## Ellipsometry with polarisation-entangled photons

Optics Express, Vol. 14, Issue 16, pp. 7037-7045 (2006)

http://dx.doi.org/10.1364/OE.14.007037

Acrobat PDF (95 KB)

### Abstract

Polarisation-entangled photon pairs from a two crystal, type-I spontaneous parametric down conversion source are used to make accurate measurements of the ellipsometric angles of a silicon dioxide film on silicon and of internal and external reflection from BK7 glass. Since our source produces an entangled state with some mixture, a novel technique based on quantum tomography was developed to estimate the components of the density matrix for the state before and after reflection from the samples. The ellipsometric angles are readily calculated from these components and experimental measurements made on the samples were found to be in good agreement with their expected values.

© 2006 Optical Society of America

## 1. Introduction

2. D. E. Aspnes, “Expanding horizons: new developments in ellipsometry and polarimetry,” Thin Solid Films **455–456**, 3–13 (2004). [CrossRef]

*ψ*and Δ, which are defined as [1]:

*r*,

_{p}*r*are the Fresnel reflection coefficients for p- and s-polarised light respectively, at a given angle of incidence. Making ellipsometric measurements over a range of wavelengths [3

_{s}3. K. Vedam, “Spectroscopic ellipsometry: a historical overview,” Thin Solid Films **313–314**, 1–9 (1998). [CrossRef]

4. T. E. Jenkins, “Multiple-angle-of-incidence ellipsometry,” J. Phys. D Appl. Phys. **32**, R45–R56 (1999). [CrossRef]

5. P. G. Kwiat, E. Waks, A. G. White, I. Appelbaum, and P. H. Eberhard, “Ultrabright source of polarization-entangled photons,” Phys. Rev. A **60**, R773–R776 (1999). [CrossRef]

6. D. F. V. James, P. G. Kwiat, W. J. Munro, and A. G. White, “Measurement of qubits,” Phys. Rev. A **64**, 052312 (2001). [CrossRef]

7. A. Aspect, “Bell’s inequality test: more ideal than ever,” Nature (London) **398**, 189–190 (1999). [CrossRef]

8. J. L. O’Brien, G. J. Pryde, A. G. White, T. C. Ralph, and D. Branning, “Demonstration of an all-optical quantum Controlled-NOT Gate,” Nature (London) **426**, 264–267 (2003). [CrossRef]

9. N. Gisin, G. Ribordy, W. Tittel, and H. Zbinden, “Quantum cryptography,” Rev. Mod. Phys. **74**, 145–195 (2002). [CrossRef]

5. P. G. Kwiat, E. Waks, A. G. White, I. Appelbaum, and P. H. Eberhard, “Ultrabright source of polarization-entangled photons,” Phys. Rev. A **60**, R773–R776 (1999). [CrossRef]

## 2. The entangled photon ellipsometer

10. A. F. Abouraddy, K. C. Toussaint, A. V. Sergienko, B. E. A. Saleh, and M. C. Teich, “Ellipsometric measurements by use of photon pairs generated by spontaneous parametric downconversion,” Opt. Lett. **26**, 1717–1719 (2001). [CrossRef]

*H*(

*V*) denotes horizontal (vertical) polarisation. Abouraddy

*et al*[10

10. A. F. Abouraddy, K. C. Toussaint, A. V. Sergienko, B. E. A. Saleh, and M. C. Teich, “Ellipsometric measurements by use of photon pairs generated by spontaneous parametric downconversion,” Opt. Lett. **26**, 1717–1719 (2001). [CrossRef]

11. A. F. Abouraddy, K. C. Toussaint, A. V. Sergienko, B. E. A. Saleh, and M. C. Teich, “Entangled-Photon Ellip-sometry,” J. Opt. Soc. Am. B **19**, 656–662 (2002). [CrossRef]

*HV*⟩ + |

*VH*⟩, with similar results. After reflection of one photon of the pair from a sample, the state will be

*θ*

_{1}and

*θ*

_{2}with respect to the horizontal and detected with single photon detectors, then the coincidence count rate

*N*, that is, the rate at which two photons are detected simultaneously, will be

_{c}*C*is a constant determined by the source brightness, detector efficiencies and any losses of photons. If the initial state is known, that is

*ε*and

*ϕ*are known, tan

*ψ*and Δ can be calculated from the coincidence count rates for 3 different polariser settings, e.g.,

*θ*= 45°,

*θ*

_{2}= 0°,45c and 90°. If the initial state is not known, a separate calibration measurement must be made with the sample removed.

