Journal of Optics

TUTORIAL Phase matching in β-barium borate crystals for spontaneous parametric down-conversion

To cite this article: Suman Karan et al 2020 J. Opt. 22 083501

View the article online for updates and enhancements.

This content was downloaded from IP address 14.139.38.209 on 01/07/2020 at 03:25 Journal of Optics

J. Opt. 22 (2020) 083501 (20pp) https://doi.org/10.1088/2040-8986/ab89e4

Tutorial Phase matching in β-barium borate crystals for spontaneous parametric down-conversion

Suman Karan1, Shaurya Aarav1,4, Homanga Bharadhwaj2,5, Lavanya Taneja1,6, Arinjoy De3,7, Girish Kulkarni1,8, Nilakantha Meher1 and Anand K Jha1

1 Department of Physics, Indian Institute of Technology Kanpur, Kanpur UP 208016, India 2 Department of Computer Science and Engg., IIT Kanpur, Kanpur UP 208016, India 3 Department of Physics, IIT Kharagpur, Kharagpur 721302, India

E-mail: [email protected] and [email protected]

Received 10 March 2020 Accepted for publication 16 April 2020 Published 29 June 2020

Abstract Spontaneous parametric down-conversion (SPDC) is the most widely used process for generating photon pairs entangled in various degrees of freedom such as polarization, time-energy, position-transverse momentum, and angle-orbital angular momentum (OAM). In SPDC, a pump photon interacts with a non-linear optical crystal and splits into two entangled photons called the signal and the idler photons. The SPDC process has been studied extensively in the last few decades for various pump and crystal configurations, and the entangled photon pairs produced by SPDC have been used in numerous experimental studies on quantum entanglement and entanglement-based real-world quantum-information applications. In this tutorial article, we present a thorough study of phase matching in BBO crystals for spontaneous parametric down-conversion and thereby also investigate the generation of entangled photons in such crystals. First, we present a theoretical derivation of two-photon wavefunction produced by SPDC in the frequency and transverse momentum bases. We then discuss in detail the effects due to various crystal and pump parameters including the length of the crystal, the angle between the optic axis and the pump propagation direction, the pump incidence angle on the crystal surface, the refraction at the crystal surfaces, and the pump propagation direction inside the crystal. These effects are extremely relevant in experimental situations. We then present our numerical and experimental results in order to illustrate how various experimental parameters affect the phase matching and thus the generation of entangled photons. Finally, using the Keywords: parametric down-conversion, phase matching, entanglement, orbital angular momentum (Some figures may appear in colour only in the online journal)

4 Present address: Department of Electrical Engineering, Princeton Univer- sity, Princeton, NJ 08544, United States of America. 5 Present address: Department of Computer Sc., University of Toronto and Vector Institute, Canada. 6 Present address: Department of Physics, University of Chicago, Chicago, IL 60 637, United States of America. 7 Present address: Department of Physics, University of Maryland, College Park, MD 20 742, United States of America. 8 Present address: Department of Physics, University of Ottawa, Ottawa, ON K1N6N5, Canada.

2040-8986/20/083501+20$33.00 1 © 2020 IOP Publishing Ltd Printed in the UK J. Opt. 22 (2020) 083501 Tutorial two-photon wavefunction in the transverse wave-vector basis, we show how to derive the two-photon wavefunction in the OAM basis and thereby calculate the two-photon angular Schmidt spectrum. We expect this article to be useful for researchers working in various capacities with entangled photons generated by SPDC in BBO crystals. Keywords: parametric down-conversion, phase matching, entanglement, orbital angular momentum (Some figures may appear in colour only in the online journal)

1. Introduction studying foundations of quantum mechanics and for exploring quantum information-based applications. 1.1. Background In non-linear optical processes, the constraints due to the conservation laws are referred to as the phase-matching con- Spontaneous parametric down-conversion (SPDC) is a ditions. In the SPDC process, the phase matching is affected second-order non-linear optical process in which a photon by various crystal as well as pump parameters and decides the of a higher frequency passes through an optical material and efficiency with which a pump photon gets down-converted and splits into two photons of lower frequencies [1]. The photon at the emission directions into which the signal and idler photons the higher frequency is referred to as the pump photon and the get generated. The SPDC phase-matching has been extens- two photons at lower frequencies are referred to as the signal ively studied in the past. It was shown by Franken et al [9] that and the idler photons. SPDC is called parametric since the in order to observe the second-order non-linear effect of spon- total energy remains conserved in the process. The constraints taneous parametric down-conversion, one not only requires of energy and momentum conservations in SPDC require that a highly intense pump field but also a non-centrosymmetric the sum of the energies of the signal and idler photons be crystal. In the context of second-harmonic generation in equal to the energy of the pump photon and that the sum KDP crystals, Giordmaine [10] and Maker [11] independently of the momenta of the signal and idler photons be equal to showed that the birefringence and dispersive properties of the the momentum of the pump photon. The SPDC process was crystal can be utilized for achieving the required phase match- first predicted theoretically by Louisell et al in 1961 [2], who ing conditions. Midwinter et al [12] extended this technique to modelled it as two oscillators coupled via a time-dependent three-wave interaction processes, such as SPDC, for achieving reactance. Later, this effect was studied by many others before the conditions for phase matching. Using the Kleinman’s sym- its first experimental observation as parametric fluorescence in metry conjecture, Midwinter et al [12] showed that the differ- a LiNbO3 crystal [3]. In this experiment, an intense argon-ion ent types of phase-matching are satisfied by different crystals laser beam was used as a pump and the parametric fluores- with appropriate symmetry properties [13]. The phase match- cence was observed over a significant portion of the visible ing conditions in SPDC also decide the polarizations that the and near-infrared spectrum through temperature tuning of the pump, the signal and the idler photons can have. For a negative crystal. A similar effect was observed in an experiment with uniaxial birefringent crystal, such as β-barium borate (BBO) ammonium dihydrogen phosphate (ADP) crystal. In this case [12, 14–17], in which the refractive index of light propagat- the spectral tuning was achieved by rotating the ADP crystal ing along the optic axis of the crystal is less than the refract- around the pump-beam direction [4]. Later, instead of treat- ive index of light propagating perpendicular to it, there are a ing SPDC as a coupled oscillator problem, Giallorenzi et al total of eight possible combinations of polarization states that [5] studied it as a scattering problem in which a pump photon the pump, signal and idler photons can take. However, due transforms into a pair of photons due to non-linear polarizabil- to phase matching constraints, only a few combinations are ity. In their study, Giallorenzi et al [5] were able to incorporate allowed. In general, when both the signal and idler photons the effects due to factors such as the anisotropy and disper- have the same polarization, the phase-matching is referred to sion of the non-linear crystal, input beam of finite spectral as type-I phase matching. On the other hand, when the signal width, etc. Around the same time, Klyshko and co-authors and idler photons have orthogonal polarizations, it is referred studied SPDC as part of their investigations of spontaneous to as type-II phase matching. Furthermore, if the directions of scattering due to quadratic and cubic terms in the expansion propagation of the down-converted signal and idler photons of the polarizability [6, 7]. They obtained formulas for the are along the propagation direction of the pump photon then it intensity of scattered light and its dependence on the fre- is known as collinear phase-matching. Alternatively, in non- quency and observation direction. Following up on the work collinear phase matching, the signal and idler photons propag- by Klyshko and co-authors, Burnham et al [8] experiment- ate in directions non-collinear with that of the pump photon. ally studied coincidences between the down-converted signal The two SPDC photons have been shown to be entangled and idler photons and thereby demonstrated for the first time in various degrees of freedom such as energy-time [18–21], that in the SPDC process the two photons are emitted simul- position-momentum [22–24], polarization [25–27], and angle- taneously. This pioneering coincidence-detection experiment angular momentum [28–30]. In the last few decades, entan- essentially marks the beginning of using SPDC photons for glement of the SPDC photons has been used for probing the

2 J. Opt. 22 (2020) 083501 Tutorial foundations of quantum mechanics. One major step in this the desired phase matching condition. On the other hand, in direction has been the ruling out of any local hidden vari- BBO crystals, broad phase matching ranges can be very easily able interpretations of quantum mechanics [31, 32] through achieved. Both the collinear and non-collinear phase matching the experimental violations of Bell’s inequality [33, 34]. Such requirements can be satisfied simply by changing the angle violations have been observed in various degrees of freedom between the optic axis of the crystal and the pump propaga- including polarization [35, 36], position and momentum [28], tion direction. Moreover, these crystals have broad transmis- time and energy [18, 37, 38], spatial-parity [39], and OAM sion ranges, have high damage threshold, and are readily avail- [40]. Using hyperentangled states of the SPDC photons, even able with various thicknesses. Therefore, a substantial number simultaneous violations of Bell’s inequalities in more than one of experiments, especially those investigating spatial correla- degree of freedom have been reported [19, 41–43]. In addi- tions of entangled photons, still employ BBO crystals for gen- tion to being used for probing the foundations of quantum erating entangled photons via SPDC. mechanics, SPDC photons have also been used for explor- Because of the wide use of BBO crystals in generating ing the entanglement-based real-world applications such as SPDC photons with a variety of phase-matching geometries, quantum information [44–48], quantum cryptography [49– several works in the past have investigated different aspects 51], quantum dense coding [44, 52], optical measurements of SPDC phase matching in BBO crystals [25, 76–84]. How- [53, 54], imaging [55–57], spectroscopy [58], quantum litho- ever, a comprehensive study of SPDC phase-matching in BBO graphy and metrology [59–61]. crystals covering all the major aspects of phase matching SPDC has been experimentally realized using a vari- worked out in one place is still lacking. Such a study would ety of bulk crystals such as lithium niobate (LiNbO3) be very useful to researchers working in various capacities [3, 62], BBO [63–65], potassium dihydrogen phosphate with entangled photons produced by SPDC in BBO crys- (KDP) [66], ammonium dihydrogen phosphate (ADP) [4, tals. In this tutorial article, we present a thorough study of 67], and Ba2NaNb5O15 [68, 69] crystals. The bulk crystals the generation of entangled two-photon fields by the SPDC such as BBO do not have very high down-conversion effi- process in BBO crystal. We discuss the SPDC phase match- ciency. Therefore, in more recent years, periodically-poled, ing in various details and study how it is affected by vari- quasi-phase matched waveguide crystals are being used to ous crystal and pump parameters. Through our simulation and achieve higher SPDC efficiency, although in a restricted experimental results, we also highlight several facts that would phase-matching range. Several experiments have reported be very useful during experimental investigations with SPDC SPDC generation using periodically-poled potassium titanyl photons. Most of the calculations presented in this article have phosphate (PPKTP) waveguide crystals [70–72]. The other been carried out with the wavefunction of the down-converted quasi-phase matched waveguide crystals that are being used photons expressed in the position or momentum bases. How- for SPDC generation include the periodically-poled lithium ever, in the last section of this article, we discuss the two- niobate (PPLN) crystals [73, 74], and the periodically-poled photon wavefunction in the orbital angular momentum (OAM) lithium tantalate (PPLT) crystals [75]. basis and show how to calculate the OAM spectrum of the In the past, although most of the foundational experiments entangled two-photon field. [18–30, 35–43] as well as experiments exploring applications of photonic entanglement [44–61] have used BBO crystals for producing entangled photons by SPDC, the more recent 1.2. Outline of the paper experimental studies that require high down-conversion effi- ciencies in limited emission directions have started using the This article is organized as follows. In section 2, we dis- quasi-phase matched periodically-poled crystals for the pur- cuss the spontaneous parametric down-conversion process in pose. Nevertheless, the current experimental studies, either BBO crystal. First, we derive the SPDC Hamiltonian and foundational or application-oriented, that do not necessarily thereby work out the two-photon state produced by SPDC. require very high down-conversion efficiencies still employ We study how different pump and crystal parameters affect BBO crystals. This is due to the fact that the quasi-phase the phase matching and thus the generation of the signal and matching process that enables the periodically poled crystals idler photons in SPDC. Our study includes effects due to sev- to have high-down-conversion efficiencies compared to the eral experimental factors such as crystal length, angle between bulk crystals does so at the expense of limiting the phase- optic axis and beam propagation direction, pump incidence matching ranges. As a consequence, although the periodic- angle on the crystal surface, refraction at the crystal surface, ally poled crystals become highly efficient for collinear down- pump propagation inside the crystal, and the location of the conversion, the quasi-phase matching in such crystals does beam waist of a Gaussian pump field. All these effects are not allow for non-collinear down-conversion. Therefore, while extremely relevant in experimental situations. In section 3, we these crystals can be used for studying temporal and polariz- discuss different polarization effects in SPDC. We investig- ation correlations of entangled photons, they are not effect- ate how phase matching conditions decide the possible com- ive in studying the spatial correlations of entangled photons binations of polarizations that the signal, idler and the pump in non-collinear emission geometries. Furthermore, the phase photons can have, and how the polarization of these photons matching in the periodically-poled crystals is achieved by affect their propagation inside the non-linear medium. In sec- changing the temperature of the crystals, and therefore these tion 4, we calculate the detection probabilities of the down- crystals require temperature tuning and stability for achieving converted photons for both type-I and type-II phase matching

