1. Introduction

Radiometry is important to a wide range of applications requiring accurate or absolute measurements of radiation spectral responsivity. These include lighting, optical sources, detectors, optical components, spectrometers, and fluorescence microscopy. Blackbody sources and electron storage rings (synchrotron radiation) are accepted primary radiation source standards in radiometry because their spectral characteristics can be calculated from fundamental physical processes and parameters [1–4]. Synchrotron radiation sources are limited to large national laboratories such as BESSY [5]; the radiometric standard based on high-temperature blackbodies and cryogenic radiometers at NIST [3, 6] is not available for in situ calibration in users’ laboratories. Spontaneous parametric down-conversion (SPDC), in which a single photon is down-converted into a correlated photon pair is an alternative primary source for use in radiometry [2, 7–9].

SPDC, also known as parametric fluorescence or parametric scattering, 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 [10–15]. It offers a convenient mechanism of measuring detector efficiency in the photon counting regime where blackbodies are not suitable. Measurements made with such a SPDC sources are intrinsically absolute, not relying on an externally calibrated radiometric standard. By replacing the specimen/samples with a SPDC source at the focal plane of a microscope spectroscopy system, one can correct the spectral response including transmittance and geometric losses due to all optical components used in the set-up. One can further identify or correct transmission losses due to reflection and absorption, and geometric losses due to limiting apertures and detector area in situ. However, the absolute spectral distribution of a SPDC source has not been characterized as comprehensively as in the case of blackbodies and electron storage rings.

Here we experimentally measure angular spectral distributions of parametric fluorescence and describe a compact set-up to obtain the absolute spectral efficiency of an imaging spectrometer equipped with a CCD camera in the visible wavelength range (450 nm to 1000 nm). In a focal area of 100 μm² in a BBO (beta-barium borate) crystal pumped by a 100 mW 405-nm pump laser, the parametric fluorescence spectral flux density at 810 nm is comparable to that from a T = 1500 K ideal blackbody source. The conical parametric fluorescence is directional and can be focused or relayed through a complex microscopic and spectroscopic system. The compact set-up is portable and can in principle be calibrated against a NIST standard radiometer. The wavelength limit is largely imposed by the responsivity of the silicon-based CCD camera. Our method can also be extended to the near infrared wavelength range (900 nm to 1700 nm) for a spectroscopy or microscopy system equipped with an InGaAs array.

Although a detailed analysis of the angle-resolved parametric fluorescence spectrum is presented here, calibration of the instrument spectral response function can be performed without a full angle-resolved spectrum as long as the scattered pump laser light is rejected properly. Using the Fourier transform imaging method and simple models described here, one can select suitable lenses and fibers, and BBO crystals with matching dimensions or numerical apertures to optimize the overall system efficiency. The method here can be also used to monitor angular distributions of SPDC and optimize the photon pair generation rates and coupling efficiency into single-channel avalanche photodiodes or photon counters via fibers.

The use of SPDC as a primary source in radiometry relies on the fact that the fundamental physical processes of SPDC are well known [4]. Since its prediction and observation in the 1960s, parametric fluorescence has become a technique for measuring second-order nonlinear susceptibilities [15–18] and for developing tunable light sources via parametric oscillation or amplification processes. In 1969, Zeldovich and Klyshko first proposed the use of parametric fluorescence (luminescence) as a nonclassical source of photon pairs in [19]. This description was experimentally verified by Burnham et al. in [20]. Signal–idler photon pairs generated by SPDC have been used to address fundamental issues of quantum theory and have found applications in quantum entanglement and quantum information processing [21–24].

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 can be met by selecting birefringent crystals with appropriate refractive indices or by designing waveguides or periodic structures of specific wavelengths. There are two major types of phase-matching schemes for parametric down-conversions: 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, KD₂PO₄) [25, 26], BBO (beta-barium borate, β-BaB₂O₄) [27, 28], and LBO (lithium niobate, LiNbO₃) [13]. Among these, LBO and BBO crystals have been studied for harmonic frequency generation, optical parametric oscillation, and generation of bi-photon states. Theoretical modeling and numerical calculation of phase-matching in these nonlinear crystals are well established [29–31].

