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Home Search Collections Journals About Contact us My IOPscienceAn optical parametric oscillator as a high-flux source of two-mode light for quantumlithographyThis article has been downloaded from IOPscience. Please scroll down to see the full text article.2009 New J. Phys. 11 113055(http://iopscience.iop.org/1367-2630/11/11/113055)View the table of contents for this issue, or go to the journal homepage for moreDownload details:IP Address: 130.39.62.90The article was downloaded on 03/12/2010 at 20:57Please note that terms and conditions apply.


New Journal of PhysicsThe open–access journal for physicsAn optical parametric oscillator as a high-flux sourceof two-mode light for quantum lithographyHugo Cable 1,2,4 , Reeta Vyas 3,4 , Surendra Singh 3and Jonathan P Dowling 11 Horace C. Hearne Jr. Institute for Theoretical Physics,Department of Physics and Astronomy, Louisiana State University,Baton Rouge, Louisiana 70803, USA2 Centre for Quantum Technologies, National University of Singapore,3 Science Drive 2, Singapore 1175433 Department of Physics, University of Arkansas, Fayetteville,Arkansas 72701, USAE-mail: cqthvc@nus.edu.sg and rvyas@uark.eduNew Journal of Physics 11 (2009) 113055 (17pp)Received 27 August 2009Published 30 November 2009Online at http://www.njp.org/doi:10.1088/1367-2630/11/11/113055Abstract. 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 Botoet al (2000 Phys. Rev. Lett. 85 2733), for etching simple patterns on multi-photonabsorbing materials with sub-Rayleigh resolution, using two-mode entangledstates of light. We consider an OPO with two down-converted modes that sharethe same frequency but differ in field polarization or direction of propagation,and derive analytical expressions for the multi-photon absorption rates whenthe OPO is operated below, near and above its threshold. Because of strongnon-classical correlations between the two modes of the OPO, the interferencepatterns resulting from the superposition of the two modes are characterized byan effective wavelength that is half of their actual wavelength. The interferencepatterns resulting when the two modes of the OPO are used for etching are alsocharacterized by an effective wavelength half that for the illuminating modes.We compare our results with those for the case of a high-gain optical amplifiersource and discuss the relative merit of the OPO.4 Authors to whom any correspondence should be addressed.New Journal of Physics 11 (2009) 1130551367-2630/09/113055+17$30.00 © IOP Publishing Ltd and Deutsche Physikalische Gesellschaft


2Contents1. Introduction 22. Method 42.1. The setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.2. Master equation and solution using the positive-P distribution . . . . . . . . . . 53. Results 83.1. Below-threshold regime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93.2. Near- and above-threshold regime . . . . . . . . . . . . . . . . . . . . . . . . 114. Discussion 13Acknowledgments 15Appendix. Fringe patterns below threshold 15References 161. IntroductionThe Rayleigh criterion states that diffraction limits the resolution of a traditional opticallithographic system, and specifies a minimum feature size of half the wavelength of theilluminating beam. A variety of classical procedures exist which can exceed this limit withcertain trade-offs, and some examples of optical and non-optical super-resolution techniquesare described in [1]–[3]. Boto et al suggested an entirely different approach in 2000in [4], and their proposal has attracted considerable research interest [5]–[11]. The keyobservation here is that all existing optical lithographic procedures assume that the illuminatinglight fields are classical. The Rayleigh limit arises in part from the fundamental photonstatistics of laser light, according to which the constituent photons are uncorrelated. Tocircumvent this, Boto et al proposed exploiting entangled states of light, and in particularpath-entangled states of light of the form |N0〉 + |0N〉 in the photon-number basis, oftentermed ‘N00N’ states. In the scheme detailed in [4], the N00N states propagate througha simple interferometer, and then interfere at a N-photon-absorbing recording material ina counter-propagating configuration. A brief calculation can illustrate the basic idea. TheN-photon absorption rate at the substrate is proportional to the expectation value of theobservable Ê (−)N Ê (+)N /N!, where Ê (+) =exp(ikx)â 1 + exp(−ikx)â 2 is the annihilation operatorfor the combined field. â 1 and â 2 denote annihilation operators for two interferometer modes, xdenotes translation across the substrate, and k denotes the wave-number 2π/λ for the light (weassume grazing incidence). For the case of classical illumination, the two interferometer modesare assumed to be in coherent states so that the absorption rate is proportional to [cos(2kx) + 1] N .For the case of N00N states, the absorption rate is proportional to [cos(2Nkx) + 1]. In principlethen, the procedure can etch a series of straight lines corresponding to an effective wavelengthof 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 ofentangled states [12]. Such super-resolution techniques exploiting non-classical states could beused in combination with existing sub-Rayleigh procedures based on classical methods. Theconcept of interferometric lithography using a multi-photon recording material with a classicalsource has been demonstrated experimentally [13]. However, considerable work remains to beNew Journal of Physics 11 (2009) 113055 (http://www.njp.org/)