3 J. Opt. 22 (2020) 083501 Tutorial conditions. Furthermore, we calculate the conditional detec- where E(r,t) stands for the electric field, B(r,t) stands for the 1 tion probability in the position basis. We then present vari- magnetic flux density, H(r,t) = µ B(r,t) is termed as magnetic ous experimental and numerical results in section 5 in order field, where µ is the magnetic permeability. One can write to highlight how various experimental parameters affect the µ = µrµ0, where µr is the relative permeability of the medium phase matching and thus the generation of entangled photons. and µ0 is called the magnetic permeability of free space D(r,t) In this context, we specifically point out that the intensity stands for the electric displacement vector. Here, the medium of the down-converted photons remains maximum for the is assumed to be magnetically isotropic, with the value of µr collinear phase-matching and decreases from the maximum equal to one, but electrically anisotropic. The electric displace- value with increasing non-collinearity. Nevertheless, the total ment vector D(r,t) inside the medium is given by [85, 86] power of the generated photons stays independent of the non- collinearity of phase matching. In section 6, we derive the two- D(r,t) = ϵ0E(r,t) + P(r,t), (5) photon wavefunction in the OAM basis and present the theor- ( , ) etical results that are useful for studying entanglement in the where P r t is termed as polarization. When the pump ( , ) = OAM basis. Finally, we present the conclusions in section 7. field strength is weak, the polarization is given by P r t ϵ χ(1) ( , ) χ(1) We would like to point out that the general formalism of 0 E r t , where is the linear susceptibility. However, section 2 involving the derivation of the interaction Hamilto- when the pump field strength is strong, the polarization has nian for SPDC and thereby the two-photon wavefunction higher-order contributions given by is applicable to SPDC process not only in BBO crystals ( ) ( ) P(r,t) = ϵ χ 1 E(r,t) + ϵ χ 2 E(r,t)E(r,t) but also in any medium that has second-order non-linearity. 0 0 (3) ··· Furthermore, the general formalism presented in section 4 + ϵ0χ E(r,t)E(r,t)E(r,t) + , for calculating the coincidence count rate, individual photon = P(1)(r,t) + P(2)(r,t) + P(3)(r,t) + ··· , (6) count rates, and the conditional detection probabilities is also applicable to SPDC process in any non-linear medium and where χ(2) is called the second-order non-linear susceptibil- not just in BBO crystals. So, while the present tutorial art- ity, χ(3) is called the third-order non-linear susceptibility, etc. icle presents a complete study of phase-matching and SPDC- P(1)(r,t) is called the linear polarization while P(2)(r,t) is photon generation in BBO crystals, the general formalism called the second-order non-linear polarization, etc. Typically, presented in this article can be used for studying the SPDC χ(2) and χ(3) are several orders of magnitude smaller than χ(1) phase-matching and thereby the entangled photons generation and as a result we see non-linear effects only at very high field in any other non-linear medium. Therefore, we expect this strengths. Using equation (3)–(6), we write the wave equation tutorial article to be useful in general to researchers working in inside a non-linear medium as various capacities with entangled photons produced via SPDC. ∇(∇ · E(r,t)) − ∇2E(r,t) = h   i ∂2 2. Spontaneous parametric down-conversion − µ ϵ 1 + χ(1) E(r,t) + P(2)(r,t) + ··· . (7) 0 ∂t2 0 2.1. Electromagnetic field inside a non-linear medium Here 1 + χ(1) = n2 with n being the refractive index of the Interaction of classical electromagnetic waves with a mater- medium. The susceptibilities above are tensor quantities, and ial medium is described by the Maxwell’s equations. These for the second-order non-linear effects we explicitly write the equations are first-order coupled differential equations. The second-order non-linear polarization as solutions to these equations provide information about the (2) (2) behavior of electric and magnetic fields inside a medium. The (P )l = ϵ0χlmn(E)m(E)n. (8) Maxwell’s equations in a medium devoid of free charges and (2) currents are given as Here, l, m, and n are cartesian indices and χlmn is the second- order susceptibility tensor [85, 87]. In this article, we mostly ∇ · D(r,t) = 0, (1) discuss the second-order non-linear optical effect of spontan- eous parametric down-conversion.

2.2. Introduction to SPDC ∇ · B(r,t) = 0, (2) There are a variety of second-order non-linear effects that take place inside a non-linear medium, such as second-harmonic generation, sum-frequency generation, and difference- ∂B(r,t) frequency generation. However, the second-order non-linear ∇ × E(r,t) = − , (3) ∂t effect that we concentrate on in this article is the spontaneous parametric down-conversion (SPDC). In SPDC, a single input photon at higher frequency, called the pump photon, interacts ∂D(r,t) with a non-linear crystal and splits into two photons of lower ∇ × H(r,t) = , (4) ∂t frequencies, called the signal and idler photons [8, 88]. The

4 J. Opt. 22 (2020) 083501 Tutorial

to the second-order non-linear term, etc. Since we are inter- ested in second-order non-linear optical effects, we will look at the second-order contribution to the energy, which is given by ˆ · 3 HI(t) = D(r,t) E(r,t)d r, Vˆ (2) · 3 = ϵ0 χ E(r,t)E(r,t) E(r,t)d r, Figure 1. (a) Difference frequency generation process. (b) ˆ V Spontaneous parametric down conversion process. (c) Energy-level diagram of SPDC. = P(2)(r,t) · E(r,t)d3r, (10) V V SPDC is a type of difference-frequency generation process where the integration extends over the volume of the non- and is depicted in figure 1(a). We note that in the process linear medium. This is the general expression for the con- of difference-frequency generation, if the field at frequency tribution to the total energy due to second-order non-linear ω2 is present in the input field, it stimulates the difference optical effects. However, we are interested only in the non- − frequency generation at ω3 = ω1 ω2, and as a result the linear optical process of parametric down-conversion. So, process is usually very efficient and is called optical paramet- using equation (8), we write HI(t) as [89, 90] ˆ ric amplification/oscillation. Even, when the frequency ω2 is (2) not present in the input field, one can still have the gener- HI(t) = ϵ0 χlmn − V ation of difference frequency at ω3 = ω1 ω2 accompanied × ( ( , )) ( ( , )) ( ( , )) 3 , by the generation of frequency ω2. However, in this case the Ep r t l Es r t m Ei r t nd r (11) generation of the difference frequency is stimulated by the where p, s, and i stand for pump, signal, and idler, respectively. presence of the vacuum mode at frequency ω2 in the input field and is therefore very inefficient. This second-order pro- In writing the above expression, we have used the vectorial form of the electric fields. This is because of the fact that the cess of generating the frequencies ω3 and ω2 from the input down-conversion process does depend on the polarizations of frequency of ω1 is referred to as the spontaneous paramet- ric down-conversion (see figure1 (b)). The term spontaneous the pump, signal and idler fields. However, working with vec- refers to the fact that the process is stimulated by the vacuum torial fields makes the calculations quite cumbersome. There- fore, our approach in this article is to first find out the allowed mode at frequency ω2 in the input field and the term paramet- ric signifies that the total energy remains conserved during combinations of polarization states that the signal, idler and the process. Figure 1(c) represents the energy-level diagram pump photons can have and then for each combination, work with the scalar fields as far as calculating the detection prob- of the SPDC process, in which the photon at frequency ω1 gets absorbed and the absorber goes to a virtual state. From abilities of the down-converted photons are concerned. To this there, it decays down to the ground state emitting photons end, we write HI(t) as − ˆ at frequencies ω2 and ω3 = ω1 ω2. The constraints due to (2) 3 energy and momentum conservation make the two down- HI(t) = ϵ0 χ Ep(r,t)Es(r,t)Ei(r,t)d r, (12) V converted photons entangled in various degrees of freedom including energy-time [19, 20], position-momentum [22, 24], where χ(2) stands for the second-order nonlinearlity, and polarization [26, 27], and angular position-orbital angular Ep(r,t), Es(r,t), and Ei(r,t) are the scalar electric fields for an momentum [28, 29]. These entangled photons generated by allowed combination of polarizations of the signal, idler and SPDC have now become very important not only for studying pump photons. the foundations of quantum mechanics but also for real-world In quantum formalism, the contribution HI(t) to the energy applications of quantum entanglement. ˆ takes the form of the interaction Hamiltonian operator HI(t) and can be written as 2.3. Hamiltonian for SPDC ˆ ˆ (2) ˆ ˆ ˆ 3 HI(t) = ϵ0 χ Ep(r,t)Es(r,t)Ei(r,t)d r. (13) In order to study SPDC in all the details, we first need to derive V the Hamiltonian for the process. For this, we start with the total ˆ ˆ ˆ (electrical plus magnetic) energy density of the electromag- Here, Ep(r,t), Es(r,t), and Ei(r,t) are the electric field operat- netic field, which is given by ors corresponding to the pump, signal and idler fields, respect- h i ively. Next, we write the fields in terms of their negative and · (1) · D(r,t) E(r,t) = ϵ0E(r,t) + ϵ0χ E(r,t) E(r,t) positive frequency parts. This procedure is equivalent to the h i complex analytic signal representation for classical fields [91]. (2) · ··· + ϵ0χ E(r,t)E(r,t) E(r,t) + (9) Although the complex analytic signal representation was intro- duced as an outstanding tool for simplifying the mathematical The first term is the contribution to the electromagnetic energy handling of classical fields, its counterpart in the context of due to the linear term. The second term is the contribution due quantum field has become a necessary ingredient for handling

5 J. Opt. 22 (2020) 083501 Tutorial creation and annihilation of photons. Following [92], we write that a classical electric field E(r,t) can be represented in terms a quasi-monochromatic quantum electric field operator Eˆ(t) as of its spectral decomposition as

