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Resolving the wave-vector and the refractive index from the coe?cient of re?ectance

V. U. Nazarov? and Y.-C. Chang

arXiv:0710.4388v1 [physics.optics] 24 Oct 2007

Research Center for Applied Sciences, Academia Sinica, 128 Section 2 Academia Road, Nankang, Taipei 115, Taiwan ? Corresponding author: nazarov@gate.sinica.edu.tw

We resolve the existing controversy concerning the selection of the sign of the normal-to-the-interface component of the wave-vector kz of an electromagnetic wave in an active (gain) medium. Our method exploits the fact that no ambiguity exists in the case of a ?lm of the active medium since its coe?cient of re?ectance is invariant under the inversion of the sign of kz . Then we show that the limit of the in?nite ?lm thickness determines a unique and physically consistent choice of the wave-vector and the refractive index. Practically important implications of the theory are identi?ed and discussed. c 2008 Optical Society of America OCIS codes: 000.2690, 260.2110, 350.5500 Recently, there has been much of debate regarding the correct selection of the sign of the normal-to-the-interface component of the wave-vector kz of an electromagnetic wave propagating in an active medium or, equivalently for the normal incidence, of the refractive index of the active medium [1–6]. Speci?cally, if one considers a system of a glass prism, a metal ?lm, and an active dielectric (the Kretschmann-Raether con?guration) as schematized in Fig. 1 (a), then the coe?cient of re?ectance is given by [7] (for the sake of simplicity, we consider nonmagnetic media with ? = 1) R= where the coe?cients rij = (kiz ?j ? kjz ?i )/(kiz ?j + kjz ?i ) describe the parallel ?eld components, kiz = ± 2π ?i ? ?0 sin2 θ, λ 1 (3) (2) r01 + r12 e2ik1z d1 1 + r01 r12 e2ik1z d1

2

,

(1)

?0 , ?1 , and ?2 are complex dielectric constants of the prism, the metal, and the active dielectric, respectively, λ is the light wave-length in vacuum, and θ is the angle of incidence. For an active medium (?′′ < 0) the choice of the sign of the square root in Eq. (3) is far from 2 evident [1–6, 8] while this choice strongly a?ects the coe?cient of re?ectance of Eq. (1). In this Letter, we show that the problem of the sign of k2z can be unambiguously and naturally resolved by ?rst considering a ?lm of the active dielectric then letting the thickness of the ?lm tend to in?nity. Let us consider a system in Fig. 1 (b) which di?ers from that in Fig. 1 (a) by the ?nite thickness d2 of the active dielectric and the presence of a semi-in?nite passive dielectric with the dielectric constant ?3 . For this system the coe?cient of re?ectance can be written as [9]

a) b)

3 2

2

d d

Incident Reflected beam

2

1

d

Incident Reflected beam

1

1

1

0

beam

0

beam

Fig. 1. (Color online) Systems under study: (a) Semi-in?nite dielectric (?0 ), metal ?lm (?1 , d1 ) and a semi-in?nite active medium (?2 ). (b) As in (a) but the active medium constitutes a ?lm of the thickness d2 , and there is a semiin?nite dielectric on the top with the dielectric function ?3 .

R= e2ik1z d1 r12 +e2ik2z d2 r23 +r01 1+e2ik2z d2 r12 r23 1 + e2ik2z d2 r12 r23 +e2ik1z d1 r01 (r12 + e2ik2z d2 r23 )

2

(4) .

This is evident [and can also be directly veri?ed with use of Eq. (2)] that Eq. (4) is invariant under the transformation k2z → ?k2z . Therefore, there is no ambiguity in the selection of the sign of k2z in the case of a ?lm of the active dielectric. Now, putting d2 to in?nity in Eq. (4), we immediately retrieve Eq. (1) if and only if k2z has a positive imaginary part regardless of whether medium 2 is active or passive. Accordingly, the branch cut in the complex plane 2 of k2z must be taken along the positive part of the real axis (0 ≤ φ < 2π, where φ is the 2 argument of k2z ), in agreement with [2] and in disagreement with [1] and [3, 10], where the branch cuts were taken along the negative imaginary (?π/2 ≤ φ < 3/2 π) and negative 2

real (?π ≤ φ < π) axes, respectively. Therefore, for an active medium the refractive index √ n = ? has negative real part. This concludes our proof as which of the two signs of k2z should be chosen in Eq. (3) to be substituted into Eq. (1).

