Angle-Resolved Spectroscopy of Parametric Fluorescence

The parametric fluorescence from a nonlinear crystal forms a conical radiation pattern. We measure the angular and spectral distributions of parametric fluorescence in a beta-barium borate crystal pumped by a 405-nm diode laser employing angle-resolved imaging spectroscopy. The experimental angle-resolved spectra and the generation efficiency of parametric down conversion are compared with a plane-wave theoretical analysis. The parametric fluorescence is used as a broadband light source for the calibration of the instrument spectral response function in the wavelength range from 450 to 1000 nm.


I. INTRODUCTION
Signal-idler photon pairs generated by spontaneous parametric down-conversion (SPDC) have been used to address fundamental issues of quantum theory and have found application in quantum entanglement and quantum information processing and metrology [1][2][3][4]. The SPDC process, also known as parametric fluorescence or parametric scattering [5][6][7][8][9][10], is a second-order optical process in which a driving pump photon is scattered into signal-idler photon pairs subject to energy and momentum conservation. This spontaneous parametric emission can be described properly only by field quantization. Since its prediction and observation in the 1960s, parametric fluorescence has become a technique for measuring second-order nonlinear susceptibilities [10][11][12][13] and for developing tunable light sources via parametric oscillation or amplification processes. In 1969, Zeldovich and Klyshko [14] first proposed the use of parametric fluorescence (luminescence) as a nonclassical source of photon pairs. This description was experimentally verified by Burnham et al. in 1970 [15].
The possible wave vectors of the signal-idler photon pairs are determined by energy and momentum conservation, a constraint referred to as phase-matching, leading to highly directional parametric emission. The phase-matching condition frequently cannot be met for specific wavelengths of interest or practical applications owing to the limited tunability of inherent dispersion of nonlinear materials. However, it can be met by selecting polarization birefringent crystals with appropriate refractive indices or by designing waveguides or periodic structures of specific wavelengths. There are two major types of polarization phase-matching schemes for parametric downconversions: Type-I, where signal-idler photons have the same polarization (co-linearly polarized photons), and Type-II, where the signal-idler photons have orthogonal polarization (cross-linearly polarized photons). Both types of parametric processes have been used to generate photon pairs, sometimes referred to as biphoton states, which exhibit correlation/entanglement for variables including polarization, momentum, time, energy, and angular momentum.
When the phase-matching condition is met, the signal and idler radiation form a conical pattern independent of the intensity of the pump source. The angular distribution of parametric fluorescence is determined by the energy of the pump, signal, and idler waves, subject to the dispersion of the crystal and walk-off angles of these three waves. The magnitude of the second-order nonlinear susceptibility χ (2) is a typical selection criterion for parametric downconversion. Many uniaxial or biaxial nonlinear crystals have been used for parametric down conversion: for example, KD*P (potassium dideuterium phosphate, KD 2 PO 4 ), BBO (beta-barium borate, β − BaB 2 O 4 ), and LBO (lithium niobate, LiNbO 3 ). In this report, we use BBO, a negative uniaxial class 3m crystal characterized by a wide range of transparency over the ultraviolet (λ ≈ 200nm) to the infrared (λ ≈ 3500nm) portion of the spectrum [16]. BBO crystals have been widely studied for harmonic frequency generation, optical parametric oscillation, and generation of bi-photon states.

