An optical parametric oscillator as a high-flux source of two-mode light for quantum lithography

We investigate the use of an optical parametric oscillator (OPO), which can generate relatively high-flux light with strong non-classical features, as a source for quantum lithography. This builds on the proposal of Boto et al (2000 Phys. Rev. Lett. 85 2733), for etching simple patterns on multi-photon absorbing materials with sub-Rayleigh resolution, using two-mode entangled states of light. We consider an OPO with two down-converted modes that share the same frequency but differ in field polarization or direction of propagation, and derive analytical expressions for the multi-photon absorption rates when the OPO is operated below, near and above its threshold. Because of strong non-classical correlations between the two modes of the OPO, the interference patterns resulting from the superposition of the two modes are characterized by an effective wavelength that is half of their actual wavelength. The interference patterns resulting when the two modes of the OPO are used for etching are also characterized by an effective wavelength half that for the illuminating modes. We compare our results with those for the case of a high-gain optical amplifier source and discuss the relative merit of the OPO.


Introduction
The Rayleigh criterion states that diffraction limits the resolution of a traditional optical lithographic system, and specifies a minimum feature size of half the wavelength of the illuminating beam. A variety of classical procedures exist which can exceed this limit with certain trade-offs, and some examples of optical and non-optical super-resolution techniques are described in [1]- [3]. Boto et al suggested an entirely different approach in 2000 in [4], and their proposal has attracted considerable research interest [5]- [11]. The key observation here is that all existing optical lithographic procedures assume that the illuminating light fields are classical. The Rayleigh limit arises in part from the fundamental photon statistics of laser light, according to which the constituent photons are uncorrelated. To circumvent this, Boto et al proposed exploiting entangled states of light, and in particular path-entangled states of light of the form |N 0 + |0N in the photon-number basis, often termed 'N 00N ' states. In the scheme detailed in [4], the N 00N states propagate through a simple interferometer, and then interfere at a N -photon-absorbing recording material in a counter-propagating configuration. A brief calculation can illustrate the basic idea. The N -photon absorption rate at the substrate is proportional to the expectation value of the observableÊ (−)NÊ (+)N /N !, whereÊ (+) =exp(ikx)â 1 + exp(−ikx)â 2 is the annihilation operator for the combined field.â 1 andâ 2 denote annihilation operators for two interferometer modes, x denotes translation across the substrate, and k denotes the wave-number 2π/λ for the light (we assume grazing incidence). For the case of classical illumination, the two interferometer modes are assumed to be in coherent states so that the absorption rate is proportional to [cos(2kx) + 1] N . For the case of N 00N states, the absorption rate is proportional to [cos(2N kx) + 1]. In principle then, the procedure can etch a series of straight lines corresponding to an effective wavelength of λ/(2N ). Variations of the method have been proposed for creating more general onedimensional (1D) and two-dimensional (2D) interference patterns by employing a family of entangled states [12]. Such super-resolution techniques exploiting non-classical states could be used in combination with existing sub-Rayleigh procedures based on classical methods. The concept of interferometric lithography using a multi-photon recording material with a classical source has been demonstrated experimentally [13]. However, considerable work remains to be 3 done to demonstrate the feasibility of using a bright non-classical source. A general program that aims to do so will require investigation of the source, imaging system and multi-photon absorption process.