ˆ ∞ ˆ ∞ ˜ −iωt ˆ ˜ˆ −iωt E(r,t) = E(r,ω)e dω, E(t) = E(ω)e dω, −∞ −∞ ˆ ˆ ˆ ∞ 0 ∞ ˜ −iωt ˆ −iωt ˆ −iωt = E(ρ,z;ω)e dω, = ˜(ω) ω + ˜(ω) ω, −∞ E e d E e d ˆ ˆ  −∞ 0 ∞ − i(q·ρ+k z) 2 −iωt = Eˆ( )(t) + Eˆ(+)(t). (14) = A a(q,ω)e z d q e dω. (17) −∞ − Here Eˆ(+)(t) and Eˆ( )(t) are the positive and negative fre- Here A is a constant, k = (k ,k ,k ) = (q,k ), r = (x,y,z) = quency parts of the electric field operator, and since the elec- x y z z (ρ,z), and a(q,ω) is the angular spectrum representation of ˆ( ) ˜ˆ(ω) = ˜ˆ†(−ω) tric field operator E t is Hermitian, we have E E . the field. We know that the spectral decomposition of afield Representing this way, we write the pump, signal and idler remains the same irrespective of whether it is represented electric field operators in the following form within classical or quantum description. Therefore, we use − the above representation to write the quantized electric fields Eˆ (r,t) = Eˆ(+)(r,t) + Eˆ( )(r,t), p p p simply by replacing a(q,ω) with the corresponding annihila- ˆ ˆ(+) ˆ(−) Es(r,t) = Es (r,t) + Es (r,t), tion operator aˆ(q,ω) [92, 94]. This way, we write the positive- ˆ ˆ(+) ˆ(−) frequency part of the signal, idler and pump field operators as Ei(r,t) = Ei (r,t) + Ei (r,t). ˆ(+) ˆ ˆ Here Ep (r,t) is the positive frequency part of the pump elec- · − ˆ(+) 2 i(qs ρ+kszz ωst) tric field operator, etc. Using the above forms for the field oper- Es (r,t) = d qsdωsAse aˆ(qs,ωs), ators, the Hamiltonian in equation (13) can be written as ˆ ˆ (+) ( ·ρ+ −ω ) ˆ h i ˆ 2 i qi kizz it Ei (r,t) = d qidωiAie aˆ(qi,ωi), ˆ (2) ˆ(+) ˆ(−) HI(t) = ϵ0 χ Ep (r,t) + Ep (r,t) ˆ ˆ V · − h ih i (+) 2 i(q ρ+kpzz ωpt) Eˆ (r,t) = d q dω A e p V(q ,ω ). (18) × ˆ(+) ˆ(−) ˆ(+) ˆ(−) 3 p p p p p p Es (r,t) + Es (r,t) Ei (r,t) + Ei (r,t) d r. (15) Here, we note that in quantizing the electromagnetic fields The resulting expression for the interaction Hamiltonian one ends up with a summation over wave-vectors. However, is a sum of eight different terms with all possible combina- in the limit when the quantization volume goes to infinity, the tions of the three fields. However, there are only two terms, discrete set of wave-vectors becomes continuous and the sum- ˆ(+) ˆ(−) ˆ(−) ˆ(−) ˆ(+) ˆ(+) mation can be replaced by an integral. In writing the form of Ep (r,t)Es (r,t)Ei (r,t) and Ep (r,t)Es (r,t)Ei (r,t), that lead to energy conserving processes and thus contrib- the field operators above, we have made use of this limiting ute appreciably to the down-conversion process. The contri- condition. We further note that the intensity of the pump field is butions due to the other six terms get averaged out when the usually several orders of magnitude larger than that of the sig- ˆ nal and idler fields. Therefore, we treat the pump field classic- interaction Hamiltonian HI(t) is integrated over time. There- fore, we neglect the contributions due to these other terms. ally and replace the annihilation operator corresponding to the ( ,ω ) Neglecting these contributions is equivalent to making the pump field by the field amplitude V qp p . With this assump- rotating-wave approximation as in the case of treating atomic tion and using the form of the field operators in equation (18), absorption and emission processes (see [93], section 2.3). we write equation (16) as We note that these approximations hold only for second- ˆ ˆ ˆ ˆ order non-linear optical processes such as SPDC and that for ˆ (2) ∗ ∗ 3 HI(t) = ϵ0χ ApAs Ai d r dωpdωsdωi the higher-order non-linear optical processes the non-energy- ˆ ˆ ˆ V
conserving terms may lead to important contributions. The × 2 2 2 d qpd qsd qiV ωp,qp effective interaction Hamiltonian for the process of parametric   down-conversion can thus be given by the following simplified × − − · − − exp i(qp qs qi) ρ + i(kpz ksz kiz)z form: − † † × i(ωs+ωi ωp)t ˆ e aˆ (qs,ωs)aˆ (qi,ωi) + H.c. (19) ˆ (2) ˆ(+) HI(t) = ϵ0 χ Ep (r,t) V (2) Here, although χ and Aj with j = p, s, and i are frequency × ˆ(−) ˆ(−) 3 Es (r,t)Ei (r,t)d r + H.c., (16) dependent quantities, they vary very slowly within the fre- quency ranges of interest for most experimental situations. So, where H.c. stands for Hermitian conjugate. Next, we need to we have taken them outside of the integral. Equation (19) is express the above Hamiltonian in terms of the photon creation the interaction Hamiltonian for spontaneous parametric down- and annihilation operators. For this purpose, we first realize conversion.

6 J. Opt. 22 (2020) 083501 Tutorial

× † † | ⟩ | ⟩ aˆ (qs,ωs)aˆ (qi,ωi) vac s vac i + H.c., (22) ´ ´ where we have substituted d3r = d2ρdz. The interaction time t0 is assumed to be much longer than the time scale over which down-conversion takes place. Therefore, the limits of time integration can be extended to −∞ and ∞ [90, 95] such that the time integration yields ˆ ∞ − i(ωs+ωi ωp)t − e dt = δ (ωs + ωi ωp). (23) −∞

Similarly, we assume that the transverse area of the non-linear crystal is much larger compared to the transverse area of the pump field, and therefore we write the space integral inequa- tion (22) as ˆ ∞  − − · 2 exp i(qp qs qi) ρ d ρ −∞ Figure 2. A typical experimental arrangement for SPDC. The pump ˆ field is along the z direction. The figure shows the laboratory frame 0 × { − − } coordinates of the non-linear crystal of thickness L. exp i(kpz ksz kiz)z dz, −L
− − = δ qp qs qi Φ(ωs, ωi,qs,qi), (24) 2.4. The generation of entangled two-photon field where We now derive the state of the entangled photons that gets generated in SPDC. Let us consider a pump field interact- ˆ 0 ing with a non-linear optical crystal of thickness L (see fig- − − Φ(ωs, ωi,qs,qi) = exp[i(kpz ksz kiz)z]dz, ure 2). We assume that the pump photon starts interacting with − − | − ⟩  L   the crystal at time t = t0 and that the state ψ( t0) of the − L − L down-converted signal and idler photons at that instant is given = L sinc (ksz + kiz kpz) exp i(ksz + kiz kpz) , | − ⟩ | ⟩ | ⟩ | ⟩ 2 2 by ψ( t0) = vac s vac i, where vac represent a vacuum mode. The state of the two photons evolves under Hamilto- is called the phase-matching function. The action of the cre- nian HI(t) of equation (19) and at the end of the interaction, ation operators in equation (22) is given by that is, at time t = 0, the two-photon state is given by † † | ⟩ | ⟩ | ⟩ | ⟩ ´ aˆ (qs,ωs)aˆ (qi,ωi) vac s vac i = qs,ωs s qi,ωi i, (25) − i 0 Hˆ (t)dt | ⟩ ℏ −t I | − ⟩ ψ(0) = e 0 ψ( t0) . (20) | ⟩ where qs,ωs s represents a state having one signal photon with transverse wave-vector q and frequency ω . The Hermitian The parametric interaction is assumed to be very weak, so s s conjugate part of |ψ ⟩ in equation (22) contains the operator the state in equation (20) can be approximated by the first two tp aˆ(q ,ω )aˆ(q ,ω ). The action of this operator on the vacuum terms of a perturbation expansion. The first term is simply the s s i i | − ⟩ state is given by initial vacuum state ψ( t0) . The second term is the state of the down-converted two-photon field and is given by | ⟩ | ⟩ aˆ(qs,ωs)aˆ(qi,ωi) vac s vac i = 0. (26) ˆ 0 | ⟩ − i ˆ | − ⟩ Therefore, using equations (23) through (26), we write equa- ψtp = ℏ HI(t)dt ψ( t0) . (21) −t tion (22) as 0 ˆ ˆ ˆ ˆ ˆ ˆ The third and the higher-order terms in the expansion are | ⟩ 2 2 2 ψtp = A dωpdωsdωi d qpd qsd qi the states of four- and higher-photon fields, and we assume × − that the probability of generation of these states are negligible. V(ωp,qp)Φ(ωs, ωi,qs,qi)δ (ωs + ωi ωp) ˆ
Substituting the expression of HI(t) from equation (19) into the × − | ⟩ | ⟩ δ q + q q q ,ωs s q ,ωi i, (27) above equation, we obtain the state of the two-photon field at s i p s i ∗ ∗ the exit face inside the non-linear crystal: = ϵ0 χ(2) ω where A iℏ ApAs Ai . Now, integrating over d p and dqp, ˆ ˆ ˆ ˆ ˆ we get ϵ 0 |ψ ⟩ = 0 χ(2) ∗ ∗ 2ρ ω ω ω ˆ ˆ ˆ ˆ tp ℏ ApAs Ai dt d dz d pd sd i i −t V | ⟩ 2 2 ˆ ˆ ˆ 0 ψtp = A dωsdωi d qsd qiV(ωs + ωi,qs + qi) × 2 2 2 (ω , ) d qpd qsd qiV p qp × Φ(ω , ω ,q ,q )|q ,ω ⟩ |q ,ω ⟩ . (28)   s i s i s s s i i i × ( − − ) · ρ + ( − − ) exp i qp qs qi i kpz ksz kiz z The above equation represents the state of the two-photon field × { − } exp i(ωs + ωi ωp)t at the exit face of the crystal produced by SPDC.

7 J. Opt. 22 (2020) 083501 Tutorial

Figure 3. Schematic diagram depicting the situation when the pump beam waist is located at a distance d behind the crystal.

2.5. Effects due to the beam waist location of a Gaussian pump beam In most situations, the pump field is taken to be in the form Figure 4. Schematic diagram illustrating effects due to the non-normal incidence of the pump field on the non-linear crystal. of a Gaussian beam and the location of the Gaussian beam waist is taken to coincide with that of the crystal. However, in many experimental situations, the location of the pump beam can be written as waist does not coincide with that of the crystal. Effects of this     |q |2w2 |q |2d kind have been studied in several works in the past [55, 81, ikpzd − p 0 − p ikpd V(qp)e = exp exp i e . (31) 82, 96, 97], and in this subsection, we present detailed calcu- 4 2kp lation illustrating how the beam-waist location of the pump field affects the SPDC process. Let us consider the situation We note that in the context of using the above spectrum for shown in figure 3, in which the non-linear crystal is at a dis- calculating the coincidence count rate in equation (81), the tance d away from the beam waist location of the Gaussian constant factor eikpd does not make any contribution. This is pump field. Our aim is to find out the form of the pumpfield the expression for V(qp) = V(qs + qi) that we will be using in V(qs + qi) that needs to be used in equation (81). In order to do equation (81) for calculating the coincidence count rates. ′ ′ this, we first define a coordinate system (x,y,z ) = (ρ,z ) such ′ that the location of the pump beam waist is at z = 0. We define 2.6. Effects due to the non-normal incidence of pump field another coordinate system (x,y,z) = (ρ,z) such that the crys- ′ on the SPDC crystal tal is located at z = 0. Thus we have z = z + d. The electric ′ ′ field amplitude V˜(ρ,z ) at a given z can be written as In experimental situations, it is quite often the case that the ˆ ˆ SPDC crystal is oriented in such a way that the pump field is 0 ′ · ′ incident on the crystal at an angle other than 90 . In this sub- V˜(ρ,z ) = V(q )ei(qp ρ+kpzz )d2q, (29) p section, we consider the effects on the two-photon coincidence count rate caused by the non-normal incidence. This situation ′ where V(qp) is the angular spectrum of the field at z = 0 and is shown in figure 4. The SPDC crystal is rotated by an angle ′ where we have used kp = (qp,kpz). Now using z = z + d, we α with respect to the pump propagation direction. Inside the ′ rewrite equation (29) in the (ρ,z) coordinate system as crystal, the z axis represents the direction of propagation of ˆ ˆ the pump field when it is incident perpendicular to the crystal, · ˜ ikpzd i(qp ρ+kpzz) 2 that is, when α = 0, while the z axis represents the direction of V(ρ,z) = V(qp)e e d q, (30) propagation of the pump beam inside the rotated crystal. The ′ angle between the optic axis and the z axis is represented by ˜(ρ, ) Here V z is the same electric field amplitude but written θp0 while θp represents the angle between the z axis and the in the (ρ,z) coordinate system. Comparing equations (29) and optic axis of the rotated crystal. Using Snell’s law of refrac- ( ) ikpzd (30), we see that V qp e can be taken as the angular spec- tion we write sin α = npe sin β, where α is the angle of incid- trum of the field at z = 0. Therefore, we find that if the angular ence of the pump field, β is the angle of refraction inside the spectrum of the pump field is known at a distance d behind the crystal, and npe is the refractive index of the pump field which crystal, the angular spectrum at the crystal plane can simply is taken to be extra-ordinary polarized. Here, we note that the ikpzd be taken as V(qp)e . pump field polarization is decided by the phase-matching con- In most cases, the pump field satisfies the paraxial ditions, and we show in the next section through the phase- ≈ approximation. Under this approximation, we write kpz matching considerations that the pump field has to be extra- − | |2 | | kp qp /2kp, where kp = kp = ωp0/c = (ωs0 + ωi0)/c. For ordinary polarized for negative uniaxial crystals such as BBO. a pump field with Gaussian spectrum, the angular spectrum Therefore, we write −| |2 2 at the beam waist is given by [98] V(qp) = exp qp w0/4 ,   − where w0 is the beam waist at z = d. Therefore the angular −1 sin α = β = sin . (32) spectrum of the field at the crystal location, that is,at z 0 npe

