Theoretical Background,
II. 2 Parametric gain analysis.
When pum ped by intense optical radiation at a frequency Vp, a nonlinear optical material can provide gain at two lower frequencies called the signal frequency, Vg, and the idler frequency, V/. These three frequencies are related by the conservation of energy relation:
= + .
[n.i]
The param etric interaction is phase-dependent, and proper phasing is required for energy to flow from the pump field to the signal and idler fields. Phase-velocity matching ensures that the relative phase of the three waves does not change with propagation through the nonlinear material. Phase- matching is described by the wave-vector (or phase) mis-match, M , which for the case of collinear propagation, can be expressed by the scalar relationship
M
= - f c s - f c i = — ---1,
[n.
2]
^ C
where k^, and k^ are the wave-vector magnitudes of the pump, signal, and idler waves, respectively, with corresponding indices of refraction given by rip, rig, and n(, and c is the velocity of light. Useful parametric gain exists in the range of signal and idler frequencies for which |AA:|<;r//, where I is the length of the nonlinear material [1]. The parametric gain is maximum near Ak = 0, ensuring that the three waves propagate synchronously w ith constructive interference. Phase-matching is often achieved by controlling the birefringence of a nonlinear crystal through temperature or angle of propagation.
An OPO requires feedback at either (or both) the signal and idler frequencies. If there is feedback at only one frequency, the device is called a singly resonant oscillator. Doubly-resonant oscillators have feedback simultaneously at the signal and idler frequencies. A further extension to these configurations is to resonate the pum p frequency within the OPO cavity. Both singly- and doubly-resonant oscillators can be formed with pum p enhanced fields. The doubly-resonant oscillator with a pum p enhanced field is often referred to as a triply-resonant oscillator where all three waves are brought to resonance. The number of waves brought to resonance w ithin the OPO cavity affects drastically the operating characteristics of the device. The important differences between the various resonant cavities include the pum p power threshold, the conversion efficiencies, and the pump frequency stability requirements.
Typically, feedback is provided by placing the nonlinear material in a cavity formed by external mirrors. This cavity can be formed within the resonator of the pump source, or external to the pum p source. In this
chapter, only external resonator designs are considered. Therefore, external cavities are analysed that involve a nonlinear material surrounded by mirrors to provide the desired feedback. However, highly reflecting coatings can be applied directly to the nonlinear material. This allows for the formation of monolithic OPO resonators, and reference to these devices is made frequently throughout the text.
Phase-matching is the major factor in determining the broad (coarse) tuning properties of an OPO, although cavity resonances have the major effect on details of frequency tuning. The conditions Vp-Vg + V/ and Ak = 0 define the phase-matching curves. The spectral width of the parametric gain is also determined by phase-matching.
Similar to the steady-state analysis of a laser resonator, when the gain exceeds the loss in an OPO, the device reaches threshold and oscillates. At threshold, the output power increases dramatically, similar to the behaviour of a laser. The generated output is coherent and collinear with the pum p laser beam. Once above threshold, the parametric oscillator converts efficiently the pump radiation to continuously tunable signal and idler fields, with the gain clamped at its threshold value (similar to the steady-state operation of a laser oscillator). Since the gain is proportional to the pump field, above threshold any additional pum p power must be diverted into power at the signal and idler fields.
The first distinction to make is between cw and pulsed operation. In general, these devices have many similarities, when considering the coarse frequency tuning properties of the OPO. However, the most significant difference is the much greater peak pump powers that are available from pulsed laser sources. This allows greater flexibility in the design of the OPO resonator, with regard to cavity resonances, cavity stability, and focusing requirements.
There are many different forms of resonators that have been used to provide cw parametric oscillation. The cavities that are analysed in this chapter are displayed in figure II. 1. Figure II. 1 (a) illustrates parametric amplification over the length of the nonlinear crystal. Figure II. 1 (b) illustrates the ring-cavity oscillator. The ring cavity provides single-pass parametric interaction, and can offer improved OPO conversion efficiency. In addition, this design offers advantages of reduced feedback to the pum p source. Figure II. 1 (c) displays the single-cavity, standing-wave OPO where
(a) «p(0) « s » ) crystal K Vp =v, + V,- |Ak|< ^ -► « g(0 -> oc^ U) > a^(/) Ak kg -/c; (W crystal M ^ 3 \ \ V \ \ \ ^
-►a
"►«J-Otft owt w a tn M crystal -4—^i
M .-►a:
o u t out ((Z) a tn■I
M M BS crystal\
M 3 \ \ \ \ \ \ \ a a o u t > a OMt Figure IL 1.(a) Schematic representation of optical parametric amplification. Optical parametric oscillators are formed by the addition of mirrors that form an optical resonator, (b) Ring-cavity oscillator with travelling- wave fields and a single-pass through the gain medium, (c) Single cavity oscillator with standing-wave fields, (d) Dual-cavity oscillator with separate optical resonators for different resonant fields; BS: beam-splitter.
the OPO fields are brought to resonance between two common cavity mirrors. Finally, figure II. 1 (d) displays the dual-cavity, standing-wave OPO resonator, which allows separate OPO resonant cavities to be formed with independent cavity length control.