## 3. Ellipsometry with mixed states

5. P. G. Kwiat, E. Waks, A. G. White, I. Appelbaum, and P. H. Eberhard, “Ultrabright source of polarization-entangled photons,” Phys. Rev. A **60**, R773–R776 (1999). [CrossRef]

*H*and

*V*polarised components in the pump beam is controlled by a waveplate and determines the value of

*ε*. The phase

*ϕ*is determined by the relative phase of the

*H*and

*V*components of the pump beam as well as details of the phase matching and the thickness of the crystals. But

*ϕ*also depends on the angle at which the photons are emitted and, for non-degenerate pairs, their wavelength. Due to the angle and wavelength dependence of

*ϕ*, the observed state will not be a pure state but will rather be a mixture of entangled states of the form |

*HH*⟩ +

*εe*|

^{iϕ}*VV*⟩ with different

*ϕ*. Apertures and bandpass filters can be used to separate out pairs of photons with similar

*ϕ*, thereby approximating a pure state. However, we found that ellipsometric measurements made with our source using Eqn (5), which assumes a pure state, gave poor results, with measured values of tan

*ψ*and Δ differing from their expected values by 3 to 15 standard deviations. We believe this is caused by the mixed nature of the state produced by our source due to the variation in

*ϕ*. We estimated experimentally that

*ϕ*varies over ~ 0.8 radians for the states detected (see Section 4 for details). The coincidence count rate Nc from which the ellipsometric parameters are calculated is nonlinear with respect to

*ϕ*and thus the value of

*N*obtained is not necessarily the same as that for a pure state with

_{c}*ϕ*equal to the average

*ϕ*for our state. For example, it can be shown that the count rates with no sample for states of this form with

*ϕ*uniformly distributed over 0.8 rad and with average 0 and

*π*/2 radians will be the same as those for pure states with

*ϕ*= 0.228 rad and

*ϕ*=

*π*/2 rad, respectively.

*= 1. Our source actually produces a continuous spectrum of pure states as*

_{i}W_{i}*ϕ*changes continuously with the opening angle and wavelength. However, the sum form of the density matrix, rather than the integral form, will be used as it makes the following discussion clearer. This has no effect on the result as the integral form can be taken as the limit of an infinite sum. For the purposes of this paper, we are interested in the amounts of |

*HH*⟩ and |

*VV*⟩ and the phase between them, so the following will be useful later:

*r*=

_{H}*r*,

_{p}*r*=

_{V}*r*, produces

_{s}*R*is a normalisation constant. The components

*ρ*

_{11},

*ρ*

_{44}then become

*ρ*is the density matrix for the state after reflection of one photon from the sample. Since tan

_{r}*ψ*= |

*r*|/|

_{p}*r*|,

_{s}*ρ*

_{41}becomes

*ρ*

_{11},

*ρ*

_{44}(or

*ρ*

_{11}/

*ρ*

_{44}) and

*ρ*

_{41}(or at least its phase) can be measured, both with and without reflection from the sample, the ellipsometric parameters tan

*ψ*and Δ can be calculated.

6. D. F. V. James, P. G. Kwiat, W. J. Munro, and A. G. White, “Measurement of qubits,” Phys. Rev. A **64**, 052312 (2001). [CrossRef]

*P*

_{1}and for photon 2 having the operator

*P*

_{2}, the operator for the two photon measurement is

*P*

_{1}⊗

*P*

_{2}[14

14. A. F. Abouraddy, A. V. Sergienko, B. E. A. Saleh, and M. C. Teich, “Quantum entanglement and the two-photon Stokes parameters,” Opt. Commun. **201**, 93–98 (2002). [CrossRef]

*ρ*

_{11}and

*ρ*

_{44}is simple. Let

*N*(

_{c}*P*

_{1}⊗

*P*

_{2}) be the coincidence count rate for analyser settings

*P*

_{1}and

*P*

_{2}with the

*H*analyser set to transmit horizontally polarised light and

*V*analyser set to transmit vertically polarised light:

*C*is a constant determined by the source brightness, detector efficiencies and any losses of photons.

