We have formulated expressions providing the number of terms (Y-G diagrams)1contributing to an n-th order susceptibility of a non-linear optical process. Obtaining an accurate count of these terms is important for diverse reasons: a) To accurately specify the number of terms needed to evaluate the nonlinear optical susceptibility either analytically or numerically, b) To help develop sequences of diagrammatic contributions which can be summed to provide analytical results for the strong interaction limit. An example is the two-level system for semiclassical and quantized fields, c) To treat two or more processes (terms) which could give rise to interference, resulting, for example, in extra resonances due to a lack of complete coherence, d) To generalize results from quantum electrodynamics to situations for which multiple photon interactions and zero point fluctuations associated with non-vacuum states need to be considered. We shall discuss the number of possible diagrams in terms of three factors, which we call a permutation factor relating to the number of possible frequencies that can be assigned to each interaction vertex, a commutation factor relating to interaction vertex assignment on the two sides of the diagram, and a state factor relating to the number of diagrams generated from an enumeration of specific material state assignments between interaction vertices. We will discuss the application of these results to several situations of interest including second harmonic generation, degenerate four-wave mixing, and resonant CARS (coherent anti-Stokes Raman scattering) in a four-level system. The relevance of the present results to experimental results which have been obtained will be discussed along with future directions which need to be addressed including the summability of diagrams under specific approximations, the calculation of expectation values, and the nonperturbative representation of processes.
We have formulated expressions providing the number of terms (Y-G diagrams)1contributing to an n-th order susceptibility of a non-linear optical process. Obtaining an accurate count of these terms is important for diverse reasons: a) To accurately specify the number of terms needed to evaluate the nonlinear optical susceptibility either analytically or numerically, b) To help develop sequences of diagrammatic contributions which can be summed to provide analytical results for the strong interaction limit. An example is the two-level system for semiclassical and quantized fields, c) To treat two or more processes (terms) which could give rise to interference, resulting, for example, in extra resonances due to a lack of complete coherence, d) To generalize results from quantum electrodynamics to situations for which multiple photon interactions and zero point fluctuations associated with non-vacuum states need to be considered. We shall discuss the number of possible diagrams in terms of three factors, which we call a permutation factor relating to the number of possible frequencies that can be assigned to each interaction vertex, a commutation factor relating to interaction vertex assignment on the two sides of the diagram, and a state factor relating to the number of diagrams generated from an enumeration of specific material state assignments between interaction vertices. We will discuss the application of these results to several situations of interest including second harmonic generation, degenerate four-wave mixing, and resonant CARS (coherent anti-Stokes Raman scattering) in a four-level system. The relevance of the present results to experimental results which have been obtained will be discussed along with future directions which need to be addressed including the summability of diagrams under specific approximations, the calculation of expectation values, and the nonperturbative representation of processes.
The theory of optical birefringence is discussed from the point of view of the scattering of polarized photons by molecules using the framework of formal scattering theory and quantum electrodynamics. In Part One, those features from noncovariant quantum electrodynamics relevant to the molecular scattering of photons are reviewed. Chapter One contains an outline of classical electrodynamics in the Coulomb gauge, including the Stokes parameter and density matrix formulations of polarization. Chapter Two deals with the quantization of the radiation field, and the occupation number representation is defined. The quantum mechanical description of polarized light is outlined in Chapter Three: operator equivalents of the density matrix and Stokes parameters are invoked, the parameters themselves being obtained from the expectation values of the Stokes operators with the state vector of the photon. The Stokes parameter and density matrix are then applied to the polarization of a general beam containing an arbitrary number of photons, and some new methods are developed. The interaction of the electromagnetic field with bound electrons is formulated in Chapter Four in terms of a canonically transformed interaction Hamiltonian in which the multipole terms are explicit. Since each term is then gauge-invariant, this Hamiltonian is particularly suited to the application of diagrammatic perturbation theory which follows from the iteration of the scattering matrix in the interaction representation (Chapter Five). Also the extension of the theory to multiphoton processes is facilitated "because the field strength is explicit in the transformed Hamiltonian. The S-matrix connects the initial and final asymptotic states of the scattering system, and in Chapter Six the state vector and density operator of the scattered photon beam are obtained from those of the incident beam. Part Two contains the application of the above methods to a variety of linear and nonlinear optical birefringence effects in the absence or presence of external fields. In Chapter Seven, general equations are obtained for the polarization changes imposed on a forward scattered photon in terms of matrix elements of the reaction operator between plane polarized photon base states; whereas Chapter Eight extends this treatment to an n-photon beam, thereby encompassing the hyperpolarizability contributions to the polarization changes