For the case of two-photon quantum lithography, one could use parametric down conversion in a medium exhibiting an χ (2) optical non-linearity as a source of photon pairs. Each of these photon pairs propagating through a Hong-Ou-Mandel interferometer with a symmetric beam splitter will yield a two-photon N 00N state at the output ports of the interferometer [14]. Subsequent quantum interference of the two modes on a two-photon absorbing material will then create a fringe pattern with fringe spacing that is half of what is achieved when a classical light source is used. However, the output of common parametric down-conversion experiments is not expected to be sufficiently bright to be useful for typical two-photon recording materials. In [5], Agarwal et al considered a strongly pumped high-gain optical parametric amplifier (OPA). By assuming a single-mode operation, it can be shown that the state generated by an unseeded OPA is the two-mode squeezed vacuum state of the form ∞ n=0 tanh n (G)|n |n in the photon-number basis. Here G is the gain parameter, which depends on the interaction volume in the crystal, the amplitude of the electric field of the pump beam, and the strength of the secondorder susceptibility χ (2) . To quantify the contrast of the interference pattern at the output, the visibility is defined as the difference of the maximum and minimum absorption rates, divided by the sum of these rates. The visibility varies between 0 and 1. Explicit calculation reveals that the visibility falls from 1 to an asymptotic value of 0.2, as the gain parameter G of the OPA is increased from 0 → ∞. This result was generalized in [9] to the case of higher-order multi-photon absorbing materials. Essentially the same behavior is seen in these cases, with the visibility falling from 1 to an asymptotic value, which is greater the higher the order of the absorption process. These predicted features have been demonstrated experimentally, using coincidence measurements at photodetectors to simulate the recording medium [11].
In this paper, we investigate the use of an optical parametric oscillator (OPO) as a practical source of non-classical light for lithography. In this case, the process of parametric amplification occurs in an optical cavity in resonance with the signal and idler modes. We will consider the signal and idler modes to have the same optical frequency but orthogonal polarizations. The signal and idler cavity modes are coupled to external propagating modes through a transmissive end mirror. Photons are created in pairs in the cavity modes, but escape independently out of the cavity on a timescale of the order of the cavity lifetime. The twin beams emerging from the OPO, corresponding to the light in the two modes, are highly correlated. Unlike the OPA, the OPO has a well-defined threshold for oscillation, and the below-, near-and above-threshold regimes require different mathematical treatments. The theory for the OPO is complicated by the need to account for mode losses, and the coupling of cavity and pump modes. To analyze the quantum lithography procedure using the light from an OPO, we will follow the approach developed in [15]- [18] to investigate the properties of light from OPOs. By using the positive-P representation [19] of the density matrix, the Markovian master equation describing the dynamics of the standard model of the OPO can be translated into a multi-variate Fokker-Planck equation for the positive-P function. The Fokker-Planck equation can be treated using the techniques from non-equilibrium classical statistical mechanics [20,21], allowing the steady-state expectation values of various observables to be evaluated. The positive-P method has not previously been applied to quantum lithography, and has crucial advantages compared to other standard approaches. For the OPO source, it leads to a Fokker-Planck equation that can Signal and idler beams emerging from the OPO are directed along different spatial paths by a polarizing beam splitter (pbs). A half-wave plate (hwp), orientated with its fast axis at an angle π/4 with respect to the horizontal plane, is placed on one spatial path. This ensures that the light in the two paths shares the same polarization. (Alternatively the OPO and pbs could be replaced by a non-collinear down-conversion source for which the signal and idler beams are degenerate both in frequency and polarization.) The two modes are then combined at a 50 : 50 beam splitter whose output is finally combined at a multiphoton-absorbing recording material.
be treated analytically. In particular, the method can be applied to the near-threshold regime, which is problematic in other approaches.
In section 2, we describe the setup for quantum lithography and present a quantum-classical correspondence based on the positive-P representation that allows us to convert the master equation for the density matrix for the OPO into a Fokker-Planck equation. A solution of the resulting Fokker-Planck and Langevin equations is presented following [17,18]. In section 3, we propagate the c-number variables corresponding to the output from the OPO through a simple interferometer and determine the rate for multi-photon absorption of arbitrary order. We then look in detail at the features of the interference patterns, focusing on the below-threshold case in section 3.1, and the near-and above-threshold cases in section 3.2. In section 4, we compare our results with those for the OPA, and comment on their similarities and differences. In addition, we address the issue of what realistic powers may be available from an OPO source below and above threshold by looking at a current experimentally realized system, and estimate concrete values for the flux expected at the output.