In this article, we first present experimental angle-resolved images and spectra of parametric fluorescence (SPDC) from a BBO crystal which is widely used for the generation of entangled biphoton states [21–24, 32]. The experimental angle-resolved spectra and the generation efficiency of parametric down conversion agree with a plane-wave theoretical analysis. Finally, we describe procedures to use parametric fluorescence as a broadband light source for the calibration of the instrument spectral response function in the wavelength range from 450 to 1000 nm.

2. 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 as illustrated in Fig. 1. This cut angle θ_m is chosen for the Type-I (e → o + o) degenerate parametric down-conversion 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 when the parametric fluorescence signal is maximized. 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.

 

Fig. 1 Schematic diagram of the crystal and laboratory frame coordinates for parametric down-conversion in a BBO crystal. The phase-matching angle θ_m is defined as the angle formed by the crystal optical axis (z′) and the pump wave vector (z). The angles θ′_s and θ_s are, respectively, internal and external angles formed by the signal and pump wave vectors. Here the incident pump wave is horizontally polarized, leading to a vertically polarized down-converted signal and idler waves.

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The pump is a violet diode laser with a TEM₀₀ linearly polarized 2-mm 1/e² 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 100 μm. 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 and spectra 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. 2. 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.

 

Fig. 2 Schematic diagram of the experimental setup. The transmitted and scattered pump photons are rejected by a miniature beam blocker, a thin-film notch filter, and long-pass filters. The angular and spectral distributions of conical parametric fluorescence are measured by an imaging spectrometer through a Fourier transform optical system. L1, L2 and L3 are convergent lenses with focal length f = 500, 100, and 100 mm, respectively; Obj is an objective (20× N.A.=0.26) with an effective focal length of 10 mm (Mitutoyo Plan Apo infinity-corrected long-working-distance objective); P1 and P2 are Glan–Taylor and Glan–Thompson polarizers; M1 and M2 are silver mirrors; HWP is a half-wave plate for λ = 405 nm; BB is a miniature pump beam blocker; NF is a notch filter (Semrock 405-nm StopLine single-notch filter); and LP represents longpass filters (a Semrock 409-nm blocking edge BrightLine long-pass filter and a Schott GG435 glass filter). When L2 is removed, a real-space fluorescence image is formed at the entrance of the spectrometer with an imaging magnification of 10×. Examples of real-space and angle-resolved fluorescence images are shown with actual dimensions.

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The fluorescence image is recorded by a CCD positioned in a conjugate imaging plane of the Fourier plane. The resultant intensity distribution is related to the Fourier transform of the intensity of the parametric fluorescence I(x, y) = |F(x, y)|², where F(x, y) is the electromagnetic field distribution at the crystal. By projecting this far-field image through the entrance slit and the first-order diffraction of a 300 lines/mm grating, we obtain angle-resolved spectra as an image by taking the spectral dispersion of the parametric fluorescence as a function of angle. The spectral resolution of 0.1 nm is determined by the dispersion of the grating and the width of the entrance slit (≈ 100μm). The spatial and angular resolutions of the system are approximately 2 μm and 2 mrad, respectively, limited by the pixel size of the liquid-nitrogen-cooled CCD camera.

