amplified total internal reflection: theory, analysis, and demonstration of existence via FDTD,” Opt. Express **16**, 1903–1913 (2008). [CrossRef] [PubMed]

*ik*

_{1z}

*z*) in the integral. The transmitted field was computed with the same equation, but with

*τ*instead of

*ρ*exp(−

*ik*

_{1z}

*z*) in the integral (see Eq. (4)). For

*z*> 0 we clearly see an evanescent decaying field, while the reflected field for

*z*< 0 is larger than unity.

*ω*

_{1}. If we insist on using only the first term of Eq. (13) in this case, a simple calculation shows that the power reflectance would have been bounded by

*k*

_{2z}and the reflected field would be discontinuous as we pass the critical angle. This is clearly a paradox, as the branch cuts were chosen arbitrarily. The dilemma is resolved by noting that the entire Eq. (13) must be used in this domain; both terms naturally coexist and cannot be separated. As we approach the critical angle, ℰ

_{bc}(0,

*t*) becomes comparable to or larger than the first term in Eq. (13), for all times. A different choice of branch cuts will alter each of these contributions, but the sum remains the same. For finite transverse dimension, the side wave’s contribution to the “reflected” field does not necessarily diverge any more; however, the intensity of the reflected field can be arbitrarily large as the dimension is increased, or if the reflections from the transverse end facets are large.

## 4. General gain media

*k*

_{2z}with negative imaginary part at an observation frequency

*ω*

_{1}<

*k*. Consider the permittivity where the complex numbers

_{x}c*N*and

*P*are located in the lower half-plane, and

*ω*

_{2}is a real constant. The longitudinal wavenumber satisfies

*ω*

_{2}=

*k*, we can tailor the frequency dependence of

_{x}c*N*=

*n*–

*iC*and

*P*=

*p*–

*iC*, where

*C*> 0. All poles and zeros are now located in the (closed) lower half-plane. For

*ω*> 0, assuming

*C*≪

*n*,

*p*, the longitudinal wavenumber can be written for real functions

*A*(

*ω*) and

*B*(

*ω*); in addition

*A*(

*ω*) > 0. Hence, for

*n*<

*p*,

*k*

_{2z}is analytic in the upper half-plane of

*ω*, and since

*k*

_{2z}→ +

*ω*/

*c*as

*ω*→ ∞,

*k*

_{2z}will be located in the forth quadrant of the complex

*k*

_{2z}-plane, i.e., Re

*k*

_{2z}> 0 and Im

*k*

_{2z}< 0 for all

*ω*> 0. A proper evanescent or “anti-evanescent” wave has |Re(

*k*

_{2z})/Im(

*k*

_{2z})| ≪ 1, so we search for values of

*ω*

_{1}satisfying this requirement. Analyzing Fig. 6, there exists an

*ω*

_{1}where |Re(

*k*

_{2z})/Im(

*k*

_{2z})| ≪ 1 for

*n*<

*ω*

_{1}<

*p*. We have hence found a medium for which

*k*

_{2z}describes an “anti-evanescent” wave in a finite frequency range.

*k*. While there are no zeros of

_{x}*k*considered above, this is not the case for all possible

_{x}*k*. Thus, also for this medium there are growing waves. The fact that the medium has large gain, and the presence of instabilities, mean that it is very challenging to observe the “anti-evanescent” response in practice. In principle, however, up to a certain time the amplitude of the instabilities can be limited by ensuring a narrowbanded spectrum of incident

_{x}*k*’s. Formally, if

_{x}*σ*is the width of the incident wave, and ℰ

*(*

_{σ}*x*,

*z*,

*t*) is the resulting electric field, lim

_{σ→∞}ℰ

*(*

_{σ}*x*,

*z*,

*t*) tends to the “anti-evanescent” response as

*t*→ ∞, while lim

_{t→∞}ℰ

*(*

_{σ}*x*,

*z*,

*t*) = ∞ for any finite

*σ*.

