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Effect of width of incident Gaussian beam on the longitudinal shifts and distortion in the reflected beam
Optics Communications 319 (2014) 1–7
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Effect of width of incident Gaussian beam on the longitudinal shifts and distortion in the reflected beam Ziauddin, Sajid Qamar n Department of Physics, COMSATS Institute of Information Technology, Islamabad, Pakistan
art ic l e i nf o
a b s t r a c t
Article history: Received 17 October 2013 Received in revised form 14 December 2013 Accepted 31 December 2013 Available online 11 January 2014
Control of the longitudinal shifts, i.e., spatial and angular Goos–Hänchen (GH) shifts, is revisited to study the effect of width of incident Gaussian beam on the shifts and distortion in the reflected beam. The beam is incident on a cavity consisted of atomic medium where each four-level atom follows N-type atom-field configuration. The atom-field interaction leads to Raman gain process which has been used earlier to observe a significant enhancement of the negative group index, i.e., in the range 103 to 104 for 23Na condensate [G.S. Agarwal, S. Dasgupta, Phys. Rev. A 70 (2004) 023802]. The negative and positive longitudinal shifts could be observed in the reflected light corresponding to the anomalous and normal dispersions of the intracavity medium, respectively. It is observed that the shifts are relatively large for small range of beam width and these became small for large width of the incident beam. It is also noticed that the magnitudes of spatial and angular GH shifts behave differently when the beam width increases. Further, distortion in the reflected beam decreases with an increase in beam width. & 2014 Elsevier B.V. All rights reserved.
Keywords: Reflected beam Effect of width of reflected beam Longitudinal shifts and distortion
1. Introduction It has already been noticed that the center of reflected beam undergoes two main types of shifts which can be categorized as longitudinal and transverse shifts [1–7]. The longitudinal shifts can again be divided into two types, i.e., spatial and angular Goos– Hänchen (GH) shifts [1,2] whereas the transverse shifts can be divided further into spatial and angular Imbert–Fedorov shifts [3,4]. The spatial longitudinal shift, first predicted by Newton, occurs when a beam is reflected with its center presents a tiny spatial shift in the plane of incidence relative to its geometrical optics position [8]. This spatial shift was measured quantitatively by Goos and Hänchen in 1947 [1] and thus named as Goos–Hänchen (GH) shift. In recent years, the study of GH shifts has gained considerable interest due its involvement in optical waveguides and microcavities [9]. Further, it has certain applications, e.g., in bio-sensors [10], optical sensing [11], plasma physics [12] and dispersive media [13]. The phenomenon of the spatial GH shift has been studied in different structures which include for example, photonic crystals, negative refractive media, plasmonics, near-field optics, nonlinear optics, atom optics and acoustics [14–23]. It is a common thought that angular longitudinal (GH) shift is different from spatial longitudinal (GH) shift. The former is due to partial reflection where reflected intensity is less than incident
n
Corresponding author. Tel.: þ 92 51 9049281. E-mail address: [email protected] (S. Qamar).
0030-4018/$ - see front matter & 2014 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.optcom.2013.12.088
intensity [24] whereas the latter is due to total reflection where reflected intensity is equal to incident intensity [2]. The angular GH shift is a small angular deviation from the law of specular reflection θinc ¼ θref [24,25] and it has been observed experimentally in optical [26] and microwave [27] regimes. Recently, dual nature of spatial and angular shifts in optical reflection has been investigated [28]. It has been noticed that the separation between spatial and angular GH shifts becomes artificial when reflection occurs from a lossy surface (e.g., a metal), however, the two shifts are mutually exclusive for lossless media. In another recent study, Luca has observed enhancement in spatial and angular GH shifts using a three-layer surface-plasmon resonance [7]. Therefore, it is instructive to revisit the problem and investigate the behavior of spatial and angular GH shifts for a system where the coherent control of these shifts could be attained. We have suggested some schemes where the coherent control of the spatial GH shift could be achieved via manipulation of the optical susceptibility of an intracavity medium [29–32]. Following the same approach, we investigate an experimentally viable scheme [33] to study influence of beam width on the magnitude of the longitudinal shifts (spatial and angular GH shifts) and distortion in the reflected light when a Gaussian-shaped beam is incident on a cavity. The cavity contains an atomic medium where each four-level atom follows N-type atom-field configuration [33]. It has been observed that the atom-field system exhibits manipulation of the negative group index (for anomalous dispersion) via a control field and thus leads to an enhancement of the group
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index in the range 103 to 104. Here, we consider both normal and anomalous dispersions via a control field to study the behavior of angular and spatial GH shifts and distortion in the reflected beam for different choices of incident beam width.
2. Model 2.1. Fundamental concept We consider a Gaussian-shaped beam having a width w is incident on a cavity which contains atomic vapor cell. The incident beam makes an angle θ with the z-axis. Each atom inside the cavity follows N-type atom-field configuration and the atom-field system Raman gain process. The cavity consists of three layers, i.e., 1, 2 and 3, see schematic in Fig. 1(a). The layers 1 and 2 are walls of the cavity made up of some dielectric materials with permittivity ε1 and each having thickness d1. The layer 3 is the intracavity atomic medium having thickness d2 and permittivity ε2. The permittivity ε2 of the intracavity medium depends on the optical susceptibility χ of the intracavity medium [29–32] via the relation
