Transmission and reflection in a perfectly amplifying and absorbing medium
چکیده
We study transmission and reflection properties of a perfectly amplifying as well as absorbing medium analytically by using the tight binding equation. Different expressions for transmittance and reflectance are obtained for even and odd values of the sample length which is the origin of their oscillatory behavior. In a weak amplifying medium, a cross-over length scale exists below which transmittance and reflectance increase exponentially and above which transmittance decays exponentially while the reflectance gets saturated. Depending on amplification transmittance and reflectance show singular behavior at the cross-over length. In a weak absorbing medium we do not find any cross-over length scale. The transmission coefficient behaves similar to that in an amplifying medium in the asymptotic limit. In a strong amplifying/absorbing medium, the transmission coefficient decays exponentially in the large length limit. In both weak as well as strong absorbing media the logarithm of the reflection coefficient shows the same behavior as obtained in an amplifying medium but with opposite sign. PACS Numbers: 42.25.Bs, 71.55.Jv, 72.10-d Typeset using REVTEX 1 In recent years the study of wave propagation in an active random medium, in the presence of absorption or amplification has attracted much interest [1-23]. Due to the absence of a conservation law for photons, light may be absorbed or amplified in the medium while phase coherence is preserved. The interplay of absorption or amplification and localization has been studied extensively by using the Helmholtz equation with an imaginary dielectric constant of an appropriate sign or by using the Schrödinger equation with imaginary potentials. Experimentally a random amplifying medium can be achieved in a turbid laser dye or a powdered laser crystal [1]. The optical propagation or magnetic excitons in solids which terminate upon reaching trapping sites can be thought of as an absorbing medium. Several results have been obtained in this area. The system plays a dual role as an amplifier or absorber and a reflector [5]. When the strength of the imaginary potential is increased beyond a certain limit both absorber and amplifying scatterer act as a reflector. Absorption suppresses the transmission just as amplification does in the large length limit [6]. A cross-over length scale exists below which the amplification enhances the transmission and above which it reduces the transmission [6,7,17,18]. In contrast, the reflectance saturates in the large length limit for both types of media [17,18]. The distribution of reflectance [8,21], transmittance [5,7] and the phase of transmittance [15] and reflectance [5,17] for both the active media has been studied by several authors analytically [3,4,16,22] as well as numerically [5,17–19]. Most of the studies have been made on active random media. For proper understanding of the effect of randomness in active media it is important to study the perfect absorbing as well as amplifying system. But much less attention has been paid in this direction [9,24,25]. So, we study here the transmission and reflection properties of both the perfect systems by using the tight binding equation, (E − ǫn)cn = V (cn−1 + cn+1). (1) Here, E is the particle energy, V is nearest-neighbor hopping amplitude, ǫn is the n-th site energy. and cn is the amplitude of the n-th site. Without any loss of generality we assume 2 V = 1. For a perfectly active medium ǫn = iη makes the Hamiltonian non-Hermitian and consequently the particle conservation fails. Here η is a fixed real number which is positive (negative) for an absorbing (amplifying) medium. In a previous study [25] it has been shown numerically with some analytical calculations that for a perfectly absorbing medium transmittance decays exponentially with the sample length but reflectance saturates after initial oscillations. On the other hand, for an amplifying medium transmittance initially increases up to a certain length through large oscillations after which it decays and the reflectance also shows an initial increasing behavior after which it saturates to a value larger than unity. In Ref. [9] a perfectly amplifying medium has been studied in the Kronig Penny model. The authors obtain exact expressions for transmission coefficient and the cross-over length. They have also studied the disorder effect on the system. However, we feel that the work on perfect amplifying/absorbing medium is not complete. Some interesting results which may help to understand the properties of active media in the presence of randomness seem to have been overlooked in previous studies [9,24,25]. So, here we perform the analytical study of perfect amplifying as well as absorbing media using Eq. 1. By using the transfer matrix method we obtain exact expressions for the transmission and reflection coefficients for both the media, described by Eq. 1. The oscillatory behavior of both transmittance and reflectance is understood from those expressions. Depending on values of η for the amplifying medium we find singular behavior in the transmission and reflection coefficients at the cross-over length. For both amplifying and absorbing media, the transmittance and reflectance are studied in different ranges of η and sample length. We now discuss the transfer matrix method [26] to calculate the transmission and reflection coefficients. The active medium consisting of N sites (n = 1 to N) is placed between two semi-infinite perfect leads with all site-energies are taken to be zero and V = 1. From Eq. 1 one can easily obtain the site-amplitude for any length of the chain from the initial ones through sequential product of transfer matrices in the following way,