8 J. Opt. 22 (2020) 083501 Tutorial

(a) (b) configuration. In this configuration, the phase matching con- straint ∆kz = 0 requires the following equation to be satisfied (see [85], section 2.3): s i s i n σ(ω )ω n σ(ω )ω n σ(ω )ω p p0 p0 = s s0 s0 + i i0 i0 , (34) c c c p p where as earlier, p, s, and i stand for pump, signal, and idler, respectively. The index σ can take two values. If the polariza- Figure 5. Phase-matching diagrams for the (a) collinear and (b) tion direction is perpendicular to the plane formed by the optic non-collinear emission geometries. The condition for perfect axis and the propagation direction then it is called ordinary phase-matching is kpz = ksz + kiz. polarized, and in that case σ = o. Alternatively, if the polar- ization direction is parallel to the plane formed by the optic From the ray diagram shown in figure 4, we obtain axis and the propagation direction then it is called extraordin-   ary polarized, and in that case σ = e [102]. Therefore, npe(ωp0) − θ = θ + β = θ + sin 1 sin α . (33) represents the refractive index of the extra-ordinary polar- p p0 p0 npe ized pump photon at frequency ωp0, etc. Combining the above equation with energy conservation equation, that is, ω = Thus we see that the non-normal incidence of the pump field p0 ω + ω , we rewrite equation (34) as affects the angle between the optic axis and the pump propaga- s0 i0 ω tion direction. Therefore, we need to use θp instead of θp0 when − i0 − npσ(ωp0) nsσ(ωs0) = [niσ(ωi0) nsσ(ωs0)]. (35) calculating kpz and subsequently the phase-matching function ωp0 Φ(q ,q ) in equation (28). s i ≥ Here, we note that ωp0 > ωs0 ωi0. We now consider all pos- sible polarization scenarios for pump, signal and idler photons 3. Polarization effects in SPDC one by one and find out the polarization scenarios that can potentially satisfy the phase matching condition. We assume 3.1. Polarization effects due to phase matching constraints normal dispersion for the down-conversion crystal, that is, we assume that the refractive index increases with frequency. The generation and emission-directions of SPDC photons are BBO crystal is an example of a down-conversion crystal that decided by the constraints imposed by the conservation of has normal dispersion for both ordinary and extraordinary energy and momentum. These constraints are referred to as polarization [103]. There can be a total of eight possible polar- the phase-matching conditions. Numerous textbooks discuss ization scenarios for the pump, signal and idler photons. For phase matching based on crystal class and permutation sym- each scenario, we represent the down-conversion process by a metries [16, 17, 99–102]. If the directions of propagation of rightarrow and the individual pump, signal and idler photons the down-converted signal and idler photons are along the only by their polarization index. For example, e → o + e rep- propagation direction of the pump photon then it is known as resents an extraordinary polarized pump down-converting into the collinear phase matching. Alternatively, in non-collinear an ordinary signal photon and an extraordinary idler photon, phase matching, the signal and idler photons propagate in dir- etc. ections non-collinear with that of the pump photon. The phase Case I: e → e + e: In this case, the pump, signal and idler matching diagrams of collinear and non-collinear SPDC are photons are all extraordinary polarized and therefore equation shown in figure 5(a) and (b), respectively. As we see from (35) can be written as equation (28), the efficiency of the SPDC process, that is,the generation probablity of the down-converted photons depends − ωi0 − npe(ωp0) nse(ωs0) = [nie(ωi0) nse(ωs0)]. (36) on the phase-matching function Φ(qs,qi) through the wave ωp0 − vector mismatch ∆kz = ksz + kiz kpz. It can be shown that 2 We note that n (ω ) > n (ω ),n (ω ). Considering ω ≥ this efficiency goes as sinc [∆kzL/2]. When ∆kz = 0, that is, pe p0 se s0 ie i0 s0 when the phase matching is perfect, the efficiency is max- ωi0 we find that the above equation cannot be satisfied. There- imum. The efficiency of the SPDC process decreases asthe fore, the phase matching condition can never be satisfied phase mismatch ∆k increases [11]. and thus this particular polarization scenario is not possible. z → Next, we recall that in deriving the Hamiltonian in equa- Case II: o o + o: In this case, the pump, signal and idler tion (19), we had considered only the scalar fields even though photons are all ordinary polarized and therefore the equation polarization plays an important role in deciding the phase- (35) can be written as matching condition. In this section, we consider the polar- ω (ω ) − (ω ) = i0 [ (ω ) − (ω )]. ization effects due to the phase matching constraints. Our npo p0 nso s0 nio i0 nso s0 (37) ωp0 approach is to first find out the possible polarization scenarios that can satisfy the phase-matching condition and for those We note that npo(ωp0) > nso(ωs0),nio(ωi0). As we assume ≥ scenarios use equation (28) for calculating the detection prob- ωs0 ωi0, we find that the equation above cannot be satisfied. abilities. In order to list the possible polarization scenarios that Therefore, the phase matching condition can never be satisfied can satisfy the phase matching, we analyze it in the collinear and thus this particular polarization scenario is not possible.

9 J. Opt. 22 (2020) 083501 Tutorial

→ Case III: e o + o: In this case, equation (35) can be This equation holds only when no < ne, that is, when the crys- written as tal is positive uniaxial. Case VIII: o → o + e: Equation (35) in this case can be − ωi0 − npe(ωp) nso(ωs0) = [nio(ωi0) nso(ωs0)]. (38) written as ωp0 − ωi0 − ω ≥ ω npo(ωp0) nso(ωs0) = [nie(ωi0) nso(ωs0)]. (46) We find that for s0 i0, the right hand side of the above ωp0 equation is negative. Now in order for the left hand side of this equation to be negative, we have to ensure that no > ne. There- Considering the same assumptions as in the above case, we fore, this scenario can indeed be satisfied by negative uniaxial write the above equation as crystals such as BBO. − ≃ Case IV: o → e + e: In this case, equation (35) can be 3no(ωs0) ne(ωs0) 0. (47) written as This equation holds only when no < ne, that is, when the crys- − ωi0 − npo(ωp0) nse(ωs0) = [nie(ωi0) nse(ωs0)]. (39) tal is positive uniaxial. In general, when both the signal and ωp0 idler photons have the same polarization, it is referred to as the type-I phase matching and when they have orthogonal polar- We find that for ω ≥ ω , the right hand side of equation (39) s0 i0 izations, it is referred to as the type-II phase matching [76, is negative. Now in order for the left hand side of this equa- 78]. Thus, the possible polarization scenarios that can satisfy tion to be negative, we have to ensure that n (ω ) < n (ω ). po p0 ie i0 the phase-matching conditions can be represented in a tabular Therefore, this scenario can be satisfied by positive uniaxial form as follows: crystals. Case V: e → o + e: Equation (35) in this case can be writ- ten as Type Positive uniaxial Negative uniaxial type-I o → e + e e → o + o ωi0 n (ω ) − n (ω ) = [n (ω ) − n (ω )]. (40) → + → + pe p0 so s0 ω ie i0 so s0 o o e e e o p0 type-II o → e + o e → o + e

We take the SPDC to be degenerate, that is, ωp0 = 2ωs0 = Thus for positive uniaxial crystals, the pump field polariz- 2ωi0 and assume that the dispersion relation is linear for both the extraordinary and ordinary polarizations. We thus ation has to be ordinary in order to satisfy the phase-matching ≈ condition. Similarly, for negative uniaxial crystals, the pump have npe(2ωs0) 2npe(ωs0). Now, substituting npe(ωs0) = field polarization needs to be extraordinary in order to satisfy nie(ωs0) = ne(ωs0) and nso(ωs0) = no(ωs0), we write the above equation as the phase matching condition. ≈ 3ne(ωs0) no(ωs0). (41) 3.2. Wave propagation inside the SPDC crystals

This equation holds only when ne < no, that is, when the crys- In the previous subsection, we found out all possible polariz- tal is negative uniaxial. ation scenarios that can satisfy the phase-matching condition. Case VI: e → e + o: Equation (35) in this case can be writ- In this subsection, we study how polarization affects the wave ten as propagation inside an SPDC crystal. We note that in what fol- ω lows we restrict our analysis only to negative uniaxial crystals n (ω ) − n (ω ) = i0 [n (ω ) − n (ω )]. (42) and for brevity we represent the refractive indices without their pe p se s0 ω io i0 se s0 p0 frequency arguments. We start by writing the electric displace- ment vector D in terms of the electric field E inside an elec- Considering the same assumptions as in the above case, we trically anisotropic medium [104]: write the above equation as      − ≃ Dx ϵxx ϵxy ϵxz Ex 3ne(ωs0) no(ωi0) 0, (43)      Dy = ϵyx ϵyy ϵyz Ey . (48) Dz ϵzx ϵzy ϵzz Ez This equation holds only when ne < no, that is, when the crys- tal is negative uniaxial. Here, (x, y, z) is the Cartesian coordinate system and D and E Case VII: o → e + o: Equation (35) in this case can be x x are the x-component of the electric displacement vector and written as electric field, etc. The× 3 3 matrix is known as the dielectric ω − i0 − tensor with εij being its elements, where i, j = x, y, z. Suppose npo(ωp0) nse(ωs0) = [nio(ωi0) nse(ωs0)]. (44) ωp0 the dielectric tensor can be written in the diagonal form in the ′ ′ ′ (x ,y ,z ) coordinate system, that is, Considering the same assumptions as in the above case, we   write the above equation as ϵ ′ 0 0  x  ϵ = 0 ϵy′ 0 , (49) − ≃ 3no(ωs0) ne(ωs0) 0. (45) 0 0 ϵz′