ε2 ¼ 1 þ χ ;
ð1Þ
where χ is a complex quantity having real and imaginary parts describing the dispersion and gain (or absorption) properties of the intracavity medium. We know that the optical susceptibility of an atomic medium can be manipulated using external controls. This leads to the enhancement of the refractive index [34,35], sub- and superluminal wave propagation [33,36–38], group delay [39–41] and the coherent control of the Goos–Hänchen shift [29–32]. For an incident Gaussian probe light, the electric field at z¼ 0 can be written as Z 1 Eix ðz; yÞ ¼ pffiffiffiffiffiffi Eðky Þeiðkz z þ ky yÞ dky ; ð2Þ 2π where 2 wy 2 Eðky Þ ¼ pffiffiffiffiffiffie½ wy ðky ky0 Þ =4 2π
is the angular spectrum of the Gaussian beam centered at y¼0 of the plane of z¼ 0, ky0 ¼ k sin θ, wy ¼ w sec θ, θ is the incident angle, kz and ky are the z- and y-components of wave number k, respectively. The expression for the reflected probe field can therefore be written as [29,42] Z 1 Erx ðz; yÞjz o 0 ¼ pffiffiffiffiffiffi rðky ÞEðky Þeið kz z þ ky yÞ dky ; ð3Þ 2π
where rðky Þ is the reflection coefficient and can be calculated using the standard characteristic matrix approach for a stratified medium as has been done in [30]. The spatial GH shift Sr can be defined using the normalized first momentum of the electric field for the reflected light, i.e., R1 yjEr ðyÞj2 dy : ð4Þ Sr ¼ R11 rx 2 1 jEx ðyÞj dy We also know that for a Gaussian incident light having a narrow width w the reflected light can be distorted. Therefore, it is also instructive to define explicit expression for the measurement in 1=2 the distortion w r =w, where w r ¼ βr cos θ and the distortion in the reflected light can be measured using the normalized second momentum of the electric field [32,42], i.e., R1 ðy S Þ2 jEr ðyÞj2 dy : ð5Þ βr ¼ 4 1R 1 rr x2 1 jE x ðyÞj dy The angular GH shift is proportional to the angular derivative of the amplitude reflectivity, i.e., jrj which can be written for a plane wave [43] as
ϕr ¼
1 djrj : jrj dky
ð6Þ
As the Gaussian beam is constructed via superposition of plane waves therefore we can write down the expression for angular GH shift for incident Gaussian beam as [44] R1 k ϕ jEðky Þj2 dky Φr ¼ R11 y r : ð7Þ 2 1 jEðky Þj dky We can also define the corresponding distortion w r =w in the 1=2 reflected light as w r ¼ ηr cos θ where R1 ðk Φr Þ2 jEðky Þj2 dky ηr ¼ 4 1 R 1y : ð8Þ 2 1 jEðky Þj dky 2.2. Atom-field interaction We assume that each atom of the intracavity atomic medium follows four-level N-type atom-field configuration [33,45], see Fig. 1(b). The energy levels are designated as ja〉, jb〉, jc〉 and jd〉. The atomic level jd〉 is coupled to the level ja〉 via a pump field E1 (frequency ν1) and level jc〉 is coupled to level jb〉 via a control field E2 (frequency ν2). The Rabi frequencies (field detunings) corresponding to the pump and control fields are Ω1 (Δ1) and Ω2 (Δ2), respectively. A Gaussian-shaped beam acting as probe pulse is incident from vacuum on the cavity. The field corresponding to the probe pulse couples the atomic transition jc〉 to ja〉 and the
Fig. 1. (a) The light is incident on the wall of the cavity from vacuum making an angle θ with the z-axis. The cavity consists of three layers, i.e., two walls of the cavity (layers 1 and 3) each having thickness d1 and intracavity atomic medium (layer 2) with thickness d2. (b) Energy-level diagram of four-level N-type atomic system.
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Fig. 2. Real (solid line) and imaginary (dashed line) parts of the susceptibility are plotted versus probe field detuning Δp for (a) Ω2 ¼ 2:8γ and (b) Ω2 ¼ 5γ. The remaining parameters are selected as Δ1 ¼ 50γ, Δ2 ¼ 0, Ω1 ¼ 4γ, γ da ¼ γ ca ¼ γ cb ¼ γ db ¼ 2γ, Γ dc ¼ 0:01γ, Γ bc ¼ Γ bd ¼ Γ ac ¼ Γ ad ¼ 2:01γ and Γ ab ¼ 4:01γ, these are the same values as has been used in [33].
corresponding Rabi frequency and probe field detuning are Ωp and Δp, respectively. The interaction Hamiltonian in the rotating-wave and dipole approximation for the atom-field interaction can be written down as ℏ V ¼ ½Ω1 e iΔ1 t a〉〈d þ Ω2 e iΔ2 t b〉〈cj 2 þ Ωp e iΔp t ja〉〈cj þ H:c:: ð9Þ The nonlinear Raman gain susceptibility χ ¼ βΩ1 D is calculated in a similar fashion as described in [33] where β ¼ (3N λp3)/(32π3) with N and λP being the atomic number density and wavelength of light, respectively, whereas D is calculated as " i 2Γ ad ½Γ ab iðΔp Δ2 Þ D¼ A ðγ þ γ ÞðΓ 2 þ Δ2 Þ da ca ad 1 # ½Γ ab iðΔp Δ2 Þ½Γ bd iðΔp Δ1 Δ2 Þ jΩ2 j2 =4 þ ; ðΓ ad þ iΔ1 Þ½½Γ dc iðΔp Δ1 Þ½Γ bd iðΔp Δ1 Δ2 Þ þ jΩ2 j2 =4 2
ð10Þ with A ¼ ðΓ ac iΔp Þ½Γ ab iðΔp Δ2 Þ þ jΩ2 j =4, where γqa and γqb (q A c; d) are the spontaneous decay rates from levels ja〉 and jb〉 to the ground levels and Γij (ij A a; b; c; d) is the dephasing rate corresponding to the coherence between levels ja〉-jd〉, ja〉-jb〉, jb〉-jd〉 and ja〉-jc〉, except Γdc which is the collisional relaxation rate. 2
3. Results and discussion We know from an earlier study [33] that the optical susceptibility of a four-level atomic system in N-type configuration can be manipulated using a control field which leads to a control over dispersion behavior of the atomic medium. For small intensity of the control field atomic medium exhibits normal dispersion (slow light behavior), however, as the intensity of the control field increases the dispersion properties of the medium change and anomalous dispersion (fast light behavior) occurs. In the former case the group index is positive whereas for the latter case the group index becomes negative. Further, the positive and negative group index can lead to positive and negative longitudinal (spatial and angular GH) shifts in the reflected beam [30–32]. Thus we can get a control over the shifts in the reflected beam via strength of the control field. In the following, we initially discuss manipulation of the Raman gain process via optical susceptibility χ of the atomic medium using control field E2 with Rabi frequency Ω2. For a more insight we also present a dressed state analysis corresponding to the atomic transition jb〉2jc〉. Next, we investigate the behavior of
the longitudinal shifts in the reflected beam when the width of the incident Gaussian beam changes and finally we study distortion in the reflected beam.