II. EXPERIMENTAL METHODS
We measure the angular distribution and photon flux of parametric fluorescence from a 3-mm thick BBO crystal. The BBO crystal is cut at an angle of θ m = 29 ± 0.5 • with respect to the optical axis. This cut angle θ m is chosen for the Type-I (e → o + o) degenerate parametric downconversion at λ = 810 nm with a pump λ p = 405 nm. The crystal is mounted on a three-axis rotary mount with the crystal's optical axis (OA) in the horizontal plane. The angle formed by the OA and the pump's propagation wave vector can be finely tuned by tilting the crystal to satisfy the phase-matching condition for various θ m near the crystal cut angle. We can thus adjust the pump and signal Poynting vectors from collinear to non-collinear and generate parametric fluorescence with varying conical emission angles.
The pump is a violet diode laser with a T EM 00 linearly polarized 2-mm 1/e 2 diameter output beam at a wavelength λ p = 405 nm (CNI Laser MLL-III-405). The pump beam is focused on the crystal through a lens (L1) with a focal length of 500 mm. Lens L1 and the objective are positioned to form a telescope such that the residual pump beam is collimated with a reduced beam radius below 100µm. By passing the pump beam through a pair of a half-wave plate (HWP) and a Glan-Taylor polarizer (P1), we can vary the incident pump intensity by rotating the HWP while maintaining the degree of linear polarization better than 99.9%.
The angle-resolved images (Fig. 3) and spectra (Fig. 6) of parametric fluorescence are measured by a Fourier transform optical system, including a 20× long-working-distance objective and an imaging spectrometer as shown in Fig. 1. The BBO crystal is positioned at the focal plane of the objective lens (effective focal length f o = 10 mm). The parametric fluorescence with an amplitude distribution F (x, y) at the crystal is collected by a 20× microscope objective with a 10-mm effective focal length (numerical aperture N.A. = 0.26). The back focal plane of the objective is the Fourier transform plane with coordinates (u, v) = (f o × sin(θ x ), f o × sin(θ y )). The collection angle is within ±15 • , limited by the objective. The objective lens converges parallel rays emanating from the crystal to the back focal plane of the objective. In this plane, the fluorescence image in the crystal is transformed into a far-field image in spatial frequency that is related to the emission angle as described above. The spatial intensity distribution of parametric fluorescence in the back focal plane of the objective lens thus corresponds to the angular distribution of radiation. This Fourier transform plane is placed at the front focal plane of lens L2 (focal length f = 100 mm). Lenses L2 and L3 are identical and separated by a distance of 2f, and they relay the Fourier transformed images to the entrance plane and then onto the charge-couple device (CCD) through the zero-order diffraction off the grating of the imaging spectrometer (PI-Acton SpectroPro 2750i, focal length 750 mm). In this way, we measure the angular distribution of parametric fluorescence. When lens L2 is removed, we project the real-space spatial intensity distribution of parametric fluorescence from the BBO crystal onto the CCD. The three-wave parametric processes are calculated according to the conservation of energy and momentum, commonly referred to as phase matching. The angle-resolved spectra of the parametric fluorescence are consistent with the tuning curves calculated for the phase-matching condition under a plane-wave approximation.
The energy conservation condition is expressed as where ω p is the frequency of the incident pump wave and ω s and ω i are the frequencies of the signal and idler waves.
The momentum conservation condition can be expressed as where k p , k s , and k i are the pump, signal, and idler wave vectors, respectively. For Type-I downconversion in a BBO, the signal and idler labels are arbitrary. In the case of degenerate down conversion, k s = k i , and Eq. (2) reduces to where n p and n s are the indices of refraction of the pump and signal, and θ s is the angle formed by the propagation directions of the signal and pump waves inside the crystal.
We use a BBO crystal cut at an angle of θ m = 29 ± 0.5 • , optimized for Type-I paramet- The extraordinary index of refraction,ñ, depends on the phase-matching angle θ m and follows the relationship:ñ In the parametric process, a pump wave of wavelength λ p creates signal waves at λ s , and angles θ s , subject to energy conservation (Eq. (1)) and momentum conservation (Eq. (2)). In the Type-I e-o-o case, n s = n o (λ s ) and n p =ñ(θ m , λ p ) (Eq. 4). Here the labeling of signal and idler waves is arbitrary (θ s = θ i ). A continuum of phase-matching functions Φ(λ s , θ s ) for parametric fluorescence can be obtained using the aforementioned equations and indices of refraction n o (λ s ) and n e (λ s ).
Indices of refraction of wavelengths ranging from 0.3 µm to 5 µm are extracted from "NIST Noncollinear Phase Matching in Uniaxial and Biaxial Crystals Program" as described in Ref. [17].
We calculate the phase matching functions (tuning curves) for down-converted signal/idler waves ranging from 430 to 1000 nm for a pump wave λ p = 405 nm.