10 J. Opt. 22 (2020) 083501 Tutorial

′ ′ ′ where (x ,y ,z ) denotes the new coordinate system, which is Since ω/c ≠ 0, the above condition requires that either obtained by rotating the (x, y, z) coordinate system by an angle 2 2 2 q ′ + q ′ + k ′ ω2 θp around the y-axis as shown in figure 6. The two coordinate x y z = , 2 2 (57) systems are related as no c      ′ or x cosθp 0 sinθp x  ′     ω y = 0 1 0 y . (50) (n q ′ )2 + (n q ′ )2 + (n k ′ )2 − ( n n )2 = 0. (58) ′ − o x o y e z o e z sinθp 0 cosθp z c We note that the relation given in equation (57) is the disper- ′ 2 ′ ′ 2 Next, we write ϵz /ϵ0 = ne and ϵx /ϵ0 = ϵy /ϵ0 = no and sion relation for ordinary polarized light field and the rela- express the electric displacement vector of equation (48) in tion given in equation (58) is the dispersion relation for the the new coordinate system as extraordinary polarized light field. Now, we need to write both      these dispersion relations in the (x, y, z) coordinate system. We ′ 2 ′ Dx n0 0 0 Ex note that the phase of a plane wave remains invariant under ′  ′   2  ′  ′ Dy = ϵ0 0 n0 0 Ey . (51) coordinate transformation, that is, k · r = k · r , which in the ′ 2 ′ Dz 0 0 ne Ez component form can be written as

We know that a monochromatic electric field E(r,t) propagat- (qxx + qyy + kzz) = qx′ (xcosθp + zsinθp) + qy′ y ′ − ing through an anisotropic medium can be written as + kz ( xsinθp + zcosθp). (59) { − } E(r,t) = E0(r) exp i(k.r ωt) , (52) Comparing the coefficients of x, y, z on both sides of the above equation, we obtain where k = qxˆx + qyˆy + kzˆz and r = xˆx + yˆy + zˆz. In the ′ ′ ′ = ′ θ − ′ θ , (x ,y ,z ) coordinate system, the field is given by qx qx cos p kz sin p (60)  ′ ′ ′ ′ − E(r ,t) = E0(r ) exp i(k .r ωt) , (53) qy = qy′ , (61) ′ ˆ′ ˆ′ ˆ′ ′ ′ ˆ′ ′ ˆ′ ′ ˆ′ where k = qx′ x + qy′ y + kz′ z and r = x x + y y + z z . The ′ ′ ′ wave equation in the (x ,y ,z ) coordinate system can be writ- ten as kz = qx′ sinθp + kz′ cosθp. (62) 2 ′ ′ ′2 ∂ D ∇ (∇ · E) − ∇ E = −µ , (54) These relations can be inverted to yield 0 ∂t2 qx′ = qx cosθp + kz sinθp, (63) ′ ′ ∂ ′ ∂ ′ ∂ ∇ = ˆ ′ + ˆ ′ + ˆ ′ where x ∂x y ∂y z ∂z . Using equation (51), we write the above wave equation as qy′ = qy, (64)     ′ ′ ′ ω 2 ϵ (k · E)k − |k |2E = − E. (55) c ϵ 0 ′ − kz = qx sinθp + kz cosθp. (65)

The above vector equation yields three simultaneous scalar Using the above relations, we write equation (57) in the (x, y, z) equations involving the three components of the electric field: coordinate system as

   q2 + q2 + k2 ω2 2 x y z noω 2 2 = . (66) − q ′ − k ′ E ′ + q ′ q ′ E ′ + q ′ k ′ E ′ = 0, n2 c2 c y z x x y y x z z o    2 noω Therefore, within the paraxial approximation, the longitudinal ′ ′ ′ − 2 − 2 ′ ′ ′ ′ qx qy Ex + q ′ k ′ Ey + qy kz Ez = 0, c x z wave vector component kz of an ordinary polarized light field    2 inside an anisotropic medium can be written as [81] neω 2 2 ′ ′ ′ ′ ′ ′ − ′ − ′ ′ r qx kz Ex + qy kz Ey + qx qy Ez = 0.   c ω 2 ω − 2 − 2 ≈ − c 2 2 kz = no qx qy no (qx + qy). (67) c c 2noω A non-trivial solution for Ex′ , Ey′ , Ez′ exists only if the determ- inant of the coefficients becomes zero, that is, only if Similarly, using equations (63)–(65), we write equation (58)
h
i in the (x, y, z) coordinate system as ω 2 ω 2 − 2 − 2 − 2 × " # " # no q ′ q ′ k ′ c c x y z 2 2 2 2 2 h
i sin θp cos θp 2 cos θp sin θp 2 qy ω 2 ′ + k + + q + − ( ′ )2 − ( )2 − ( ′ )2 = . 2 2 z 2 2 x 2 c none noqx noqy nekz 0 (56) ne no ne no ne

11 J. Opt. 22 (2020) 083501 Tutorial

Figure 7. Experimental set up for the generation of SPDC photon pairs. The pump beam waist wo is located at a distance d from the front face of the crystal. The photons are detected at a distance z from the crystal.

Figure 6. The coordinate representation of the non-linear crystal in ′ ′ ′ both lab frame (x, y, z) and rotated frame (x ,y ,z ). The rotated 4. Calculating the detection probabilities of the coordinate system is rotated by an angle θp around the y-axis. generated entangled photons

    In this section, we derive explicit expressions for the phase- 1 1 ω 2 + − θ − = . matching function for various possible polarization scenarios. 2 2 sin2 pqxkz 0 (68) ne no c We restrict our analysis to the BBO crystal which is a negat- ive uniaxial crystal. We note that the dispersion relations for 2 The above equation is in the form of akz + bkz + c = 0. There- the ordinary and extraordinary polarized light inside the BBO fore, solving for kz and using the paraxial approximation, the crystal are given by [103] longitudinal wave vector component kz of the extraordinary 0.0184 polarized light field inside an anisotropic medium can be writ- n2 = 2.7405 + − 0.0155λ2, (74) ten as [81] o λ2 − 0.0179 q
= −α + η ω 2 − β2 2 − γ2 2, kz qx c qx qy 0.0128   n2 = 2.3730 + − 0.0044λ2. (75) ≈ −α + η ω − c β2 2 + γ2 2 , e 2 − qx c 2ηω qx qy (69) λ 0.0156 Here λ is the wavelength of light in µm. Also, for brevity, we where show the refractive indices without their corresponding fre- (n2 − n2)sinθ cosθ quency arguments. α = o e p p , (70) 2 2 2 2 no sin θp + ne cos θp 4.1. Calculating the coincidence and the individual photon count rates n n β = o e , (71) In this article, we are primarily interested in studying the two- 2 2 2 2 no sin θp + ne cos θp photon wavefunction in the position and linear momentum bases. Therefore, in what follows, we assume that the signal, idler and pump fields are monochromatic with central frequen- n cies given by ω , ω , and ω , respectively. With this assump- γ =q o , (72) s0 i0 p0 tion and using equation (28), we write the state of the down- n2 sin2 θ + n2 cos2 θ o p e p converted two-photon field at the exit face of the crystal as ˆ ˆ |ψ ⟩ = A d2q d2q V(q + q )Φ(q ,q )|q ⟩ |q ⟩ . (76) n n tp s i s i s i s s i i η =q o e . (73) 2 2 2 2 no sin θp + ne cos θp Here, we do not explicitly show the frequency arguments. We now calculate the probability of detecting the signal and idler ≡ Here θp is the angle between the optic axis and the pump photons at a distance z from the crystal at locations rs (ρs,z) ≡ propagation direction, that is, the z-axis. By changing the angle and ri (ρi,z), respectively, in coincidence (see figure 7). The θp, one can generate entangled photons with various different coincidence count rate Rsi(rs,ri), which is the probability per 2 phase matching conditions. (unit time) that a signal photon is detected at position (ρs,z)

12 J. Opt. 22 (2020) 083501 Tutorial

and an idler photon is detected at position (ρi,z), is given by vice versa. Thus the signal and idler photon count rates at loc- [92]: ations (ρs,z) and (ρi,z), respectively, can be written as ˆ Rsi(rs,ri) = Rsi(ρs,ρi,z) 2 Rs(ρs,z) = Rsi(ρs,ρi,z)d ρi, (82) − (−) (+) = α α ⟨ψ |Eˆ( )(r )Eˆ (r )Eˆ (r )Eˆ(+)(r )|ψ ⟩, s i tp s s i i i i s s tp 2 ⟨ | ⟨ | |ˆ(+) ˆ(+) | ⟩ ˆ = αsαi vac s vac i Es (rs)Ei (ri) ψtp , (77) 2 Ri(ρi,z) = Rsi(ρs,ρi,z)d ρs. (83) Here αs and αi denote the quantum efficiencies of the detect- ˆ(+) ors kept at locations (ρs,z) and (ρi,z), respectively. Es (rs) ˆ(+) 4.2. Calculating the conditional detection probability and Ei (ri) denote the positive-frequency parts of the elec- tric field operators at detection locations (ρs,z) and (ρi,z), In addition to the coincidence and individual photon count respectively. The two field operators can be written as rates, in many experimental situations, one is also interested in the conditional detection probabilities of individual photons. ˆ(+)( ) = ˆ(+)(ρ , ), Es rs Eˆs s z For example, it may of interest in certain experiment to calcu- · − 2 late the detection probability of the signal photon as a function 2 iksz i(qs ρs qs z/2ks) = d qse e aˆ(qs), (78) of position ρs given that the idler photon is already detected at position ρi0. In fact, quite often, by detecting an idler at a particular position the region over which the signal photon ˆ(+) ˆ(+) could be found becomes very localized. The width of this E (ri) = E (ρi,z), i ˆi region is called the position-correlation width. The position- · − 2 2 ikiz i(qi ρi qi z/2ki) = d qie e aˆ(qi). (79) correlation of entangled photons have been studied and util- ized in several experimental situations [57, 105–107]. The 2 | |2 2 | |2 | | | | position-correlation width is calculated as follows. For the Here qs = qs , qi = qi , ks = ks = ωs0/c, and ki = ki = fixed idler position (ρi0,z) the coincidence count rate can be ωi0/c, where c is the speed of light. We note that in writing ˆ(+) ˆ(+) written using equation (81) as the electric field operators Es (ρs,z) and Ei (ρi,z) above, we have made use of the paraxial approximation. Substituting ˆ ˆ from equations (76), (78) and (79) in equation (77), we obtain (ρ ,ρ , ) = α α 2 2 i(ks+ki)z the following expression for the coincidence count rate: Rsi s i0 z s i A d qsd qi e

ˆ ˆ ˆ ˆ × ( + )Φ( , ) V qs qi qs qi 2 2 2 ′ 2 ′ Rsi(ρ ,ρ ,z) = αsαi A d q d q d q d q    2 s i s i s i | |2 | |2 × · · − qs z − qi z exp i qs ρs + qi ρi0 , ( + ) 2k 2k × ei ks ki zV(q + q )Φ(q ,q ) s i   s i s i  ′ 2 ′ 2 (84) ′ ′ |q | z |q | z × exp i q · ρ + q · ρ − s − i s s i i 2k 2k s i where r = (ρ ,z) and r = (ρ ,z). The width of 2 s s i0 i0 R (ρ ,ρ ,z) is the position-correlation width of the signal × ⟨ | ⟨ | ˆ( ′ )ˆ( ′)| ⟩ | ⟩ . si s i0 vac s vac ia qs a qi qs s qi i (80) photon at a plane z away from the crystal. We note that in a sim- ilar manner, one can also calculate the momentum-correlation ′ ′ | ⟩ | ⟩ ′ − width. Now using the relation aˆ(qs)aˆ(qi) qs s qi, i = δ(qs ′ − | ⟩ | ⟩ Equation (81) for the coincidence count rate and equa- qs)δ(qi qi) vac s vac i, we write equation (80) as tions (82) and (83) for photon count rates at the signal and ˆ ˆ idler locations have been derived for a very generic pump (ρ ,ρ , ) = α α 2 2 i(ks+ki)z Rsi s i z s i A d qsd qi e field and the phase matching function. We now calculate the phase-matching function Φ(qs,qi) for both type-I and type-II × V(q + q )Φ(q ,q ) phase-matching conditions. Using the expression for Φ(q ,q ) s i s i s i    2 in a particular phase-matching condition, one can calculate the |q |2z |q |2z × · ρ + · ρ − s − i . coincidence count rate using equation (81) and the individual exp i qs s qi i 2ks 2ki photon count rates using equations (82) and (83). (81) 4.3. Calculating the detection probabilities for type-I phase In addition to the coincidence count rate, we also calculate the matching photon count rates of the individual signal and idler photons at locations (ρs,z) and (ρi,z), respectively. The photon count In type-I phase matching, the pump field is extraordinary rate at the signal location (ρs,z) is obtained by integrating the polarised and the down-converted fields are ordinary polar- coincidence count rates over all possible idler locations, and ised. Therefore, with paraxial approximation, the z-component