3.1. Manipulation of the Raman gain process via control field E2 We consider that the atom is initially in level jd〉 and a far detuned pump field E1 is applied between levels jd〉 and ja〉 the corresponding Rabi frequency is Ω1. Due to the large detuning Δ1 of the pump field, the atom does not make transition to the level ja〉, however, in the presence of probe field Ep, atom makes Raman transition to jc〉. This leads to a single Raman gain peak (for low strength of the control field E2) corresponding to the probe field detuning Δp. In the presence of strong control field single Raman gain peak splits into two gain peaks. This is due to the Stark splitting of the atomic levels jc〉 and jb〉 into sublevels. This must be noticed that no splitting of the levels ja〉 and jd〉 takes place due to the large detuning associated with the pump field E1 corresponding to the atomic transition between ja〉 and jd〉. For single gain peak, we have observed normal dispersion whereas anomalous dispersion has been observed corresponding to the two gain peaks. In Fig. 2, we plot real and imaginary parts of the susceptibility (χ) versus probe field detuning Δp for two choices of control field Rabi frequency, i.e., Ω2 is equal to (a) 2.8γ and (b) 5γ. The remaining parameters are selected as Δ1 ¼ 50γ , Δ2 ¼ 0, Ω1 ¼ 4γ , γ da ¼ γ ca ¼ γ cb ¼ γ db ¼ 2γ , Γ dc ¼ 0:01γ , Γ bc ¼ Γ bd ¼ Γ ac ¼ Γ ad ¼ 2:01γ and Γ ab ¼ 4:01γ with the Einstein A coefficient for the D2 transition for 23Na is 2π 9:795 106 s 1 [33]. For Ω2 ¼ 2:8γ , the intracavity atomic medium exhibits normal dispersion and a single gain peak appears in the spectrum at the central frequency of the probe pulse corresponding to the detuning Δ1 ¼ 50γ , see Fig. 2(a). However, when the control field becomes large enough, i.e., Ω2 ¼ 5γ , the single gain peak splits into a doublet and dispersion properties of the intracavity medium become anomalous, see Fig. 2(b). This is due to the splitting of the atomic levels jc〉 and jb〉 into sublevels. The separation between the maxima of the two gain peaks is directly proportional to the magnitude of the Rabi frequency Ω2 which is the separation between the two sublevels. This implies that the control field creates two dressed states and the gain doublet arises as a consequence of Raman transitions to these states (sublevels of jc〉). Here, we present a dressed state analysis for the atomic transition jb〉 to jc〉 which is coupled to control field E2 having the corresponding Rabi frequency Ω2. We can write the reduced Hamiltonian Hc as H c ¼ ℏΔ2 jb〉〈bjþ ℏΩ2 jb〉〈cjþ H:c:
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Diagonalization of this Hamiltonian leads to the eigenvalue equation λ Δ2 λ Ω ¼ 0 with the corresponding eigenvalues qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi λ1;2 ¼ ðΔ2 7 Δ22 þ4Ω22 Þ=2. The corresponding dressed states are 2
2 2
jb〉 ¼ Sjα〉 Cjβ〉 and jc〉 ¼ Sjα〉 þ Cjβ 〉; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 2 where S ¼ λ1 = λ1 þ Ω2 and C ¼ Ω2 = λ1 þ Ω2 . This exhibits that due to the presence of strong control field the Raman transition from jd〉 to jc〉 becomes transition from jd〉 to the dressed states jα〉 and jβ 〉, see Fig. 3. As a result the two Raman gain peaks emerge with their centers corresponding to eigenvalues λ1;2 . 3.2. Behavior of longitudinal shifts versus width of the incident Gaussian-shaped beam For an arbitrary choice of incident angle θ ¼ 0.13 rad, we study the behavior of longitudinal (spatial and angular GH) shifts versus width of the incident Gaussian beam. We consider that the intracavity atomic medium exhibits normal dispersion (Ω2 ¼ 2:8γ ). We select probe field detuning Δp ¼ 50γ , thickness of each wall of the cavity as d1 ¼ 0:2 μm and length of the intracavity medium as d2 ¼ 5 μm whereas the remaining parameters are the same as in Fig. 2(a). We would like to mention that our choice of probe field detuning Δp ¼ 50γ corresponds to Δ1 and is required for Raman transition from jd〉 to jc〉. Using Eqs. (4) and (7), we plot spatial and angular GH shifts (Sr and Φr, in units of λp which is in optical domain and we keep it fixed as in [47]) versus width (w) of the incident Gaussian beam, see Fig. 4(a). Initially, spatial GH shift Sr increases with beam width w and becomes maximum for w ffi 5λp, i.e., approximately equal to 7 10 3 λp . Then it decreases exponentially with further increase
in w and becomes approximately equal to 200 10 6 λp at w Z 60λp . For further increase in w, the change in spatial shift Sr is insignificant and can be considered constant for higher values of w, see inset of Fig. 4(a). This behavior of the spatial GH shift is in accordance with some earlier experimental and theoretical observations in frustrated total reflection [46] and in weakly absorbing media [32,47]. For angular shift, we notice that as beam width w increases the angular shift Φr decreases and gets a minimum value of 3 10 3 λp which is remained almost constant for w Z 10λp . The analysis of spatial and angular shifts shows that the shifts are large for small values of the beam width, however, these shifts decrease with an increase in beam width. Next, we change the strength of the control field and select Ω2 ¼ 5γ . For this value of the Rabi frequency Ω2 the dispersion of the intracavity medium changes from normal to anomalous, see Fig. 2(b). For anomalous dispersion, we expect negative shifts in the reflected beam. We again plot Sr and Φr versus w of the incident Gaussian beam keeping all the other parameters same as in the previous case of normal dispersion. We observe negative GH shifts (both spatial and angular) in the reflected beam. We notice that the magnitude of the negative spatial GH shift, initially, increases with w and becomes maximum, i.e., 10λp at w E 5λp. Further increase in w leads to a sharp decrease in the magnitude of the negative spatial shift and for w Z 60λp it becomes 35 10 2 λp and remained almost constant for higher values of beam width, i.e., w Z 60λp , see inset of Fig. 4 (b). This is again in accordance with earlier observations [32,46,47]. For angular GH shift, the magnitude of the shift is almost zero for small values of the beam width. However, it increases slightly with an increase in beam width w and becomes maximum at around w ¼ 60λp and is remained almost constant for w Z 60λp . From these plots, we can observe that the two shifts are positive and negative for normal and anomalous dispersions, respectively. However, we can also notice that the spatial shift is much larger than the angular shift for small values of beam width, i.e., r 10λp ( r 30λp ) for normal (anomalous) dispersion. The two shifts are equal when w is approximately equal to 15λp (35λp) for normal (anomalous) dispersion. However, the magnitude of angular shift is larger than spatial shift for beam width w Z 15λp ð Z 35λp Þ for normal (anomalous) dispersion. 3.3. Influence of incident beam width on distortion in the reflected beam
Fig. 3. Dressed states picture: due to the presence of strong control field atomic levels jb〉 and jc〉 split into sublevels jα〉 and jβ〉.
Since a Gaussian beam is constructed via superposition of plane waves with each component having same wave-vector (κ ¼ 2π =λ
Fig. 4. Spatial and angular GH shifts are plotted versus Gaussian beam width w for (a) normal dispersion, i.e., Ω2 ¼ 2:8γ and (b) anomalous dispersion, i.e., Ω2 ¼ 5γ. The thickness of each walls of the cavity is d1 ¼ 0:2 μm and thickness of the intracavity medium is d2 ¼ 5 μm. The incident angle of the probe pulse is θ ¼ 0:13 rad and Δp ¼ 50γ. The remaining parameters are the same as in Fig. 2.