B. Angle-Resolved Imaging
We adopt the plane-wave analysis developed in Refs. [7,18,19] to determine the angular distribution of parametric fluorescence. The effects of a finite pump beam size have also been considered in, for example, Refs. [20,21]. Under a plane-wave approximation, the parametric We can determine the angular distribution of parametric fluorescence using the following phasematching function for a finite crystal length L and a pump Gaussian beam profile with a 1/e 2 radius W [17,22,23]: The mismatch wave vector, ∆k, is decomposed into longitudinal (ẑ k p ) and transverse (in x-y plane) parts: ∆k z and κ = ∆k x + ∆k y . The phase-matching tolerances can be considered in terms of the angular spread (∆θ s ) and spectral bandwidth (∆λ s ), defined as the full-width-athalf-maximum (FWHM) for the above function. For a pump wave with a focal radius W ≈ 50µm, only the tolerances from the sinc 2 part can be measured in our optical system. For this situation, considering a Taylor series expansion of ∆k z near the perfect phase-matching point (∆k z = 0), we can analytically deduce the angular spread (∆θ FWHM ) and spectral bandwidth (∆λ FWHM ) for the degenerate case (k s = k i ): and where ∆k z = |k p − 2 k s cos(θ s )|, k s = n s ω s /c = 2π n s /λ s , and θ s 1.
The parametric fluorescence signal of angle-resolved images at λ = 810 nm are shown in Fig. 3.
These false-color images, taken through a 1-nm band-pass filter, represent the angular intensity distributions of parametric fluorescence at λ = 810 ± 0.5 nm for the phase-matching angle θ m = 28.6 • , 28.8 • , 29.1 • , and 29.4 • . The BBO crystal is cut at the designed angle with about 1 • tolerance.
To determine the phase-matching angle θ m with better precision, we first set the phase-matching angle for the collinear case by comparing the simulated and experimental angular distributions.
We then deduce the phase-matching angle for the non-collinear case from the tilting angle of the crystal relative to that for the collinear case. The conical signal angle (θ s ) of degenerate parametric fluorescence at λ = 810 nm increases with θ m . In Fig. 4, we plot the angular spread (∆θ FWMH ) as a function of the inverse of the signal angle (1/θ s ). The angular spread decreases with θ s for small angles when sin(θ s ) ≈ θ s . We attribute the discrepancy between experimental data and Eq.
(6) to a limited experimental angular resolution (≈ 2 mrad), finite pump beam size and spatial coherence, and the birefrigent walk-off.

C. Angle-Resolved Spectroscopy
The parametric fluorescence flux per unit frequency is where d ef f is the effective second-order nonlinear coefficient, L the interacting crystal length,

D. Fluorescence Photon Flux
The integrated parametric fluorescence photon flux can be obtained by the integrating over ξ and κ s in Eq. (7). For values of θ m or ω s such that the parametric fluorescence cone has a radius sufficiently large with negligible emission at the cone center, the resultant integrated fluorescence flux is [18] The efficiency η s ≡ N s /N p is a coefficient depending largely on the material properties such as the second-order nonlinear coefficient, interacting crystal length, and index of refraction for the pump wave. Assuming a bandwidth ∆λ = 1 nm and L = 3 mm, the efficiency coefficient η s = 1.3 × 10 −10 for the degenerate parametric fluorescence at λ s = 810 nm under λ p = 405 nm. Specifically, we evaluate η s for θ m = 29.12 • (Fig. 6g). We integrate the photon flux of simulated angle-resolved spectra for λ s = 809.5 → 810.5 nm, θ s = 0 → 7.5 • , and φ = 0 → 2π. The total parametric fluorescence photon flux, including both degenerate signal and idler waves, is 2 N s ≈ 4.2 × 10 7 /s .  Table I.

E. Instrument Spectral Response Function
Parametric fluorescence spectra can also be used to calibrate the instrument spectral response function (ISRF) of the imaging spectroscopy system. According to Eq. (8), which is valid for noncollinear cases, we can deduce a parameter S ≡ N s × λ 4 s λ 2 i = 2 (2π) 4h c d ef f 2 L N p dλ s /( 0 n p 2 ) [18]. S is a wavelength-independent constant for a given pump wavelength and geometry. We define a generalized spectral function, S(λ s ) ≡ N s (λ s ) λ 4 s λ 2 i , for both calculated and experimental angle-resolved spectra. The calculated S sim (λ s ) exhibit less than 1% variation between λ = 460 and 1000 nm. Experimentally, S * exp (λ s ) = N * s (λ) λ 4 s λ 2 i can be determined from the integration of an angle-resolved spectrum N * s (λ, θ s ) over θ s . S exp (λ) represents the relative ISRF of the imaging spectroscopy system, including the optical components, grating, and CCD camera along the fluo-rescence collection optical path. The value of the ISRF at a fixed wavelength can then be used to determine the absolute ISRF across the parametric fluorescence wavelength range. The effective efficiency, defined as (# of photo-generated electrons / # of photons), is ≈20% at λ = 810 nm in our experiments. S * exp (λ) and S sim (λ) for phase-matching angles θ m = 29.1 • and 29.4 • are shown in Fig. 7. The stray scattered pump laser signal becomes increasingly difficult to subtract from these angle-resolved spectra, leading to a distorted S * exp (λ). We thus use S * exp (λ) at θ m = 29.1 • to deduce the ISRF of our optical setup shown in Fig. 1.

