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A Physical Explanation of Excess Quantum Noise Due to Non-Orthogonal Modes

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A Physical Explanation of Excess Quantum Noise Due to Non-Orthogonal Modes OPEN ACCESS Related content - Nonlinear behaviour of a cascade J''JJ A physical explanation of excess quantum noise vector laser A Kul'minskii, R Vilaseca and R Corbalán due to non-orthogonal modes - Polarization and transverse-mode selection in quantum-well vertical-cavity surface-emitting lasers: index- and gain- To cite this article: A M van der Lee et al 2001 New J. Phys. 3 2 guided devices J Martín-Regalado, S Balle, M San Miguel et al. - Polarization phenomena in a View the article online for updates and enhancements. laser coherently pumped by a linearly polarized field R Vilaseca, C Serrat, A Kul'minskii et al. This content was downloaded from IP address 170.106.202.226 on 25/09/2021 at 04:05 A physical explanation of excess quantum noise due to non-orthogonal modes AMvanderLee,MPvanExter,NJvanDruten† and J P Woerdman Huygens Laboratory, Leiden University, PO Box 9504, Leiden, The Netherlands E-mail: [email protected] New Journal of Physics 3 (2001) 2.1–2.15 (http://www.njp.org/) Received 15 November 2000; online 23 March 2001 Abstract. We introduce a physical model of excess quantum noise in a laser with non-orthogonal polarization modes. We discuss the cause of the polarization excess quantum noise in terms of an injected wave excitation factor. Within this context the excess noise is due to the fact that the lasing mode is not the polarization state that experiences the highest gain. 1. Introduction When first introduced by Klaus Petermann in 1979 for gain-guided lasers, the Petermann K- factor was controversial [1]. The K-factor (or excess noise factor) is the factor by which the quantum noise is enhanced with respect to the level set by the classical formula (of 1958) of Schawlow and Townes [2]. It was felt by some that since the Schawlow–Townes formula was set in the ‘stone’ of quantum mechanics, Petermann’s result had to be in violation of fundamental principles. This controversy was later solved by Haus and Kawakami [3], who pointed out that the noise terms for different propagating modes in a loss- or gain-guided cavity are correlated and so were still in agreement with the fundamental principles of quantum mechanics. In 1989 Siegman showed that the excess noise factor is not just a property of gain- or loss-guided lasers but of any laser cavity with non-orthogonal modes [4]. The loss of the laser cavity during a round trip makes the propagation of the light non-unitary, which can lead to non-orthogonal modes. However, the relationship between loss and non-orthogonality is a very subtle one. For different cavity configurations with the same average losses, the K-factor can have totally different values. The transverse eigenmodes of the most common laser resonator, the stable cavity, are also practically orthogonal and thus do not show an excess noise factor. This in contrast to an unstable resonator were non-orthogonal transverse eigenmodes appear naturally. † Present address: Department of Applied Physics, Delft University of Technology, Lorentzweg 1, 2628 CJ Delft, The Netherlands. New Journal of Physics 3 (2001) 2.1–2.15 PII: S1367-2630(01)19094-4 1367-2630/01/000002+15$30.00 © IOP Publishing Ltd and Deutsche Physikalische Gesellschaft 2.2 Experiments measuring the quantum-limited linewidth of such lasers have demonstrated large excess-noise factors [5, 6]. Our present interest is in non-orthogonal polarization modes. In a stable cavity with orthogonal transverse modes, one can make the polarization modes non-orthogonal; this gives a two-mode model system for studying excess quantum noise (in contrast to the many transverse modes in an unstable resonator). Large polarization K-factors have also been demonstrated experimentally [7, 8]. The polarization non-orthogonality has also proven to be a useful model system to demonstrate that excess quantum noise is coloured [9] and that excess quantum noise can seriously hinder intensity noise squeezing below the standard quantum limit [10]. In both situations, the transverse-mode case and the polarization-mode case, the excess noise factor is satisfactorily described by mode-non-orthogonality theory [4]. This theory formally connects non-orthogonal modes to excess noise, but it does not give a clear view on the physical origin of the excess noise factor. For an unstable resonator a more physical picture of excess noise exists, in terms of the injected-wave excitation (IWE) factor [11]. In this picture the origin of excess quantum noise is attributed to the ‘preamplification’ of the quantum noise before it ends up in the lasing mode. A key role is played by the adjoint of the lasing mode, which is defined by the biorthogonality relation; it is the mode that