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Fig. 5. Plots of distortion in reflected light corresponding to the spatial (a–d) and angular (e–h) shifts versus incident angle θ for two choices of width w of the incident Gaussian pulse, i.e., 20λp and 100λp. For normal and anomalous dispersions, we consider Ω2 is equal to 2:8γ and 5γ, respectively. The remaining parameters are the same as in Fig. 4 for normal and anomalous dispersions.
where λ is the wavelength) with a Gaussian amplitude spectrum; each of its angular spectrum components has a different amplitude. The amplitude of the component decreases when its angular spectrum departs from the centered one. When a Gaussian beam is reflected from a surface of a weakly absorbing medium, each of its plane wave components undergoes a different phase shift and amplitude change. This may lead to a distortion in the reflected beam. The beam width of Gaussian light plays an important role as it determines different modes [47]. The number of modes is different for different beam widths of Gaussian light and therefore the profiles of the reflected beam are also different. In Ref. [47] the profiles of the power density of the reflected beams have been plotted at incident angle θ ¼ 76:551 with different beam widths. It has been observed that the number of modes varies with changing the width of the Gaussian beam. This is due to the fact that for small width of the Gaussian beam the reflected beam is highly distorted and there exist many number of modes. By increasing
the beam width of the Gaussian light the number of modes decreases and the distortion decreases as well. It has been reported by Wang et al. [47] that for the beam width of 50λp, the reflected beam splits into three beams which reduces to two for 100λ. They have observed that the reflected beam does not split when the beam width is 200λp, however, the distortion still exists. The distortion vanishes when the beam width becomes 500λp and a near perfect Gaussian-shaped beam is reflected. Their conclusion was that larger the beam width is the less important the missing mode is and therefore more perfect Gaussian-shaped beam is reflected. We study the distortion in the reflected beam for spatial and angular GH shifts by considering two different values of width of incident Gaussian beam. To measure the distortion in the reflected light for spatial GH shift, we use Eq. (5) and plot the ratio wr =w (distortion) versus incident angle θ of the Gaussian beam for normal and anomalous dispersions of the intracavity medium. We consider two different values of beam width, i.e., w ¼ 20λp and
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w ¼ 100λp while keeping all the other parameters same as in Fig. 4 for normal and anomalous dispersions. Corresponding to the spatial GH shift, we plot distortion in the reflected beam versus incident angle for normal dispersion of the intracavity medium, see Fig. 5(a) and (b). The distortion of light decreases for higher values of beam width of Gaussian light, compare Fig. 5(a) and (b). Similarly, we plot the distortion in the reflected beam versus incident angle for anomalous dispersion of the intracavity medium. We again observe that distortion decreases for higher values of beam width, see Fig. 5(c) and (d). We notice that the distortion in the reflected light for the anomalous dispersion is relatively high as compared to the normal dispersion. It is due to the fact that anomalous dispersion leads to negative group index of the intracavity medium and therefore the light is highly distorted. Similarly in Fig. 5(e)–(h), we plot distortion in the reflected beam corresponding to the angular GH shift and again for the same values of beam width w as in the case of spatial GH shift, i.e., 20λp and 100λp. We use Eq. (8) and plot the distortion versus incident angle θ for the normal and anomalous dispersions of the intracavity medium, see Fig. 5(e)–(h). We observe similar behavior of distortion in the reflected beam for normal and anomalous dispersions as we have observed previously in the case of spatial GH shift, i.e., distortion decreases with an increase in beam width w. However, for normal dispersion of the intracavity medium the distortion corresponding to the angular shift increases with incident angle which is different from the case of spatial shift, compare Fig. 5(e) and (f) with (a) and (b), respectively. This analysis implies that the distortion decreases with an increase in width of the incident pulse. For narrow width of the Gaussian beam, the beam penetrates more into the medium and as the penetration depth is proportional to the magnitude of the shift therefore a relatively large shift appears. The magnitude of the shift decreases sharply with increasing width of the incident Gaussian beam. However, it becomes very small and remained constant for large enough value of the beam width. It is due to the fact that for large width of the Gaussian light the characteristic of the Gaussian beam is similar to the uniform plane wave.
4. Summary We have proposed a scheme to study the effect of width of incident Gaussian beam on the two longitudinal shifts, i.e., spatial and angular GH shifts. We have considered that a Gaussian beam having width w is incident on a cavity from vacuum (with permittivity ε0) making an angle θ with the z-axis. The cavity consists of three layers, i.e., two walls of the cavity (layers 1 and 3) having thickness d1 each and an intracavity medium (layer 2) of length d2. The walls of the cavity are made of some dielectric material with permittivity ε1 whereas the intracavity medium is an atomic medium consisted of four-level atoms. The permittivity of the intracavity medium is ε2 which is directly related to the optical susceptibility χ of the intracavity medium. We have considered two strong driving fields (pump and control fields) and a weak Gaussian probe beam. The atom-field configuration thus follows an N-type system and exhibits Raman gain process. The normal and anomalous dispersions of the intracavity medium can be attained via the control field. We have studied the effect of beam width on the spatial and angular GH shifts in the reflected beam corresponding to the normal and anomalous dispersions. As expected we have observed that both shifts are positive (negative) for normal (anomalous) dispersion of the intracavity medium. We have fixed the incident angle as θ ¼0.13 rad and studied the behavior of these shifts for
different choices of width of the Gaussian incident beam. The spatial shift is large for small values of beam width and becomes tiny as the width increases. However, these tiny shifts become almost independent of beam width for w Z60λp . We have also noticed that the magnitudes of the two shifts behave differently for different ranges of beam width. Next, we have changed the behavior of the intracavity gain medium from normal to anomalous dispersions via the control field and investigated again the behavior of the two shifts (spatial and angular) versus w for a fixed incident angle θ ¼ 0:13 rad. We have observed negative shifts in the reflected beam. We have observed that the two shifts initially have increased with width w and then have decreased for much higher values of w. This is in agreement with previous experimental observations [46,47]. For anomalous dispersion of the intracavity medium, we have noticed that the spatial GH shift has become maximum at beam width w ¼ 5λp while the angular GH shift has become maximum at beam width w ¼ 50λp . We have also studied the distortion in the reflected probe light beam in spatial and angular GH shifts and have noticed that the distortion in the reflected beam for the two shifts has decreased for higher values of beam widths. It has been observed that a relatively large amplitude of the shifts could be observed in the reflected beam for small beam widths, however, this accompanies relatively large distortion in the reflected beam. Here we would also like to mention that in all of our analyses we have considered the incident angle θ ¼0.13 rad, in fact, this is one possible choice of the incident angle. We know from earlier studies [29–31] that cavity behaves like Febry–Perot resonator and multiple GH shifts could be observed at different incident angles.
