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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 18, Iss. 5 — Mar. 1, 2010
  • pp: 4310–4315
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Observation of spontaneous parametric down-conversion excited by high brightness blue LED

G. Tamošauskas, J. Galinis, A. Dubietis, and A. Piskarskas  »View Author Affiliations


Optics Express, Vol. 18, Issue 5, pp. 4310-4315 (2010)
http://dx.doi.org/10.1364/OE.18.004310


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Abstract

We report on what is to our knowledge the first observation of the parametric fluorescence in bulk nonlinear crystals excited by commercial high-brightness incoherent blue LED.

© 2010 Optical Society of America

1. Introduction

Since the first consideration 1961 [1

1. W. H. Louisell, A. Yariv, and A. E. Siegman, “Quantum fluctuations and noise in parametric processes,” Phys. Rev. 124, 1646–1654 (1961). [CrossRef]

] and experimental observation in 1967 [2

2. S. E. Harris, M. K. Oshman, and R. L. Byer “Observation of tunable optical parametric fluorescence,” Phys. Rev. Lett. 18, 732–734 (1967). [CrossRef]

], spontaneous parametric down-conversion (parametric fluorescence) in media with χ(2) nonlinearity attracts a steadily growing interest in connection with fundamental studies and applications in quantum optics [3-10

3. J. G. Rarity, K. D. Ridley, and P. R. Tapster, “Absolute measurement of detector quantum efficiency using parametric downconversion,” Appl. Opt. 26, 4616–4619 (1987). [CrossRef] [PubMed]

]. In the process of spontaneous parametric down-conversion, three optical fields with different frequencies, i.e. pump, signal and idler, become coupled through the second-order nonlinearity of the medium, satisfying energy and momentum conservation. The parametric fluorescence could be regarded as a spontaneous parametric scattering process, where a single pump photon ”splits” into signal and idler photons, which are often termed as a biphoton field [11

11. D. N. Klyshko, Photons and Nonlinear Optics (Nauka, Moscow, 1980).

] and whose power scales linearly with the pump power. A conventional approach for excitation of the parametric fluorescence employs a phase-matched nonlinear medium and a laser source of high spatial and temporal coherence. Photon generation via parametric down-conversion process is accessed in many different operating regimes, ranging from a single photon [12

12. S. A. Castelleto and R. E. Scholten, “Heralded single photon sources: a route towards quantum communication and photon standards,” Eur. Phys. J. Appl. Phys. 41, 181–194 (2008). [CrossRef]

] to broadband [13

13. M. B. Nasr, S. Carrasco, B. E. A. Saleh, A. V. Sergienko, M. C. Teich, J. P. Torres, L. Torner, D. S. Hum, and M. M. Fejer, “Ultrabroadband biphotons generated via chirped quasi-phase-matched optical parametric down-conversion,” Phys. Rev. Lett. 100, 183601 (2008). [CrossRef] [PubMed]

] emission and in a variety of the nonlinear media using different interaction geometries, for brief overview see [14

14. J. Peřina Jr., M. Centini, C. Sibilia, and M. Bertolotti, “Photon-pair generation in random nonlinear layered structures,” Phys. Rev. A 80, 033844 (2009). [CrossRef]

] and references therein. An alternative possibility, which has not been experimentally considered up to date, is to excite the parametric fluorescence using spatially and temporally incoherent light sources. This possibility is offered by the advent of high-brightness and low-cost light sources, such as light emitting diodes (LEDs) emitting in the green-blue spectral region, which recently became commercially available [15

15. F. A. Ponce and D. P. Bour, “Nitride-based semiconductors for blue and green light-emitting devices,” Nature 386, 351–359 (1997). [CrossRef]

]. In what follows, we demonstrate how the parametric fluorescence with appreciable power and controllable spatial coherence properties is generated in lithium iodate (LiIO3), potassium dihydrophosphate (KDP) and beta-barium borate (BBO) crystals in type I and II phase-matching configurations, using an incoherent high-brightness blue LED as a pump source.