of the forward scattered beam. It is then demonstrated that the fundamental process in optical birefringence can be interpreted as an infinitesimal unitary transformation of the polarization spinor. Also the S-matrix approach to birefringence is compared with the methods used in crystal optics, and with the classical treatments of the scattering of polarized light by molecules. The general equations obtained for the polarization changes of the forward scattered beam are subsequently interpreted in the light of the physical circumstances appropriate to the bire- fringence effect of interest. In Chapters Nine, Ten, and Eleven, diagrammatic perturbation theory is used to obtain explicit expressions, in terms of molecular polarizability and rotation tensors, for the linear and nonlinear contributions to the polarization changes involved in natural optical activity, and in birefringence phenomena induced by static and optical electric and magnetic fields. The nonforward scattering of polarized photons by molecules is discussed in Part Three. General equations are calculated in Chapter Twelve for the azimuth, ellipticity, and depolarization ratio of the Rayleigh beam at any scattering angle and for any polarization of the incident beam. Explicit expressions in terms of molecular quantities are then obtained for the polarization at scattering angles of 90° and almost zero from both optically inactive and optically active molecules. It is demonstrated that a determination of the precise polarization state of the scattered beam can provide useful molecular information additional to that obtained from depolarization ratios alone. This is because the depolarization ratio depends on only the diagonal elements of the polarization density matrix and so does not convey all the information about the system. Raman scattering processes are contained in the formulation and the polarization of the Raman light at any scattering angle can be obtained. This is shown to be particularly useful for predicting the polarization of stimulated Raman light for any Raman-active liquid placed within a laser cavity. By extending the nonforward scattering formulation to incorporate multiphoton processes, equations are obtained in Chapter Thirteen for the ellipticity, azimuth, and depolarization ratio of the second harmonic wave at any scattering angle as a function of the incident polarization. Several specific situations are subsequently investigated. Finally, it is shown in Chapter Fourteen that the near-forw- ard Rayleigh and Raman scattering formulations provide an explanation for the observed polarization, as a function of the initial polarization, of the self-trapped filaments that arise from the self-focusing of an intense laser beam.
The Liouville equation for the density matrix can be recast so that the external field terms which couple density matrix elements are second order in the field. This approach is shown to have a profound effect on the analysis of strong field resonant nonlinear optical problems. First, in the case where parity or appropriate field restrictions apply, there is a reduction of the number of coupled equations which must be solved by at least a factor of two. Second, this reformulation naturally leads to radiative renormalization terms which are directly related to saturation, Stark shifts and Rabi splittings. The renormalization is equivalent to a Dyson equation analysis. We use this simple formalism to obtain exact solutions to a number of nonlinear optical problems including cases where there are two or more strong, resonant fields. These problems include: 1) two levels, two strong fields, 2) three levels, two strong fields, and 3) four levels, three and four strong fields. Discussed in detail are the two level case and the three and four level cases where weak fields couple the ground state to other states. We use these results to discuss the physics of optically induced transparency, population beats, optical switching in quantum wells, and fully resonant coherent anti-Stokes Raman scattering (CARS).
This thesis presents a study of photon generation and conversion processes in nonlinear optics. Our results extend beyond the first order perturbative regime, in which only pairs of photons are generated in spontaneous parametric down-conversion or four wave mixing. They also allow us to identify the limiting factors for achieving unit efficiency frequency conversion, and to correctly account for the propagation of bright nonclassical photon states generated in the nonlinear material. This is done using the Magnus expansion, which allows us to incorporate corrections due to time-ordering effects in the time evolution of the state, while also respecting the quantum statistics associated with squeezed states (in photon generation) and single photons or coherent states (in photon conversion). We show more generally that this expansion should be the preferred strategy when dealing with any type of time-dependent Hamiltonian that is a quadratic form in the creation and annihilation operators of the fields involved. Using the Magnus expansion, simple figures of merit to estimate the relevance of these time-ordering corrections are obtained. These quantities depend on the group velocities of the modes involved in the nonlinear process and provide a very simple physical picture of the interactions between the photons at different times. These time-ordering effects are shown to be important only when the photons and pump beam overlap within the nonlinear material for a significant amount of time. We also discuss the possibility of preparing Fock states with many photons using parametric down-conversion sources and realistic photon-number-resolving detectors. After discussing photon generation processes, we study photon conversion processes in which a nonlinear material and a classical strong pump field are used modify the frequency of a second incoming low-intensity field. We show that time-ordering effects limit the efficiency of this conversion process, re-examine strategies to get around this problem, and show that it should be possible to have a clear observation of modified upconversion probabilities due to these effects, and thus have an experimental demonstration of time-ordering effects in nonlinear optics.