In a parametric scattering process, only about one out of 10¹⁰ incident photons is parametrically down-converted. It is essential to prevent the transmitted and scattered pump photons from entering the spectrometer. For Type-I phase matching in a negative uniaxial BBO crystal (e-o-o case), the polarization of the high-frequency pump (e-wave) is orthogonal to the polarization of the signal and idler (o-waves). Thus, the transmitted and scattered pump photons can be suppressed by approximately 4–5 orders of magnitude by a pair of polarizers (P1 and P2) with orthogonal polarization orientations for pump and signal/idler waves, respectively. The pump photons are further rejected by a thin-film notch filter (Semrock 405-nm StopLine single-notch filter) and two longpass filters (a Semrock 409-nm blocking edge BrightLine long-pass filter and a Schott GG435 glass filter). The filters are arranged in the sequence shown in Fig. 2 to suppress fluorescence from filters induced by the transmitted violet pump laser. The combination of polarizers and filters allows for a rejection of pump photons by approximately a factor of 10¹⁰. This rejection ratio of 10¹⁰ can be further improved above 10¹⁴ by a miniature beam blocker made of a ∼ 0.5 mm-diameter silver-paste dot on a microscope cover positioned in front of the notch filter (NF)/objective. Depending on the signal wavelength, the measured angle-resolved spectra may still contain residual transmitted and scattered pump and background fluorescence from filters. We measure such a background emission spectrum in the absence of Type-I e-o-o parametric fluorescence by rotating the BBO crystal 90 degrees azimuthally.

3. Experimental results and modeling

3.1. Phase matching

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

The momentum conservation condition can be expressed as

Down-converted signal/idler photons are co-linearly polarized but orthogonal to the polarization of the pump wave. The wavelengths and wave vectors of the parametric fluorescence are determined by the phase-matching angle θ_m, the angle formed by the optical axis of the crystal (z′-axis), and the wave vector of the pump wave (z-axis) as shown in Fig. 1. By tilting the crystal, we can vary the phase-matching condition from collinear to non-collinear, leading to a conical angle up to 5 degrees for degenerate parametric fluorescence near 810 nm.

For Type-I phase matching, the incident pump photons are subject to the extraordinary index of refraction ñ(θ_m), while the down-conversion photons are subject to the ordinary index of refraction. The extraordinary index of refraction, ñ, depends on the phase-matching angle θ_m and follows the relationship:

In the parametric process, a pump wave of wavelength λ_p creates signal waves at λ_s, and angles θ_s. 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 [29]. 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.

3.2. Angle-resolved imaging

We adopt the plane-wave analysis developed in [12,30,33] to determine the angular distribution of parametric fluorescence. The effects of a finite pump beam size have also been considered, for example, in [34, 35]. Under a plane-wave approximation, the parametric fluorescence forms a conical angular distribution. The diameter and axis of the conical emission are determined by the wavelengths of the pump, signal, and idler waves, the Poynting vector walk-off angles of the three interacting waves, and the dispersion of the nonlinear crystal. In such a parametric process, the angular spread of the conical emission is determined by the conservation of transverse and longitudinal momenta of interacting waves. The transverse momentum induced by focusing the pump into the crystal contributes to a finite angular spread of the cones. In our experiments, we use a lens with a long focal length (f = 500 mm), resulting in a focal spot with a 1/e² radius larger than 100 μm. Thus, for the experiments reported here, the effects of walk-off and finite pump beam size are negligible compared with the spectral and angular resolution of the optical system.

We can determine the angular distribution of parametric fluorescence using the following phase-matching function for a finite crystal length L and a pump Gaussian beam profile with a 1/e² radius W[29, 36, 37]:

The mismatch wave vector, Δk, is decomposed into longitudinal (ẑ || k_p) and transverse (in x–y plane) parts: Δk_z and $\kappa =\sqrt{\mathrm{\Delta }{k}_x^2+\mathrm{\Delta }{k}_y^2}$. The phase-matching tolerances can be considered in terms of the angular spread (Δθ_s) and spectral bandwidth (Δλ_s), defined as the full-width-at-half-maximum (FWHM) for the above function. For a pump wave with a focal radius W ≈ 50μm, only the tolerances from the sinc² part can be measured in our optical system. For this situation, considering a Taylor series expansion of Δk_z near the perfect phase-matching point (Δk_z = 0), we can analytically deduce the angular spread (Δθ_FWHM) and spectral bandwidth (Δλ_FWHM) for the degenerate case (k_s = k_i):

The angle-resolved images of parametric fluorescence 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 (Δθ_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. (5) to a limited experimental angular resolution (≈ 2 mrad), finite pump beam size and spatial coherence, and the birefrigent walk-off.