The setup
The setup for a quantum lithography experiment is illustrated in figure 1. We start at the OPO source. The modes corresponding to the signal and idler have frequency ω and we denote the corresponding creation operators asâ † 1 andâ † 2 . The signal and idler modes are assumed to experience linear losses characterized by the decay rate γ , which arises from transmission, absorption and scattering losses. We assume the pump field to be a classical field of frequency 2ω and of normalized amplitude denoted by ε (assumed to be given by a positive value). The cavity pump mode is subject to linear loss with decay rate γ 3 and we assume that γ 3 γ . The creation operator corresponding to the cavity pump mode isâ † 3 . The Hamiltonian describing this system in the interaction picture can be written aŝ where κ is the mode-coupling constant determined by the strength of the optical nonlinearity andĤ loss describes the interaction of the cavity modes with the modes of the loss reservoir. Light emitted by the OPO into the external signal and idler modes, assumed to have orthogonal polarizations, is separated into two spatial paths by a pbs. A hwp placed in one path ensures the polarizations are made parallel. These modes are combined at a symmetric 50 : 50 beam splitter and the output modes of the beam splitter are finally combined at the multi-photon absorbing material with the help of mirrors so that the interfering beams are counter-propagating at grazing incidence over the area of interest. Two classical beams incident on a single-photon absorbing material would generate a fringe pattern of the form ∝ [1 + cos(2kx)], where k denotes the optical wave number for the signal and idler modes, and x denotes translation across the substrate. We shall denote the optical phase difference 2kx by φ. When the recording medium is a p-photon absorber and the illuminating beams have statistics which need not be classical, the absorption rate is given by , whereρ is the density matrix for the state of the illuminating field,Ê (+) andÊ (−) are, respectively, the positive and negative frequency components of the field at the absorber, and σ ( p) is a generalized crosssection for the process. This result holds whenever the optical field may be considered stationary, quasi-monochromatic and resonant [22,23]. It can be seen that the absorption process will be strongly influenced by the statistical properties of the light [24].

Master equation and solution using the positive-P distribution
Following the standard procedure and using the Hamiltonian (1) for the OPO, the Markovian master equation for the reduced density operator for the pump, signal and idler fields is given by [21,25] ∂ρ ∂t = whereĤ I here excludes theĤ loss term of equation (1), which couples the cavity modes to reservoir modes. The effect ofĤ loss is contained in the decay terms in equation (2). We now map this master equation into a classical Fokker-Planck equation using the positive-P representation introduced by Gardiner and Drummond [19]. For the current problem this representation is defined as follows, where α 1 , α 1 * , α 2 , α 2 * , α 3 and α 3 * are six independent complex variables, and we have written α≡(α 1 , α 1 * , α 2 , α 2 * , α 3 , α 3 * ). It should be emphasized that the asterisks in the variable indices 6 do not correspond to complex conjugation. The complex variables α i and α i * correspond to the mode annihilation and creation operators via the relationsâ i |α i =α i |α i and α i * |â † i =α i * α i * |. The distribution function P( α) may be assumed to have the mathematical properties of a probability density function: it is real valued, positive and normalized to one when integrated over the full domain D of α. For the master equation, equation (2), the corresponding positive-P function satisfies the following equation, This equation has the form of a multivariate Fokker-Planck equation [26], and the use of the positive-P distribution ensures that the diffusion matrix is positive. Then the Langevin equations corresponding to equation (4) can be written as where the ξ i are real-valued, white-noise, stochastic variables with ξ j (t) = 0 and for all values of the indices.