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Effect of width of incident Gaussian beam on the longitudinal shifts and distortion in the reflected beam Ziauddin, Sajid Qamar n Department of Physics, COMSATS Institute of Information Technology, Islamabad, Pakistan
art ic l e i nf o
a b s t r a c t
Article history: Received 17 October 2013 Received in revised form 14 December 2013 Accepted 31 December 2013 Available online 11 January 2014
Control of the longitudinal shifts, i.e., spatial and angular Goos–Hänchen (GH) shifts, is revisited to study the effect of width of incident Gaussian beam on the shifts and distortion in the reflected beam. The beam is incident on a cavity consisted of atomic medium where each four-level atom follows N-type atom-field configuration. The atom-field interaction leads to Raman gain process which has been used earlier to observe a significant enhancement of the negative group index, i.e., in the range 103 to 104 for 23Na condensate [G.S. Agarwal, S. Dasgupta, Phys. Rev. A 70 (2004) 023802]. The negative and positive longitudinal shifts could be observed in the reflected light corresponding to the anomalous and normal dispersions of the intracavity medium, respectively. It is observed that the shifts are relatively large for small range of beam width and these became small for large width of the incident beam. It is also noticed that the magnitudes of spatial and angular GH shifts behave differently when the beam width increases. Further, distortion in the reflected beam decreases with an increase in beam width. & 2014 Elsevier B.V. All rights reserved.
Keywords: Reflected beam Effect of width of reflected beam Longitudinal shifts and distortion
1. Introduction It has already been noticed that the center of reflected beam undergoes two main types of shifts which can be categorized as longitudinal and transverse shifts [1–7]. The longitudinal shifts can again be divided into two types, i.e., spatial and angular Goos– Hänchen (GH) shifts [1,2] whereas the transverse shifts can be divided further into spatial and angular Imbert–Fedorov shifts [3,4]. The spatial longitudinal shift, first predicted by Newton, occurs when a beam is reflected with its center presents a tiny spatial shift in the plane of incidence relative to its geometrical optics position [8]. This spatial shift was measured quantitatively by Goos and Hänchen in 1947 [1] and thus named as Goos–Hänchen (GH) shift. In recent years, the study of GH shifts has gained considerable interest due its involvement in optical waveguides and microcavities [9]. Further, it has certain applications, e.g., in bio-sensors [10], optical sensing [11], plasma physics [12] and dispersive media [13]. The phenomenon of the spatial GH shift has been studied in different structures which include for example, photonic crystals, negative refractive media, plasmonics, near-field optics, nonlinear optics, atom optics and acoustics [14–23]. It is a common thought that angular longitudinal (GH) shift is different from spatial longitudinal (GH) shift. The former is due to partial reflection where reflected intensity is less than incident
n
Corresponding author. Tel.: þ 92 51 9049281. E-mail address: [email protected] (S. Qamar).
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intensity [24] whereas the latter is due to total reflection where reflected intensity is equal to incident intensity [2]. The angular GH shift is a small angular deviation from the law of specular reflection θinc ¼ θref [24,25] and it has been observed experimentally in optical [26] and microwave [27] regimes. Recently, dual nature of spatial and angular shifts in optical reflection has been investigated [28]. It has been noticed that the separation between spatial and angular GH shifts becomes artificial when reflection occurs from a lossy surface (e.g., a metal), however, the two shifts are mutually exclusive for lossless media. In another recent study, Luca has observed enhancement in spatial and angular GH shifts using a three-layer surface-plasmon resonance [7]. Therefore, it is instructive to revisit the problem and investigate the behavior of spatial and angular GH shifts for a system where the coherent control of these shifts could be attained. We have suggested some schemes where the coherent control of the spatial GH shift could be achieved via manipulation of the optical susceptibility of an intracavity medium [29–32]. Following the same approach, we investigate an experimentally viable scheme [33] to study influence of beam width on the magnitude of the longitudinal shifts (spatial and angular GH shifts) and distortion in the reflected light when a Gaussian-shaped beam is incident on a cavity. The cavity contains an atomic medium where each four-level atom follows N-type atom-field configuration [33]. It has been observed that the atom-field system exhibits manipulation of the negative group index (for anomalous dispersion) via a control field and thus leads to an enhancement of the group
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index in the range 103 to 104. Here, we consider both normal and anomalous dispersions via a control field to study the behavior of angular and spatial GH shifts and distortion in the reflected beam for different choices of incident beam width.
2. Model 2.1. Fundamental concept We consider a Gaussian-shaped beam having a width w is incident on a cavity which contains atomic vapor cell. The incident beam makes an angle θ with the z-axis. Each atom inside the cavity follows N-type atom-field configuration and the atom-field system Raman gain process. The cavity consists of three layers, i.e., 1, 2 and 3, see schematic in Fig. 1(a). The layers 1 and 2 are walls of the cavity made up of some dielectric materials with permittivity ε1 and each having thickness d1. The layer 3 is the intracavity atomic medium having thickness d2 and permittivity ε2. The permittivity ε2 of the intracavity medium depends on the optical susceptibility χ of the intracavity medium [29–32] via the relation
ε2 ¼ 1 þ χ ;
ð1Þ