3done to demonstrate the feasibility of using a bright non-classical source. A general programthat aims to do so will require investigation of the source, imaging system and multi-photonabsorption process.For the case of two-photon quantum lithography, one could use parametric downconversion in a medium exhibiting an χ (2) optical non-linearity as a source of photon pairs. Eachof these photon pairs propagating through a Hong–Ou–Mandel interferometer with a symmetricbeam splitter will yield a two-photon N00N state at the output ports of the interferometer [14].Subsequent quantum interference of the two modes on a two-photon absorbing material willthen create a fringe pattern with fringe spacing that is half of what is achieved when a classicallight source is used. However, the output of common parametric down-conversion experimentsis 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 anunseeded OPA is the two-mode squeezed vacuum state of the form ∑ ∞n=0 tanhn (G)|n〉|n〉 in thephoton-number basis. Here G is the gain parameter, which depends on the interaction volume inthe crystal, the amplitude of the electric field of the pump beam, and the strength of the secondordersusceptibility χ (2) . To quantify the contrast of the interference pattern at the output, thevisibility is defined as the difference of the maximum and minimum absorption rates, dividedby the sum of these rates. The visibility varies between 0 and 1. Explicit calculation revealsthat the visibility falls from 1 to an asymptotic value of 0.2, as the gain parameter G of theOPA is increased from 0 → ∞. This result was generalized in [9] to the case of higher-ordermulti-photon absorbing materials. Essentially the same behavior is seen in these cases, withthe visibility falling from 1 to an asymptotic value, which is greater the higher the order ofthe absorption process. These predicted features have been demonstrated experimentally, usingcoincidence measurements at photodetectors to simulate the recording medium [11].In this paper, we investigate the use of an optical parametric oscillator (OPO) as a practicalsource of non-classical light for lithography. In this case, the process of parametric amplificationoccurs in an optical cavity in resonance with the signal and idler modes. We will consider thesignal and idler modes to have the same optical frequency but orthogonal polarizations. Thesignal and idler cavity modes are coupled to external propagating modes through a transmissiveend mirror. Photons are created in pairs in the cavity modes, but escape independently outof the cavity on a timescale of the order of the cavity lifetime. The twin beams emergingfrom the OPO, corresponding to the light in the two modes, are highly correlated. Unlikethe OPA, the OPO has a well-defined threshold for oscillation, and the below-, near- andabove-threshold regimes require different mathematical treatments. The theory for the OPOis complicated by the need to account for mode losses, and the coupling of cavity and pumpmodes. To analyze the quantum lithography procedure using the light from an OPO, we willfollow the approach developed in [15]–[18] to investigate the properties of light from OPOs. Byusing the positive-P representation [19] of the density matrix, the Markovian master equationdescribing the dynamics of the standard model of the OPO can be translated into a multi-variateFokker–Planck equation for the positive-P function. The Fokker–Planck equation can be treatedusing the techniques from non-equilibrium classical statistical mechanics [20, 21], allowing thesteady-state expectation values of various observables to be evaluated. The positive-P methodhas not previously been applied to quantum lithography, and has crucial advantages comparedto other standard approaches. For the OPO source, it leads to a Fokker–Planck equation that canNew Journal of Physics 11 (2009) 113055 (http://www.njp.org/)


5absorption and scattering losses. We assume the pump field to be a classical field of frequency2ω and of normalized amplitude denoted by ε (assumed to be given by a positive value). Thecavity pump mode is subject to linear loss with decay rate γ 3 and we assume that γ 3 ≫ γ . Thecreation operator corresponding to the cavity pump mode is â † 3. The Hamiltonian describing thissystem in the interaction picture can be written as() ( )Ĥ I =i¯hκ â † 1↠2â3−â 1 â 2 â † 3+ i¯hγ 3 ε â † 3 −â 3 + Ĥ loss , (1)where κ is the mode-coupling constant determined by the strength of the optical nonlinearityand Ĥ 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 orthogonalpolarizations, is separated into two spatial paths by a pbs. A hwp placed in one path ensuresthe polarizations are made parallel. These modes are combined at a symmetric 50 : 50 beamsplitter and the output modes of the beam splitter are finally combined at the multi-photonabsorbing material with the help of mirrors so that the interfering beams are counter-propagatingat grazing incidence over the area of interest. Two classical beams incident on a single-photonabsorbing material would generate a fringe pattern of the form ∝ [1 + cos(2kx)], where kdenotes the optical wave number for the signal and idler modes, and x denotes translationacross the substrate. We shall denote the optical phase difference 2kx by φ. When the recordingmedium is a p-photon absorber and the illuminating beams have statistics which need not beclassical, the absorption rate is given by σ (p) tr[ ˆρ(Ê (−) (x)) p (Ê (+) (x)) p ], where ˆρ is the densitymatrix for the state of the illuminating field, Ê (+) and Ê (−) are, respectively, the positive andnegative frequency components of the field at the absorber, and σ (p) is a generalized crosssectionfor 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 bestrongly influenced by the statistical properties of the light [24].2.2. Master equation and solution using the positive-P distributionFollowing the standard procedure and using the Hamiltonian (1) for the OPO, the Markovianmaster equation for the reduced density operator for the pump, signal and idler fields is givenby [21, 25]∂ ˆρ∂t = 1 [ĤI , ˆρ ] + γi ¯h2∑(2â i ˆρâ † i − â † i âi ˆρ − ˆρâ † i âi) + γ 3 (2â 3 ˆρâ † 3 − ↠3â3 ˆρ − ˆρâ 3â3), † (2)i=1where Ĥ I here excludes the Ĥ loss term of equation (1), which couples the cavity modes toreservoir modes. The effect of Ĥ loss is contained in the decay terms in equation (2). We now mapthis master equation into a classical Fokker–Planck equation using the positive-P representationintroduced by Gardiner and Drummond [19]. For the current problem this representation isdefined as follows,∫ˆρ=3∏Di=1d 2 α i (d) 2 |α i 〉 〈α i∗ |α i∗ P (⃗α) , (3)〈α i∗ |α i 〉where α 1 , α 1∗ , α 2 , α 2∗ , α 3 and α 3∗ are six independent complex variables, and we have written⃗α≡(α 1 , α 1∗ , α 2 , α 2∗ , α 3 , α 3∗ ). It should be emphasized that the asterisks in the variable indicesNew Journal of Physics 11 (2009) 113055 (http://www.njp.org/)