Since it is assumed that the pump mode loss is much greater than the signal or idler loss (γ 3 γ ), the pump field can be adiabatically eliminated. Settingα 3 =α 3 * = 0 we find 7 Using these results in equation (5), we obtain where time has been scaled in terms of the cavity lifetime (τ = γ t). Parameter n 0 = 2γ γ 3 /κ 2 is proportional to the square of the number of photons in the cavity at threshold and it sets the scale for the number of photons necessary to explore the nonlinearity of interaction. σ = 2γ 3 ε/κ is a dimensionless measure of the pump field amplitude, so that the threshold condition εκ = γ gives σ = n 0 .
Using the procedure set out in [16,17], the dimensionality of this set of Langevin equations can be reduced from eight to four. Accordingly, we introduce four real-valued variables u 1 ,u 2 ,u 3 and u 4 by  The variables u i may be interpreted as scaled pseudo-quadrature variables. An exact distribution for the positive-P function for the OPO may now be written down [17], which is valid below-, near-and above-threshold regimes. For n 0 ∼ 10 6 − 10 8 , which are typical values for laboratory systems [27], the positive-P function is given to a very good approximation by where u = (u 1 , u 2 , u 3 , u 4 ) . Parameters a 1 , a 2 correspond to the pump strength and are given by where r = σ/n 0 = κε/γ . It can be seen from equation (9) that any moment of the form, u n 1 1 u n 2 2 u n 3 3 u n 4 4 = d 4 u u n 1 1 u n 2 2 u n 3 3 u n 4 4 P( u), has the value zero if any of the n i is odd. The distribution of equation (9) can be used to evaluate the steady-state expectation values for any normally ordered product of the field operators. In the next section, we will compute the absorption rates for multi-photon recording media, exploiting the symmetries of P( u) and making approximations valid for the different regimes of operation of the OPO.

Results
In this section, we compute the multi-photon absorption rates at the recording medium for a quantum-lithographic process, by computing expectation values using the positive-P distribution presented in equation (9). We first propagate the signal and idler variables α 1 , α 1 * and α 2 , α 2 * through the imaging apparatus as described in section 2. By causing the signal and idler fields to interfere at a symmetric 50 : 50 beam splitter, the fields at the output are given by Propagating the fields to the multi-photon recording material, the combined field at a location corresponding to an optical phase difference of φ, is given by β 3 = (β 1 e iφ + β 2 ) and β 3 * = (β 1 * e −iφ + β 2 * ). Substituting equation (11) in these expressions, we obtain The rate of p-photon absorption then is given by the average quantity, Substituting equation (8), we also have, We first evaluate the fringe pattern for a one-photon absorber, given by I 1 3 (φ). Inspecting the symmetries of the positive-P function, equation (9), we see that symmetries exist between the variables u 1 and u 3 , as well as u 2 and u 4 , so that P(u 1 , u 2 , u 3 , u 4 ) = P(u 3 , u 2 , u 1 , u 4 ) and P(u 1 , u 2 , u 3 , u 4 ) = P(u 1 , u 4 , u 3 , u 2 ). It follows immediately that This equation implies that there is no dependence on φ either below or above threshold and the illumination of the substrate is uniform across the surface in the ensemble-averaged sense. Next we consider the case of multi-photon absorbing materials.

