An optical parametric oscillator as a high-flux source of two-mode ...

Home Search Collections Journals About Contact us My IOPscience**An** **optical** **parametric** **oscillator** **as** a **high**-**flux** **source** **of** **two**-**mode** light for quantumlithographyThis article h**as** been downloaded from IOPscience. Ple**as**e 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 w**as** downloaded on 03/12/2010 at 20:57Ple**as**e note that terms and conditions apply.

New Journal **of** PhysicsThe open–access journal for physics**An** **optical** **parametric** **oscillator** **as** a **high**-**flux** **source****of** **two**-**mode** light for quantum lithographyHugo Cable 1,2,4 , Reeta Vy**as** 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** Arkans**as**, Fayetteville,Arkans**as** 72701, USAE-mail: cqthvc@nus.edu.sg and rvy**as**@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-cl**as**sical 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 **mode**s 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-cl**as**sical correlations between the **two** **mode**s **of** the OPO, the interferencepatterns resulting from the superposition **of** the **two** **mode**s are characterized byan effective wavelength that is half **of** their actual wavelength. The interferencepatterns resulting when the **two** **mode**s **of** the OPO are used for etching are alsocharacterized by an effective wavelength half that for the illuminating **mode**s.We compare our results with those for the c**as**e **of** a **high**-gain **optical** amplifier**source** 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. M**as**ter 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 **optical**lithographic system, and specifies a minimum feature size **of** half the wavelength **of** theilluminating beam. A variety **of** cl**as**sical procedures exist which can exceed this limit withcertain trade-**of**fs, 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 h**as** attracted considerable research interest [5]–[11]. The keyobservation here is that all existing **optical** lithographic procedures **as**sume that the illuminatinglight fields are cl**as**sical. The Rayleigh limit arises in part from the fundamental photonstatistics **of** l**as**er 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 b**as**is, **of**tentermed ‘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 b**as**ic 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 **mode**s, xdenotes translation across the substrate, and k denotes the wave-number 2π/λ for the light (we**as**sume grazing incidence). For the c**as**e **of** cl**as**sical illumination, the **two** interferometer **mode**sare **as**sumed to be in coherent states so that the absorption rate is proportional to [cos(2kx) + 1] N .For the c**as**e **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 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-cl**as**sical states could beused in combination with existing sub-Rayleigh procedures b**as**ed on cl**as**sical methods. Theconcept **of** interferometric lithography using a multi-photon recording material with a cl**as**sical**source** h**as** been demonstrated experimentally [13]. However, considerable work remains to beNew Journal **of** Physics 11 (2009) 113055 (http://www.njp.org/)

