Optics and Spectroscopy, Vol. 89, No. 5, 2000, pp. 737–741. Translated from Optika i Spektroskopiya, Vol. 89, No. 5, 2000, pp. 801–805. Original Russian Text Copyright © 2000 by Ivanov, Sementsov.
PHYSICAL AND QUANTUM OPTICS
Negative Shift of a Light Beam Reflected from the Interface between Optically Transparent and Resonant Media O. V. Ivanov and D. I. Sementsov Ul’yanovsk State University, Ul’yanovsk, 432700 Russia Received December 27, 1999
Abstract—The lateral shift of a light beam reflected from the interface between an optically transparent dielectric and a medium for which the frequency dependences of the real and imaginary parts of the permittivity have a resonant form is investigated. At certain radiation frequencies and angles of incidence, a negative displacement of the reflected beam along the interface takes place. © 2000 MAIK “Nauka/Interperiodica”.
INTRODUCTION The reflection of bounded beams is a well-known problem of optics, which involves, in particular, the lateral displacement of a beam reflected from an interface [1]. The effect of a spatial shift of a beam in the process of total internal reflection (TIR), called the Goos– Hanchen (GH) shift [2], has been studied in many areas of physics, such as acoustics, plasma physics, quantum mechanics, and surface physics. At the present time, this effect continues to be discussed for new classes of structures [3, 4], in particular, waveguide structures [5], for new experimental schemes of its measurement [6] based on the models that refine and work out the classical variants of the theory in detail [7]. The GH shift is of the order of the wavelength; therefore, the smallness of the shift for optical wavelengths impedes direct measurements of it in the case of a single reflection from an interface. However, for a microwave range, the determination of the GH shift in experiments in the regime of a single reflection presents no difficulties, and the GH effect has been studied in detail in the above range [8, 9]. One can enhance the GH effect by bringing the frequency of the incident wave closer to the resonance frequencies of the media located on different sides of the interface. The influence of the resonant absorption on the GH effect under the total internal reflection of a laser beam from the glass– cesium vapor interface close to a cesium absorption line was studied experimentally in [10]. As in the microwave range, close to an absorption line, the magnitude of the lateral shift of the reflected beam can increase by an order or more. In this paper, we study the peculiarities of the shift of a light beam reflected from the plane interface between transparent and resonant media and analyze the conditions of realization of the negative GH shift.
MODELS OF THE STRUCTURE AND THE LIGHT BEAM For the structure being investigated, consider a plane interface between a transparent dielectric with the real permittivity ε1, which is assumed to be constant in the frequency range being studied, and a resonant medium for which the frequency dependence of the complex permittivity has the form [11] ( ε 0 – ε ∞ )ν 0 ε 2(ν) = ε ∞ + ------------------------------, 2 2 ν 0 – ν – iνg 2
(1)
where ν0 is the resonance frequency; g is the resonance line width; and ε0 and ε∞ are, respectively, the static and high-frequency permittivities of the medium. Many optical materials have a similar frequency dependence of the permittivity close to their absorption lines. The presence of color centers in optically transparent alkalihaloid crystals can also lead to the analogous dependence of ε(ν). To determine the magnitude of the lateral shift of a light beam reflected from the interface of two media, we will use the two-wave representation within the framework of which the simplest wave packet can be formed by two plane waves propagating at slightly different angles. Therefore, we assume that a light beam, represented by two plane monochromatic waves propagating at close angles ϕ± = ϕ ± ∆ϕ/2 and having amplitudes and wave vectors equal in absolute values, is incident from a nonresonant medium onto a boundary with a resonant medium at the angle ϕ. The field of each of these waves can be represented as (i)
E ± = A exp [ i ( 2πνt – k 1± r ) ],
(2)
where k1± = (k1 sin ϕ±, 0, k1 cos ϕ±), k1 = k0 ε 1 , k0 = 2πν/c, and c is the velocity of light in a vacuum. The interference pattern, which can be interpreted as a set of rays separated by the distance 2π/k1∆ϕ having in this
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where kzj = k 0 ε j – k xj and kxj = k0 ε 1 sinϕ are the components of the complex wave vector kj for waves propagating in the coating (j = 1) and resonant (j = 2) media. In the case under consideration, the quantities r and t are complex, which leads to the dependence of phases of the reflected and transmitted waves on the angle of incidence of the light beam. We represent the reflection and transmission coefficients of the corresponding waves (5) in the form 2
d (r)
S1 kr k1 qr
q1
ε1 ε2
∆(r)
r = ρ exp ( iη ),
Fig. 1. Scheme of the displacement of the wave packet reflected from the interface between transparent and resonant media.