*ω*= 0. While the medium is causal in principle, the medium might be easier to realize if the pole is moved slightly away from the origin, into the lower half-plane. It turns out that this modification does not alter the permittivity function signifiantly, in the frequency range of interest. Also, if desired, the behavior at

*ω*= ∞ can be adjusted along the lines described in Ref. [17

**73**, 026605 (2006). [CrossRef]

*t*when the transients have died out. Only a single

*k*has been excited. The reflection amplitude is 0.98, and the transmitted field is an exponentially increasing function of

_{x}*z*. While realizable in principle, the example is highly unrealistic: To observe a behavior similar to that in Fig. 7,

*t*must be at least of the order of 10

^{2}(

*k*)

_{x}c^{−1}; otherwise the transients would disturb the picture. Any realistic gain medium has finite thickness. However, to act as a semi-infinite medium, the thickness

*d*of the gain medium must satisfy

*d*>

*ct*, or

*k*≳ 10

_{x}d^{2}, such that the light has not reached the back end. With the “anti-evanescent” growth rate in Fig. 7, this would imply unphysically large fields (or in practice, nonlinear gain saturation). Hence, if the “anti-evanescent” behavior is to be observed experimentally, one would need to construct a medium where the transients die out rapidly, and/or a medium which leads to a sufficiently small |Im

*k*

_{2z}|. At the same time the medium must violate the conditions in Sec. 3; that is, it must have large gain and/or large dispersion for some frequencies.

## 5. Conclusion

*k*

_{2z}in the second and fourth quadrant of the complex plane) can be attained with a suitably engineered medium. In other words, evanescent gain may or may not be the case, dependent on the detailed permittivity function. This demonstrates the fact that the sign of

*k*

_{2z}cannot be determined from the electromagnetic parameters at a single frequency, but must be identified from the entire frequency domain dependence, after a check of possible non-analytic points (instabilities) in the upper half-plane of complex frequency.

## A. Finite incident beam and finite size medium

*k*and frequency

_{x}*ω*. Thus we can treat a causal excitation of each

*k*separately.

_{x}*x*,

*t*) be the incident TE field at the interface between the high-index medium and the active low index medium. Performing a Laplace transform

*t*→

*ω*followed by a Fourier transform

*x*→

*k*, we obtain the transformed field

_{x}*E*(

*k*,

_{x}*ω*). The inverse transform is given by

*E*(

*k*,

_{x}*ω*′ +

*iγ*) is absolute integrable with respect to

*k*and

_{x}*ω*′. This is the case assuming that the incident field is sufficiently smooth with respect to

*t*and

*x*. For example, taking the incident wave to be

*a*(

*x*)

*e*

^{iKxx}

*b*(

*t*)

*e*

^{−iω1t}, the transformed field becomes

*E*(

*k*,

_{x}*ω*′ +

*iγ*) =

*A*(

*k*–

_{x}*K*)

_{x}*B*(

*ω*′ –

*ω*

_{1}+

*iγ*), where

*A*is the Fourier transform of

*a*, and

*B*is the Laplace transform of

*b*. Here we assume that

*b*(

*t*) = 0 for

*t*< 0. If

*a*and

*b*are continuous,

*A*and

*B*are absolute integrable.

*C*and

*γ*. Then the transforms

*t*→

*ω*followed by

*x*→

*k*exist, and we can express the total field in the form (18). The total field is determined using the wave equation. In order to consider each mode

_{x}*k*separately, we interchange the order of integration for each term in the wave equation. To do so, we require the second order derivatives with respect to

_{x}*t*and

*x*to be continuous.

*k*, given a sufficiently smooth incident field. For this solution, the Fresnel equations show that the reflection and transmission coefficients tend to zero and unity, respectively, as |

_{x}*ω*′| → ∞ or |

*k*| → ∞. Therefore the reflected and transmitted field in the (

_{x}*ω*,

*k*)-domain adopt any absolute integrability property from the incident field.

_{x}*u*(

*t*)exp(

*ik*–

_{x}x*iω*

_{1}

*t*) is not continuous. Hence, strictly speaking, the above described method cannot be used. However, by smoothing the discontinuity around

*t*= 0, we can make the field and its second order derivative continuous. This modification will not affect the discussion in general, since a slower transient will reduce the bandwidth. Thus the side waves are excited weaker, such that inequality (14) is satisfied with an even larger margin.