13 J. Opt. 22 (2020) 083501 Tutorial of the pump, signal, and idler fields are given by   2 2 2 2 ωi0 − c | |2 ω c β q + γ q kiz = nio q . (92) − p0 − p px p py c 2nioωi0 i kpz = αpqpx + ηp , (85) c 2ηpωp0 The phase matching function in this case is given by → Φ(q ,q ) Φeeo(q ,q ) ωs0 − c 2 2 s i s  i    ksz = nso ω (qsx + qsy), (86) c 2nso s0 L L = L sinc ∆k exp i∆k , (93) 1z 2 1z 2

k = n ωi0 − c (q2 + q2 ). (87) where iz io c 2nioωi0 ix iy − Here ∆k1z = ksz + kiz kpz ω c   ( 2 − 2 ) θ θ = −α q + η s0 − β2q2 + γ2q2 npo npe sin p cos p s sx s η ω s sx s sy αp = , c 2 s s0 2 2 θ + 2 2 θ ω ω npo sin p npe cos p i0 − c | |2 − p0 + nio qi + αp(qsx + qix) ηp nponpe ω β = , c 2nio i0 c p 2   2 θ + 2 2 θ c 2 2 2 2 npo sin p npe cos p + β (q + q ) + γ (q + q ) . (94) 2η ω p sx ix p sy iy q npo p p0 γp = , 2 2 θ + 2 2 θ npo sin p npe cos p Here Φeeo(qs,qi) represents the phase matching function. We α ,β ,γ ,η q nponpe note that p p p p are calculated by putting in the values ηp = . ω α ,β ,γ ,η 2 2 2 2 of no and ne at the pump frequency p0 while s s s s are nop sin θp + npe cos θp calculated by putting in the values of no and ne at the signal frequency ω . The angle θ remains the same in each case. The phase-matching function in this case is given by s0 p Case-2 :If the signal is extraordinary polarized and the idler Φ(q ,q ) → Φ (q ,q ) is ordinary, then we have s i eoo s i      L L β2 2 + γ2 2 = L sinc ∆k exp i∆k , (88) − ωp0 − c p qpx p qpy z 2 z 2 kpz = αpqpx + ηp , (95) c 2ηpωp0 where

− k = n ωs0 − c |q |2, (96) ∆kz = ksz + kiz kpz, sz so c 2nsoωs0 s ω ω ω   s0 i0 − p0 c 2 2 2 2 = nos + noi ηp + βp qpx + γp qpy c c c 2ηpωp0   c c β2 2 + γ2 2 + α ( + ) − | |2 − | |2. ωi0 c i qix i qiy p qsx qix qs qi (89) k = −α q + η − . (97) 2nosωs0 2noiωi0 iz i ix i c 2ηiωi0 Φ ( , ) Here eoo qs qi represents the phase matching function when The phase matching function in this case is given by the pump field is extraordinary polarized while both the signal and idler fields are ordinary polarized. Φ( , ) → Φ ( , ) qs qi  eoe qs qi = ∆ L ∆ L , L sinc k2z 2 exp i k2z 2 (98) 4.4. Calculating the detection probabilities for type-II phase matching where − In type-II phase matching, the pump field is extraordinary ∆k2z = ksz + kiz kpz, polarized while the polarizations of the signal and idler fields ω c ω = s0 − | |2 − α + η i0 are orthogonal to each other. Hence there are two possibilit- nso qs iqix i c 2nsoωs0  c ies for the signal and idler polarizations in this type of phase c β2q2 + γ2q2 ω matching. − i ix i iy + α (q + q ) − η p0 η ω p sx ix p Case-1 : If the signal is extraordinary polarized and the  2 i i0  c 2 2 2 2 idler is ordinary, then we have c βp (qsx + qix) + γp (qsx + qix) + . (99)   2ηpωp0 β2 2 + γ2 2 − ωp0 − c p qpx p qpy kpz = αpqpx + ηp , (90) Φ c 2ηpωp0 Here eoe is the phase matching function for pump, signal and idler being extraordinary, ordinary and extraordinary polarized   light respectively. We note that αp,βp,γp,ηp are calculated by ω c β2q2 + γ2q2 putting in the values of n and n at the pump frequency ω = −α + η s0 − s sx s sy , o e p0 ksz sqsx s (91) while α ,β ,γ ,η are calculated by putting in the values of n c 2ηsωs0 i i i i o

14 J. Opt. 22 (2020) 083501 Tutorial

Figure 9. (a)–(d) The experimental signal photon intensity obtained using an EMCCD camera kept at z = 35 mm for various different Figure 8. (a)–(c) The experimental signal photon intensity obtained ◦ ◦ using an EMCCD camera kept at z = 35 mm for various different values of θp. (a) θp = 40.48 ,α = 19.11 (b) θp = ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ values of θp. (a) θp = 28.64 ,α = 0.33 (b) θp = 28.74 ,α = 0.49 40.99 ,α = 19.94 (c) θp = 41.40 ,α = 20.60 (d) θp = ◦ ◦ ◦ ◦ (c) θp = 28.95 ,α = 0.82 . The corresponding numerical plots are 41.78 ,α = 21.23 . The corresponding numerical plots are shown shown in (d)–(f) respectively. in (e)–(h) respectively. and ne at the idler frequency ωi0. We note that in experimental situations, both of the above cases of type-II phase matching are satisfied at the same time. Therefore, the coincidence and numerically evaluate equations (81) and (82) for various dif- θ the individual count rates have contributions due to both the ferent values of p and with both type-I and type-II phase- matching functions as given in equations (88), (93), and (98). phase-matching functions, Φeoe(qs,qi) and Φeeo(qs,qi). ◦ We find that for θp < 28.6 , no phase-matching gets satisfied. Figure 8 shows the two-dimensional images of down- 5. Experimental and numerical results conversion output obtained using an EMCCD camera and also the numerically calculated two-dimensional plots R(ρs,z) for In this section, we present results of our experimental and z = 35 mm at various different values of θp. Figures 8(a)–(c) numerical studies. We evaluate equations (81)–(83) in order to are the experimental plots while (d)–(f) are the corresponding study how the generation of entangled photons depend on vari- numerical plots. We see that the collinear type-I phase match- ◦ ous experimental parameters. In most cases, we also present ing starts around θp = 28.6 and the output pattern looks like our experimental observations in order to highlight how it a circular blob. When θp is increased further the circular blob matches with the theory. In our experiments, we use a 405 nm, turns into an annulus with non-collinear phase matching. The 100 mW continuous wave laser as the pump field. The pump radius of the annulus increases as θp is increased. We see a power of 100 mW ensured that we were working within the very good match between theory and experiment. weak down-conversion limit, in which case the probability of We find that the type-II phase matching starts around θp = ◦ producing a four-photon state is negligibly small compared to 40.48 in the form of two off-axis blobs. Figure 9 shows the that of producing a two-photon state. The pump field is in the two-dimensional images of down-conversion output obtained form of a Gaussian beam having a beam-waist of 388 µm. For using an EMCCD camera and also the numerically calculated β the non-linear optical crystal, we use a 2 mm thick -barium two-dimensional plot R(ρs,z) for z = 35 mm at various differ- borate (BBO) crystal. By putting suitable wavelength filters, ent values of θp for type-II phase matching. Figures 9(a)–(d) we make sure that we have degenerate down-conversion with are the experimental plots while (e)–(h) are the corresponding ◦ the central wavelength of both the signal and idler photons numerical plots. We note that at θp = 40.48 both the blobs being 810 nm. The distance between the crystal and the beam have orthogonal polarizations and that one blob corresponds waist location turns out to be d = 107.8 cm. For recording to the signal photon while the other one corresponds to the the down-converted photons, we use an Andor iXon Ultra idler photon. As θp is increased, the two blobs open up and × EMCCD camera having 512 512 pixels and kept at z = 35 become two separate annuli. When θp is increased even further mm. For our numerical experiments, we use the same value of the two annuli start overlapping with each other. We find that the experimental parameters as mentioned above. the match between theory and experiment is not as good as in the case of type-I phase-matching. This is because of the fact that the BBO crystal that we are using in our experiments is 5.1. Effect of θp on the down-conversion output meant to be used for type-I phase matching. Using it for type-II First of all we study how the angle θp between the phase matching means rotating the crystal about the y-axis by optic axis and the pump propagation direction affects the large angles which causes a large background observed in our down-conversion output. For a range of values of θp, we experiments. Nevertheless, as far as the overall θp dependence

15 J. Opt. 22 (2020) 083501 Tutorial

Figure 10. Numerically calculated total output power as a function of θp for (a) type-I phase matching and (b) type-II phase matching. ◦ type-I phase matching takes place for θp > 28.6 while type-II ◦ phase matching takes place for θp > 40.48 . is concerned, the match between the theory and experiment seems quite good. Figure 11. (a) Numerically calculated maximum intensity of signal photons as a function of θp for type-I phase matching condition at z = 35 mm. Collinear to non-collinear phase matching is achieved by increasing θp angle. (b) The intensity profile obtained with 5.2. Effect of θp on the total output power of the ◦ collinear phase matching at θp = 28.64 . (c) The intensity profile down-converted photons ◦ obtained at θp = 28.74 . Dashed circles are the area over which the intensity is calculated. Next, we numerically study how the output power of the down- converted photons depend on the angle θp between the optic axis and the pump propagation direction. For this, we numer- phase matching condition. For this purpose, first we numer- ically evaluate equations (82) and (83) for various different ically calculate the signal photon intensity R(ρ ,z) for vari- values of θ such that it covers both type-I and type-II phase s p ous different values of θ . Then, as shown in figure 11, we matching scenarios. For each value of θ , we calculate the p p select small regions on the intensity plots to find the maximum total output power of the down-converted photons by integrat- intensity. Figure 11(a) shows the variation of intensity as a ing over the two-dimensional space. In figure 10, we plot this function of θ . We find that the photons have maximum intens- power as a function of θ . While figure 10(a) shows this plot p p ity in the collinear emission geometry. As the phase matching over θ values that give type-I phase matching, figure 10(b) p becomes more and more non-collinear, the intensity becomes shows this plot over θ values that give type-II phase match- p lesser. This result shows that the intensity and thus the photon- ing. emission efficiency is higher in collinear phase matching com- We note that in figure 10(a) the output power is nearly ◦ ◦ pared to the non-collinear phase matching. Thus, in experi- zero for θ < 28.6 . This means that for θ < 28.6 , the phase p p ments in which more intense photon source is required the col- matching condition is not satisfied. We get type-I phase match- ◦ linear phase matching should be preferred. Here, as illustrated ing for θ > 28.6 . We note that once the total output power p in the previous section, we note that the total output power reaches its maximum, a further increase in θ value does not p remains the same irrespective of whether the collinear or the change the output power, which then becomes almost constant non-collinear phase matching is satisfied. as a function of θp. In figure 10(b), the total output power is ◦ ◦ nearly zero for θp < 40.48 . Here we note that for θp > 28.6 , type-I phase matching is always satisfied. However, the size of 5.4. Effect of crystal length on down-conversion ◦ the anuulus becomes very large. So, for θ > 40.48 , we cal- p The length of the crystal L directly affects the phase matching culate the contribution to the total power only due to the type- ◦ in SPDC. Figure 12 shows the down-conversion intensity II phase matching. We find that as for θ > 40.48 , the total p pattern for three different values of the crystal thickness L for output power due to type-II phase matching starts to increase ◦ type-I phase matching condition at θ = 28.74 . We find that L and once it reaches its maximum value it stays constant as a p does not affect the overall non-collinearity of emission. How- function for θ . Thus for both type-I and type-II phase match- p ever, it does affect the width of the annulus, which becomes ing, we find that the total output power after reaching its max- broader as L becomes thinner. This means that as the crystal imum value becomes independent of θ , even though the out- p becomes thinner, the phase-matching becomes more relaxed. put intensity pattern changes. We can also see this through equation (24) that as L increases the sinc term become narrower. Figure 13 shows the down- conversion intensity pattern for three different values of the 5.3. Variation of type-I intensity with θp crystal thickness L for type-II phase matching condition at ◦ In this subsection, we study how the intensity of down- θp = 41.78 . Just as in the case of type-I phase matching, we converted photons changes as a function of θp for the type-I find that the overall effect of increasing L is just the tightening