2. Experimental setup

The experimental setup is schematically depicted in Fig. 1(a). The whole setup is assembled in a single black box and consists of three compartments isolated by the optical filters. The pump source with polarization, beam shape and optical power modulators is assembled in the first section. High-brightness blue LED (LD W5AP-3V8A-35, OSRAM) emitting a total 2.5 W power into a wide (~ 2π sr) solid angle serves as a pump source. The emission spectrum of the LED has FWHM of 24 nm and central wavelength of 457 nm, as illustrated in Fig. 1(b). The Glan prism GP is mounted on a rotation stage to linearly polarize the LED radiation. The lenses L1 (f1 = +22 mm), L2 (f2 = +200 mm) and the variable apertures D1 and D2 are used to shape the pump beam in the following way. The LED crystal with the emitting area of 2.2 mm2 is imaged by the lens L1 onto the plane of the aperture D2. In this geometrical arrangement the aperture D1 shapes the spatial spectrum of the pump beam, while the aperture D2 allows to vary the diameter and power of the pump beam on the input face of the nonlinear crystal without further impact on its far-field distribution. The color glass filter F1 is used to block the long-wave radiation with λ > 560 nm.

Fig. 1. (a) Experimental setup. Lenses L1, L2 and apertures D1, D2 serve for LED beam shaping, F1 and F2 are the color filters, GP is the Glan prism for polarization control, L3 is the imaging lens, FP is the film polarizer. (b) Emission spectrum of the LED and spectral detection range of the detection system in terms of quantum efficiency.

We estimate the longitudinal (temporal) coherence length of the LED radiation in the order of ~ 9 μm (~ 30 fs), as being inversely proportional to its spectral bandwidth. The original light beam, emitted by the LED has poor quality (high divergence), as defined in terms of the beam propagation factor M2 ~ 104. In the experiment, the LED beam quality is improved by shaping its spatial spectrum to a rectangular profile, with a divergence of 14 mrad, that yields M2 in the range of 40 – 200, depending on the size of the aperture D2. Obviously, this improvement is achieved at the cost of the LED output power, with the maximum available pump power at the nonlinear crystal input of 0.53 mW.

The second compartment consists of the nonlinear crystal, which serves as a generator of the parametric fluorescence. The crystal is mounted on a motorized rotation stage, with rotation in the phase matching (θ) plane. In the experiment we used three different nonlinear crystals: 20-mm-thick LiIO3, 20-mm-thick KDP, both cut for type I phase matching, and 8-mm-thick BBO cut for type II phase matching.

The third compartment consists of the detection system, which employs high (16-bit) dynamic range CCD camera (ANDOR DV420A-OE), thermoelectrically cooled down to -50°C. The parametric fluorescence is collected by means of a composite lens L3 (f3 = +54.6 mm), which forms an angular distribution pattern onto the plane of the CCD sensor. The film polarizer FP (type 3M, American Polarizers Inc.) is used to analyze the polarization of the parametric fluorescence. The spectral detection range around the degeneracy (λ = 914 nm) is set as follows. The short-wave cutoff is set by combining a color filter and a dichroic mirror, indicated as F2 in Fig. 1(a), while the long-wave cutoff is imposed by the natural absorption limit of the silicon detector. This combination yields a band pass of 106 nm (at FWHM) around the degeneracy, and the quantum efficiency of the entire detection system illustrated in Fig. 1(b) is obtained by combining the data of the filter transmittance and the quantum efficiency of the CCD camera provided by the manufacturer. The CCD camera is mounted on the automated vertical translation stage, which allows recording the angular distribution of the parametric fluorescence in the angular window of ±230 mrad and ±170 mrad with respect to θ and ϕ axes of the nonlinear crystal.

3. Results and discussion

Figure 2 shows examples of the angular distributions of the parametric fluorescence excited in type I phase matching LiIO3, and KDP crystals, recorded with the pump power of 0.21 mW, and in type II phase matching BBO crystal, recorded with the pump power of 77 μW. The crystal offset Δθ is defined with respect to the scalar phase matching angle at the degeneracy θs for the central pump wavelength (λ = 457 nm), that is 35.6°, 41.9° and 36.9° for LiIO3, KDP and BBO crystals, respectively. In type I phase matching configuration (LiIO3 and KDP crystals) due to inseparable polarization, wavelength and direction of the signal and idler waves around the degeneracy, the parametric fluorescence is emitted as a single cone, whose angular diameter gradually increases with the crystal offset. The apparent cone thickness is defined by the dispersion characteristics of the particular nonlinear crystal, spatial and temporal spectra of the pump and spectral range of detection. In type II phase matching (BBO crystal) the parametric fluorescence is emitted as a pair of cones, whose axes are mutually shifted in the phase matching plane. The leftmost cone represents the e-polarized wave, while the rightmost cone – the o-polarized wave. For illustrative reasons these were recorded with the film polarizer adjusted at 45° with respect to the Glan polarizer. This case might be of particular interest, since polarization-entangled photon states are expected to occur at the crossing points of the emission cones [Fig. 2(i)]. The color coding in Fig. 2 represents the detected number of photons per second emitted into μ sr solid angle, which is derived after careful noise subtraction.