Below-threshold regime
Inspecting the positive-P distribution, equation (9), for the below-threshold case, κε γ , we find that a 1 and a 2 are large negative quantities, and the distribution can be approximated by For a typical OPO with n 0 ∼ 10 6 this approximation is very good for the parameter r ranging from 0 to 0.99. The four variables u i are Gaussian and independent. Their even-order moments are given by and their odd-order moments vanish. In order to compute the multi-photon absorption rates I p 3 (φ), defined by equations (13) and (14), a suitable grouping of terms must be found. To this end we define the function so that I p 3 can be expressed as To evaluate the moments needed in this expression we exploit the fact that u i are uncorrelated variables. Moreover, since the pairs of variables u 1 and u 3 , and u 2 and u 4 , share the same distribution, we conclude . Expanding an arbitrary moment k of F(u 1 , u 4 , φ) and using the observations of the preceding paragraph, we find By substituting the expressions for the even moments of u 1 and u 4 in equation (17) and the form of parameters a 1 and a 2 defined by equation (10), this sum can be evaluated as Using these relations, we arrive at a compact expression for the p-photon absorption rate, From an examination of the form of the absorption rates I p 3 (φ) in equation (22), we conclude that for a single photon absorber ( p = 1), the absorption rate is constant and no fringe pattern is created, as already seen following equation (15). For a multi-photon ( p 2) absorber, the rate I p 3 (φ) is given as a sum of even powers of cos(φ). The corresponding fringe pattern therefore has terms with periods corresponding to cos(2nφ) where 2n p. As p increases, there is an increasing contribution from higher-power terms, resulting in sharper interference patterns. The fringe patterns for p = 1, . . . , 6 are plotted in figures 2(a) and (b), for the pump ratio r = 0.15 and 0.9, respectively. Figure 3 shows fringe visibilities as r ranges from 0 to 0.95. It is seen that in each case with p 2, the visibility is close to 1 when the OPO is operated far below threshold, but as the pump power is increased the visibilities fall steadily. In a lithographic process, it is possible to compensate for unwanted constant exposure by using a substrate with greater depth, and visibilities as little as 0.2 are often considered adequate in practice [5]. This criterion is satisfied in all cases considered here with p 2. Equation (22) is listed explicitly in table 1 of appendix for the cases of p = 1, . . . , 6. The corresponding results for an OPA were reported in the work of Agarwal et al [9]. A comparison of our results with equations (19)- (22) of [9] shows that the two sets of formulae for the multi-photon absorption rates coincide in identifying r with tanh(G) in the analysis of Agarwal et al, where G represents the single-pass gain for the OPA.
The pump ratio range 0 r < 1 for the OPO (below threshold), corresponds to the OPA singlepass gain range 0 G < ∞. Visibilities for the sub-threshold OPO are listed in table 2 of the appendix for p = 1, . . . , 6, which agree with the corresponding results reported for the OPA over the corresponding range of the parameter G.
A striking feature of the fringe patterns, predicted for both an OPO source and an OPA source, is the increasing visibility with the order p of the absorption process. In the analysis above for the OPO, we have identified the origin of this improvement to be the contributions of higher spatial frequency interference terms. This improvement of visibility with the order of absorption is not a quantum effect; other interference-based setups with classical fields can show similar improvements. An example is provided by [28], which looked at a higher-order generalization of a standard Hanbury Brown-Twiss setup. The authors consider intensity correlations at two or more photodetectors placed in different locations and excited by two distant light sources. As is well known, the visibility of second-order intensity interference cannot exceed 50% if the light sources are classical. However, Agafonov et al have shown theoretically and experimentally that the visibilities (given classical sources) can be much greater for higher-order measurements, which record coincidences at three or four photodetectors.

Near-and above-threshold regime
As the OPO is pumped more strongly and passes through threshold, pump ratio r exceeds unity and with that the nature of the positive-P distribution changes significantly. Inspecting the pump parameters a 1 and a 2 , defined by equation (10), we see that while a 2 continues to take large negative values of increasing size as r increases, a 1 is zero at threshold and takes positive values for pump ratio r > 1. The positive-P distribution, given by equation (9), may now be well approximated by the expression below, which is valid both near and far above threshold for values of n 0 greater than 10 4 [18], where N denotes a normalization factor for the u 1 , u 3 component, defined by and erfc(z) ≡ (2/ √ π ) ∞ z e −s 2 ds is the complementary error function. It can be seen that the variables u 2 and u 4 are independent Gaussian variables with pump parameter a 2 in the belowthreshold regime. However, the variables u 1 and u 3 with pump parameter a 1 are now strongly coupled. As a consequence, the decomposition given by equation (19), used to evaluate I p (φ) in the below-threshold case, can no longer be applied.