6do not correspond to complex conjugation. The complex variables α i and α i∗ correspond to themode annihilation and creation operators via the relations â 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 aprobability density function: it is real valued, positive and normalized to one when integratedover the full domain D of ⃗α. For the master equation, equation (2), the corresponding positive-Pfunction satisfies the following equation,∂ P∂t=[ ∂∂α 1(γ α 1 − κα 2∗ α 3 ) + ∂∂α 2(γ α 2 − κα 1∗ α 3 )+ ∂ (γ α 1∗ − κα 2 α 3∗ ) +∂ (γ α 2∗ − κα 1 α 3∗ )∂α 1∗ ∂α 2∗+ ∂ (γ 3 α 3 + κα 1 α 2 − γ 3 ε) +∂ (γ 3 α 3∗ + κα 1∗ α 2∗ − γ 3 ε)∂α 3 ∂α 3∗+ κ ∂2 α 3+ κ∂2 α]3∗P. (4)∂α 1 ∂α 2 ∂α 1∗ ∂α 2∗This equation has the form of a multivariate Fokker–Planck equation [26], and the use of thepositive-P distribution ensures that the diffusion matrix is positive. Then the Langevin equationscorresponding to equation (4) can be written asdα 1dtdα 2dt= −γ α 1 + κα 2∗ α 3 + √ κα 3[ξ 1 (t) + iξ 2 (t)]√2,= −γ α 2 + κα 1∗ α 3 + √ κα 3[ξ 1 (t) − iξ 2 (t)]√2,dα 3dtdα 1∗dtdα 2∗dt= −γ 3 α 3 − κα 1 α 2 + γ 3 ε,= −γ α 1∗ + κα 2 α 3∗ + √ κα 3∗[ξ 3 (t) − iξ 4 (t)]√2,= −γ α 2∗ + κα 1 α 3∗ + √ κα 3∗[ξ 3 (t) + iξ 4 (t)]√2,(5)dα 3∗dt= −γ 3 α 3∗ − κα 1∗ α 2∗ + γ 3 ε,where the ξ i are real-valued, white-noise, stochastic variables with 〈ξ j (t)〉 = 0 and〈ξ i (t)ξ j (t ′ )〉 = δ i j δ(t − t ′ ) 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. Setting ˙α 3 = ˙α 3∗ = 0 we findα 3 = ε − κ γ 3α 1 α 2 ,α 3∗ = ε − κ γ 3α 1∗ α 2∗ .(6)New Journal of Physics 11 (2009) 113055 (http://www.njp.org/)