Fig. 2. Angular distributions of the parametric fluorescence excited (a)–(c) in LiIO3, (d)–(f) in KDP, (g)–(i) in BBO crystals, recorded at different crystal offset Δθ from the scalar phase matching angle.

Figure 3 plots the number of detected photons and the calculated spectral flux of the parametric fluorescence in type I phase matching LiIO3 and KDP crystals of equal thickness (20 mm). The plot suggests a linear dependence of the detected photon number versus the pump power, as expected from the very nature of the parametric fluorescence process. The number of photons was obtained by integrating the recorded parametric fluorescence images over a virtual aperture that contained 99% of the parametric fluorescence power. Here we note that the detected photon numbers in both crystals are reasonably above the camera detection limit, that is estimated as 20 photons/s. The spectral flux is evaluated taking into account the spectral response function of the detector, shown in Fig. 1(b). Dashed lines in Fig. 3 show the calculated spectral flux of the parametric fluorescence using plane and monochromatic wave model [16

16. R. L. Byer and S. E. Harris, “Power and bandwidth of spotaneous parametric emission,” Phys. Rev. 168, 1064–1068 (1968). [CrossRef]

, 17

17. A. Joobeur, B. E. A. Saleh, and M. C. Teich, “Spatiotemporal coherence properties of entangled light beams by parametric down-conversion,” Phys. Rev. A 50, 3349–3361 (1994). [CrossRef] [PubMed]

], which was ad hoc adapted for the case of incoherent pump with the particular spectral and angular properties as used in the experiment. Specifically, the parametric fluorescence power is calculated according to equation

Fig. 3. Number of detected photons and spectral flux of the parametric fluorescence in LiIO3 and KDP crystals of 20 mm thickness, measured at equal crystal offset Δθ = +0.5°. Solid lines show the linear fit, dashed lines represent the results of numerical simulation.
Ps=L20π/2π/2π/2π/2Pωpϕθβq(ωs)sinc2(ΔkL2)dϕdθdωs,
(1)

where L is the nonlinear crystal length, q(ωs) is the spectral response function of the detector, θ and ϕ are the relevant angles within the nonlinear crystal, ω is the frequency, n is the refractive index, the subscripts p, s, i denote pump, signal and idler waves, respectively. β is the nonlinear coupling coefficient expressed as

β=ωs4ωideffh̅ns4π3ε0c5ninp,
(2)

where deff is the effective nonlinearity of the medium. Δk is the phase mismatch, defined as

Δk=kpkski+kskp2ki(ϕ2+θ2),
(3)

where k = /c is the wavenumber. The pump is described as a superposition of individual plane and monochromatic waves with random phases, and whose power is expressed as

Pωpϕθ=S(ωp)SϕθP0,
(4)

where functions S(ωp) and S(ϕ, θ) denote the spectral density in frequency and space domains, respectively, and P0 is the total pump power. The relevant parameters of the nonlinear crystals were taken from [18

18. D. N. Nikogosyan, Nonlinear Optical Crystals: A Complete Survey (Springer, New York, 2005).

].

Fig. 4. Effect of the pump coherence on the angular properties of the parametric fluorescence. Central cross-sections of the parametric fluorescence cones excited (a) in LiIO3 crystal with different spectral bandwidth and (b) in KDP crystal with different divergence of the pump. Points show the experimental data, lines represent the results of numerical simulation.