16 J. Opt. 22 (2020) 083501 Tutorial

2.8 (a) (b) (c)

0.0 (mm) y

-2.8 1.0 (d)µ =28.64o (e)µ =28.74o (f) µ =28.84o p p p L=1 mm L=2 mm L=3 mm 0.4 Figure 12. Numerically simulated down-conversion intensity Intensity (a.u) 0.0 patterns for type-I phase matching at different crystals lengths for θp ◦ -2.8 0.0 2.8-2.80.0 28. 8-2. 0.0 2.8 = 28.74 . (a) L = 1 mm, (b) L = 2 mm, (c) L = 3 mm. x(mm) x(mm) x(mm)

Figure 14. Numerical plots of conditional probability of signal ◦ photon with fixed idler position for (a) θp = 28.64 , (b) (a) (b) (c) ◦ ◦ θp = 28.74 , (c) θp = 28.84 the corresponding one dimensional plots along the x direction are shown in (d)–(f) respectively.

6.1. Introduction to the orbital angular momentum of a photon It is known that Laguerre–Gaussian (LG) modes are the L=1 mm L=2 mm L=3 mm solutions to the paraxial Helmholtz equation. The LG modes form a family of orthogonal modes that have a well defined Figure 13. Numerically simulated down-conversion intensity orbital angular momentum. The amplitude of an LG mode − patterns for type-II phase matching at different crystals lengths for ilϕ ◦ has an azimuthal phase dependence e , where l is called θp = 41.78 . (a) L = 1 mm, (b) L = 2 mm, (c) L = 3 mm. the azimuthal mode index. Allen et al showed that an LG mode with index l possesses an orbital angular momentum of lℏ per photon [109]. Thus, within quantum mechanics, l has of the phase-matching condition which causes the two annuli the interpretation that a field with an LG amplitude distribu- to become narrower. tion carries an orbital angular momentum of lℏ per ℏω photon energy [109]. The index p represents the radial mode index and

5.5. Effect of θp on the position-correlation width decides how the field amplitude gets distributed in the radial direction. In this subsection, we evaluate the position-correlation width for type-I SPDC for various θp values. Using the 6.2. Derivation of the two-photon wavefunction expression given in equation (84), we numerically evaluate the conditional probabilities and plot them in figure 14. In order to The two photon wavefunction generated by SPDC process can get the correct scaling for the pixel size in numerical simula- be written in the in LG basis as XX tions, we use the fact that the position-correlation width of the , |ψ ⟩ = Cls ps |l ,p ⟩ |l ,p ⟩ , (100) signal (idler) photon for a fixed position of the idler (signal) tp li,pi s s s i i i photon at any plane is twice the pump beam size at that plane ls,ps li,pi (see [108]). In figure 14, we find that for the collinear case | , ⟩ the correlation area forms a circle but as the non-collinearity where ls ps s represents the state of a signal photon in the increases the correlation area no longer remains symmetric. LG mode having orbital angular mode index ls and the radial mode index ps. Using the momentum basis representation of Figures 14(a)–(c) show the plots of two-dimensional condi- ⟨ | ⟩ l LG mode, that is, q l,p = LGp(q) [110] in equation (76), we tional probablility at various θp values. Figures 14(d)–(f) are ls,ps the one-dimensional cuts along the x-direction. Using these write Cli,pi as [111, 112] one dimensional plots, we calculate the correlation width ˆ ˆ along the x direction and find it to be 847 µm, 581 µm, and ls,ps = 2 2 ( + ) Cli,pi A d qsd qiV qs qi 401 µm, respectively, at the three θp values.  ∗  ∗ × Φ( , ) ls ( ) li ( ) . qs qi LGps qs LGpi qi (101) 6. Two-photon wavefunction in the OAM basis Now using the polar coordinate in transverse momentum We have so far mostly discussed the two-photon wavefunc- space, that is qs = (qsx,qsy) = (ρscosϕs,ρssinϕs), qi = tions in the position and momentum bases. The other set (qix,qiy) = (ρicosϕi,ρisinϕi), dqs = ρsdρsdϕs and dqi = of bases that have more recently become very important for ρ ρ ϕ ls,ps id id i we can write Cli,pi as quantum information is the angular-position and OAM basis. ˆ ˆ ∞ ˆ ˆ There are several quantum information applications that have π ls,ps = (ρ ,ρ ,ϕ ,ϕ )Φ(ρ ,ρ ,ϕ ,ϕ ) Cli,pi V s i s i s i s i been proposed based on the OAM of photons. 0 −π

17 J. Opt. 22 (2020) 083501 Tutorial

 ∗  ∗ × ls (ρ ,ϕ ) li (ρ ,ϕ ) ρ ρ ρ ρ ϕ ϕ . LGps s s LGpi i i s id sd id sd i 7. Conclusion (102) SPDC is the most widely used process for generating photon- pairs entangled in various degrees of freedom such as position- 6.3. Calculation of the angular Schmidt spectrum momentum, energy-time, polarization, and orbital angular momentum. In SPDC, a pump photon interacts with a non- The probability of detecting signal and idler photons with linear optical crystal and splits into two entangled photons ℏ ℏ OAM values ls and li respectively, can be represented as called the signal and idler photons. The SPDC process has X∞ X∞ been studied extensively in the past few decades for various , ls = | ls ps |2. pump and crystal configurations, and the photon pairs pro- Pli Cli,pi (103) ps=0 pi=0 duced by SPDC have been used in numerous experimental studies on quantum entanglement and entanglement-based Now using equation (102), we can get real-world quantum-information applications. In this tutorial ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ article, we have presented a thorough study of phase matching X∞ X∞ ∞ π in BBO crystals and have thereby studied the entangled photon ls P = V(ρs,ρi,ϕs,ϕi) li generation by SPDC. We have discussed in details the effects 0 −π ps=0 pi=0 on phase-matching due to various crystal and pump paramet- × ∗(ρ′ ,ρ′,ϕ′ ,ϕ′)Φ(ρ ,ρ ,ϕ ,ϕ )Φ∗(ρ′ ,ρ′,ϕ′ ,ϕ′) V s i s i s i s i  s i s i ers such as the length of the crystal, the angle between the ∗ ′ ∗ ′ × ls (ρ ) ls (ρ ) li (ρ ) li (ρ ) optic axis of the crystal and the pump propagation direction, LGps s LGps s LGpi i LGpi i ′− ′− the pump incidence angle on the crystal surface, the refrac- × ils(ϕs ϕs)+ili(ϕi ϕi) e tion at the crystal surfaces, and the pump propagation direc- × ′ ′ ′ ′ ′ ′ ρsρsρiρidρsdρsdρidρidϕsdϕsdϕidϕi,(104) tion inside the crystal. We have then presented theoretical and experimental results illustrating how various phase matching ls (ρ ,ϕ ) = where we have used the relations LGps s s conditions affect the generation of photon pairs by SPDC. We ls ilsϕs LGp (ρs) e etc. Using the identity shown in [113], have also highlighted a few results that could be very useful P∞s l ∗l ′ 1 2 − ′2 p=0(LG)p(ρ)(LG)p (ρ ) = π δ(ρ ρ ), in the above equa- for experimental investigations with SPDC photons. Finally, tion, we get we have derived the two-photon wavefunction in the orbital ˆ ˆ ˆ ˆ ˆ ˆ angular momentum basis and have calculated the two-photon ∞ π 1 ∗ ′ ′ angular Schmidt spectrum. Pls = V(ρ ,ρ ,ϕ ,ϕ )V (ρ ,ρ ,ϕ ,ϕ ) li 2 s i s i s i s i 4π 0 −π ∗ ′ ′ ′− ′− × ils(ϕs ϕs)+ili(ϕi ϕi) Φ(ρs,ρi,ϕs,ϕi)Φ (ρs,ρi,ϕs,ϕi)e Acknowledgment × ′ ′ ρsρidρsdρidϕsdϕsdϕidϕi, (105) We acknowledge very fruitful discussions with P H Souto Ribeiro and C H Monken. We further acknowledge financial which essentially can be represented as support through the research Grant No. EMR/2015/001931 from the Science and Engineering Research Board (SERB), ˆ ˆ ∞ ˆ ˆ 1 π Department of Science and Technology, Government of India Pls = ρ ρ V(ρ ,ρ ,ϕ ,ϕ ) li 2 s i s i s i and through the research Grant No. DST/ICPS/QuST/Theme- 4π 0 −π 1/2019 from the Department of Science and Technology, Gov- 2 − − ernment of India. S K thanks the University Grant Commis- × ilsϕs iliϕi Φ(ρs,ρi,ϕs,ϕi)e dϕsdϕi dρsdρi. (106) sion (UGC), Government of India for financial support. NM acknowledges Indian Institute of Technology Kanpur for the Considering the conservation of orbital angular momentum in postdoctoral fellowship. − the SPDC process, we can show that ls = li = l for Gaussian pump beam [29]. In this situation, the corresponding probabil- ORCID iDs l ity P−l is the angular Schmidt spectrum Sl, which can be writ- ten as Suman Karan  https://orcid.org/0000-0003-0301-0020 Nilakantha Meher  https://orcid.org/0000-0002-5057-5350 ˆ ˆ ∞ ˆ ˆ 1 π Anand K Jha  https://orcid.org/0000-0002-5186-6264 = ρ ρ (ρ ,ρ ,ϕ ,ϕ ) Sl 2 s i V s i s i 4π 0 −π

2 References − − ×Φ(ρ ,ρ ,ϕ ,ϕ )e il(ϕs ϕi)dϕ dϕ dρ dρ . (107) s i s i s i s i [1] Klyshko D N 1988 Photons Nonlinear Optics (New York: Routledge) (https://doi.org/10.1201/9780203743508) θ [2] Louisell W H, Yariv A and Siegman A E 1961 Phys. Rev. The width of this spectrum depends on p and hence depends 124 1646–54 on the rotation of the crystal. This widthP is characterized by [3] Harris S E, Oshman M K and Byer R L 1967 Phys. Rev. Lett. 2 the angular Schmidt number Kα = 1/ l Sl [114]. 18 732–4