 

Fig. 3 Angular intensity distribution of parametric fluorescence at 810 nm. (a)–(d) Angle-resolved images of parametric fluorescence for λ = 810 ± 0.5 nm and θ_m as indicated. The external angle θ_s is formed by the pump and signal wave vectors in air [see also Fig. 1]. The color palette represents the intensity of the single radiation. Parametric fluorescence is spectrally filtered through a 1-nm bandpass filter with central wavelength λ = 800 nm. The phase-matching angle is adjusted by tilting the BBO crystal with respect to the pump wave vector. The collinear phase-matching angle is set to θ_m = 28.6° according to a theoretical calculation using indices of refraction given in [29]. (e)–(g) Experimental (blue dashed line) and theoretical (red solid line) cross-sections. The non-collinear phase-matching angle relative to the collinear one can be determined experimentally by measuring the tilting angle of the crystal surface with respect to the pump wave vector.

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Fig. 4 Angular spreads Δθ_FWHMof parametric fluorescence at 810 nm. Δθ_FWHM is plotted as a function of the inverse of the signal angle (1/θ_s). Experimental data are represented with error bars as solid circles. The red solid line is the theoretical curve according to Eq. 5, while the dashed line is the theoretical curve including a finite angular resolution of 2 mrad. Selected experimental angular intensity profiles for A (θ_s = 1.3° = 0.023 rad), B (2.8° = 0.049 rad), and C (3.7° = 0.065 rad) are shown in the inset.

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3.3. Angle-resolved spectroscopy

The parametric fluorescence flux per unit frequency is

Experimental angle-resolved spectra $N_s^{*}\left({\lambda }_s,{\theta }_s\right)$ and corresponding calculated N_s(λ_s, θ_s) are shown in Figs. 5 (a)–(d). The tuning curves, corresponding to the perfect phase-matching condition, are shown as white dashed lines on the experimental spectra. The experimental parameters are N_p = 1.63 × 10¹⁷/s (P_p = 80 mW), L = 3 mm, and W² = 60 μm ×30 μm. The indices of refraction n_s, n_i, and n_p are evaluated using the database in [29] (n_s,i ≈ 1.66 at 810 nm). The effective nonlinear coefficient of a BBO crystal can be deduced from its d-matrix using d_eff = d₃₁ sin(θ_m + ρ) − d₂₂ cos(θ_m + ρ) sin(3ϕ) for Type-I phase matching. θ_m is the phase-matching angle, ϕ the azimuthal angle, and ρ the birefringent walk-off angle. We use d_eff =1.75 pm/V for λ_p = 405 nm adopting from [18, 24, 38].

 

Fig. 5 Angle-resolved spectra of parametric fluorescence. (a)–(d) Experimental angle-resolved parametric fluorescence images (left panel) and spectra (right panel) for a phase-matching angle θ_m = 28.6°, 28.8°, 29.1°, and 29.4°, respectively. (e)–(h) Theoretical fluorescence flux calculated according to Eq. (6). The tuning curves for the perfect phase-matching condition are indicated by the white dashed lines on experimental imaging spectra. The color palette represents the calculated photon flux with Δλ_s = 1 nm and Δθ_s = 0.5 mrad on a logarithmic scale.

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Computationally, the angular distributions of parametric fluorescence are calculated for signal wavelengths from 430 to 1000 nm with step size δλ = 1 nm and δθ_s = 0.03° ≈ 0.5 mrad. Selected calculated angular distributions of fluorescence flux per 1 nm are shown in Figs. 5(e)–5(h) for θ_m = 28.6°, 28.8°, 29.1°, 29.4°. The shortest signal wavelength that appears in the simulation is ≈ 440 nm, limited by the availability of refractive indices between λ = 0.3 μm and 5 μm [31]. Experimentally, parametric fluorescence with a wavelength as short as 431 nm can be observed near θ_s = 0.