is orthogonal to all other cavity eigenmodes (apart from the lasing mode). In a cavity with non-orthogonal transverse modes, a wave which has the same transverse mode profile as the adjoint mode excites the lasing mode most efficiently (even more efficient than the eigenmode itself). In an unfolded lens guide picture of an unstable resonator as depicted in figure 1, the physical reason of this excitation advantage is easily seen (this figure was taken from [12]). Before the adjoint mode ends up as the lasing mode it has a few loss-free round trips, whereas the eigenmode experiences loss each round trip due to its magnification. It has been shown that this excitation advantage is mathematically equivalent to the K-factor that follows from mode-non-orthogonality theory [4]. The purpose of this paper is to develop an analogous IWE picture for the non-orthogonal polarization modes, in order to clarify the physical origin of the excess quantum noise in this case. We can derive analytical results for the polarization dynamics, as the polarization state space consists of only two modes, contrary to the manifold of modes in transverse mode space. The article starts in section 2 with the basic equations that describe the polarization evolution and the separate polarization anisotropies, which together produce the non-orthogonal modes. Furthermore, we calculate the IWE factor for two distinct regimes in which the laser can operate; these regimes are the single-mode regime (section 3) and a peculiar two-mode regime (section 4). We end the article with conclusions (section 5). 2. Introducing the basic elements In order to create non-orthogonal polarization modes one needs, in essence, two kinds of polarization anisotropies. The first is a polarization anisotropic loss and the second is a polarization anisotropic dispersion. When only one of the two is present the eigenmodes are orthogonal; only the mixture of the two can yield non-orthogonal modes. Experimentally, two ways have been demonstrated to create non-orthogonal polarization modes. One is to use linear dichroism, i.e. a difference in loss of two orthogonal linear polarization states, in combination with circular birefringence, i.e. a difference in the refractive index of the σ± polarized states [7]. The other method is to use linear dichroism in New Journal of Physics 3 (2001) 2.1–2.15 (http://www.njp.org/) 2.3 unstable resonator equivalent lensguide a) b) Figure 1. A confocal unstable resonator is depicted as a series of lenses that form a equivalent lens guide. A resonator round trip is then equivalent to moving one period through the lens guide. In (a) the eigenmode is depicted, in (b) the adjoint mode. In each round trip the eigenmode spills over the outcoupling mirror, whereas the adjoint mode is focused to its diffraction limit before it starts to diverge and becomes the eigenmode. So the adjoint mode has a number of loss- free round trips; this makes injection in this mode more efficient than injection in the eigenmode. combination with (misaligned) linear birefringence, i.e. a difference in refractive index between two orthogonal linearly polarized states [8]. We have experimentally used the first method and this article will treat the polarization dynamics arising from a linear dichroism and a circular birefringence (see [7] for the experimental implementation of linear dichroism and circular birefringence). We will model the laser as a linear amplifier with a fixed gain, the gain being equal to the losses. As the saturation is neglected this describes a laser at threshold. We choose the σ± states as a basis. The time evolution of the electric field is given by the following equation (see [13] for further discussion): d E E + =(M + gI) + (1) dt E− E− where E+ and E− are the (complex) amplitudes of the circular polarizations σ+ and σ−, respectively. The polarization evolution is separated into two parts. The (2 × 2) matrix M contains the polarization-anisotropic part of the evolution (the trace of M is zero). The real and imaginary parts of the eigenvalues of M determine the relative loss and the relative frequency of the eigenmodes respectively (where relative means, with respect to the loss and frequency of the modes in the absence of M). The isotropic part of the evolution is given by gI, where g is a constant and I is the unity matrix. The value of g is fixed by the condition that the lasing mode of the cavity has a net gain of zero. As the trace of M is zero, this implies that g ≤ 0. The dynamics of two complex mode amplitudes has, in principle, four degrees of freedom. As we are not interested in the overall phase factor, we are left with three remaining degrees of freedom which can be chosen as the intensity I and the two polarization parameters θ and χ, New Journal of Physics 3 (2001) 2.1–2.15 (http://www.njp.org/) 2.4 a) χ θ Zσ b) + 2 χ 2θ X Y σ - Figure 2.
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