Saturation effects in coherent anti-Stokes Raman scattering (CARS) spectroscopy are discussed. The discussion is limited to Raman-resonant CARS (${\ensuremath{\omega}}_{l}\ensuremath{-}{\ensuremath{\omega}}_{s}\ensuremath{\simeq}{\ensuremath{\omega}}_{21}$, where ${\ensuremath{\omega}}_{l,s}$ are the frequencies of the pump fields with powers Pl,s, and ${\ensuremath{\omega}}_{21}$ is the frequency of the Raman transition $|1〉\ensuremath{\rightarrow}|2〉$) with the possible addition of a one-photon resonance (${\ensuremath{\omega}}_{l}\ensuremath{\simeq}{\ensuremath{\omega}}_{31}$ is the frequency of the electronic transition $|1〉\ensuremath{\rightarrow}|3〉$). For these cases we show that the CARS polarization pCARS is proportional to the off-diagonal density-matrix element ${\ensuremath{\rho}}_{21}$. In order to determine ${\ensuremath{\rho}}_{21}$, we use Laplace transforms to solve the Bloch equations for the effective two-level system ∣1〉 and ∣2〉 when ${\ensuremath{\omega}}_{l}$ is far from resonance, or for the three-level system ∣1〉,∣2〉, and ∣3〉, when ${\ensuremath{\omega}}_{l}\ensuremath{\simeq}{\ensuremath{\omega}}_{31}$. The steady-state expression for pCARS in the former case gives ${\mathcal{p}}_{\mathrm{CARS}}\ensuremath{\propto}{P}_{l}{P}_{s}^{\frac{1}{2}}$ at low powers and ${\mathcal{p}}_{\mathrm{CARS}}\ensuremath{\propto}{P}_{l}^{0}{P}_{s}^{\frac{\ensuremath{-}1}{2}}$ at high powers. In the three-level system, we show that when the pressure is low and at least one field is weak, the slow time dependence of ${\ensuremath{\rho}}_{21}$ must be considered. When one field is strong and the other weak, the CARS spectrum is Stark split. When Pl is high, for example, ${\mathcal{p}}_{\mathrm{CARS}}\ensuremath{\propto}{P}_{l}^{0}{P}_{s}^{\frac{1}{2}}$ for ${\ensuremath{\omega}}_{s}\ensuremath{\simeq}{\ensuremath{\omega}}_{32}$ and ${\mathcal{p}}_{\mathrm{CARS}}\ensuremath{\propto}{({P}_{l}{P}_{s})}^{\frac{1}{2}}$ when ${\ensuremath{\omega}}_{s}\ensuremath{\simeq}{\ensuremath{\omega}}_{32}\ifmmode\pm\else\textpm\fi{}{V}_{13}$ where V13 is the one-photon Rabi frequency for the $|1〉\ensuremath{\rightarrow}|3〉$ transition. The Wilcox-Lamb approximation is used to reduce the three-level Bloch equations to rate equations containing one- and two-photon terms. When the fields are so weak that both one- and two-photon terms are small compared to the decay terms, the usual expression for pCARS is reproduced. If only the direct two-photon processes are important, the effective-two-system results are reproduced. When both fields are intense and nearly resonant, the steady state is rapidly achieved. The results for the case where one field is much stronger than the other are essentially the same as those for one strong and one weak field. When the fields are of comparable strength, ${\mathcal{p}}_{\mathrm{CARS}}\ensuremath{\propto}{P}_{l}^{\frac{1}{2}}{P}_{s}^{0}$ for ${\ensuremath{\omega}}_{l}\ensuremath{\simeq}{\ensuremath{\omega}}_{31}$ and ${\ensuremath{\omega}}_{s}\ensuremath{\simeq}{\ensuremath{\omega}}_{32}$, and the CARS spectrum is split into five components.
In earlier chapters we have considered in detail the susceptibility formalism of nonlinear optics, which is perhaps the most familiar approach and has a wide range of application. Starting from the constitutive relations of Chapter 2, the susceptibility formalism is quite general. In many practical applications, a particular phenomenon can be described accurately by a single order of nonlinearity, and the susceptibility then provides a useful and convenient description. However, this is not always the case. Some of the most interesting phenomena in nonlinear optics involve close resonance with the transition frequencies of the medium, and perhaps also the use of very intense optical fields. As remarked in §4.5, in these circumstances the resonant susceptibilities display mathematical divergences which are clearly unphysical. Strictly, these divergences occur only because higher-order nonlinearities have been neglected. Successive orders of nonlinearity take into account such effects as saturation, power broadening and level shifts (optical Stark effect). For intense fields or very close resonance, the contributions from several orders of nonlinearity may be comparable in magnitude. Therefore, despite its generality, the susceptibility formalism does not necessarily provide the most practical approach for the description of resonant processes. This chapter is concerned with the problem of deriving alternative and more manageable descriptions of resonant nonlinear processes, illustrated by examples. For much of the chapter, the nonlinear medium is treated as a two-level system (described in §6.2).
Starting with well known formulae for nth-order atomic susceptibilities, irreducible spherical tensor techniques and Racah algebra are used to put these in a form suitable for the treatment of complex nonlinear processes, e.g. where fine and hyperfine interactions can be included and the interacting fields can have quite general polarizations. By way of illustration the susceptibilities are calculated in detail for third harmonic generation and Raman scattering in alkali vapours, taking into account the effect of spin-orbit coupling.