The experimental angle-resolved parametric fluorescence images are shown in the left panel of Fig. 5. These images are taken through a notch filter and longpass filters (cutoff wavelength of 420 nm) as shown in Fig. 2. Experimental angle-resolved spectra are acquired by spectrally resolving the parametric fluorescence across the fluorescence cone center through the entrance slit with an opening of 100 μm, corresponding to Δθ_x ≈ 0.6°. Angle-resolved fluorescence images [Figs. 5(a)–5(d), left panel] are taken for the zero-order diffraction of a holographic grating with 1800 lines/mm, while angle-resolved spectra (right panel) are dispersed by a grating with 300 lines/mm and a blaze wavelength of 1000 nm. The angular resolution is about 2 mrad, while the spectral resolution is about 0.1 nm. Residual stray or scattered pump laser light can be measured by rotating the BBO crystal 90° azimuthally. Such background ’noise’ is subtracted. Note that the measured CCD intensity is subject to the nonuniform spectral sensitivity and collection efficiency of the optical system including the effects of lens coating, optical filters, and gratings, and the spectral response of the CCD camera. The spectra branches out from θ_s = 0 at λ ≈ 433 nm. The secondary weak arcs in the wavelength range above 870 nm outside of the expected parametric fluorescence peaks are due to the second-order diffraction of the grating for the fluorescence from approximately 435 nm to 500 nm. The inner arcs branching from 435 nm and closing near 530 nm are due to parametric fluorescence from a Type-II parametric process (e → o + e). The turning curves for such Type-II parametric fluorescence are indicated by black dashed-doted lines in experimental spectral images.

3.4. Fluorescence photon flux

The integrated parametric fluorescence photon flux can be obtained by the integrating over ξ and κ_s in Eq. (6) [30]. 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 (i.e. non-collinear casese), the resultant integrated fluorescence flux is

The wavelength of parametric fluorescence generated here ranges from ≈ 430 nm to above 1000 nm. The collection efficiency and spectral response of the optical systems could vary more than an order of magnitude in such a broad wavelength range. Therefore we consider the degenerate parametric fluorescence at λ = 810 nm to compare the experimentally determined fluorescence photon flux with the theoretically integrated photon flux [Eq. (7)].

The integrated degenerate parametric fluorescence flux at λ_s = 810 nm as a function of the incident pump flux are shown in Fig. 6. First, we determine the overall collection efficiency and the spectral response of the imaging and spectroscopy system by passing a laser beam of λ = 810 nm with a known photon flux through the same optical path as the parametric fluorescence. We then integrate the signal in angle-resolved images as shown in Fig. 3. Taking into account the system response at λ = 810 nm, we can determine the absolute fluorescence flux experimentally within roughly 20% error. The relative fluorescence fluxes, determined by the linearity of the CCD camera, is within 1% error. The absolute value of the pump flux is known to within roughly 10% error. The relative pump fluxes, as varied by a combination of a half-wave plate and a polarizer and limited by the linearity of the power meter, are known within a few percent. Thus we can investigate the pump flux (power) dependence of parametric fluorescence flux with precision. Fluorescence signal flux is linearly proportional to the pump flux over two order of magnitude, confirming that the dominant signal is spontaneous parametric fluorescence as described by Eq. (7). The slopes correspond to the efficiency coefficients. Considering both signal and idler fluorescence, we determined η = 2η_s to be approximately 2.8 × 10⁻¹⁰. Experimental and theoretical values of η = 2η_s for selected phase-matching angles are listed in Table 1.