To proceed further in this case, we look in detail at the moments for the u i that arise in computing I p (φ) above threshold. The moments for the Gaussian variables u 2 and u 4 are as before given by equation (17). By expressing u 1 and u 3 in polar coordinates, it follows that a general moment of the form u 2s 1 u 2t 3 u 2m 2 u 2n 4 can be expressed as where B(·, ·) denotes the Beta function, defined by B(S, T ) ≡ (S) (T )/ (S + T ), which arises from the integration over the angular component associated with u 1 and u 3 . The integration over the remaining radial component leads to the R(·) contribution defined by for s + t 1, and integrating by parts, we get a recursion relation for R(·). Here we list the first few radial functions, Below threshold (r = κε/γ < 1), the parameter a 1 is negative. It is zero at threshold (r = κε/γ = 1) and positive above threshold (r = κε/γ > 1). As a 1 increases from negative to positive values, the OPO goes through a phase transition and the intensities of the signal and idler modes increase very rapidly. For typical values of n 0 = 10 6 − 10 8 the region of phase transition is very narrow and corresponds to a small change, approximately, 0.99 → 1.01 in r . The statistics of the OPO change dramatically [17,18] over this range. We find that the behavior of visibility for p-photon absorption also changes significantly as the operating point of the OPO changes from below to above threshold.
For the OPO operation much above threshold (a 1 1), we can considerably simplify the expression for I p 3 (φ). It follows from equations (24) and (27) that u 2s 1 u 2t 3 B(s + 1/2, t + 1/2)a s+t 1 /π . Since powers of n 0 appear only in the denominator for moments of u 2 and u 4 , and in the expansion of I p (φ) the powers of the variables u i satisfy s + t + m + n = p, it follows that all contributions from variables u 2 and u 4 can be neglected. Within this approximation,  and we find the absorption rate The p-photon absorption rates for much of the above threshold operation of the OPO are therefore seen to generate fringe patterns, which are largely independent of the strength of the pump. Figure 4 shows how the fringe visibility changes near the threshold regime for the two-photon case. Note that as r changes from 0.97 → 1.03 the parameter a 1 changes from −42 → +42. As expected, when much above threshold the visibility is almost constant.

Discussion
In conclusion, we have found that an OPO source can be used to generate fringe patterns with an effective wavelength half of the actual wavelength of the signal and idler modes that interfere at a p-photon absorbing medium ( p 2). For the case of two-photon absorption, the visibility of the predicted fringe pattern falls from a maximum of 1 below threshold to an asymptotic value of 0.2 at high pump powers. Similar behavior is found for high-order absorption processes, and the asymptotic value increases with the order p, for example to 0.43 for p = 3 and 0.63 for p = 4. Below threshold, the forms of the fringe patterns depend strongly on the pump power. As the OPO threshold is approached, these patterns evolve toward an asymptotic limit, becoming largely independent of the strength of the pump high above threshold. Comparing this with the results reported in [9], we find that the fringe patterns generated by using an OPO operating below threshold (0 r < 1) are similar to fringe patterns generated by an OPA operated with a corresponding gain in the range (0 G < ∞). In the case of an OPA there is no cavity and no above-threshold regime, and the process of optical parametric amplification occurs in propagating modes. The agreement between the two results is not surprising, since [9] considers an idealized model of an OPA by disregarding photon losses, and assuming a single-mode operation by ignoring all but one down-converted spatial mode. In the OPO model, mode losses are included in the decay rates and the presence of a cavity naturally provides a single mode selection. The reparameterization of the dynamics of the OPO in terms of the four pseudo-quadrature variables u i above sheds some light on why the changes in fringe patterns and visibilities are observed. Far below threshold, all four of the u i contribute approximately equally to the absorption process, and behave as independent variables. As the pump power is increased and threshold is approached, parameter a 1 tends to 0. Since a 1 appears in the denominator of each of the moments of variables u 1 and u 3 , they make a growing contribution compared to u 2 and u 4 . Above threshold, variables u 1 and u 3 make the primary contribution to the absorption process, and the effects of u 2 and u 4 can be neglected. Variables u 1 and u 3 are also strongly coupled in this regime. The results of these changes are expressed in the compact general formulae for the multi-photon absorption rates, equations (22) and (28), involving a sum over powers of products of terms [r + cos(φ)] k [r − cos(φ)] p−k . Below threshold parameter r < 1 the effect of increasing the order p of the multi-photon absorption process is to suppress the contributions of powers of r in favor of powers of cos(φ) around its maxima, sharpening the fringe pattern. As threshold is approached, i.e. r → 1 − , the contribution of cos(φ) relative to r decreases, and the fringe patterns assume a fixed form. Hence the visibility saturates above threshold.