3done to demonstrate the fe**as**ibility **of** using a bright non-cl**as**sical **source**. A general programthat aims to do so will require investigation **of** the **source**, imaging system and multi-photonabsorption process.For the c**as**e **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. Each**of** 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** **mode**s on a **two**-photon absorbing material willthen create a fringe pattern with fringe spacing that is half **of** what is achieved when a cl**as**sicallight **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 **as**suming 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 b**as**is. 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 contr**as**t **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 **as**ymptotic value **of** 0.2, **as** the gain parameter G **of** theOPA is incre**as**ed from 0 → ∞. This result w**as** generalized in [9] to the c**as**e **of** **high**er-ordermulti-photon absorbing materials. Essentially the same behavior is seen in these c**as**es, withthe visibility falling from 1 to an **as**ymptotic value, which is greater the **high**er the order **of**the absorption process. These predicted features have been demonstrated experimentally, usingcoincidence me**as**urements 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-cl**as**sical light for lithography. In this c**as**e, the process **of** **parametric** amplificationoccurs in an **optical** cavity in resonance with the signal and idler **mode**s. We will consider thesignal and idler **mode**s to have the same **optical** frequency but orthogonal polarizations. Thesignal and idler cavity **mode**s are coupled to external propagating **mode**s through a transmissiveend mirror. Photons are created in pairs in the cavity **mode**s, but escape independently out**of** 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** **mode**s, are **high**ly correlated. Unlikethe OPA, the OPO h**as** 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 pump**mode**s. 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 m**as**ter equationdescribing the dynamics **of** the standard **mode**l **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 cl**as**sical statistical mechanics [20, 21], allowing thesteady-state expectation values **of** various observables to be evaluated. The positive-P methodh**as** not previously been applied to quantum lithography, and h**as** 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 **as**sume the pump field to be a cl**as**sical field **of** frequency2ω and **of** normalized amplitude denoted by ε (**as**sumed to be given by a positive value). Thecavity pump **mode** is subject to linear loss with decay rate γ 3 and we **as**sume 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 † 1â† 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 **mode**s with the **mode**s **of** the loss reservoir.Light emitted by the OPO into the external signal and idler **mode**s, **as**sumed to have orthogonalpolarizations, is separated into **two** spatial paths by a pbs. A hwp placed in one path ensuresthe polarizations are made parallel. These **mode**s are combined at a symmetric 50 : 50 beamsplitter and the output **mode**s **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 cl**as**sical 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 **mode**s, and x denotes translationacross the substrate. We shall denote the **optical** ph**as**e difference 2kx by φ. When the recordingmedium is a p-photon absorber and the illuminating beams have statistics which need not becl**as**sical, 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,qu**as**i-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. M**as**ter equation and solution using the positive-P distributionFollowing the standard procedure and using the Hamiltonian (1) for the OPO, the Markovianm**as**ter 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 − â† 3â3 ˆρ − ˆρâ 3â3), † (2)i=1where Ĥ I here excludes the Ĥ loss term **of** equation (1), which couples the cavity **mode**s toreservoir **mode**s. The effect **of** Ĥ loss is contained in the decay terms in equation (2). We now mapthis m**as**ter equation into a cl**as**sical 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 emph**as**ized that the **as**terisks 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 the**mode** 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 **as**sumed 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 m**as**ter 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 h**as** 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 **as**dα 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, stoch**as**tic 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 **as**sumed 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 h**as** 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 me**as**ure **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),h**as** 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** operation**of** 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** ph**as**e 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 c**as**e **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 c**as**e, 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 **as**p∑( )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 form**of** 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 pincre**as**es 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 h**as** terms with periods corresponding to cos(2nφ) where 2n p. As pincre**as**es, there is an incre**as**ing contribution from **high**er-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 c**as**ewith p 2, the visibility is close to 1 when the OPO is operated far below threshold, but **as**the pump power is incre**as**ed 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 **of**ten considered adequate in practice [5]. This criterion is satisfiedin all c**as**es considered here with p 2. Equation (22) is listed explicitly in table 1 **of** appendixfor the c**as**es **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 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-p**as**s 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 singlep**as**sgain 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 incre**as**ing 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** **high**er spatial frequency interference terms. This improvement **of** visibilitywith the order **of** absorption is not a quantum effect; other interference-b**as**ed setups withcl**as**sical fields can show similar improvements. **An** example is provided by [28], which lookedat a **high**er-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 **source**s. As is well known, the visibility **of** second-order intensityinterference cannot exceed 50% if the light **source**s are cl**as**sical. However, Agafonov et alhave shown theoretically and experimentally that the visibilities (given cl**as**sical **source**s) canbe much greater for **high**er-order me**as**urements, which record coincidences at three or fourphotodetectors.3.2. Near- and above-threshold regimeAs the OPO is pumped more strongly and p**as**ses 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** incre**as**ing size **as** r incre**as**es, 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 c**as**e, can no longer be applied.To proceed further in this c**as**e, 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 **as**before 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 **as**sociated 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 incre**as**es from negative topositive values, the OPO goes through a ph**as**e transition and the intensities **of** the signal andidler **mode**s incre**as**e very rapidly. For typical values **of** n 0 = 10 6 − 10 8 the region **of** ph**as**etransition 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 point**of** 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 **of**the pump. Figure 4 shows how the fringe visibility changes near the threshold regime for the**two**-photon c**as**e. 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 **mode**s that interfere ata p-photon absorbing medium (p 2). For the c**as**e **of** **two**-photon absorption, the visibility **of**the predicted fringe pattern falls from a maximum **of** 1 below threshold to an **as**ymptotic value**of** 0.2 at **high** pump powers. Similar behavior is found for **high**-order absorption processes,and the **as**ymptotic value incre**as**es 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 **as**ymptotic 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 c**as**e **of** an OPA there isno cavity and no above-threshold regime, and the process **of** **optical** **parametric** amplificationoccurs in propagating **mode**s. The agreement between the **two** results is not surprising, since [9]considers an idealized **mode**l **of** an OPA by disregarding photon losses, and **as**suming 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 **mode**l,**mode** losses are included in the decay rates and the presence **of** a cavity naturally provides **as**ingle **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 incre**as**ed andthreshold 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 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** products**of** terms [r + cos(φ)] k [r − cos(φ)] p−k . Below threshold parameter r < 1 the effect **of** incre**as**ingthe 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 thresholdis approached, i.e. r → 1 − , the contribution **of** cos(φ) relative to r decre**as**es, and the fringepatterns **as**sume a fixed form. Hence the visibility saturates above threshold.The OPO **source** h**as** several experimental advantages. **An** OPO **source** would haveimproved temporal and spatial coherence compared to an OPA. The signal and idler **mode**sat the outputs **of** the OPO are collimated because **of** the use **of** a cavity, and **high**er powerscan be obtained than is the c**as**e for the OPA, both **of** which are important in light **of** the smallcross-sections for typical multi-photon absorption processes. Collimated, **high** power outputsalso incre**as**e 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 **high**er-order absorption process will incre**as**e 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 cl**as**sical l**as**er beam at 800 nm **as** the **source**, w**as** implementedwith a lithographic recording material b**as**ed 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 **high**er. Using a cl**as**sical ph**as**e-shifted-grating method,an approximately **two**-fold enhancement **of** resolution w**as** 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 ph**as**e-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 **mode**s. 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** ph**as**e 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 **mode**s 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-b**as**ed recording material that can respond at lower powers can takeadvantage **of** a **high**er-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 thresholdB**as**ed on the general formula for the absorption rates below threshold, equation (22), thefollowing tables explicitly list the absorption rates and the fringe visibilities corresponding t**of**ringe 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, Nag**as**ako E M and Bentley S J 2001 Phys. Rev. Lett. 86 1389[6] D’**An**gelo 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, Gl**as**ser 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] Vy**as** R and Singh S 1989 Phys. Rev. A 40 5147[16] Vy**as** R 1992 Phys. Rev. A 46 395[17] Vy**as** R and Singh S 1995 Phys. Rev. Lett. 74 2208[18] Vy**as** 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-Cl**as**sical Fields 1st edn vol 2 (Berlin:Springer)New Journal **of** Physics 11 (2009) 113055 (http://www.njp.org/)

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