*d*. For

*t*<

*d*/

*c*the fields will be the same as if the finite-size medium were replaced by a semi-infinite medium.

## B. Instabilities in infinite media

20. P. A. Sturrock, “Kinematics of growing waves,” Phys. Rev. **112**, 1488–1503 (1958). [CrossRef]

*t*< 0. It is not clear whether this is possible, not even in principle, since perturbations in the remote past would not die out but rather increase exponentially.

*d*, there will be no instabilities provided the gain is sufficiently weak. Examples of such configurations include optical amplifiers, and laser resonators with pumping below threshold. When there are no instabilities, we can turn on the pump in remote past such that the perturbations have died out before

*t*= 0. For 0 <

*t*<

*d*/

*c*we can still regard the medium as semi-infinite, since, as seen from Fig. 8, it makes no difference.

## C. Determining the reflected time-domain field

*F*,

*ω*

_{0}, and Γ are positive parameters, describing the resonance strength, frequency, and bandwidth, respectively. The physical, time-domain reflected field at

*z*=

*x*= 0 is given by the inverse Laplace transform (12), repeated for convenience here: The field can be interpreted by evaluating integral (21) by a generalized version of the residue theorem. We here recognize that integrating along path −∞ +

*iγ*to +∞ +

*iγ*, is the same as integrating around all branch cuts and poles. The denominator

*k*

_{1z}+

*k*

_{2z}does not have any zeros, provided the permittivity

*ɛ*

_{1}can be considered constant and larger than unity in the frequency range of interest. Thus we only need to consider the branch cuts extending from branch points of

*k*

_{1z}and

*k*

_{2z}, and the pole at

*ω*=

*ω*

_{1}. Note that the branch cuts are arbitrary, as long as they extend from the branch points. We let all branch cuts lie parallel to the imaginary axis, towards Im

*ω*= −∞. See illustration in Fig. 9. The branch points of

*k*

_{1z}are located far away from (and below) the real frequency axis, provided the medium’s bandwidth is sufficiently large. The wavenumber

*k*

_{2z}has two branch points in the upper half-plane, located immediately above

*ω*= ±

*k*. In addition there are four branch points located below the real frequency axis, with imaginary parts −Γ/2; two simple zeros and two simple poles. The integrals around the latter four branch cuts decay with time constant at most 2/Γ. Thus, for

_{x}c*t*≳ 2/Γ, the only contributing terms are the residue of the pole at

*ω*

_{1}, and the contribution ℰ

_{bc}(0,

*t*) from the two remaining branch cuts of

*k*

_{2z}: Here

*k*

_{1z}and

*k*

_{2z}have been evaluated at the frequency

*ω*

_{1}. We write ℰ

_{bc}(0,

*t*) = ℰ

_{bc−}(0

*,t*) + ℰ

_{bc+}(0,

*t*), where ℰ

_{bc−}(0,

*t*) and ℰ

_{bc+}(0,

*t*) are the contributions from the branch cuts in the left and right half-planes, respectively.

*F*≪ 1, Γ ≪

*ω*

_{0}and

*k*, the branch cut in the right half-plane extends from approximately

_{x}c*ω*=

*k*+

_{x}c*iδ*to

*ω*=

*k*–

_{x}c*i*∞, where

*δ*≤

*F*Γ. Then, for

*t*≳ 2/Γ Here subscripts l and r indicate that

*ρ*(

*ω*) is discontinuous when crossing the branch cut, denoting the left and right side of the branch cut respectively. We further define

*f*

_{l,r}(

*ω*) =

*k*

_{2z}/

*k*

_{1z}. Since

*k*

_{2z}is small in the vicinity of

*k*, by first order approximation

_{x}c*ρ*

_{l,r}(

*ω*) = 1 – 2

*f*

_{l,r}(

*ω*), where

*f*

_{r}(

*ω*) = −

*f*

_{l}(

*ω*). The integral (23) can now be simplified: In order to obtain a manageable expression for