The spatial and temporal coherence of the parametric fluorescence is crucial for many experiments in quantum optics and it is generally set by the coherence properties of the pump source [19

19. S. Cialdi, F. Castelli, and M. G. A. Paris, “Properties of entangled photon pairs generated by a CW laser with small coherence time: theory and experiment,” J. Mod. Opt. 56, 215–225 (2009). [CrossRef]

, 20

20. P. S. K. Lee, M. P. van Exter, and J. P. Woerdman, “How focused pumping affects type-II spontaneous parametric down-conversion,” Phys. Rev. A 72, 033803 (2005). [CrossRef]

]. Figure 4 provides examples how the coherence properties of the parametric fluorescence are modified by changing temporal [Fig. 4(a)] and spatial [Fig. 4(b)] coherence properties of the pump. Here we plot a central cross-section of the parametric fluorescence cones in the phase matching plane, with their thickness serving as a measure of coherence. Figure 4(a) shows the effect of the time bandwidth of the pump beam on the angular width of the parametric fluorescence excited in LiIO3 crystal. The time bandwidth of the pump is modified by means of an interference filter, which has a FWHM transmission bandwidth of 7 nm, and is inserted in the pump beam path in front of the aperture D2. Figure 4(b) shows how the angular distribution of the parametric fluorescence power in KDP crystal is affected by changing the pump beam divergence from 14 mrad to 38 mrad by means of the aperture D1.

4. Conclusion

In conclusion, our results suggest that high-brightness LEDs could be used as an attractive alternative to laser sources for excitation of well-detectable parametric fluorescence in bulk nonlinear crystals. We show that the spatial coherence properties of the parametric fluorescence could be controlled by simple means, e.g. by shaping the spatial and temporal spectra of an incoherent pump beam, offering a possibility to observe quantum optics effects, as suggested by earlier observation [21

21. P. Kumar, O. Aytür, and J. Huang, “Squeezed-light generation with an incoherent pump,” Phys. Rev. Lett. 64, 1015–1018 (1990). [CrossRef] [PubMed]

]. The parameters of the LED-pumped parametric fluorescence might be notably improved using advanced LED beam shaping techniques [22

22. J. Y. Joo, C. S. Kang, S. S. Park, and S.-K. Lee, “LED beam shaping lens based on the near-field illumination,” Opt. Express 17, 23449–23458 (2009), http://www.opticsinfobase.org/oe/abstract.cfm?uri=oe-17-26-23449. [CrossRef]

], so making it a versatile source for many experiments in quantum optics, for instance, for studies of quantum-classical transitions and for testing of quantum communication networks, to mention a few.

References and links

1.

W. H. Louisell, A. Yariv, and A. E. Siegman, “Quantum fluctuations and noise in parametric processes,” Phys. Rev. 124, 1646–1654 (1961). [CrossRef]

2.

S. E. Harris, M. K. Oshman, and R. L. Byer “Observation of tunable optical parametric fluorescence,” Phys. Rev. Lett. 18, 732–734 (1967). [CrossRef]

3.

J. G. Rarity, K. D. Ridley, and P. R. Tapster, “Absolute measurement of detector quantum efficiency using parametric downconversion,” Appl. Opt. 26, 4616–4619 (1987). [CrossRef] [PubMed]

4.

E. C. Cheung, K. Koch, G. T. Moore, and J. M. Liu, “Measurements of second-order nonlinear optical coefficients from the spectral brightness of parametric fluorescence,” Opt. Lett. 19, 168–170 (1994). [CrossRef] [PubMed]

5.

B. E. A. Saleh, B. M. Jost, H.-B. Fei, and M. C. Teich, “Entangled-photon virtual-state spectroscopy,” Phys. Rev. Lett. 80, 3483–3486 (1998). [CrossRef]

6.

T. Jennewein, C. Simon, G. Weihs, H. Weinfurter, and A. Zeilinger, “Quantum cryptography with entangled photons,” Phys. Rev. Lett. 84, 4729–4732 (2000). [CrossRef] [PubMed]

7.

A. N. Boto, P. Kok, D. S. Abrams, S. L. Braunstein, C. P. Williams, and J. P. Dowling, “Quantum interferometric optical litography: exploiting entanglement to beat the diffraction limit,” Phys. Rev. Lett. 85, 2733–2736 (2000). [CrossRef] [PubMed]

8.

K. C. Toussaint, G. Di Giuseppe, K. J. Bycenski, A. V. Sergienko, B. E. A. Saleh, and M. C. Teich, “Quantum ellipsometry using correlated-photon beams,” Phys. Rev. A 70, 023801 (2004). [CrossRef]

9.