*ρ*

_{41}) requires the real and imaginary parts of

*ρ*

_{41}. These can be found from the following combinations of coincidence count rate measurements:

*D*,

*A*,

*R*and

*Z*are analysers set to transmit diagonally, anti-diagonally, right hand and left hand circularly polarised light, respectively. The ellipsometric parameters can thus be calculated from 10 coincidence count rate measurements each for (i) the state with reflection of one photon from the sample, and (ii) without reflection from the sample (calibration). The advantage of this technique over that presented in Section 2 is that it will work for any mixed state where the density matrix has significant

*ρ*

_{11},

*ρ*

_{44}and

*ρ*

_{41}components, that is, where a significant fraction of the pairs are in states of the form |

*HH*⟩ +

*εe*|

^{iϕ}*VV*⟩ with limited variation in

*ε*and

*ϕ*. These can be easily prepared [5

**60**, R773–R776 (1999). [CrossRef]

*HV*⟩ or |

*VH*〉 components which can occur if the crystals of the source are misaligned [15

15. A. G. White, D. F. V. James, P. H. Eberhard, and P. G. Kwiat, “Nonmaximally Entangled States: Production, Characterization and Utilization,” Phys. Rev. Lett. **83**, 3103–3107 (1999). [CrossRef]

## 4. Apparatus

**60**, R773–R776 (1999). [CrossRef]

*β*barium borate (BBO), 0.1 mm thick, cut with the optic axis at 29° to the crystal face. These are pumped by a Toptica iBeam-405 laser, nominally 50 mW at 405.3 nm which was supplied as a self-contained unit housing the violet diode, collimating optics and control electronics. The laser is temperature controlled and power stabilised, with internal optics that include an anamorphic prism pair to circularise the output beam. The maximum pump power was measured at 40.3 mW. Two kinematically-mounted mirrors steer the beam and a half-wave plate controls the polarisation state of the pump beam and thus the relative amounts of |

*HH*⟩ and |

*VV*⟩ pairs in the state produced [5

**60**, R773–R776 (1999). [CrossRef]

15. A. G. White, D. F. V. James, P. H. Eberhard, and P. G. Kwiat, “Nonmaximally Entangled States: Production, Characterization and Utilization,” Phys. Rev. Lett. **83**, 3103–3107 (1999). [CrossRef]

16. T. B. Pittman, D. V. Strekalov, D. N. Klyshko, M. H. Rubin, A. V. Sergienko, and Y. H. Shih, “Two-photon geometric optics,” Phys. Rev. A **53**, R2804–R2815 (1996). [CrossRef]

*f*= 300 mm) and reference (

*f*= 400 mm) arm are placed at their respective focal lengths from the crystals. The apertures, which were made of black card, are approximately 2 mm wide and 6 mm high and are placed 570 mm apart in the sample arm and 700 mm apart in the reference arm. The detectors have 25 mm diameter,

*f*= 35 mm achromatic lenses in front of them and are approximately 1.2 m from the crystals. Polarisation analysers placed in each arm can be set to transmit any polarisation state (elliptical as well as linear) and consist of quarter- and half-wave plates mounted in Newport SR50CC rotation stages, followed by fixed polarising beam splitter cubes. Custom bandpass filters centred on 810 nm and with 10 nm FWHM are used to reject unwanted light. The photons are detected with single photon counting modules and coincidences detected by a time to amplitude converter and recorded by a PC. The coincidence window is approximately 10 ns.