*f*

_{l}(

*ω*), it is useful to express

*p*, and zeros denoted by subscripts

*k*and

_{x}c*ω*

_{0}(indicating the location along the real frequency axis),

*δ*= Im(

*ω*

_{kxc}) and

*ω*= Im(

_{i}*ω*), and recognizing that (

*ω*–

*ω*

_{ω0})/(

*ω*–

*ω*) ≈ 1 at

_{p}*ω*=

*ω*

_{kxc}, Eq. (25) can be simplified:

*k*–

_{x}c*ω*

_{1}≫ Γ, we can now find an upper bound of integral (24) by noting that

*ω*considered. We can estimate ℰ

_{i}_{bc−}(0,

*t*) similarly, yielding the bound Consequently for 2/Γ ≲

*t*≲ 1/

*F*Γ, the field is well described by the first term in Eq. (22).

## References and links

1. | G. N. Romanov and S. S. Shakhidzhanov, “Amplification of electromagnetic field in total internal reflection from a region of inverted population,” JETP Lett. |

2. | A. A. Kolokolov, “Reflection of plane-waves from an amplifying medium,” JETP Lett. |

3. | P. R. Callary and C. K. Carniglia, “Internal-reflection from an amplifying layer,” J. Opt. Soc. Am. |

4. | W. Lukosz and P. P. Herrmann, “Amplification by reflection from an active medium,” Opt. Commun. |

5. | R. F. Cybulski and C. K. Carniglia, “Internal-reflection from an exponential amplifying region,” J. Opt. Soc. Am. |

6. | R. F. Cybulski and M. P. Silverman, “Investigation of light amplification by enhanced internal-reflection .1. theoretical reflectance and transmittance of an exponentially nonuniform gain region,” J. Opt. Soc. Am. |

7. | A. A. Kolokolov, “Determination of the reflection coefficient of a plane monochromatic wave,” J. Commun. Technol. Electron. |

8. | A. A. Kolokolov, “Fresnel formulas and the principle of causality,” Phys. Usp. |

9. | J. Fan, A. Dogariu, and L. J. Wang, “Amplified total internal reflection,” Opt. Express |

10. | K. J. Willis, J. B. Schneider, and S. C. Hagness, “Amplified total internal reflection: theory, analysis, and demonstration of existence via FDTD,” Opt. Express |

11. | A. Siegman, “Fresnel reflection, Lenserf reflection, and evanescent gain,” Opt. Photon. News |

12. | C. J. Koester, “Laser action by enhanced total internal reflection,” IEEE J. Quantum Electron. |

13. | B. Y. Kogan, V. M. Volkov, and S. A. Lebedev, “Superluminescence and generation of stimulated radiation under internal-reflection conditions,” JETP Lett. |

14. | S. A. Lebedev, V. M. Volkov, and B. Y. Kogan, “Value of the gain for light internally reflected from a medium with inverted population,” Opt. Spectrosc. |

15. | M. P. Silverman and J. R. F. Cybulski, “Investigation of light amplification by enhanced internal reflection. Part II. Experimental determination of the single-pass reflectance of an optically pumped gain region,” J. Opt. Soc. Am. |

16. | B. Nistad and J. Skaar, “Causality and electromagnetic properties of active media,” Phys. Rev. E |

17. | J. Skaar, “Fresnel equations and the refractive index of active media,” Phys. Rev. E |

18. | A. Lakhtakia, J. B. Geddes III, and T. G. Mackay, “When does the choice of the refractive index of a linear, homogeneous, isotropic, active, dielectric medium matter?” Opt. Express |

19. | L. V. Ahlfors, |

20. | P. A. Sturrock, “Kinematics of growing waves,” Phys. Rev. |

21. | R. J. Briggs, |

**OCIS Codes**

(140.3380) Lasers and laser optics : Laser materials

(250.4480) Optoelectronics : Optical amplifiers

(260.2110) Physical optics : Electromagnetic optics

(160.3918) Materials : Metamaterials

**ToC Category:**

Physical Optics

**History**

Original Manuscript: July 1, 2011

Revised Manuscript: August 22, 2011

Manuscript Accepted: September 25, 2011

Published: October 14, 2011

**Citation**

Jon Olav Grepstad and Johannes Skaar, "Total internal reflection and evanescent gain," Opt. Express **19**, 21404-21418 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-22-21404