R. Thew and N. Gisin, “Quantum communication,” Nat. Photon. 1, 165–171 (2007). [CrossRef]

10.

I. P. Degiovanni, M. Genovese, V. Schettini, M. Bondani, A. Andreoni, and M. G. A. Paris, “Monitoring the quantum-classical transition in thermally seeded parametric down-conversion by intensity measurements,” Phys. Rev. A 79, 063836 (2009). [CrossRef]

11.

D. N. Klyshko, Photons and Nonlinear Optics (Nauka, Moscow, 1980).

12.

S. A. Castelleto and R. E. Scholten, “Heralded single photon sources: a route towards quantum communication and photon standards,” Eur. Phys. J. Appl. Phys. 41, 181–194 (2008). [CrossRef]

13.

M. B. Nasr, S. Carrasco, B. E. A. Saleh, A. V. Sergienko, M. C. Teich, J. P. Torres, L. Torner, D. S. Hum, and M. M. Fejer, “Ultrabroadband biphotons generated via chirped quasi-phase-matched optical parametric down-conversion,” Phys. Rev. Lett. 100, 183601 (2008). [CrossRef] [PubMed]

14.

J. Peřina Jr., M. Centini, C. Sibilia, and M. Bertolotti, “Photon-pair generation in random nonlinear layered structures,” Phys. Rev. A 80, 033844 (2009). [CrossRef]

15.

F. A. Ponce and D. P. Bour, “Nitride-based semiconductors for blue and green light-emitting devices,” Nature 386, 351–359 (1997). [CrossRef]

16.

R. L. Byer and S. E. Harris, “Power and bandwidth of spotaneous parametric emission,” Phys. Rev. 168, 1064–1068 (1968). [CrossRef]

17.

A. Joobeur, B. E. A. Saleh, and M. C. Teich, “Spatiotemporal coherence properties of entangled light beams by parametric down-conversion,” Phys. Rev. A 50, 3349–3361 (1994). [CrossRef] [PubMed]

18.

D. N. Nikogosyan, Nonlinear Optical Crystals: A Complete Survey (Springer, New York, 2005).

19.

S. Cialdi, F. Castelli, and M. G. A. Paris, “Properties of entangled photon pairs generated by a CW laser with small coherence time: theory and experiment,” J. Mod. Opt. 56, 215–225 (2009). [CrossRef]

20.

P. S. K. Lee, M. P. van Exter, and J. P. Woerdman, “How focused pumping affects type-II spontaneous parametric down-conversion,” Phys. Rev. A 72, 033803 (2005). [CrossRef]

21.

P. Kumar, O. Aytür, and J. Huang, “Squeezed-light generation with an incoherent pump,” Phys. Rev. Lett. 64, 1015–1018 (1990). [CrossRef] [PubMed]

22.

J. Y. Joo, C. S. Kang, S. S. Park, and S.-K. Lee, “LED beam shaping lens based on the near-field illumination,” Opt. Express 17, 23449–23458 (2009), http://www.opticsinfobase.org/oe/abstract.cfm?uri=oe-17-26-23449. [CrossRef]

OCIS Codes
(230.3670) Optical devices : Light-emitting diodes
(270.5585) Quantum optics : Quantum information and processing

ToC Category:
Quantum Optics

History
Original Manuscript: December 15, 2009
Revised Manuscript: January 18, 2010
Manuscript Accepted: January 30, 2010
Published: February 17, 2010

Citation
G. Tamošauskas, J. Galinis, A. Dubietis, and A. Piskarskas, "Observation of spontaneous parametric down-conversion excited by high brightness blue LED," Opt. Express 18, 4310-4315 (2010)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-5-4310


