Surface spontaneous parametric down-conversion
Jan Perina Jr, Antonin Luks, Ondrej Haderka, Michael Scalora
aa r X i v : . [ qu a n t - ph ] A ug Surface spontaneous parametric down-conversion
Jan Peˇrina Jr., Anton´ın Lukˇs, Ondˇrej Haderka
Joint Laboratory of Optics of Palack´y University and Institute of Physics of Academy of Sciences of the Czech Republic,17. listopadu 50A, 772 07 Olomouc, Czech Republic
Michael Scalora
Charles M. Bowden Research Center, RD&EC, Redstone Arsenal, Bldg 7804, Alabama 35898-5000, USA
Surface spontaneous parametric down-conversion is predicted as a consequence of continuity re-quirements for electric- and magnetic-field amplitudes at a discontinuity of χ (2) nonlinearity. Ageneralization of the usual two-photon spectral amplitude is suggested to describe this effect. Ex-amples of nonlinear layered structures and periodically-poled nonlinear crystals show that surfacecontributions to spontaneous down-conversion can be important. When studying the process of second-harmonic gen-eration under considerable phase mismatch more thanthirty years ago, the generation of second-harmonic fieldfrom a boundary between two homogeneous media thatdiffer by values of χ (2) nonlinearity has been discovered[1, 2]. The surface second-harmonic field arises here asa consequence of continuity requirements for projectionsof electric- and magnetic-field vector amplitudes into theplane of the boundary. Physically, a pumping field atfrequency ω creates a step profile of nonlinear polariza-tion at frequency 2 ω and with wave vector 2 k ( ω ) thatbecomes the source of the usual volume second-harmonicfield. The wave vector of the surface second-harmonicfield is k (2 ω ) in agreement with dispersion properties ofthe nonlinear material. This effect is even found in non-linear media with negative index of refraction as the nu-merical solution of nonlinear Maxwell equations revealedin [3]. The studied parametric effect should be distin-guished from resonant surface second-harmonic genera-tion.Spontaneous parametric down-conversion (SPDC) [4]belongs together with second-harmonic generation to χ (2) processes. This poses the question about surface effectsin SPDC. In volume SPDC, photon pairs are generatedfrom the vacuum state, due to quantum fluctuations (orquantum noise) inherent in this state. In this case, a non-linear material responds to the presence of optical fieldsthrough quantum nonlinear polarization that acts as asource of new fields. In a close vicinity of the boundary,the interacting fields as well as the nonlinear polarizationare modified in order to comply with natural fields’ con-tinuity requirements at the boundary. This results in thegeneration of additional photon pairs from the area ofthe boundary (several wavelengths thick) that constitutesurface SPDC.Our study of surface SPDC is organized as follows.Nonlinear Heisenberg equations are derived first to treatSPDC inside the nonlinear medium. Nonlinear correc-tions to electric- and magnetic-field amplitudes occurnaturally at boundaries and give additional, i.e. surface,contributions to SPDC. Subsequently, the derivation ofquantities characterizing the emitted photon pairs is ad-dressed. Finally, two important examples are discussed. Adopting the quantization of energy flux [5, 6] we de-scribe the process of SPDC involving the signal, idler,and pump fields by the Heisenberg equations with an ap-propriate interaction momentum operator ˆ G int [5]:ˆ G int ( z ) = 4 ǫ d eff A√ π X α,β,γ = F,B Z dω s Z dω i h E ( − ) p α ( z, ω s + ω i ) ˆ E (+) s β ( z, ω s ) ˆ E (+) i γ ( z, ω i ) + h . c . i . (1)The positive-frequency part of an electric-field amplitudeˆ E (+) m α can be expressed using annihilation operator ˆ a m α as follows ( m = p, s, i ; α = F, B ):ˆ E (+) m α ( z, ω m ) = i s ¯ hω m ǫ c A n m ( ω m ) ˆ a m α ( z, ω m ); (2)ˆ E ( − ) m α = ( ˆ E (+) m α ) † . Subscript F ( B ) indicates a field prop-agating forward (backward), i.e. along + z ( − z ) axis.Symbol ǫ means permittivity of vacuum, d eff is effectivenonlinear