where χ is a complex quantity having real and imaginary parts describing the dispersion and gain (or absorption) properties of the intracavity medium. We know that the optical susceptibility of an atomic medium can be manipulated using external controls. This leads to the enhancement of the refractive index [34,35], sub- and superluminal wave propagation [33,36–38], group delay [39–41] and the coherent control of the Goos–Hänchen shift [29–32]. For an incident Gaussian probe light, the electric field at z¼ 0 can be written as Z 1 Eix ðz; yÞ ¼ pffiffiffiffiffiffi Eðky Þeiðkz z þ ky yÞ dky ; ð2Þ 2π where 2 wy 2 Eðky Þ ¼ pffiffiffiffiffiffie½ wy ðky ky0 Þ =4 2π
is the angular spectrum of the Gaussian beam centered at y¼0 of the plane of z¼ 0, ky0 ¼ k sin θ, wy ¼ w sec θ, θ is the incident angle, kz and ky are the z- and y-components of wave number k, respectively. The expression for the reflected probe field can therefore be written as [29,42] Z 1 Erx ðz; yÞjz o 0 ¼ pffiffiffiffiffiffi rðky ÞEðky Þeið kz z þ ky yÞ dky ; ð3Þ 2π
where rðky Þ is the reflection coefficient and can be calculated using the standard characteristic matrix approach for a stratified medium as has been done in [30]. The spatial GH shift Sr can be defined using the normalized first momentum of the electric field for the reflected light, i.e., R1 yjEr ðyÞj2 dy : ð4Þ Sr ¼ R11 rx 2 1 jEx ðyÞj dy We also know that for a Gaussian incident light having a narrow width w the reflected light can be distorted. Therefore, it is also instructive to define explicit expression for the measurement in 1=2 the distortion w r =w, where w r ¼ βr cos θ and the distortion in the reflected light can be measured using the normalized second momentum of the electric field [32,42], i.e., R1 ðy S Þ2 jEr ðyÞj2 dy : ð5Þ βr ¼ 4 1R 1 rr x2 1 jE x ðyÞj dy The angular GH shift is proportional to the angular derivative of the amplitude reflectivity, i.e., jrj which can be written for a plane wave [43] as
ϕr ¼
1 djrj : jrj dky
ð6Þ
As the Gaussian beam is constructed via superposition of plane waves therefore we can write down the expression for angular GH shift for incident Gaussian beam as [44] R1 k ϕ jEðky Þj2 dky Φr ¼ R11 y r : ð7Þ 2 1 jEðky Þj dky We can also define the corresponding distortion w r =w in the 1=2 reflected light as w r ¼ ηr cos θ where R1 ðk Φr Þ2 jEðky Þj2 dky ηr ¼ 4 1 R 1y : ð8Þ 2 1 jEðky Þj dky 2.2. Atom-field interaction We assume that each atom of the intracavity atomic medium follows four-level N-type atom-field configuration [33,45], see Fig. 1(b). The energy levels are designated as ja〉, jb〉, jc〉 and jd〉. The atomic level jd〉 is coupled to the level ja〉 via a pump field E1 (frequency ν1) and level jc〉 is coupled to level jb〉 via a control field E2 (frequency ν2). The Rabi frequencies (field detunings) corresponding to the pump and control fields are Ω1 (Δ1) and Ω2 (Δ2), respectively. A Gaussian-shaped beam acting as probe pulse is incident from vacuum on the cavity. The field corresponding to the probe pulse couples the atomic transition jc〉 to ja〉 and the
Fig. 1. (a) The light is incident on the wall of the cavity from vacuum making an angle θ with the z-axis. The cavity consists of three layers, i.e., two walls of the cavity (layers 1 and 3) each having thickness d1 and intracavity atomic medium (layer 2) with thickness d2. (b) Energy-level diagram of four-level N-type atomic system.
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Fig. 2. Real (solid line) and imaginary (dashed line) parts of the susceptibility are plotted versus probe field detuning Δp for (a) Ω2 ¼ 2:8γ and (b) Ω2 ¼ 5γ. The remaining parameters are selected as Δ1 ¼ 50γ, Δ2 ¼ 0, Ω1 ¼ 4γ, γ da ¼ γ ca ¼ γ cb ¼ γ db ¼ 2γ, Γ dc ¼ 0:01γ, Γ bc ¼ Γ bd ¼ Γ ac ¼ Γ ad ¼ 2:01γ and Γ ab ¼ 4:01γ, these are the same values as has been used in [33].
corresponding Rabi frequency and probe field detuning are Ωp and Δp, respectively. The interaction Hamiltonian in the rotating-wave and dipole approximation for the atom-field interaction can be written down as ℏ V ¼ ½Ω1 e iΔ1 t a〉〈d þ Ω2 e iΔ2 t b〉〈cj 2 þ Ωp e iΔp t ja〉〈cj þ H:c:: ð9Þ The nonlinear Raman gain susceptibility χ ¼ βΩ1 D is calculated in a similar fashion as described in [33] where β ¼ (3N λp3)/(32π3) with N and λP being the atomic number density and wavelength of light, respectively, whereas D is calculated as " i 2Γ ad ½Γ ab iðΔp Δ2 Þ D¼ A ðγ þ γ ÞðΓ 2 þ Δ2 Þ da ca ad 1 # ½Γ ab iðΔp Δ2 Þ½Γ bd iðΔp Δ1 Δ2 Þ jΩ2 j2 =4 þ ; ðΓ ad þ iΔ1 Þ½½Γ dc iðΔp Δ1 Þ½Γ bd iðΔp Δ1 Δ2 Þ þ jΩ2 j2 =4 2
ð10Þ with A ¼ ðΓ ac iΔp Þ½Γ ab iðΔp Δ2 Þ þ jΩ2 j =4, where γqa and γqb (q A c; d) are the spontaneous decay rates from levels ja〉 and jb〉 to the ground levels and Γij (ij A a; b; c; d) is the dephasing rate corresponding to the coherence between levels ja〉-jd〉, ja〉-jb〉, jb〉-jd〉 and ja〉-jc〉, except Γdc which is the collisional relaxation rate. 2
3. Results and discussion We know from an earlier study [33] that the optical susceptibility of a four-level atomic system in N-type configuration can be manipulated using a control field which leads to a control over dispersion behavior of the atomic medium. For small intensity of the control field atomic medium exhibits normal dispersion (slow light behavior), however, as the intensity of the control field increases the dispersion properties of the medium change and anomalous dispersion (fast light behavior) occurs. In the former case the group index is positive whereas for the latter case the group index becomes negative. Further, the positive and negative group index can lead to positive and negative longitudinal (spatial and angular GH) shifts in the reflected beam [30–32]. Thus we can get a control over the shifts in the reflected beam via strength of the control field. In the following, we initially discuss manipulation of the Raman gain process via optical susceptibility χ of the atomic medium using control field E2 with Rabi frequency Ω2. For a more insight we also present a dressed state analysis corresponding to the atomic transition jb〉2jc〉. Next, we investigate the behavior of
the longitudinal shifts in the reflected beam when the width of the incident Gaussian beam changes and finally we study distortion in the reflected beam.