7Using these results in equation (5), we obtain√dα 1dτ = −α 1 + (σ − 2α ( )1α 2 ) α 2∗ σ − 2α 1 α 2 ξ1 + iξ 2+√ ,n 0 n 0 2dα 2dτ = −α 2 + (σ − 2α 1α 2 ) α 1∗n 0+dα 1∗dτ = −α 1∗ + (σ − 2α 1∗α 2∗ ) α 2n 0+dα 2∗dτ = −α 2∗ + (σ − 2α 1∗α 2∗ ) α 1n 0+√σ − 2α 1 α 2( )ξ1 − iξ 2√ ,n 0 2√σ − 2α 1∗ α 2∗( )ξ3 − iξ 4√ ,n 0 2√( )σ − 2α 1∗ α 2∗ ξ3 + iξ 4√ ,n 0 2where time has been scaled in terms of the cavity lifetime (τ = γ t). Parameter n 0 = 2γ γ 3 /κ 2is proportional to the square of the number of photons in the cavity at threshold and it sets thescale for the number of photons necessary to explore the nonlinearity of interaction. σ = 2γ 3 ε/κis a dimensionless measure of the pump field amplitude, so that the threshold condition εκ = γgives σ = n 0 .Using the procedure set out in [16, 17], the dimensionality of this set of Langevin equationscan be reduced from eight to four. Accordingly, we introduce four real-valued variables u 1 ,u 2 ,u 3and u 4 by⎛ ⎞⎛⎞ ⎛ ⎞α 1 ( ) 1 1 i i u 1⎜ α 2⎟ σ2 1/4⎝α 1∗⎠ = ⎜1 1 −i −i⎟ ⎜u 2⎟8n 0⎝1 −1 −i i⎠⎝u 3⎠ . (8)α 2∗ 1 −1 i −i u 4The variables u i may be interpreted as scaled pseudo-quadrature variables. An exact distributionfor 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 laboratorysystems [27], the positive-P function is given to a very good approximation by{ (P(⃗u) ∝ exp a 1 u21 + ) ( u2 3 + a2 u22 + ) 1[ (u u2 4 −212+ u2 2 + u2 3 + ) 2 ( u2 4 + 4 u21 + ) ( u2 3 u22 + 4) ] }u2 , (9)where ⃗u = (u 1 , u 2 , u 3 , u 4 ) . Parameters a 1 , a 2 correspond to the pump strength and are given bya 1 = √ 2n 0 (r − 1),a 2 = − √ 2n 0 (r + 1),where r = σ/n 0 = κε/γ . It can be seen from equation (9) that any moment of the form,∫∫∫∫〈un 1〉1 un 22 un 33 un 44 = d 4 u u n 11 un 22 un 33 un 44 P(⃗u),has the value zero if any of the n i is odd. The distribution of equation (9) can be used to evaluatethe steady-state expectation values for any normally ordered product of the field operators. In thenext section, we will compute the absorption rates for multi-photon recording media, exploitingthe symmetries of P(⃗u) and making approximations valid for the different regimes of operationof the OPO.New Journal of Physics 11 (2009) 113055 (http://www.njp.org/)(7)(10)


83. ResultsIn this section, we compute the multi-photon absorption rates at the recording mediumfor a quantum-lithographic process, by computing expectation values using the positive-Pdistribution 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 andidler fields to interfere at a symmetric 50 : 50 beam splitter, the fields at the output are given byβ 1 = (−α 1 + iα 2 )√2,β 2 = (−α 2 + iα 1 )√2,β 1∗ = (−α 1∗ − iα 2∗ )√2,(11)β 2∗ = (−α 2∗ − iα 1∗ )√2.Propagating the fields to the multi-photon recording material, the combined field at a locationcorresponding 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 obtainβ 3 = √ 1 [α1 (−e iφ + i) + α 2 (ie iφ − 1) ]2β 3∗ = 1 √2[α1∗ (−e −iφ − i) + α 2∗ (−ie −iφ − 1) ] .The rate of p-photon absorption then is given by the average quantity,(12)I p3 (φ)=〈β p 3 (⃗u)β p 3∗(⃗u)〉. (13)Substituting equation (8), we also have,β 3∗ β 3 = 1 [(u2c 2 1 − u 2 ) cos( φ2)+ i (u 3 − u 4 ) sin( φ2)][( )( )]φ φ× (u 1 + u 2 ) cos − i (u 3 + u 4 ) sin . (14)2 2We first evaluate the fringe pattern for a one-photon absorber, given by I3 1 (φ). Inspecting thesymmetries of the positive-P function, equation (9), we see that symmetries exist betweenthe 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 ) andP(u 1 , u 2 , u 3 , u 4 ) = P(u 1 , u 4 , u 3 , u 2 ). It follows immediately thatI 1 3 (φ) = r√ 2n 0(〈u21 〉−〈u2 2〉). (15)This equation implies that there is no dependence on φ either below or above threshold and theillumination of the substrate is uniform across the surface in the ensemble-averaged sense. Nextwe consider the case of multi-photon absorbing materials.New Journal of Physics 11 (2009) 113055 (http://www.njp.org/)