coefficient, A transverse area of the fields, c speed of light in vacuum, and h . c . replaces the hermitian-conjugated terms. Symbol k m α is a wave vector, ω m fre-quency, and n m index of refraction of field m α .The Heisenberg equations, e.g., for the signal-field op-erators ˆ a s α ( z, ω s ) can then be derived assuming equal-space commutation relations [7]: d ˆ a s α ( z, ω s ) dz = ik s α ( ω s )ˆ a s α ( z, ω s )+ X β,γ = F,B Z dω i g ( ω s , ω i ) E (+) p β (0 , ω s + ω i ) × exp[ ik p β ( ω s + ω i ) z ]ˆ a † i γ ( z, ω i ) , α = F, B ; (3)Coupling constant g , g ( ω s , ω i ) = 2 id eff √ ω s ω i / ( c √ π p n s ( ω s ) n i ( ω i )), is linearly proportional to nonlinear co-efficient d eff .The solution of Eq. (3) for annihilation operatorsˆ a s α ( z, ω s ) up to the first power of g gives us the formulafor operator ˆ E (+) s α defined in Eq. (2):ˆ E (+) s α ( z, ω s ) = i s ¯ hω s ǫ c A n s ( ω s ) exp[ ik s α ( ω s ) z ] × h ˆ a s α (0 , ω s ) + X β,γ = F,B Z dω i g ( ω s , ω i ) × E (+) p β (0 , ω s + ω i ) exp[ i ∆ k β,αγ ( ω s , ω i ) z/ × z sinc[∆ k β,αγ ( ω s , ω i ) z/ a † i γ (0 , ω i ) i ; α = F, B ; (4)sinc( x ) = sin( x ) /x and ∆ k β,αγ ( ω s , ω i ) = k p β ( ω s + ω i ) − k s α ( ω s ) − k i γ ( ω i ).The positive-frequency magnetic-field amplitude op-erators ˆ H (+) s α can be derived using the formula H (+) s α ( z, ω s ) = − i/ ( ω s µ ) ∂E (+) s α ( z, ω s ) /∂z ( µ denotespermeability of vacuum) provided that the electric-field[magnetic-field] amplitude E s α [ H s α ] is polarized along+ x [+ y ] axis. The obtained operator ˆ H (+) s α can be de-composed into two parts denoted as ˆ H (+)Fr s α and ˆ H (+)nFr s α :ˆ H (+) s α ( z, ω s ) = ˆ H (+)Fr s α ( z, ω s ) + ˆ H (+)nFr s α ( z, ω s ) , (5)ˆ H (+)Fr s α ( z, ω s ) = k s α ( ω s ) ω s µ ˆ E (+) s α ( z, ω s ) , (6)ˆ H (+)nFr s α ( z, ω s ) = s ¯ hcn s ( ω s )2 µ ω s A X β,γ = F,B Z dω i g ( ω s , ω i ) × E (+) p β ( ω s + ω i ) exp[ ik p β ( ω s + ω i ) z ] × exp[ − ik i γ ( ω i ) z ]ˆ a † i γ (0 , ω i ) , α = F, B. (7)By definition, the magnetic-field amplitude operatorˆ H (+)Fr s α ( z, ω s ) is linearly proportional to the electric-field amplitude operator ˆ E (+) s α ( z, ω s ). The remainingmagnetic-field operator ˆ H (+)nFr s α is of purely nonlinear ori-gin and the usual derivation of Fresnel relations does nottake it into account. Standard approaches to nonlinearinteractions thus do not involve this nonlinear term andso they neglect surface effects. We note that the ’nonlin-ear’ magnetic-field operator ˆ H (+)nFr s α occurs as a classicalfield amplitude also in the description of stimulated para-metric processes (e.g., in difference-frequency generation)and yields surface contributions to these processes.The electric- and magnetic-field amplitudes E m α ( z, ω m ) and H m α ( z, ω m ) originating in the nonlinearinteraction and written in Eqs. (4) and (5—7) have toobey continuity requirements at the input and outputboundaries of the nonlinear medium. We illustrate ourapproach to this problem considering the signal field atthe input boundary ( z = 0). Four electric and magneticfields are involved in the continuity requirements at thisboundary (see Fig. 1): two at the linear left-hand side[denoted by superscript (0)] and two at the nonlinearright-hand side. Because the magnetic-field amplitudes H s F and H s B inside the nonlinear medium have alsononlinear contributions H nFr s F and H nFr s B given in Eq. (7)additional (surface) amplitude corrections δE s F and δE (0) s B [together with δH s F and δH (0) s B ] in the fieldsleaving the boundary naturally occur. The amplitude ✲ E (0) s F H (0) s F E (0) s B H (0) s B ✛ ✛ ✲ E s F + δE s F H Fr s F + H nFr s F + δH s F E s B + δE s B H Fr s B + H nFr s B + δH s B z = 0 FIG. 1: Scheme showing electric and magnetic fields at theinput boundary. For details, see the text. corrections δE (0) s B and δH (0) s B of the outgoing field outsidethe nonlinear medium can be involved in the fields obey-ing Fresnel relations [8] at the expense of introductionof fictitious