3.1. Manipulation of the Raman gain process via control field E2 We consider that the atom is initially in level jd〉 and a far detuned pump field E1 is applied between levels jd〉 and ja〉 the corresponding Rabi frequency is Ω1. Due to the large detuning Δ1 of the pump field, the atom does not make transition to the level ja〉, however, in the presence of probe field Ep, atom makes Raman transition to jc〉. This leads to a single Raman gain peak (for low strength of the control field E2) corresponding to the probe field detuning Δp. In the presence of strong control field single Raman gain peak splits into two gain peaks. This is due to the Stark splitting of the atomic levels jc〉 and jb〉 into sublevels. This must be noticed that no splitting of the levels ja〉 and jd〉 takes place due to the large detuning associated with the pump field E1 corresponding to the atomic transition between ja〉 and jd〉. For single gain peak, we have observed normal dispersion whereas anomalous dispersion has been observed corresponding to the two gain peaks. In Fig. 2, we plot real and imaginary parts of the susceptibility (χ) versus probe field detuning Δp for two choices of control field Rabi frequency, i.e., Ω2 is equal to (a) 2.8γ and (b) 5γ. The remaining parameters are selected as Δ1 ¼ 50γ , Δ2 ¼ 0, Ω1 ¼ 4γ , γ da ¼ γ ca ¼ γ cb ¼ γ db ¼ 2γ , Γ dc ¼ 0:01γ , Γ bc ¼ Γ bd ¼ Γ ac ¼ Γ ad ¼ 2:01γ and Γ ab ¼ 4:01γ with the Einstein A coefficient for the D2 transition for 23Na is 2π 9:795 106 s 1 [33]. For Ω2 ¼ 2:8γ , the intracavity atomic medium exhibits normal dispersion and a single gain peak appears in the spectrum at the central frequency of the probe pulse corresponding to the detuning Δ1 ¼ 50γ , see Fig. 2(a). However, when the control field becomes large enough, i.e., Ω2 ¼ 5γ , the single gain peak splits into a doublet and dispersion properties of the intracavity medium become anomalous, see Fig. 2(b). This is due to the splitting of the atomic levels jc〉 and jb〉 into sublevels. The separation between the maxima of the two gain peaks is directly proportional to the magnitude of the Rabi frequency Ω2 which is the separation between the two sublevels. This implies that the control field creates two dressed states and the gain doublet arises as a consequence of Raman transitions to these states (sublevels of jc〉). Here, we present a dressed state analysis for the atomic transition jb〉 to jc〉 which is coupled to control field E2 having the corresponding Rabi frequency Ω2. We can write the reduced Hamiltonian Hc as H c ¼ ℏΔ2 jb〉〈bjþ ℏΩ2 jb〉〈cjþ H:c:
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Diagonalization of this Hamiltonian leads to the eigenvalue equation λ Δ2 λ Ω ¼ 0 with the corresponding eigenvalues qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi λ1;2 ¼ ðΔ2 7 Δ22 þ4Ω22 Þ=2. The corresponding dressed states are 2
2 2
jb〉 ¼ Sjα〉 Cjβ〉 and jc〉 ¼ Sjα〉 þ Cjβ 〉; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 2 where S ¼ λ1 = λ1 þ Ω2 and C ¼ Ω2 = λ1 þ Ω2 . This exhibits that due to the presence of strong control field the Raman transition from jd〉 to jc〉 becomes transition from jd〉 to the dressed states jα〉 and jβ 〉, see Fig. 3. As a result the two Raman gain peaks emerge with their centers corresponding to eigenvalues λ1;2 . 3.2. Behavior of longitudinal shifts versus width of the incident Gaussian-shaped beam For an arbitrary choice of incident angle θ ¼ 0.13 rad, we study the behavior of longitudinal (spatial and angular GH) shifts versus width of the incident Gaussian beam. We consider that the intracavity atomic medium exhibits normal dispersion (Ω2 ¼ 2:8γ ). We select probe field detuning Δp ¼ 50γ , thickness of each wall of the cavity as d1 ¼ 0:2 μm and length of the intracavity medium as d2 ¼ 5 μm whereas the remaining parameters are the same as in Fig. 2(a). We would like to mention that our choice of probe field detuning Δp ¼ 50γ corresponds to Δ1 and is required for Raman transition from jd〉 to jc〉. Using Eqs. (4) and (7), we plot spatial and angular GH shifts (Sr and Φr, in units of λp which is in optical domain and we keep it fixed as in [47]) versus width (w) of the incident Gaussian beam, see Fig. 4(a). Initially, spatial GH shift Sr increases with beam width w and becomes maximum for w ffi 5λp, i.e., approximately equal to 7 10 3 λp . Then it decreases exponentially with further increase
in w and becomes approximately equal to 200 10 6 λp at w Z 60λp . For further increase in w, the change in spatial shift Sr is insignificant and can be considered constant for higher values of w, see inset of Fig. 4(a). This behavior of the spatial GH shift is in accordance with some earlier experimental and theoretical observations in frustrated total reflection [46] and in weakly absorbing media [32,47]. For angular shift, we notice that as beam width w increases the angular shift Φr decreases and gets a minimum value of 3 10 3 λp which is remained almost constant for w Z 10λp . The analysis of spatial and angular shifts shows that the shifts are large for small values of the beam width, however, these shifts decrease with an increase in beam width. Next, we change the strength of the control field and select Ω2 ¼ 5γ . For this value of the Rabi frequency Ω2 the dispersion of the intracavity medium changes from normal to anomalous, see Fig. 2(b). For anomalous dispersion, we expect negative shifts in the reflected beam. We again plot Sr and Φr versus w of the incident Gaussian beam keeping all the other parameters same as in the previous case of normal dispersion. We observe negative GH shifts (both spatial and angular) in the reflected beam. We notice that the magnitude of the negative spatial GH shift, initially, increases with w and becomes maximum, i.e., 10λp at w E 5λp. Further increase in w leads to a sharp decrease in the magnitude of the negative spatial shift and for w Z 60λp it becomes 35 10 2 λp and remained almost constant for higher values of beam width, i.e., w Z 60λp , see inset of Fig. 4 (b). This is again in accordance with earlier observations [32,46,47]. For angular GH shift, the magnitude of the shift is almost zero for small values of the beam width. However, it increases slightly with an increase in beam width w and becomes maximum at around w ¼ 60λp and is remained almost constant for w Z 60λp . From these plots, we can observe that the two shifts are positive and negative for normal and anomalous dispersions, respectively. However, we can also notice that the spatial shift is much larger than the angular shift for small values of beam width, i.e., r 10λp ( r 30λp ) for normal (anomalous) dispersion. The two shifts are equal when w is approximately equal to 15λp (35λp) for normal (anomalous) dispersion. However, the magnitude of angular shift is larger than spatial shift for beam width w Z 15λp ð Z 35λp Þ for normal (anomalous) dispersion. 3.3. Influence of incident beam width on distortion in the reflected beam
Fig. 3. Dressed states picture: due to the presence of strong control field atomic levels jb〉 and jc〉 split into sublevels jα〉 and jβ〉.
Since a Gaussian beam is constructed via superposition of plane waves with each component having same wave-vector (κ ¼ 2π =λ
Fig. 4. Spatial and angular GH shifts are plotted versus Gaussian beam width w for (a) normal dispersion, i.e., Ω2 ¼ 2:8γ and (b) anomalous dispersion, i.e., Ω2 ¼ 5γ. The thickness of each walls of the cavity is d1 ¼ 0:2 μm and thickness of the intracavity medium is d2 ¼ 5 μm. The incident angle of the probe pulse is θ ¼ 0:13 rad and Δp ¼ 50γ. The remaining parameters are the same as in Fig. 2.