18 J. Opt. 22 (2020) 083501 Tutorial

[4] Magde D and Mahr H 1967 Phys. Rev. Lett. 18 905–7 [46] Walther P, Resch K J, Rudolph T, Schenck E, Weinfurter H, [5] Giallorenzi T G and Tang C L 1968 Phys. Rev. 166 Vedral V, Aspelmeyer M and Zeilinger A 2005 Nature 225–33 434 169 [6] Klyshko D N and Zel dovich B Y 1969 JETP Lett 9 40–3 [47] Tittel W and Weihs G 2001 Quantum Info. Comput. 1 3–56 [7] Klyshko D 1969 Sov. Phys. JETP 28 522 [48] Irvine W T M, Lamas Linares A, de Dood M J A and [8] Burnham D C and Weinberg D L 1970 Phys. Rev. Lett. Bouwmeester D 2004 Phys. Rev. Lett. 92 047902 25 84–7 [49] Tittel W, Brendel J, Zbinden H and Gisin N 2000 Phys. Rev. [9] Franken P A, Hill A E, Peters C W and Weinreich G 1961 Lett. 84 4737–40 Phys. Rev. Lett. 7 118–19 [50] Jennewein T, Simon C, Weihs G, Weinfurter H and [10] Giordmaine J A 1962 Phys. Rev. Lett. 8 19–20 Zeilinger A 2000 Phys. Rev. Lett. 84 4729–32 [11] Maker P D, Terhune R W, Nisenoff M and Savage C M 1962 [51] Sergienko A V, Atatüre M, Walton Z, Jaeger G, Saleh B E A Phys. Rev. Lett. 8 21–2 and Teich M C 1999 Phys. Rev. A 60 R2622 [12] Midwinter J E and Warner J 1965 British J. Appl. Phys. [52] Mattle K, Weinfurter H, Kwiat P G and Zeilinger A 1996 16 1135–42 Phys. Rev. Lett. 76 4656 [13] Kleinman D A 1962 Phys. Rev. 126 1977–9 [53] Sergienko A V, Shih Y H and Rubin M H 1995 J. Opt. Soc. [14] Nikogosyan D N 1991 Appl. Phys. A 52 359–68 Am. B 12 859–62 [15] Yariv A 1988 Quantum Electronics (New York: Wiley) [54] Dauler E, Jaeger G, Muller A, Migdall A and Sergienko A [16] Nye J F 1985 Physical Properties of Crystals: Their 1999 J. Res. Natl Inst. Stand. Technol. 104 1 Representation by Tensors and Matrices (Oxford: Oxford [55] Pittman T B, Shih Y H, Strekalov D V and Sergienko A V University Press) 1995 Phys. Rev. A 52 R3429 [17] Zernike F and Midwinter J E 1973 Applied Nonlinear Optics [56] Da Angelo M, Kim Y H, Kulik S P and Shih Y 2004 Phys. (New York: Wiley) Rev. Lett. 92 233601 [18] Kwiat P G, Steinberg A M and Chiao R Y 1993 Phys. Rev. A [57] Strekalov D V, Sergienko A V, Klyshko D N and Shih Y H 47 R2472 1995 Phys. Rev. Lett. 74 3600–3 [19] Strekalov D V, Pittman T B, Sergienko A V, Shih Y H and [58] Kira M, Koch S W, Smith R P, Hunter A E and Cundiff S T Kwiat P G 1996 Phys. Rev. A 54 R1 2011 Nat. Phys. 7 799 [20] Ali Khan I and Howell J C 2006 Phys. Rev. A 73 031801 [59] Boto A N, Kok P, Abrams D S, Braunstein S L, Williams C P [21] Ali-Khan I, Broadbent C J and Howell J C 2007 Phys. Rev. and Dowling J P 2000 Phys. Rev. Lett. Lett. 98 060503 85 2733–6 [22] Law C K and Eberly J H 2004 Phys. Rev. Lett. 92 127903 [60] D’Angelo M, Chekhova M V and Shih Y 2001 Phys. Rev. [23] Howell J C, Bennink R S, Bentley S J and Boyd R W 2004 Lett. 87 013602 Phys. Rev. Lett. 92 210403 [61] Matthews J C F, Zhou X Q, Cable H, Shadbolt P J, [24] Walborn S P and Monken C H 2007 Phys. Rev. A 76 Saunders D J, Durkin G A, Pryde G J and Oa Brien J L 062305 2016 Npj Quantum Information 2 16023 [25] Rubin M H, Klyshko D N, Shih Y H and Sergienko A V [62] Klishko D and Krindach D 1968 Sov. JETP 27 371 1994 Phys. Rev. A 50 5122–33 [63] Kwiat P G, Mattle K, Weinfurter H, Zeilinger A, [26] Kwiat P G, Eberhard P H, Steinberg A M and Chiao R Y Sergienko A V and Shih Y 1995 Phys. Rev. Lett. 1994 Phys. Rev. A 49 3209–20 75 4337–41 [27] Walborn S P, P´adua S and Monken C H 2003 Phys. Rev. A [64] Vanherzeele H and Chen C 1988 Appl. Opt. 27 2634–6 68 042313 [65] Cheng L, Bosenberg W and Tang C 1988 Appl. Phys. Lett. [28] Rarity J G and Tapster P R 1990 Phys. Rev. Lett. 64 2495–8 53 175–7 [29] Mair A, Vaziri A, Weihs G and Zeilinger A 2001 Nature [66] Akhmanov S, Fadeev V, Khokhlov R and Chunaev O 1967 412 313 ZhETF Pisma Redaktsiiu 6 575 [30] Leach J et al 2010 Science 329 662–5 [67] Giallorenzi T G and Tang C L 1968 Appl. Phys. Lett. [31] Bohm D 1952 Phys. Rev. 85 166–79 12 376–8 [32] Bohm D 1952 Phys. Rev. 85 180–93 [68] Kruger J S and Gleason T J 1970 J. Appl. Phys. 41 3903–4 [33] Bell J S 1964 Physics 1 195 [69] Byer R L, Harris S E, Kuizenga D J, Young J F and [34] Clauser J F, Horne M A, Shimony A and Holt R A 1969 Feigelson R S 1969 J. Appl. Phys. 40 444–5 Phys. Rev. Lett. 23 880–4 [70] Fiorentino M, Spillane S M, Beausoleil R G, Roberts T D, [35] Ou Z Y and Mandel L 1988 Phys. Rev. Lett. 61 50–3 Battle P and Munro M W 2007 Opt. Express 15 7479–88 [36] Shih Y H and Alley C O 1988 Phys. Rev. Lett. 61 2921–4 [71] Trojek P and Weinfurter H 2008 Appl. Phys. Lett. 92 211103 [37] Franson J D 1989 Phys. Rev. Lett. 62 2205–8 [72] Steinlechner F, Gilaberte M, Jofre M, Scheidl T, Torres J P, [38] Brendel J, Mohler E and Martienssen W 1991 Phys. Rev. Pruneri V and Ursin R 2014 J. Opt. Soc. Am. B Lett. 66 1142–5 31 2068–76 [39] Yarnall T, Abouraddy A, Saleh B and Teich M 2007 Phys. [73] Tanzilli S, De Riedmatten H, Tittel H, Zbinden H, Baldi P, Rev. Lett. 99 170408 De Micheli M, Ostrowsky D B and Gisin N 2001 Electron. [40] Leach J, Jack B, Romero J, Ritsch-Marte M, Boyd R W, Lett. 37 26–8 Jha A K, Barnett S M, Franke-Arnold S and Padgett M J [74] Martin A, Issautier A, Herrmann H, Sohler W, 2009 Opt. Express 17 8287–93 Ostrowsky D B, Alibart O and Tanzilli S 2010 New J. [41] Cinelli C, Barbieri M, Perris R, Mataloni P and Martini F D Phys. 12 103005 2005 Phys. Rev. Lett. 95 240405 [75] Leng H Y, Yu X Q, Gong Y X, Xu P, Xie Z D, Jin H, [42] Yang T, Zhang Q, Zhang J, Yin J, Zhao Z, Zukowski˙ M, Zhang C and Zhu S N 2011 Nat. Commun. 2 429 Chen Z B and Pan J W 2005 Phys. Rev. Lett. 95 240406 [76] Boeuf N, Branning D, Chaperot I, Dauler E, Guerin S, [43] Barreiro J T, Langford N K, Peters N A and Kwiat P G 2005 Jaeger G, Muller A and Migdall A 2000 Opt. Eng. Phys. Rev. Lett. 95 260501 39 1016–24 [44] Bouwmeester D, Pan J W, Mattle K, Eibl M, Weinfurter H [77] Monken C, Ribeiro P S and P´adua S 1998 Phys. Rev. A and Zeilinger A 1997 Nature 390 575 57 R2267 [45] Pittman T B, Jacobs B C and Franson J D 2002 Phys. Rev. A [78] Kurtsiefer C, Oberparleiter M and Weinfurter H 2001 J. Mod. 66 042303 Opt. 48 1997–2007

19 J. Opt. 22 (2020) 083501 Tutorial

[79] Molina-Terriza G, Minardi S, Deyanova Y, Osorio C I, [98] Milonni P W a and Eberly J H 2010 Laser Physics (New Hendrych M and Torres J P 2005 Phys. Rev. A 72 065802 York: Wiley) [80] Schneeloch J and Howell J C 2016 J. Opt. 18 053501 [99] New G 2011 Introduction to Nonlinear Optics (Cambridge: [81] Walborn S P, Monken C, P´adua S and Ribeiro P S 2010 Phys. Cambridge University Press) Rep. 495 87–139 [100] Yariv A V and Yeh P V 2003 Optical Waves in Crystals: [82] Rubin M H 1996 Phys. Rev. A 54 5349 Propagation and Control of Laser Radiation (New York: [83] Pittman T B, Strekalov D V, Klyshko D N, Rubin M H, Wiley) Sergienko A V and Shih Y H 1996 Phys. Rev. A [101] Hutchings D C, Sheik-Bahae M, Hagan D J and 53 2804–15 Van Stryland E W 1992 Opt. Quantum Electron. 24 1–30 [84] Klyshko D N, Chekhova M and Kulik S 2011 Physical [102] Peter E and Powers J W H 2017 Fundamentals of Nonlinear Foundations of Quantum Electronics (Singapore: World Optics (Boca Raton, FL: CRC Press) Scientific) [103] Eimerl D, Davis L, Velsko S, Graham E K and Zalkin A 1987 [85] Boyd R 2003 Nonlinear Optics (Edinburgh: Academic) J. Appl. Phys. 62 1968–83 [86] Bloembergen N 1980 J. Opt. Soc. Am. 70 1429–36 [104] Ghatak A K and Thyagarajan K 1989 Optical Electronics [87] Fowles G R 1975 Introduction to Modern Optics (New York: (Cambridge: Cambridge University Press) Dover) 2nd edn [105] Grayson T P and Barbosa G A 1994 Phys. Rev. A 49 2948–61 [88] Hong C K and Mandel L 1985 Phys. Rev. A 31 2409–18 [106] Joobeur A, Saleh B E A and Teich M C 1994 Phys. Rev. A [89] Kleinman D A 1968 Phys. Rev. 174 1027–41 50 3349–61 [90] Ou Z Y, Wang L J and Mandel L 1989 Phys. Rev. A [107] Ribeiro P H S, P´adua S, Machado da Silva J C and 40 1428–35 Barbosa G A 1994 Phys. Rev. A 49 4176–9 [91] Mandel L 1967 J. Opt. Soc. Am. 57 613–17 [108] Jha A K and Boyd R W 2010 Phys. Rev. A 81 013828 [92] Glauber R J 1963 Phys. Rev. 130 2529–39 [109] Allen L, Beijersbergen M W, Spreeuw R J C and [93] Loudon R 2000 The Quantum Theory of Light (Oxford: Woerdman J P 1992 Phys. Rev. A 45 8185–9 Oxford University Press) [110] Torres J P, Alexandrescu A and Torner L 2003 Phys. Rev. A [94] Grynberg G, Aspect A and Fabre C 2010 Introduction to 68 050301 Quantum Optics: From the Semi-Classical Approach to [111] Miatto F M, Yao A M and Barnett S M 2011 Phys. Rev. A Quantized Light (Cambridge: Cambridge University 83 033816 Press) [112] Yao A M 2011 New J. Phys. 13 053048 [95] Grice W P and Walmsley I A 1997 Phys. Rev. 56 1627–34 [113] Jha A K, Agarwal G S and Boyd R W 2011 Phys. Rev. A [96] Belinskii A and Klyshko D 1994 Sov. JETP 78 259–62 84 063847 [97] Monken C H, Ribeiro P H S and P´adua S 1998 Phys. Rev. A [114] Kulkarni G, Taneja L, Aarav S and Jha A K 2018 Phys. Rev. 57 3123–6 A 97 063846

20