Acknowledgments
We thank Brage Golding and John A. McGuire for discussions. This work was supported by grant DMR-0955944 from the National Science Foundation and by internal Strategic Initiative Projects of the College of Natural Science at Michigan State University.

IV. APPENDIX: CALCULATION OF FLUORESCENCE FLUX
Here we describe the integration over ξ of Eq. (7) for the calculation of the angular distribution of fluorescence flux here. The mismatch wave vector ∆k = k s + k i − k p can be decomposed into longitudinal (∆k zẑ ) and transverse (κ) parts: Here k s = n s ω s /c, k i = n i ω i /c, and k p = n p ω p /c are the wave numbers for the signal, idler, and pump, respectively. κ s (κ i ) is the transverse wave vector of the signal (idler) wave. The phase-matching function Φ in Eq. (5) is a function of three variables: ω s , κ s and κ i . Considering energy conservation ω p = ω s + ω i , we can carry out the integration over ξ for two independent variables, ω s and κ s . Using ξ = ξ s + ξ i = (κ s + κ i ) W , we rewrite Eq. (7) as N s (ω s , κ s ) =h d ef f 2 ω s ω i ω p L 2 N p 8π 4 c 3 0 n s n i n p dω s d 2 κ s d 2 ξ i exp − 1 2 |ξ s + ξ i | 2 · sinc 2 1 2 L ∆k z We further simplify the numerical integration by (a) considering that the sinc 2 term is a constant for a given set of ω s and κ s , and (b) applying a saddle-point approximation. Under a pump wave vector alongẑ, the phase-matching condition is met mostly for k s + k i = 0; i.e., ϕ ≈ 0. The Gaussian term of the integrand can thus be separated and integrated with a saddle-point approximation for ϕ ≈ 0 and ξ s ≈ ξ i : The integrand is a maximum at ξ s = ξ i , where pairs of signal and idler photons are emitted in approximately opposite conical directions. Applying the above two approximations, we obtain where d 2 κ s = κ s dκ s dφ, φ is the azimuthal angle.
It can be expressed as N s (λ s , θ s ), a function of experimentally measurable signal angle θ s and wavelength λ s by considering that κ s = k s sin θ s and ω s = 2πc/λ s : The equation for N s (λ s , θ s ) above, together with ω p = ω s + ω i and k p = k s + k i , is used in the numerical integration to obtain the theoretical angular distribution of parametric fluorescence shown in Fig. 6.
[       7)). The black and red curves are S sim or S * exp for θ m = 29.4 • and 29.1 • , respectively. The value of S sim is a constant with 1% standard deviation across wavelengths from 500 to 1000 nm, validating the calculated angular fluorescence spectra and Eq. (8). S * exp (λ) = N * s (λ) λ s 4 λ i 2 , where N * s (λ) is the integration over θ s ≈ −15 to 15 • of the experimental imaging spectra N * s (λ s , θ s ). S * exp (λ) represents a relative instrument spectral response function (ISRF) of the optical spectroscopy system, including optical filters and a Glan-Thompson polarizer along the path of the fluorescence , a liquid-nitrogencooled CCD (PI-Acton Spec-10:400BR), and a 300g/mm plane ruled reflectance grating with 1000-nm blaze wavelength (PI-Acton 750-1-030-1). The polarization of the fluorescence is vertically polarized. The absolution ISRF (or collection efficiency) can be deduced by calibrating S * exp (λ) at a fixed wavelength such as λ = 810 nm in our experiments.