93.1. Below-threshold regimeInspecting the positive-P distribution, equation (9), for the below-threshold case, κε ≪ γ , wefind that a 1 and a 2 are large negative quantities, and the distribution can be approximated by(√ ) (√ ) (√ ) (√ )|a1 |P(⃗u) ≈π e−|a 1|u 2 |a1 |1π e−|a 1|u 2 |a2 |3π e−|a 2|u 2 |a2 |2π e−|a 2|u 2 4 . (16)For a typical OPO with n 0 ∼ 10 6 this approximation is very good for the parameter r rangingfrom 0 to 0.99. The four variables u i are Gaussian and independent. Their even-order momentsare given by〈u2k1〉 〈 〉= u2k(2k)!3 =k! (4 |a 1 |) , k〈 〉 〈 〉(17)u2k2 = u2k(2k)!4 =k! (4 |a 2 |) , kand their odd-order moments vanish.In order to compute the multi-photon absorption rates I p3(φ), defined by equations (13)and (14), a suitable grouping of terms must be found. To this end we define the functionso that I p3F(u i , u j , φ) = [u i cos(φ/2) − iu j sin(φ/2)] 2 , (18)can be expressed asp∑( )I p3 (φ) = (r√ 2n 0 ) p p(−1) 〈 k p−k F k (u 1 , u 4 , φ) 〉 〈 F p−k (u 2 , u 3 , φ) 〉 . (19)k=0To evaluate the moments needed in this expression we exploit the fact that u i are uncorrelatedvariables. Moreover, since the pairs of variables u 1 and u 3 , and u 2 and u 4 , share the samedistribution, we conclude(−1) p−k 〈 F p−k (u 2 , u 3 , φ) 〉 = 〈 F p−k (u 3 , u 2 , φ + π) 〉l=0= 〈 F p−k (u 1 , u 4 , φ + π) 〉 .Expanding an arbitrary moment k of F(u 1 , u 4 , φ) and using the observations of the precedingparagraph, we find〈F (u1 , u 4 , φ) k〉 k∑( )2k 〈u 〉 〈 〉2l=2l1 u2k−2l4 (−1) (k−l) cos 2l (φ/2) sin 2k−2l (φ/2). (20)By substituting the expressions for the even moments of u 1 and u 4 in equation (17) and the formof parameters a 1 and a 2 defined by equation (10), this sum can be evaluated as〈F (u1 , u 4 , φ) k〉 (2k)!=k!4 k ( √ 2n 0 ) k (1 − r 2 ) [r + k cos(φ)]k . (21)Using these relations, we arrive at a compact expression for the p-photon absorption rate,[ ]3 (φ) = r p(p)!4(1 − r 2 )I pp∑k=0(2k)!(2p − 2k)!k! 2 (p − k)! 2 [r + cos(φ)] k [r − cos(φ)] (p−k) . (22)From an examination of the form of the absorption rates I p3(φ) in equation (22), weconclude that for a single photon absorber (p = 1), the absorption rate is constant and noNew Journal of Physics 11 (2009) 113055 (http://www.njp.org/)


10(a)(b)p3I (φ)/ I p 3 (0)1.00.80.60.40.2p = 1p = 2p = 60 1 2 3 4 5 6φpI 3 (φ)/ I p 3 (0)1.0 p = 10.8 p = 20.6 p = 30.4 p 40.25p = 60 1 2 3 4 5 6φFigure 2. Absorption rates I p3(φ) as functions of φ for p = 1, . . . , 6 for the OPOoperating below threshold for pump ratios (a) r = 0.15 and (b) r = 0.9. As pincreases the fringes get sharper.Visibility V1.00.80.60.40.2p = 6p = 5p = 4p = 3p = 20.00.0 0.2 0.4 0.6 0.8 1.0 rFigure 3. Fringe visibility, defined by V = [I p 3 (φ max) − I p3 (φ min)]/[I p 3 (φ max) +I p3 (φ min)] (0 V 1) for p = 2, . . . , 6 as a function of pump ratio r for theOPO operating below threshold.fringe pattern is created, as already seen following equation (15). For a multi-photon (p 2)absorber, the rate I p3(φ) is given as a sum of even powers of cos(φ). The corresponding fringepattern therefore has terms with periods corresponding to cos(2nφ) where 2n p. As pincreases, there is an increasing contribution from higher-power terms, resulting in sharperinterference 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 casewith p 2, the visibility is close to 1 when the OPO is operated far below threshold, but asthe pump power is increased the visibilities fall steadily. In a lithographic process, it is possibleto compensate for unwanted constant exposure by using a substrate with greater depth, andvisibilities as little as 0.2 are often considered adequate in practice [5]. This criterion is satisfiedin all cases considered here with p 2. Equation (22) is listed explicitly in table 1 of appendixfor the cases of p = 1, . . . , 6. The corresponding results for an OPA were reported in the workof Agarwal et al [9]. A comparison of our results with equations (19)–(22) of [9] shows thatthe two sets of formulae for the multi-photon absorption rates coincide in identifying r withtanh(G) in the analysis of Agarwal et al, where G represents the single-pass gain for the OPA.New Journal of Physics 11 (2009) 113055 (http://www.njp.org/)