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Fig. 5. Plots of distortion in reflected light corresponding to the spatial (a–d) and angular (e–h) shifts versus incident angle θ for two choices of width w of the incident Gaussian pulse, i.e., 20λp and 100λp. For normal and anomalous dispersions, we consider Ω2 is equal to 2:8γ and 5γ, respectively. The remaining parameters are the same as in Fig. 4 for normal and anomalous dispersions.
where λ is the wavelength) with a Gaussian amplitude spectrum; each of its angular spectrum components has a different amplitude. The amplitude of the component decreases when its angular spectrum departs from the centered one. When a Gaussian beam is reflected from a surface of a weakly absorbing medium, each of its plane wave components undergoes a different phase shift and amplitude change. This may lead to a distortion in the reflected beam. The beam width of Gaussian light plays an important role as it determines different modes [47]. The number of modes is different for different beam widths of Gaussian light and therefore the profiles of the reflected beam are also different. In Ref. [47] the profiles of the power density of the reflected beams have been plotted at incident angle θ ¼ 76:551 with different beam widths. It has been observed that the number of modes varies with changing the width of the Gaussian beam. This is due to the fact that for small width of the Gaussian beam the reflected beam is highly distorted and there exist many number of modes. By increasing
the beam width of the Gaussian light the number of modes decreases and the distortion decreases as well. It has been reported by Wang et al. [47] that for the beam width of 50λp, the reflected beam splits into three beams which reduces to two for 100λ. They have observed that the reflected beam does not split when the beam width is 200λp, however, the distortion still exists. The distortion vanishes when the beam width becomes 500λp and a near perfect Gaussian-shaped beam is reflected. Their conclusion was that larger the beam width is the less important the missing mode is and therefore more perfect Gaussian-shaped beam is reflected. We study the distortion in the reflected beam for spatial and angular GH shifts by considering two different values of width of incident Gaussian beam. To measure the distortion in the reflected light for spatial GH shift, we use Eq. (5) and plot the ratio wr =w (distortion) versus incident angle θ of the Gaussian beam for normal and anomalous dispersions of the intracavity medium. We consider two different values of beam width, i.e., w ¼ 20λp and
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w ¼ 100λp while keeping all the other parameters same as in Fig. 4 for normal and anomalous dispersions. Corresponding to the spatial GH shift, we plot distortion in the reflected beam versus incident angle for normal dispersion of the intracavity medium, see Fig. 5(a) and (b). The distortion of light decreases for higher values of beam width of Gaussian light, compare Fig. 5(a) and (b). Similarly, we plot the distortion in the reflected beam versus incident angle for anomalous dispersion of the intracavity medium. We again observe that distortion decreases for higher values of beam width, see Fig. 5(c) and (d). We notice that the distortion in the reflected light for the anomalous dispersion is relatively high as compared to the normal dispersion. It is due to the fact that anomalous dispersion leads to negative group index of the intracavity medium and therefore the light is highly distorted. Similarly in Fig. 5(e)–(h), we plot distortion in the reflected beam corresponding to the angular GH shift and again for the same values of beam width w as in the case of spatial GH shift, i.e., 20λp and 100λp. We use Eq. (8) and plot the distortion versus incident angle θ for the normal and anomalous dispersions of the intracavity medium, see Fig. 5(e)–(h). We observe similar behavior of distortion in the reflected beam for normal and anomalous dispersions as we have observed previously in the case of spatial GH shift, i.e., distortion decreases with an increase in beam width w. However, for normal dispersion of the intracavity medium the distortion corresponding to the angular shift increases with incident angle which is different from the case of spatial shift, compare Fig. 5(e) and (f) with (a) and (b), respectively. This analysis implies that the distortion decreases with an increase in width of the incident pulse. For narrow width of the Gaussian beam, the beam penetrates more into the medium and as the penetration depth is proportional to the magnitude of the shift therefore a relatively large shift appears. The magnitude of the shift decreases sharply with increasing width of the incident Gaussian beam. However, it becomes very small and remained constant for large enough value of the beam width. It is due to the fact that for large width of the Gaussian light the characteristic of the Gaussian beam is similar to the uniform plane wave.
4. Summary We have proposed a scheme to study the effect of width of incident Gaussian beam on the two longitudinal shifts, i.e., spatial and angular GH shifts. We have considered that a Gaussian beam having width w is incident on a cavity from vacuum (with permittivity ε0) making an angle θ with the z-axis. The cavity consists of three layers, i.e., two walls of the cavity (layers 1 and 3) having thickness d1 each and an intracavity medium (layer 2) of length d2. The walls of the cavity are made of some dielectric material with permittivity ε1 whereas the intracavity medium is an atomic medium consisted of four-level atoms. The permittivity of the intracavity medium is ε2 which is directly related to the optical susceptibility χ of the intracavity medium. We have considered two strong driving fields (pump and control fields) and a weak Gaussian probe beam. The atom-field configuration thus follows an N-type system and exhibits Raman gain process. The normal and anomalous dispersions of the intracavity medium can be attained via the control field. We have studied the effect of beam width on the spatial and angular GH shifts in the reflected beam corresponding to the normal and anomalous dispersions. As expected we have observed that both shifts are positive (negative) for normal (anomalous) dispersion of the intracavity medium. We have fixed the incident angle as θ ¼0.13 rad and studied the behavior of these shifts for
different choices of width of the Gaussian incident beam. The spatial shift is large for small values of beam width and becomes tiny as the width increases. However, these tiny shifts become almost independent of beam width for w Z60λp . We have also noticed that the magnitudes of the two shifts behave differently for different ranges of beam width. Next, we have changed the behavior of the intracavity gain medium from normal to anomalous dispersions via the control field and investigated again the behavior of the two shifts (spatial and angular) versus w for a fixed incident angle θ ¼ 0:13 rad. We have observed negative shifts in the reflected beam. We have observed that the two shifts initially have increased with width w and then have decreased for much higher values of w. This is in agreement with previous experimental observations [46,47]. For anomalous dispersion of the intracavity medium, we have noticed that the spatial GH shift has become maximum at beam width w ¼ 5λp while the angular GH shift has become maximum at beam width w ¼ 50λp . We have also studied the distortion in the reflected probe light beam in spatial and angular GH shifts and have noticed that the distortion in the reflected beam for the two shifts has decreased for higher values of beam widths. It has been observed that a relatively large amplitude of the shifts could be observed in the reflected beam for small beam widths, however, this accompanies relatively large distortion in the reflected beam. Here we would also like to mention that in all of our analyses we have considered the incident angle θ ¼0.13 rad, in fact, this is one possible choice of the incident angle. We know from earlier studies [29–31] that cavity behaves like Febry–Perot resonator and multiple GH shifts could be observed at different incident angles.
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