amplitude corrections δE s B and δH s B ofthe field impinging at the boundary from its nonlinearside. A detailed analysis then results in two equationsfor the surface amplitude corrections of fields inside thenonlinear medium:0 = δE s F (0) − δE s B (0) , H nFr s F (0) + δH s F (0) + H nFr s B (0) − δH s B (0) . (8)The positive-frequency parts of surface amplitude-correction operators δ ˆ E (+) s α and δ ˆ H (+) s α occurring in thequantum form of Eqs. (8) can be expressed usingannihilation-operator corrections δ ˆ a s α similarly to thecorresponding amplitude operators ˆ E (+) s α and ˆ H (+) s α inEqs. (2) and (6). The solution of Eqs. (8) for δ ˆ a s F and δ ˆ a s B then takes the form: δa s F (0 , ω s ) = δa s B (0 , ω s ) = ik s ( ω s ) X β,γ = F,B Z dω i g ( ω s , ω i ) E (+) p β (0 , ω s + ω i )ˆ a † i γ (0 , ω i ) . (9)Similar considerations appropriate for the outputboundary leaves us finally with an expression for opera-tors ˆ a s α ( L, ω s ) valid up to the first power of g ( L standsfor the length of nonlinear medium):ˆ a s α ( L, ω s ) = ˆ a free s α ( L, ω s ) + X β,γ = F,B Z dω i F sα,βγ ( ω s , ω i )ˆ a free † i γ ( L, ω i ) , α = F, B. (10)Operators ˆ a free s α ( L, ω s ) correspond to free-field linearpropagation, i.e. without photon-pair generation. Theidler-field amplitudes can be analyzed along the samevein.The generalized two-photon spectral amplitudes F s and F i defined in Eq. (10) describe properties of a gener-ated photon pair and are composed of two contributions: F mα,βγ = F vol α,βγ + F m, surf α,βγ ; α, β, γ = F, B. (11)Two-photon spectral amplitude F vol of the volume con-tribution has the well-known form: F vol α,βγ ( ω s , ω i ) = g ( ω s , ω i ) E (+) p α (0 , ω s + ω i ) × exp[ ik p α ( ω s + ω i ) L ] exp[ − i ∆ k α,βγ ( ω s , ω i ) L/ × L sinc[∆ k α,βγ ( ω s , ω i ) L/ α, β, γ = F, B. (12)On the other hand, surface contributions F m, surf to thetwo-photon spectral amplitudes can be expressed as: F m, surf α,βγ ( ω s , ω i ) = V mα,βγ ( ω s , ω i ) F vol α,βγ ( ω s , ω i ) , (13)where V mα,βγ ( ω s , ω i ) = ∆ k α,βγ ( ω s , ω i ) k m ( ω m ) ; α, β, γ = F, B. (14)The structure of surface contributions as described bythe two-photon amplitudes F s, surf and F i, surf resemblesthat of the volume contribution as the formula in Eq. (13)indicates. The physical interpretation is as follows. At aboundary, the only restriction for photon-pair generationis imposed by the conservation of energy. However, themutual interference of two-photon amplitudes originatingat the input and output boundaries leads to the resultthat resembles the usual phase-matching conditions. Wenote that lim L → F m, surf = 0.We further consider photon pairs with both photonspropagating forward and use operators ˆ a m ( ω m ) ( m = s, i ) defined outside the nonlinear medium. The jointsignal-idler photon-number density n ( ω s , ω i ) at the out-put plane of the nonlinear medium is given as: n ( ω s , ω i ) = Dh ˆ a † s ( ω s )ˆ a s ( ω s )ˆ a † i ( ω i )ˆ a i ( ω i ) + h . c . iE / . (15)Symbol h i denotes averaging over the initial signal- andidler-field vacuum state. Introducing two-photon spec-tral amplitudes ˜ F s and ˜ F i (transmission coefficients t m describe the output boundary),˜ F m ( ω s , ω i ) = t s ( ω s ) t i ( ω i ) F mF,F F ( ω s , ω i ) , m = s, i, (16)we arrive at the following formula: n ( ω s , ω i ) = Re { ˜ F s ∗ ( ω s , ω i ) ˜ F i ( ω s , ω i ) } . (17)As this example illustrates, a generalization of the usualformalism based on a two-photon spectral amplitude canbe given providing formulas for all physical quantitiescharacterizing photon pairs.The volume interaction among the forward-propagating pump, signal, and idler fields dominates inbulk nonlinear crystals several mm long. According toour model, the surface contributions can be approxi-mately included into the usual formalism working witha two-photon spectral amplitude Φ vol (see, e.g., [9, 10])using the formal substitution:Φ( ω s , ω i ) ←− q V sF,F F ( ω s , ω i ) × q V iF,F F ( ω s , ω i ) Φ vol ( ω s , ω i ); (18) S sv o l , S ss u r f ( a . u . ) s / p0 S sv o l + s u r f / S sv o l a b FIG. 2: Signal-field spectra S vol s (solid curve denoted as a)and S surf s (solid curve denoted as b) of volume and surfaceSPDC, respectively, and ratio S vol+surf s /S vol s of the spectrawith ( S vol+surf s ) and without ( S vol s ) the inclusion of surfaceSPDC (dashed curve); S s ( ω s ) = ¯ hω s R dω i n ( ω s , ω i ). V is defined in Eq. (14). If the nonlinear interaction isperfectly phase-matched [∆ k F,F F ( ω s , ω i ) = 0] the sur-face contributions at central frequencies are zero.Contrary to the bulk nonlinear crystals, surface SPDCcannot be neglected in nonlinear layered structures com-posed of layers typically several hundreds of nm long. Inthis case all possible nonlinear interactions as describedby the momentum operator ˆ G int in Eq. (1) give appre-ciable contributions. Fulfilment of phase-matching condi-tions is not important here, because ∆ kl ≪ π ( l denotesa typical length of one layer). Surface SPDC similarly asvolume SPDC from an individual layer is weak but bothof them are highly enhanced by constructive interferenceof fields from different layers. A generalization of thepresented theory to layered structures is straightforwardfollowing the work presented in [11, 12].As an example, we consider a structure composed of 25layers of nonlinear GaN 117 nm thick that sandwich 24linear layers of AlN 180 nm thick and studied previouslyin [11]. Volume SPDC gives efficient photon-pair gener-ation at degenerate signal- and idler-field frequencies forthe signal-field emission angle 14 deg [11] (see Fig. 2) as-suming a normally incident pump field at λ p = 664 . V mα,βγ definedin Eq. (14). Whenever the lengths of nonlinear layers areless or comparable to the coherence length of the non-linear process, we observe appreciable contributions ofsurface terms. For example, the coherence length equalsapproximately 1 µ m for GaN in our case.Surface effects give also an important contribution tophoton-pair generation rates in periodically-poled non-linear materials with sufficiently short poling periods. N v o l + s u r f / N v o l - p0 (10 -6 m) n l - ( m - ) FIG. 3: Relative contribution N vol+surf /N vol − • ) and in-verse Λ − of the optimum poling period giving quasi-phase-matching for λ s = λ i = 2 λ p (solid curve) as they dependon cw pump-field wavelength λ p in LiNbO N = R dω s R dω i n ( ω s , ω i ). Here, as an example, we consider frequency-degenerateSPDC in periodically-poled LiNbO with the optical axisperpendicular to the direction of collinearly-propagatingfields; their polarizations are parallel to the optical axis.Whereas the surface effects contribute to photon-pairgeneration rate N vol+surf only by several percent for thepump wavelength λ p = 1 µ m, the increase of photon-pairgeneration rate N by 50 % is observed for λ p = 0 . µ m(see Fig. 3). As the curves in Fig. 3 indicate the rela-tive contribution N vol+surf /N vol − / Λ nl of poling periodthat is linearly proportional to the density of surfaces pera unit length. We note that domains shorter than 1 µ m can be fabricated using light-induced domain engineering[14].Surface SPDC occurs also in nonlinear wave-guidingstructures, i.e. under the condition of total reflection andpresence of evanescent waves. A detailed analysis hasshown that the formulas presented above remain validalso in this case provided that we consider propagationconstants β instead of wave vectors. Coupling constant g then involves the overlap integral over mode functionsof the interacting fields. This is particularly interestingfor nonlinear photonic-band-gap fibers.Surface SPDC is by no means restricted to 1D non-linear structures: even greater relative contributions areexpected in 2D and 3D nonlinear samples. Surface ef-fects will also affect stimulated χ (2) processes like second-harmonic or second-subharmonic generation when stud-ied under comparable conditions. Qualitatively, they willeffectively enhance the nonlinearity. 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