11The pump ratio range 0 r < 1 for the OPO (below threshold), corresponds to the OPA singlepassgain range 0 G < ∞. Visibilities for the sub-threshold OPO are listed in table 2 of theappendix for p = 1, . . . , 6, which agree with the corresponding results reported for the OPAover the corresponding range of the parameter G.A striking feature of the fringe patterns, predicted for both an OPO source and anOPA source, is the increasing visibility with the order p of the absorption process. In theanalysis above for the OPO, we have identified the origin of this improvement to be thecontributions of higher spatial frequency interference terms. This improvement of visibilitywith the order of absorption is not a quantum effect; other interference-based setups withclassical fields can show similar improvements. An example is provided by [28], which lookedat a higher-order generalization of a standard Hanbury Brown—Twiss setup. The authorsconsider intensity correlations at two or more photodetectors placed in different locations andexcited by two distant light sources. As is well known, the visibility of second-order intensityinterference cannot exceed 50% if the light sources are classical. However, Agafonov et alhave shown theoretically and experimentally that the visibilities (given classical sources) canbe much greater for higher-order measurements, which record coincidences at three or fourphotodetectors.3.2. Near- and above-threshold regimeAs the OPO is pumped more strongly and passes through threshold, pump ratio r exceedsunity and with that the nature of the positive-P distribution changes significantly. Inspectingthe pump parameters a 1 and a 2 , defined by equation (10), we see that while a 2 continues to takelarge negative values of increasing size as r increases, a 1 is zero at threshold and takes positivevalues for pump ratio r > 1. The positive-P distribution, given by equation (9), may now bewell approximated by the expression below, which is valid both near and far above thresholdfor values of n 0 greater than 10 4 [18],P(⃗u) ≃(√|a2 |π e−|a 2|u 2 2) (√ )|a2 | (π e−|a 2|u 2 4 Ne − 1 2(u 2 1 +u2 3 −a 1) 2) , (23)where N denotes a normalization factor for the u 1 , u 3 component, defined by√2N = [π3/2erfc( − a 1 / √ 2) ],and erfc(z) ≡ (2/ √ π) ∫ ∞e −s2 ds is the complementary error function. It can be seen that thezvariables u 2 and u 4 are independent Gaussian variables with pump parameter a 2 in the belowthresholdregime. However, the variables u 1 and u 3 with pump parameter a 1 are now stronglycoupled. 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 incomputing I p (φ) above threshold. The moments for the Gaussian variables u 2 and u 4 are asbefore given by equation (17). By expressing u 1 and u 3 in polar coordinates, it follows that ageneral moment of the form 〈u 2s1 u2t 3 u2m 2 u2n 4〉 can be expressed as〈u2s1 u2t 3〉〈u2m2〉〈 〉u2n4 = R (2 s + 2t + 1, a1 ) B(s + 1 2 , t + 1 2New Journal of Physics 11 (2009) 113055 (http://www.njp.org/))2N(2m)!m! (4 |a 2 |) m (2n)!n!(4|a 2 |) n , (24)


12where B(·, ·) denotes the Beta function, defined by B(S, T ) ≡ Ɣ(S)Ɣ(T )/ Ɣ(S + T ), whicharises from the integration over the angular component associated with u 1 and u 3 . Theintegration over the remaining radial component leads to the R(·) contribution defined byR(S, a 1 ) ≡ ∫ ∞ρ S e − 1 2 (ρ2 −a 1 ) 2 dρ. Writing0R (2s + 2t + 1, a 1 ) =∫ ∞0ρ 2s+2t−1 (ρ 2 − a 1 )e − 1 2[ρ 2 −a 1] 2 dρ + a 1 R (2s + 2t − 1, a 1 ) , (25)for s + t 1, and integrating by parts,⎧∫ ∞ρ ( ) ⎨112s+2t−1 ρ 2 − a 1 e− 2[ρ 1 2 −a 1] 2 dρ = 2 e− 2 a2 1 (if s + t = 1),0⎩(s + t − 1)R(2s + 2t − 3, a 1 ) (if s + t > 1),we get a recursion relation for R(·). Here we list the first few radial functions,(26)R (1, a 1 ) = 12π N ,)+ a 11R (3, a 1 ) = 1 (−2 exp 1 2 a2 12π N ,R (5, a 1 ) = a (12 exp − 1 )2 a2 1+ (1 + a 2 1 ) 1R (7, a 1 ) =(1 + 1 2 a2 1R (9, a 1 ) =( 52 a 1 + 1 2 a3 12π N ,) (exp − 1 )2 a2 1+ ( )3a 1 + a13 1)(exp − 1 )2 a2 1+2π N ,(3 + 6a21+ a1)42π N.(27)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 topositive values, the OPO goes through a phase transition and the intensities of the signal andidler modes increase very rapidly. For typical values of n 0 = 10 6 − 10 8 the region of phasetransition is very narrow and corresponds to a small change, approximately, 0.99 → 1.01in r. The statistics of the OPO change dramatically [17, 18] over this range. We find that thebehavior of visibility for p-photon absorption also changes significantly as the operating pointof the OPO changes from below to above threshold.For the OPO operation much above threshold (a 1 ≫ 1), we can considerably simplify theexpression for I p3(φ). It follows from equations (24) and (27) that 〈u2s1 u2t 3〉 ≃ B(s + 1/2, t +1/2)a1 s+t /π. Since powers of n 0 appear only in the denominator for moments of u 2 andu 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 thisapproximation,β p 3 β p 3∗ ≃ (r√ 2n 0 ) p [ u 2 1 cos2 (φ/2) + u 2 3 sin2 (φ/2) ] p,New Journal of Physics 11 (2009) 113055 (http://www.njp.org/)