(i)
sin ϕ c ε1 2 = ----------- A p, s [ 1 + cos ( ∆ϕq 1 r ) ] 0 , 8π cos ϕ
(3)
where Ap, s are the amplitudes of the waves with p- and s-polarization, the vector q1 is orthogonal to the vector k1 and lies in the plane of incidence of the wave packet, and q1 = k1. The distribution of the energy density in the interference pattern is determined by the expression ε 2 w p, s = -----1- A p, s { 1 + cos [ ∆ϕk 1( x cos ϕ – z sin ϕ) ] }.(4) 4π
THEORETICAL ANALYSIS The coefficients of reflection and transmission of the interface for p- and s-waves determining the ratios of the corresponding components of the field of the reflected (r) and transmitted (t) total wave at the interface to the incident (i) total wave are given by the expressions (r)
(t)
2ε 1 k z2 Ex t p = ------- = -----------------------------, (i) ε 1 k z2 + ε 2 k z1 Ex
(r)
k z1 – k z2 Ey -, r s = ------- = -----------------(i) k z1 + k z2 Ey (t)
2k z1 Ey -, t s = ------- = -----------------(i) k z1 + k z2 Ey
(6)
(5)
2
2
η = arctan ( r'' ⁄ r' ),
2
2
χ = arctan ( t'' ⁄ t' ).
ρ=
r' + r'' ,
τ=
t' + t'' ,
In view of (6), the Poynting vector of the reflected total wave can be written as c ε (r) 2 2 S p, s = -----------1 A p, s ρ p, s 4π sin ϕ ∂η p, s , × 1 + cos ∆ϕ q r r – ------------ 0 ∂ϕ – cos ϕ
(7)
where the vector qr = (kz1, 0, kx1) is orthogonal to the vector kr and qr = kr = k1. As follows from (7), the reflected total wave is presented by the set of rays displaced in the direction of the vector qr by the amount 1 ∂η p, s (r) d p, s = ---- ----------k 1 ∂ϕ
Figure 1 presents the energy distribution of the light field in the wave packets of the incident and reflected waves.
ε 1 k z2 – ε 2 k z1 Ex r p = ------- = -----------------------------, (i) ε 1 k z2 + ε 2 k z1 Ex
t = τ exp ( iχ ),
where the amplitudes and phases of the corresponding coefficients are expressed in terms of their real and imaginary parts
approximation the meaning of the width of the wave packet, will be the resulting field of these waves. The averaged (over a period) flux of the energy transferable by the wave packet from two waves incident on the interface in the direction of the mean wave vector k1 = (k1+ + k1–)/2 is determined by the Poynting vector which can be represented in the following manner: S p, s
2
(8)
with respect to the rays reflected from the interface without any phase shift (Fig. 1). The beam displacement along the surface is determined by the quantity ∆(r) = d(r)/cosϕ. When the angle of incidence differs from the critical angle ϕ ≠ ϕcr = arcsin ε '2 ⁄ ε 1 (ε2 ≠ kx1/k0), at which the TIR begins, the Poynting vector for the transmitted total wave can be represented as
(t) Sp
k x1 ε 2' 2 k z1 c 2 2 , = ----------- A p τ p F( x, z) ----------------2 0 8πk 0 ε 1 k z2 ε '2 k 'z2 + ε ''k 2 '' z2
OPTICS AND SPECTROSCOPY
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NEGATIVE SHIFT OF A LIGHT BEAM REFLECTED FROM THE INTERFACE ε2 16
ds(r) /λ0 2
ε'2
(‡)
739
(‡)
8 1
1
ε''2 2 3
ϕcr
0 dp(r) /λ0 2
0 –8 –16 Rp, s 1.0
(b) 1
ε'2
1
(b) 3
ϕBr
0
3
30
3
60
90 ϕ, deg
0.5
1
2
–1
2 2
0 –0.1
–2
k x1 c 2 2 = ----------- A s τ s F( x, z) 0 , 8πk 0 k z2 '
(9)
∂χ p, s , - – q 'r + 2 exp ( ∆ϕq ''tz z ) cos ∆ϕ ----------t ∂ϕ where qt = (kz1, 0, –kx1kz1/kz2) is a complex vector, whose real part determines the beam width (2π/ q 't ∆ϕ) and the imaginary part determines the beam spreading to a plane wave. For the lateral shift of the transmitted wave in the direction of the vector q 't , we obtain an expression analogous to the expression for the shift of the reflected wave:
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0.05
∆ν/ν0
The direction of propagation of the transmitted total wave is determined by the angle α, which, for waves of the corresponding polarizations, is specified in the following manner: k x1 ε '2 -, tan α p = ----------------------------ε '2 k 'z2 + ε ''k 2 '' z2
'' z ) 1 + exp ( 2∆ϕq tz'' z ) F( x, z) = exp ( 2k z2
1 ∂χ p, s (t) -. d p, s = ---- ----------q t' ∂ϕ
0
1
Fig. 3. Frequency dependence of (a) the real (solid curve) and imaginary (dashed curve) parts of the permittivity of the resonant medium and (b) the reflection coefficient for (solid curves) p- and (dashed curves) s-polarized waves at the angles of incidence ϕ = (1) 0°, (2) 50°, and (3) 80°.