The OPO source has several experimental advantages. An OPO source would have improved temporal and spatial coherence compared to an OPA. The signal and idler modes at the outputs of the OPO are collimated because of the use of a cavity, and higher powers can be obtained than is the case for the OPA, both of which are important in light of the small cross-sections for typical multi-photon absorption processes. Collimated, high power outputs also increase the speed at which a substrate may be imaged-a critical factor in the large-scale production of, say, computer chips.
We end by looking at the requirements for driving a high-order absorption process in an experimental implementation of quantum lithography. In our proposal, there are competing demands on the pump power and gain. At lower intensities, the absorption of strongly correlated photon pairs emitted by the OPO in the recording material leads to a halving of the fringe spacing and to visibilities approaching one. However, the need for high intensities at the output to drive a higher-order absorption process will increase the probability for absorption of photons from different pairs, thereby reducing the visibility of the fringe pattern created.
In the recent experiment reported in [13] an interferometric lithographic setup, as illustrated in figure 1 but with a classical laser beam at 800 nm as the source, was implemented with a lithographic recording material based on polymethyl-methacrylate (PMMA). PMMA is transparent across the visible spectrum, but absorbs strongly at ultra-violet frequencies.
High-visibility fringe patterns were demonstrated, which were consistent with multi-photon absorption of order three and higher. Using a classical phase-shifted-grating method, an approximately two-fold enhancement of resolution was demonstrated. The experiment established a window of viable pulse energies, 80-135 µJ, for the lithographic process. We turn to a candidate OPO source, reported in [29], which we estimate would achieve a further doubling of resolution beyond that produced by the phase-shifted-grating method, for example. In the most recent experimental configuration of the OPO, the continuous-wave pump power at threshold is 225 mW. 35% above threshold, i.e. for parameter ratio r = 1.35, the output power is 50 mW for each of the signal and idler modes. Taking the parameter n 0 = 10 8 and using equations (22) and (28), find that just below threshold, the power in each of the signal Table 1. p-photon absorption rates as functions of the optical phase difference φ for the pump ratio r varying from far below to near threshold (from 0 to approximately 0.99). 45r 6 (1 − r 2 ) 6 [5 cos 6 (φ) + 90r 2 cos 4 (φ) + 120r 4 cos 2 (φ) + 16r 6 ] and idler modes can be expressed as (5/|a 1 ) µW for pump parameter a 1 −10. Clearly the above-threshold regime is more favorable for driving the multi-photon absorption process. However, a PMMA-based recording material that can respond at lower powers can take advantage of a higher-visibility fringe pattern and strong quantum correlations below threshold by using a longer exposure time. (φ min )], for the interference patterns with a p-photon absorbing recording material. The third column lists the limiting values for the visibility as the pump ratio r ranges from far below to near threshold (from 0 to approximately 0.99). p Visibility V (r ) 1 → 0.77 6 1 − 32r 6 5 + 90r 2 + 120r 4 + 32r 6 1 → 0.87