13Figure 4. Fringe visibility for p = 2 as a function of pump ratio r as the operatingpoint of the OPO changes from below threshold to above threshold for n 0 = 10 6 .and we find the absorption rateI p (φ) = (r √ 2n 0 ) pp∑( ) 〈 〉 ( ) ( )p φ φus2s1 u2p−2s 3cos 2s sin 2p−2s2 2s=0=(a1 r √ ) p p∑2n 08s=0(2s)! (2p − 2s)!s! 2 (p − s)! 2 [1 + cos(φ)] s [1 − cos(φ)] p−s . (28)The p-photon absorption rates for much of the above threshold operation of the OPO aretherefore seen to generate fringe patterns, which are largely independent of the strength ofthe pump. Figure 4 shows how the fringe visibility changes near the threshold regime for thetwo-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.4. DiscussionIn conclusion, we have found that an OPO source can be used to generate fringe patterns with aneffective wavelength half of the actual wavelength of the signal and idler modes that interfere ata p-photon absorbing medium (p 2). For the case of two-photon absorption, the visibility ofthe predicted fringe pattern falls from a maximum of 1 below threshold to an asymptotic valueof 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 0.63for p = 4. Below threshold, the forms of the fringe patterns depend strongly on the pumppower. 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. Comparingthis with the results reported in [9], we find that the fringe patterns generated by using anOPO operating below threshold (0 r < 1) are similar to fringe patterns generated by an OPAoperated with a corresponding gain in the range (0 G < ∞). In the case of an OPA there isno cavity and no above-threshold regime, and the process of optical parametric amplificationoccurs 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 aNew Journal of Physics 11 (2009) 113055 (http://www.njp.org/)


14single-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 asingle mode selection.The reparameterization of the dynamics of the OPO in terms of the four pseudo-quadraturevariables u i above sheds some light on why the changes in fringe patterns and visibilitiesare observed. Far below threshold, all four of the u i contribute approximately equally to theabsorption process, and behave as independent variables. As the pump power is increased andthreshold is approached, parameter a 1 tends to 0. Since a 1 appears in the denominator of each ofthe 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 inthis regime. The results of these changes are expressed in the compact general formulae for themulti-photon absorption rates, equations (22) and (28), involving a sum over powers of productsof terms [r + cos(φ)] k [r − cos(φ)] p−k . Below threshold parameter r < 1 the effect of increasingthe order p of the multi-photon absorption process is to suppress the contributions of powers ofr in favor of powers of cos(φ) around its maxima, sharpening the fringe pattern. As thresholdis approached, i.e. r → 1 − , the contribution of cos(φ) relative to r decreases, and the fringepatterns assume a fixed form. Hence the visibility saturates above threshold.The OPO source has several experimental advantages. An OPO source would haveimproved temporal and spatial coherence compared to an OPA. The signal and idler modesat the outputs of the OPO are collimated because of the use of a cavity, and higher powerscan be obtained than is the case for the OPA, both of which are important in light of the smallcross-sections for typical multi-photon absorption processes. Collimated, high power outputsalso increase the speed at which a substrate may be imaged—a critical factor in the large-scaleproduction of, say, computer chips.We end by looking at the requirements for driving a high-order absorption process in anexperimental implementation of quantum lithography. In our proposal, there are competingdemands on the pump power and gain. At lower intensities, the absorption of strongly correlatedphoton pairs emitted by the OPO in the recording material leads to a halving of the fringespacing and to visibilities approaching one. However, the need for high intensities at the outputto drive a higher-order absorption process will increase the probability for absorption of photonsfrom different pairs, thereby reducing the visibility of the fringe pattern created.In the recent experiment reported in [13] an interferometric lithographic setup, as illustratedin figure 1 but with a classical laser beam at 800 nm as the source, was implementedwith a lithographic recording material based on polymethyl-methacrylate (PMMA). PMMAis transparent across the visible spectrum, but absorbs strongly at ultra-violet frequencies.High-visibility fringe patterns were demonstrated, which were consistent with multi-photonabsorption of order three and higher. Using a classical phase-shifted-grating method,an approximately two-fold enhancement of resolution was demonstrated. The experimentestablished a window of viable pulse energies, 80–135 µJ, for the lithographic process. Weturn to a candidate OPO source, reported in [29], which we estimate would achieve a furtherdoubling 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 powerat threshold is 225 mW. 35% above threshold, i.e. for parameter ratio r = 1.35, the outputpower is 50 mW for each of the signal and idler modes. Taking the parameter n 0 = 10 8 andusing equations (22) and (28), find that just below threshold, the power in each of the signalNew Journal of Physics 11 (2009) 113055 (http://www.njp.org/)


15Table 1. p-photon absorption rates as functions of the optical phase differenceφ for the pump ratio r varying from far below to near threshold (from 0 toapproximately 0.99).p1I p3 (φ)r 2(1 − r 2 )23456r 2(1 − r 2 ) 2 [cos2 (φ) + 2r 2 ]3r 4(1 − r 2 ) 3 [3 cos2 (φ) + 2r 2 ]3r 4(1 − r 2 ) 4 [3 cos4 (φ) + 24r 2 cos 2 (φ) + 8r 4 ]15r 6(1 − r 2 ) 5 [15 cos4 (φ) + 40r 2 cos 2 (φ) + 8r 4 ]45r 6(1 − r 2 ) 6 [5 cos6 (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 theabove-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 takeadvantage of a higher-visibility fringe pattern and strong quantum correlations below thresholdby using a longer exposure time.AcknowledgmentsJPD and HC would like to acknowledge the Army Research Office, the Defense AdvancedResearch Projects Agency, and the Intelligence Advanced Research Projects Activity. HCfurther acknowledges support for this work from the National Research Foundation andMinistry of Education, Singapore. We thank Olivier Pfister for helpful discussions, and thereferees for their constructive comments.Appendix. Fringe patterns below thresholdBased on the general formula for the absorption rates below threshold, equation (22), thefollowing tables explicitly list the absorption rates and the fringe visibilities corresponding tofringe patterns for p-photon absorbers with p ranging from 1 to 6.New Journal of Physics 11 (2009) 113055 (http://www.njp.org/)