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References

  1. W. H. Louisell, A. Yariv, and A. E. Siegman, "Quantum fluctuations and noise in parametric processes," Phys. Rev. 124,1646-1654 (1961). [CrossRef]
  2. S. E. Harris, M. K. Oshman, and R. L. Byer, "Observation of tunable optical parametric fluorescence," Phys. Rev. Lett. 18,732-734 (1967). [CrossRef]
  3. J. G. Rarity, K. D. Ridley, and P. R. Tapster, "Absolute measurement of detector quantum efficiency using parametric down conversion," Appl. Opt. 26,4616-4619 (1987). [CrossRef] [PubMed]
  4. E. C. Cheung, K. Koch, G. T. Moore, and J. M. Liu, "Measurements of second-order nonlinear optical coefficients from the spectral brightness of parametric fluorescence," Opt. Lett. 19,168-170 (1994). [CrossRef] [PubMed]
  5. B. E. A. Saleh, B. M. Jost, H.-B. Fei, and M. C. Teich, "Entangled-photon virtual-state spectroscopy," Phys. Rev. Lett. 80,3483-3486 (1998). [CrossRef]
  6. T. Jennewein, C. Simon, G. Weihs, H. Weinfurter, and A. Zeilinger, "Quantum cryptography with entangled photons," Phys. Rev. Lett. 84, 4729-4732 (2000). [CrossRef] [PubMed]
  7. A. N. Boto, P. Kok, D. S. Abrams, S. L. Braunstein, C. P. Williams, and J. P. Dowling, "Quantum interferometric optical lithography: exploiting entanglement to beat the diffraction limit," Phys. Rev. Lett. 85,2733-2736 (2000). [CrossRef] [PubMed]
  8. K. C. Toussaint, G. Di Giuseppe, K. J. Bycenski, A. V. Sergienko, B. E. A. Saleh, and M. C. Teich, "Quantum ellipsometry using correlated-photon beams," Phys. Rev. A 70,023801 (2004). [CrossRef]
  9. R. Thew and N. Gisin, "Quantum communication," Nat. Photon. 1,165-171 (2007). [CrossRef]
  10. I. P. Degiovanni, M. Genovese, V. Schettini, M. Bondani, A. Andreoni, and M. G. A. Paris, "Monitoring the quantum-classical transition in thermally seeded parametric down-conversion by intensity measurements," Phys. Rev. A 79,063836 (2009). [CrossRef]
  11. D. N. Klyshko, Photons and Nonlinear Optics (Nauka, Moscow, 1980).
  12. S. A. Castelleto and R. E. Scholten, "Heralded single photon sources: a route towards quantum communication and photon standards," Eur. Phys. J. Appl. Phys. 41,181-194 (2008). [CrossRef]
  13. M. B. Nasr, S. Carrasco, B. E. A. Saleh, A. V. Sergienko, M. C. Teich, J. P. Torres, L. Torner, D. S. Hum, and M. M. Fejer, "Ultrabroadband biphotons generated via chirped quasi-phase-matched optical parametric down conversion," Phys. Rev. Lett. 100,183601 (2008). [CrossRef] [PubMed]
  14. J. Peřina, Jr., M. Centini, C. Sibilia, and M. Bertolotti, "Photon-pair generation in random nonlinear layered structures," Phys. Rev. A 80,033844 (2009). [CrossRef]
  15. F. A. Ponce and D. P. Bour, "Nitride-based semiconductors for blue and green light-emitting devices," Nature 386,351-359 (1997). [CrossRef]
  16. R. L. Byer and S. E. Harris, "Power and bandwidth of spontaneous parametric emission," Phys. Rev. 168,1064-1068 (1968). [CrossRef]
  17. A. Joobeur, B. E. A. Saleh, and M. C. Teich, "Spatiotemporal coherence properties of entangled light beams by parametric down-conversion," Phys. Rev. A 50,3349-3361 (1994). [CrossRef] [PubMed]
  18. D. N. Nikogosyan, Nonlinear Optical Crystals: A Complete Survey (Springer, New York, 2005).
  19. S. Cialdi, F. Castelli, and M. G. A. Paris, "Properties of entangled photon pairs generated by a CW laser with small coherence time: theory and experiment," J. Mod. Opt. 56,215-225 (2009). [CrossRef]
  20. P. S. K. Lee, M. P. van Exter, and J. P. Woerdman, "How focused pumping affects type-II spontaneous parametric down-conversion," Phys. Rev. A 72,033803 (2005). [CrossRef]
  21. P. Kumar, O. Aytür, and J. Huang, "Squeezed-light generation with an incoherent pump," Phys. Rev. Lett. 64,1015-1018 (1990). [CrossRef] [PubMed]
  22. J. Y. Joo, C. S. Kang, S. S. Park, and S.-K. Lee, "LED beam shaping lens based on the near-field illumination," Opt. Express 17,23449-23458 (2009), http://www.opticsinfobase.org/oe/abstract.cfm?uri=oe-17-26-23449. [CrossRef]

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