Fig. 2. Lateral displacement of a reflected beam as a function of the angle of incidence onto the interface of media with the parameters ε1 = 4; ε '2 = 2; and ε ''2 = (1) 0.002, (2) 0.2, and (3) 0.6. (a) s-polarization, (b) p-polarization.
(t) Ss
–0.05
3 2
3
2000
k x1 -. tan αs = -----k 'z2
(11)
When the wave passes through the interface of two transparent dielectrics, tan α p = tan α s = kx1/kz2 ; i.e., αp, s is the ordinary angle of refraction. In the case of the total internal reflection of a wave from a dielectric with ε2 < ε1, the angles αp = αs = 90°. In this case, the refracted wave propagates along the interface and is a surface wave. The presence of a surface energy flux of the transmitted wave is assumed to result in a lateral displacement of the reflected beam along the interface in the direction of propagation of the surface flux. We will consider a beam displacement as positive if the direction of the displacement coincides with the direction of the longitudinal components of the incident beam.
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ds(r)/λ0 3
1.0
(‡)
0.5 2 0 –0.1 dp(r)/λ0
1 –0.05
0
0.05
1
0.1 ∆ν/ν0 (b)
1
0
∆ν/ν0
0 2
–1 3 –2
Fig. 4. Lateral displacement of a beam reflected from the vacuum–resonant medium interface as a function of the detuning from the resonance frequency for the case of (a) pand (b) s-polarization and different angles of incidence ϕ = (1) 0°, (2) 50°, and (3) 80°.
As follows from (9), for a p-polarized wave, when ε 2' < 0, the transmitted flux component parallel to the interface is negative. In this case, the displacement of the reflected beam along the interface is also negative. (t) For a wave with s-polarization, we always have S x > 0; therefore, at the boundary with an absorbing medium, only a positive displacement of the reflected s-wave must take place. NUMERICAL ANALYSIS Based on relations (5) and (8), the calculated dependences of the quantities characterizing the processes of reflection of a light flux from the interface were obtained. Figure 2 demonstrates the dependences of displacements, normalized to the wavelength λ0 = 2π/k0, for reflected (a) s- and (b) p-polarized waves on the angle of incidence onto the interface between the transparent dielectric with ε1 = 4 and the optically less dense absorbing dielectric with ε '2 = 2 and ε ''2 = 0.002, 0.2, and 0.6 (curves 1–3). One can see from the figure that the displacement of the s-wave is positive for any angle of incidence and has a maximum for the angles of incidence close to the critical angle. For a p-wave, the behavior of the curves is similar in the region of TIR; however, at the angles of incidence close to the Brewster angle, a negative shift is observed. The larger the
absorption in the medium, the wider and less pronounced are the maxima of both the positive and negative displacement. In the case of a medium without absorption, the peak of the negative shift at the Brewster angle is infinitely narrow. Figure 3 shows the dependences of (a) the real and imaginary parts of the permittivity of the resonance layer ε2 and (b) the reflection coefficient R = |r|2 for waves of p- and s-type on the detuning from the resonance frequency ∆ν = ν – ν0 in the region of optical resonance. To construct the above dependences, we chose the following values for parameters: ε0 = 5.8, ε∞ = 5.3, and g/ν0 = 2 × 10–3; ε1 = 1; and the angles of incidence ϕ = 0°, 50°, 80° (curves 1–3). From the presented dependences, it follows that, in the region of negative values of ε '2 , the reflection coefficient is close to unity for waves with both polarizations and at any angles of incidence. Close to the frequency νph ≈ ν0 ε 0 ⁄ ε ∞ , where ε '2 = 0, one can