16Table 2. Fringe visibility, defined by V = [I p 3 (φ max) − I p3 (φ min)]/[I p 3 (φ max) +I p3 (φ min)], for the interference patterns with a p-photon absorbing recordingmaterial. The third column lists the limiting values for the visibility as the pumpratio r ranges from far below to near threshold (from 0 to approximately 0.99).p Visibility V (r) V (0 + ) → V (1 − )1 0 —2 1 − 4r 21 → 0.20[1 + 4r 2 ]3 1 − 4r 2[3 + 4r 2 ]4 1 −5 1 −16r 4[3 + 24r 2 + 16r 4 ]16r 4[15 + 40r 2 + 16r 4 ]1 → 0.431 → 0.631 → 0.776 1 −32r 6[5 + 90r2+ 120r 4 + 32r 6]1 → 0.87References[1] Mack C A 2006 Field Guide to Optical Lithography vol FG06 (Bellingham, WA: SPIE Press)[2] Yablonovitch E and Vrijen R B 1999 Opt. Eng. 38 334[3] Schift H 2008 J. Vac. Sci. Technol. B 26 458[4] Boto A N, Kok P, Abrams D S, Braunstein S L, Williams C P and Dowling J P 2000 Phys. Rev. Lett.85 2733[5] Agarwal G S, Boyd R W, Nagasako E M and Bentley S J 2001 Phys. Rev. Lett. 86 1389[6] D’Angelo M, Chekhova M V and Shih Y 2001 Phys. Rev. Lett. 87 013602[7] Boyd R W, Chang H J, Shin H and O’Sullivan-Hale C 2005 Quantum Communications and QuantumImaging III vol 5893 (San Diego, CA: SPIE) pp 58930G.1–G.4[8] Fukutake N 2005 J. Mod. Opt. 53 719[9] Agarwal G S, Chan K W, Boyd R W, Cable H and Dowling J P 2007 J. Opt. Soc. Am. B 24 270[10] Tsang M 2007 Phys. Rev. A 75 043813[11] Sciarrino F, Vitelli C, De Martini F, Glasser R, Cable H and Dowling J P 2008 Phys. Rev. A 77 012324[12] Kok P, Boto A N, Abrams D S, Williams C P, Braunstein S L and Dowling J P 2001 Phys. Rev. A 63 063407[13] Chang H J, Shin H, O’Sullivan-Hale M N and Boyd R W 2006 J. Mod. Opt. 53 2271[14] Hong C K, Ou Z Y and Mandel L 1987 Phys. Rev. Lett. 59 2044[15] Vyas R and Singh S 1989 Phys. Rev. A 40 5147[16] Vyas R 1992 Phys. Rev. A 46 395[17] Vyas R and Singh S 1995 Phys. Rev. Lett. 74 2208[18] Vyas R and Singh S 2006 ICQO ed J Banerji, P K Panigrahi and R P Singh (Delhi: Macmillan India Ltd)[19] Drummond P D and Gardiner C W 1980 J. Phys. A: Math. Gen. 13 2353[20] Carmichael H J 2008 Statistical Methods in Quantum Optics 2. Non-Classical Fields 1st edn vol 2 (Berlin:Springer)New Journal of Physics 11 (2009) 113055 (http://www.njp.org/)


17[21] Gardiner C W and Zoller P 2004 Quantum Noise, a Handbook of Markovian and Non-Markovian QuantumStochastic Methods with Applications to Quantum Optics 3rd edn (Berlin: Springer)[22] Mollow B R 1968 Phys. Rev. 175 1555[23] Agarwal G S 1970 Phys. Rev. A 1 1445[24] Qu Y and Singh S 1992 Opt. Commun. 90 111[25] Carmichael H J 1998 Statistical Methods in Quantum Optics 1. Master equations and Fokker–Planckequations 1st edn vol 1 (Berlin: Springer)[26] Risken H 1984 The Fokker–Planck equation, Methods of Solution and Applications vol 1 (Berlin: Springer)[27] Holliday G S and Singh S 1987 Opt. Commun. 62 289[28] Agafonov I N, Chekhova M V, Iskhakov T S and Penin A N 2008 Phys. Rev. A 77 053801[29] Jing J, Feng S, Bloomer R and Pfister O 2006 Phys. Rev. A 74 041804New Journal of Physics 11 (2009) 113055 (http://www.njp.org/)

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