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Journal of Modern Optics

Comment on Noiseless amplification in cavity-based optical systems with an internal two-photon process. II. Self-frequency-doubling laser and second-harmonic generation, self-down-converting laser`

Andrew G. White, Timothy C. Ralph, Hans‐A. Bachor

Published Mar 1, 1997• 1 citation
Publisher

Abstract

Abstract We consider a cavity containing both a χ(2)-nonlinear material and a laser two-level medium for the cases of self-frequency-doubling and self-down-converting configurations. For both situations, the squeezing properties of the two fields are analysed in detail, while special attention is devoted to the problem of ‘noiseless amplification’ of a small monochromatic signal injected in the lower frequency mode.

DOI
10.1080/09500349708232928

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Abstract We consider a cavity containing both a χ(2)-nonlinear material and a laser two-level medium for the cases of self-frequency-doubling and self-down-converting configurations. For both situations, the squeezing properties of the two fields are analysed in detail, while special attention is devoted to the problem of ‘noiseless amplification’ of a small monochromatic signal injected in the lower frequency mode.
We present the analysis of a laser in which an intractivity χ (2) -nonlinear element couples the lasing mode to its second harmonic. The cavity is resonant for both modes. We investigate the potential for squeezing in this system by linearizing the quantum fluctuations around the steady states of the problem. In principle, more than 50% noise reduction below the vacuum level is possible in the second-harmonic field outside the cavity. The maximum squeezing, which is neither pure amplitude nor pure phase squeezing, occurs at a finite frequency-offset from the cavity resonance.
We present a theoretical investigation of the potential for squeezing of a doubly resonant cavity that contains a laser medium as well as a ${\mathrm{\ensuremath{\chi}}}^{(2)}$ nonlinearity. If the lasing mode is coupled to its second harmonic, i.e., for a self-frequency-doubling laser, we find more than 60% squeezing in the up-converted mode outside the cavity. In the opposite case of a ``self-down-converting laser,'' i.e., when the lasing mode is coupled to its subharmonic, a perfectly squeezed vacuum in the down-converted mode develops into a less squeezed state with a nonvanishing coherent amplitude as the pump rate is increased. Our analysis extends beyond the first stable regime of the classical dynamics and compares results for different laser models.
The authors consider the problem of second harmonic generation in an optical cavity which contains both the laser and the nonlinear crystal. They calculate the spectrum of the amplitude fluctuations in the output field, for the mode which is generated by the laser. They focus on the case that the damping constant of this mode is much larger than all the other relaxation rates of the system, and show that in this limit the spectrum is correctly described by the universal normal form for two-photon processes. Finally, they show in general that in the framework of linearised theories the spectrum of the intensity fluctuations in the output field reduces exactly to the spectrum of squeezing of the amplitude quadrature component.
In the previous chapter we examined nonlinear propagation in a two-level medium, which leads to amplification or absorption of the radiation field. As we will see further on in this volume, when the medium is placed in an optical cavity one realizes a laser in the case of amplification or an optically bistable system in the absorptive configuration. On the other hand, the nonlinear response of the atomic polarization to the electric field leads to further fascinating scenarios [29, 51–60]. For example, a field with central ω 0 can generate harmonic components with central frequencies 2 ω 0 , 3 ω 0 etc., processes that are called second-harmonic generation, third-harmonic generation etc. Furthermore, the interaction of two fields with central frequencies ω 1 and ω 2 , with ω 1 > ω 2 , can generate components with central frequency ω 1 + ω 2 (sum-frequency generation) and ω 1 − ω 2 (difference-frequency generation). Such phenomena have not yet appeared in this volume because we have assumed that the electric field corresponds to a narrow frequency band around a central frequency ω 0 only, whereas to describe them it is necessary to assume that the electric field is the superposition of a number of frequency bands. In this chapter we will provide a description of the basic elements which are necessary to describe these phenomena and we will discuss a small number of them for the case of materials with a quadratic nonlinearity, referring to textbooks in nonlinear optics for the description of other phenomena. The following chapter is devoted, instead, to the case of materials with cubic nonlinearities. The starting point of nonlinear optics is the expansion of the atomic polarization in powers of the electric field, when the electric field is not too intense. In our treatment in Chapter 5 we saw that the expansion of the nonlinear susceptibility contains only even powers of the electric field (see Eq. (5.27)), and therefore the expansion of the polarization includes only odd powers.