1.2

1.0

Reflectance

0.8 Gain in active medium No gain

0.6

0.4

0.2 0 10 20 30 40 50 60 70 80 90

Angle

Fig. 2. (Color online) Coe?cient of re?ectance of the system of a glass prism, silver ?lm, and the dye of cresyl violet with (?2 = 1.85 ? 9 × 10?6 i, red solid line) and without (?2 = 1.85 + 0 × i, black dashed line) gain. Other parameters are those from [11] (see text).

In order to provide an illustrative example which helps to elucidate a number of practically instructive points, in Figs. 2 - 4 we present numerical results for the coe?cient of re?ectance of the system of a glass prism, a silver ?lm, and the dye of cresyl violet. Parameters as taken from [11] are: ?0 = 2.25, ?1 = ?18 + 0.7 i, ?2 = 1.85 ? 9 × 10?6 i, d1 = 39 nm, and λ = 633 nm. In Fig. 2 the coe?cient of re?ectance of Eq. (1) of the system with the semi-in?nite active dielectric (red solid line) is plotted and compared with that of a passive dielectric (black dashed line). The sharp dip in the spectrum with the minimum at 73.1? is due to the surface plasmon polariton associated with silver ?lm. At lower angles of incidence, the re?ectance is mainly greater than one for the system with the active dielectric, obviously due to the gain in the latter. It must be noted that the value of ?′′ = ?9 × 10?6 is small enough that the 2 corresponding spectrum is indistinguishable from that of the system with ?′′ = 0? , where 2 0? stands for the zero limit from the negative side. An important point is that there is no contradiction in re?ectance of the systems with ?′′ = 0+ and ?′′ = 0? to di?er ?nitely 2 2 ′′ (and considerably) as is seen in Fig. 2, since ?2 = 0? combined with in?nite thickness of the active medium results in the ?nite overall gain (mathematically, the limit of in?nite thickness should be taken ?rst at ?nite negative ?′′ , then the latter should be put to zero). 2 3

1.4

1.2

Reflectance

1.0

0.8

0.6

9 cm

d =

2

5 cm 1 cm 0

0.4

0.2 0 10 20 30 40 50 60 70 80 90

Angle

Fig. 3. (Color online) Convergence of the coe?cient of re?ectance of the system with a ?nite ?lm of the active dielectric to that with the semi-in?nite one. Parameters are those from [11] (see text).

In order to understand what ?lms of the active dielectric are thick enough to be considered as semi-in?nite, in Fig. 3 we plot the coe?cient of re?ectance of Eq. (4) for a number of ?lm thicknesses. The striking result of the saturation of the spectrum of a ?lm to that of the semi-in?nite medium occurring at the ?lm thickness as huge as about 9 cm is simply due to the tiny ?′′ = ?9 × 10?6 , and this should serve as a warning when weakly active dielectric 2 ?lms are modeled with semi-in?nite media. In Fig. 4 we plot the re?ectance of Eq. (1) obtained with three di?erent branch cuts in the 2 complex plane of k2z . These are compared with the coe?cient of re?ectance of the system with a ?lm of active dielectric (d2 = 9 cm), for which the selection of the branch cut does not make a di?erence, and which, therefore, at large ?lm thickness can serve as a criterion of the validity of the branch cut chosen in the semi-in?nite case. In agreement with our formal proof, the spectrum calculated with the branch cut along the positive real axis (Im kz > 0) (red dashed line) practically coincides at all angles with that of the ?lm (black solid line). At the same time, the cut along the negative imaginary axis [1] (green dashed line) results in a loss instead of the gain below the critical angle, while the cut along the negative real axis [3, 10] (blue dashed line) leads to the incorrect re?ectance in the range of the surface plasmon polariton. We point out a two-fold signi?cance of deriving the wave-vector in a half-space from that for a ?nite ?lm: Firstly, this method provides a mathematically rigorous and simple procedure to resolve the sign of the wave-vector. Secondly, in a more general perspective, in experiment there evidently exists no semi-in?nite medium but only (maybe thick) ?lms. A half-space is 4

1.2

1.0

Reflectance

0.8

Film(position of the cut is arbitrary) Cut along positive real axis (correct) Cut along negative imaginary axis (incorrect) Cut along negative real axis (incorrect)