II. 2. (i) The nonlinear susceptibility.
When an electromagnetic wave, E, propagates through a dielectric material, it induces a polarization, P. For most electromagnetic waves, where the electric field is considerably smaller than the intra-atomic field, the response of the induced polarization will be linear in the electric field, and can be expressed as follows:
P = , [n.3]
where Sq is the permittivity of free space, and Xs is the (linear) electrical
susceptibility. The electrical susceptibility is a frequency dependent, complex quantity, of which the real part is related to the refractive index {\Xs\~^^ and is responsible for reflection / refraction, and the imaginary part is related to the absorption properties of the material, and is only significant close to an atomic resonance. Equation II. 3 represents the regime of linear optics.
W hen the electromagnetic wave has an electric field strength comparable to the intra-atomic field, as provided by the high intensities associated with laser radiation, the response to the field can no longer be described by the linear relation of equation II. 3. Now, it is necessary to expand the induced polarization in powers of E. The relation between the electric field and the polarization can be expressed as the following power series expansion:
P = +...) , [n.4]
where ' Xs ' X^f^' etc. are the first, second, third, etc., -order susceptibilities. (The scalar notation in the above equation is used only to simplify the analysis; the susceptibilities are actually tensors.) From equation II. 4, the response of the polarization to the field is no longer linear.
and is referred to simply as nonlinear. By splitting this relation into linear and nonlinear parts (P^^), the polarization can be described as follows:
P = e„;t;WE+P^^ . [H.5] The nonlinear effects due to the quadratic term (a tensor of rank three) involve the mixing of frequencies in, for example, second harm onic generation, and sum and difference frequency mixing. For the even term to be present, the m edium m ust lack a centre of inversion (non- centrosymmetric). The cubic term gives rise to third harmonic generation,
Raman scattering, four wave mixing, and others. The following analysis is restricted to materials which possess a substantial second order nonlinear susceptibility. Terms of higher order, which will be considerably weaker, are neglected.
Following the analysis outlined in reference [2], as extended in reference [3], with the inclusion of the factor such that the units of are in m /V , the second-order nonlinear polarization can be expressed in tensor notation as follows:
P^^ ( Vp ) = : E( V, ) E( V; ) , [E. 6 (a)] P^^(vs) = eo;i:?':E*(vi)E(vp) , [n.6(b)] P^^{vi) = eozi^^:E’(Vs)E(vp) . [n.6 (c)]
Normally, the measure of nonlinearity is determined by second harmonic generation experiments. Therefore, it is convenient to express the equations in terms of nonlinear coefficients, d p , instead of such that = 2d p , where X(^ are the components of As these d p coefficients obey the same crystal symmetry as the piezo-electric coefficients (although different in magnitude), it is also convenient to use the condensed notation [4] that is applied to the piezo-electric coefficients. In this case, the second order nonlinear susceptibilities are specified in the principal axis system (XYZ) of the piezo-electric coefficients. The Z-axis is generally adopted as the polar axis. This condensed notation takes the form
dijk=^il{j,k) / [H. 7] where
/(1,1) = 1, /(2,2) = 2, /(3,3)-3, /(2,3) = 4, /(1,3) = 5, and /(1,2) = 6.
and where the subscripts z, and k are labelled 1, 2, and 3, respectively. They refer to the axes X, Y, and Z, respectively. The nonlinear polarization equations in terms of the d tensor for the three frequency interactions are given now by [1]
p(vp) = e„2d:E(vs)E(vi) , [n .8 (a )]
P(Vs) = e„2rf:E*(vi)E(vp) , [n.8 (b)] P(v;) = fio2d:E*(Vs)E(vp) , [n.8(c)] where the notation implies a vector sum over the axial components. The analysis is simplified further by calculating an effective nonlinear coefficient, d ^ , which is the net response of the nonlinearity, taking into account the relation between the E field polarization vectors and the symmetry of the d tensor. The term, d ^ , is a measure of the coupling between the three interacting fields, involving projection of all three fields on to the d tensor. The use of d ^ allows for scalar notation.