 

Fig. 6 Parametric fluorescence flux at 810nm. Integrated degenerate parametric fluorescence flux at λ_s = 810 nm as a function of the incident pump flux for the collinear case and three phase-matching angles θ_m of the angle-resolved images shown in Fig. 3. Signal flux is linearly proportional to the pump flux over two order of magnitude, confirming that the dominant signal is spontaneous parametric fluorescence as described by Eq. (7). The slopes η = N_s/N_p are 1.2 × 10⁻¹⁰ for θ_m = 28.6°, 2.6 × 10⁻¹⁰ for θ_m = 28.8°, and 2.8 × 10⁻¹⁰ for θ_m = 29.1° and 29.4°.

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Table 1. Parametric fluorescence efficiency η = 2N_s/N_p for 810 ± 0.5 nm

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3.5. Calculation of fluorescence flux

In this section we describe the integration over ξ of Eq. (6) for the calculation of the angular distribution of fluorescence flux here. The mismatch wave vector Δk = k′_s + k′_i − k′_p can be decomposed into longitudinal (Δk_zẑ) and transverse (κ) parts:

N_s(ω_s, κ_s, ϕ) is isotropic in ϕ. 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 sinθ_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. 5.

4. 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. (7), which is valid for non-collinear cases, we can deduce a parameter $S\equiv N_s\times {\lambda }_s^4{\lambda }_i^2=2{\left(2\pi \right)}^4\hslash c{d_{\mathit{eff}}}^{2}L{N}_{p}d{\lambda }_s/\left({\epsilon }_0{n_p}^2\right)$[30]. S is a wavelength-independent constant for a given pump wavelength and geometry. We define a generalized spectral function, $S\left({\lambda }_s\right)\equiv N_s\left({\lambda }_s\right){\lambda }_s^4{\lambda }_i^2$, 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, $S_{\mathit{exp}}^{*}\left({\lambda }_s\right)=N_s^{*}\left(\lambda \right){\lambda }_s^4{\lambda }_i^2$ can be determined from the integration of an angle-resolved spectrum $N_s^{*}\left(\lambda ,{\theta }_s\right)$ 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 fluorescence 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 approximately 20% at λ = 810 nm in our experiments. $S_{\mathit{exp}}^{*}\left(\lambda \right)$ 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 $S_{\mathit{exp}}^{*}\left(\lambda \right)$ due to overlapping of parametric fluorescence and scattered pump laser near θ = 0. We thus use $S_{\mathit{exp}}^{*}\left(\lambda \right)$ at θ_m = 29.1° to deduce the ISRF of our optical setup shown in Fig. 2.

 

Fig. 7 Instrument spectral response function. The normalized spectral density function S(λ) for θ_m = 29.1° and 29.4°. S_sim is determined from the integration over θ_s = −7.5° to 7.5° of the calculated angle-resolved spectra as shown in Fig. 5 [see also Eq. (6)]. The black and red curves are $S_{\mathit{exp}}^{*}$ 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. (7). $S_{\mathit{exp}}^{*}\left(\lambda \right)=N_s^{*}\left(\lambda \right){{\lambda }_s}^4{{\lambda }_i}^2$, where $N_s^{*}\left(\lambda \right)$ is the integration over θ_s ≈ −15° to 15° of the experimental imaging spectra $N_s^{*}\left({\lambda }_s,{\theta }_s\right)$. $S_{\mathit{exp}}^{*}\left(\lambda \right)$ 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-nitrogen-cooled 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 absolute ISRF (or collection efficiency) can be deduced by calibrating $S_{\mathit{exp}}^{*}\left(\lambda \right)$ at a fixed wavelength such as λ = 810 nm in our experiments.

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5. Conclusion

The angular and spectral distributions of parametric fluorescence in a BBO crystal pumped by a 405-nm diode laser are measured by 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. Parametric fluorescence in a BBO crystal pump by a 405-nm diode laser 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, limited by the spectral sensitivity of the silicon-based CCD camera in our spectroscopy system. Parametric fluorescence can be used as a pseudo standard light source for the calibration of instrument spectral response function of a spectroscopy system.