0.6

0.4

0.2 0 10 20 30 40 50 60 70 80 90

Angle

Fig. 4. (Color online) Implications of the selection of the three di?erent branch 2 cuts in kz complex plane. Red dashed line corresponds to the branch cut along positive real axis (0 ≤ φ < 2π). Green dotted line corresponds to the branch cut along negative imaginary axis (?π/2 ≤ φ < (3/2)π) [1]. Blue dasheddotted line corresponds to the branch cut along negative real axis (?π ≤ φ < π) [3, 10]. Black solid line corresponds to a ?lm of the active dielectric (d2 = 9 cm). Other parameters are those from [11] (see text).

a convenient abstraction with properties being the limit of the corresponding properties of a ?lm thick enough so that a further increase of the ?lm’s thickness does not change results of an experiment. Therefore, our results for semi-in?nite active media are automatically applicable to the interpretation of experimental data with su?ciently thick ?lms. This is not the case with other choices of the branch cut in the complex plane of k2z [1,3,10] as has been already discussed in conjunction with Fig. 4. Ref. [12] performs a numerical analysis of a wave-packet propagation in a system considered in [2]. Correcting errors of [3] by the same authors, [12] recon?rms the conclusion of [2] that the refractive index of the active medium considered in the latter reference has a positive imaginary part. This conclusion is in agreement with results of the present work. In particular, [12] does not ?nd the pulse refracted into the active medium to grow unboundedly which perfectly con?rms the positive imaginary part of the refractive index. In conclusion, by considering a semi-in?nite active (gain) dielectric as a limiting case of its ?nite-thickness ?lm, we have resolved the existing controversy in determining the normal-tothe-interface component of the wave-vector kz and, equivalently, of the sign of the refractive index of the active dielectric. Speci?cally, Im kz must be nonnegative for both active and 5

passive media, and, accordingly, the refractive index of an active medium has a negative real part. We have shown that this is physical that the coe?cient of re?ectance can be discontinuous with respect to a continuous change of the dielectric constant of a semi-in?nite medium from that of a passive to an active one. We have applied the theory to the system of a glass prism, silver ?lm, and the weakly active dye of cresyl violet in Kretschmann-Raether con?guration, having numerically illustrated the above general concepts. References 1. S. A. Ramakrishna and O. J. Martin, “Resolving the wave vector in negative refractive index media,” Opt. Lett. 30, 2626–2628 (2005). 2. Y.-F. Chen, P. Fischer, and F. W. Wise, “Negative refraction at optical frequencies in nonmagnetic two-component molecular media,” Phys. Rev. Lett. 95, 067402 (2005). 3. T. G. Mackay and A. Lakhtakia, “Comment on “negative refraction at optical frequencies in nonmagnetic two-component molecular media”,” Phys. Rev. Lett. 96, 159701 (2006). 4. Y.-F. Chen, P. Fischer, and F. W. Wise, “Chen, Fischer, and Wise Reply,” Phys. Rev. Lett. 96, 159702 (2006). 5. S. A. Ramakrishna, “Comment on “negative refraction at optical frequencies in nonmagnetic two-component molecular media”,” Phys. Rev. Lett. 98, 059701 (2007). 6. Y.-F. Chen, P. Fischer, and F. W. Wise, “Chen, Fischer, and Wise Reply,” Phys. Rev. Lett. 98, 059702 (2007). 7. H. Raether, Surface plasmons on smooth and rough surfaces and on gratings (SpringerVerlag, Berlin, 1988), Springer Tracts in Modern Physics, Vol. 111. 8. J. Skaar, “On resolving the refractive index and the wave vector,” Opt. Lett. 31, 3372– 3374 (2006). 9. Eq. (4) for the re?ectance of a four-component system can be derived in just the same way as Eq. (1) for a three-component system by applying Maxwell’s equations and the standard boundary conditions at interfaces. 10. J. Wei and M. Xiao, “Electric and magnetic losses and gains in determining the sign of refractive index,” Opt. Commun. 270, 455–464 (2007). 11. J. Seidel, S. Grafstrom, and L. Eng, “Stimulated emission of surface plasmons at the interface between a silver ?lm and an optically pumped dye solution,” Phys. Rev. Lett. 94, 177401 (2005). 12. J. B. Geddes III, T. G. Mackay, and A. Lakhtakia, “On the refractive index for a nonmagnetic two-component medium: resolution of a controversy,” Opt. Commun. 280, 120–125 (2007).

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