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### References

- G. N. Romanov and S. S. Shakhidzhanov, “Amplification of electromagnetic field in total internal reflection from a region of inverted population,” JETP Lett.16, 209–211 (1972).
- A. A. Kolokolov, “Reflection of plane-waves from an amplifying medium,” JETP Lett.21, 312–313 (1975).
- P. R. Callary and C. K. Carniglia, “Internal-reflection from an amplifying layer,” J. Opt. Soc. Am.66, 775–779 (1976). [CrossRef]
- W. Lukosz and P. P. Herrmann, “Amplification by reflection from an active medium,” Opt. Commun.17, 192–195 (1976). [CrossRef]
- R. F. Cybulski and C. K. Carniglia, “Internal-reflection from an exponential amplifying region,” J. Opt. Soc. Am.67, 1620–1627 (1977). [CrossRef]
- R. F. Cybulski and M. P. Silverman, “Investigation of light amplification by enhanced internal-reflection .1. theoretical reflectance and transmittance of an exponentially nonuniform gain region,” J. Opt. Soc. Am.73, 1732–1738 (1983). [CrossRef]
- A. A. Kolokolov, “Determination of the reflection coefficient of a plane monochromatic wave,” J. Commun. Technol. Electron.43, 837–845 (1998).
- A. A. Kolokolov, “Fresnel formulas and the principle of causality,” Phys. Usp.42, 931–940 (1999). [CrossRef]
- J. Fan, A. Dogariu, and L. J. Wang, “Amplified total internal reflection,” Opt. Express11, 299–308 (2003). [CrossRef] [PubMed]
- K. J. Willis, J. B. Schneider, and S. C. Hagness, “Amplified total internal reflection: theory, analysis, and demonstration of existence via FDTD,” Opt. Express16, 1903–1913 (2008). [CrossRef] [PubMed]
- A. Siegman, “Fresnel reflection, Lenserf reflection, and evanescent gain,” Opt. Photon. News21, 38–45 (2010). [CrossRef]
- C. J. Koester, “Laser action by enhanced total internal reflection,” IEEE J. Quantum Electron.2, 580–584 (1966). [CrossRef]
- B. Y. Kogan, V. M. Volkov, and S. A. Lebedev, “Superluminescence and generation of stimulated radiation under internal-reflection conditions,” JETP Lett.16, 100–101 (1972).
- S. A. Lebedev, V. M. Volkov, and B. Y. Kogan, “Value of the gain for light internally reflected from a medium with inverted population,” Opt. Spectrosc.35, 565–566 (1973).
- M. P. Silverman and J. R. F. Cybulski, “Investigation of light amplification by enhanced internal reflection. Part II. Experimental determination of the single-pass reflectance of an optically pumped gain region,” J. Opt. Soc. Am.73, 1739–1743 (1983). [CrossRef]
- B. Nistad and J. Skaar, “Causality and electromagnetic properties of active media,” Phys. Rev. E78, 036603 (2008). [CrossRef]
- J. Skaar, “Fresnel equations and the refractive index of active media,” Phys. Rev. E73, 026605 (2006). [CrossRef]
- A. Lakhtakia, J. B. Geddes, and T. G. Mackay, “When does the choice of the refractive index of a linear, homogeneous, isotropic, active, dielectric medium matter?” Opt. Express15, 17709–17714 (2007). [CrossRef] [PubMed]
- L. V. Ahlfors, Complex Analysis (McGraw-Hill International Editions, 1979).
- P. A. Sturrock, “Kinematics of growing waves,” Phys. Rev.112, 1488–1503 (1958). [CrossRef]
- R. J. Briggs, Electron-Stream Interactions with Plasmas (MIT Press, 1964).

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