see a sharp decrease in R to virtually zero. Figure 4 demonstrates the dependences of the nor(r) malized shift of the reflected wave packet d p, s /λ0 on the detuning from the resonance frequency ∆ν for pand s-polarized waves and different angles of incidence. The numerical analysis shows that, according to the above stated, for an s-wave, only a positive displacement is realized, with a maximum at ν ≈ νph, which corresponds to the TIR region. At normal inci(r) dence, GH displacement is absent, i.e., d p, s = 0 (curve 1). For a p-wave, a positive displacement also takes place in the TIR region, and a negative one takes place in the region of negative values of ε 2' and in the frequency region corresponding to the Brewster reflection. Thus, for curve 2, the Brewster reflection takes place at ∆ν ≈ 0.06ν0, and for curve 3, at ∆ν ≈ −0.01ν0. However, relationships (9) for the direction of flux of the transmitted wave show that, in the case of the incidence at the Brewster angle, the longitudinal component of the transmitted wave flux is positive. From this it follows that, in this case, the negative displacement of the wave at the interface is not connected with the presence of a negative near-surface flux in the reflecting medium. In our opinion, the reason for this anomaly is that for the angles of incidence close to the Brewster angle, the amplitude of the reflected wave is close to zero. This can lead to a change in the beam shape, for example, to its splitting into two beams when precisely at the Brewster angle no reflection occurs. This change in the beam shape cannot be described by the single parameter of displacement in the two-wave approximation. The reasons for the negative displacement during the Brewster reflection will be investigated later on. OPTICS AND SPECTROSCOPY
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NEGATIVE SHIFT OF A LIGHT BEAM REFLECTED FROM THE INTERFACE
CONCLUSIONS Thus, the numerical analysis shows that the displacement of the beam reflected from the interface between transparent and resonant media is negative for p-polarized waves when the real part of the permittivity is negative ( ε '2 < 0). In this case, as it follows from the expressions obtained for the transmitted wave flux, its longitudinal component is negative. If the wave is incident at the Brewster angle, the displacement is also negative; however, it cannot be related to the presence of a near-surface flux, whose longitudinal component does not coincide, in this case, with the direction of displacement of the reflected beam. REFERENCES 1. L. M. Brekhovskikh, Waves in Layered Media (Nauka, Moscow, 1973; Academic, New York, 1980). 2. F. Goos and H. Hanchen, Ann. Phys. 1 (6), 333 (1947). 3. Bradly M. Jost, Abdul-Azee R. Al-Rashed, and Bahaa E. A. Saleh, Phys. Rev. Lett. 81, 2233 (1998).
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4. S. B. Borisov, N. N. Dadoenkova, I. L. Lyubchanskiœ, and M. I. Lyubchanskiœ, Opt. Spektrosk. 85, 246 (1998) [Opt. Spectrosc. 85, 225 (1998)]. 5. Guided-Wave Optoelectronics, Ed. by T. Tamir (SpringerVerlag, Berlin, 1990; Mir, Moscow, 1991). 6. F. Bretenaker, A. Floch, and L. Dutriaux, Phys. Rev. Lett. 68, 931 (1992). 7. P. D. Kukharchik, V. M. Serdyuk, and I. A. Titovitskiœ, Zh. Tekh. Fiz. 69 (4), 74 (1999) [Tech. Phys. 44, 417 (1999)]. 8. J. J. Cowan and B. Anicin, J. Opt. Soc. Am. 67, 1307 (1977). 9. A. Puri and J. L. Birman, J. Opt. Soc. Am. A 3, 543 (1986). 10. E. Pfleghaar, A. Marseille, and A. Weis, Phys. Rev. Lett. 70, 2281 (1993). 11. V. M. Agranovich and V. L. Ginzburg, Crystal Optics with Spatial Dispersion, and Excitons (Nauka, Moscow, 1979; Springer-Verlag, New York, 1984).
Translated by N. Reutova