*ε*= 1 and

*ϕ*= 0 as possible [17

17. D. Dehlinger and M. W. Mitchell, “Entangled photon apparatus for the undergraduate laboratory,” Am. J. Phys. **70**, 898–902 (2002). [CrossRef]

*ε*= 1 the coincidence fringe visibility in the diagonal basis is cos

*ϕ*[5

**60**, R773–R776 (1999). [CrossRef]

*N*for a pure state is linear with respect to cos

_{c}*ϕ*and therefore a mixed state for which the average value of cos

*ϕ*is the same as that for the pure state will give the same value for Nc and thus the same fringe visibility. The average value of cos

*ϕ*for

*ϕ*uniformly distributed between -0.4 rad and 0.4 rad (0.974) is close to the visibility obtained. This suggests that the values of

*ϕ*for the pure states contributing to the mixed state may vary over ~ 0.8 rad. This is obviously an approximate estimate and neglects the possibility that

*ϕ*is non-uniformly distributed, or that the average value has not been adjusted exactly to zero as well as any other effects that might reduce the visibility. A Bell’s inequality measurement was also made [7

7. A. Aspect, “Bell’s inequality test: more ideal than ever,” Nature (London) **398**, 189–190 (1999). [CrossRef]

18. D. Dehlinger and M. W. Mitchell, “Entangled photons, nonlocality, and Bell inequalities in the under graduate laboratory,” Am. J. Phys. **70**, 903–910 (2002). [CrossRef]

*S*of

*S*= 2.760 ± 0.005. This is close to the maximum allowed value of

*S*= 2 √2 = 2.828 and in clear violation of the Bell inequality,

*S*≥ 2.

## 5. Experimental results

_{2}on a silicon substrate. Three separate, independent calibration measurements were made, labelled M1, M2 and M3. After each calibration, 10 coincidence count measurements were made with appropriate settings of the waveplates at each of three angles of incidence, 40.0°, 50.0° and 60.0°. This sample has reasonable reflectivity at 810 nm and, at 40.0° angle of incidence, provided around 65 to 75 cps with the polarisers in the

*H*⊗

*H*setting and approximately 90 cps for the

*V*⊗

*V*setting. As a consequence, we used counting times of 200 s for each coincidence count rate measurement and for the calibration measurements. The measurements used to calculate tan

*ψ*were repeated to give a total counting time of 400 s. Since entangled photon coincidences follow Poisson statistics [20

20. T. S. Larchuk, M. C. Teich, and B. E. A. Saleh, “Statistics of Entangled-Photon Coincidences in Parametric Downconversion,” Ann. NY Acad. Sci. **755**, 680–686 (1995). [CrossRef]

*ψ*and Δ are one standard deviation errors calculated from the Poisson statistics (shot noise) of the coincidence counts. These errors are expected to be optimistic since all other sources of error are neglected, of which uncertainty in the angle of incidence due to inaccurate sample alignment and imperfect beam collimation, is the likely major contributor. The rotation stages have an absolute accuracy of 0.035° and were used to determine the axes of the waveplates in a separate experiment. Standard error propagation methods were used to determine the uncertainty in tan

*ψ*and Δ from the Poisson statistics of the coincidence count measurements.

_{2}(

*n*= 1.453) and Si (

*n*= 3.685 - 0.006

*i*) at 810 nm were both obtained from curve fitting to data in [21], while the thickness of the SIO

_{2}film, which was determined ellipsometrically at 632.8 nm by the manufacturer, was given on the calibration certificate. We find that the majority of the values of tan

*ψ*and Δ are indeed within one standard deviation of their expected values. One notable exception is the value of Δ at 60.0° for calibration M2 and which we believe arises from a misalignment. The value of the phase

*ϕ*of the entangled state varies across the beam so that if the apparatus is misaligned and part of the beam is undetected, the measured average value of

*ϕ*will change. Since the apparatus has to be realigned to change the angle of incidence or perform a calibration, this is an obvious source of error in Δ. This sensitivity to misalignment could be greatly reduced by employing compensating crystals that remove the variation in

*ϕ*across the beam [22

22. J. Altepeter, E. Jeffrey, and P. Kwiat, “Phase-compensated ultra-bright source of entangled photons,” Opt. Express **13**, 8951–8959 (2005). [CrossRef] [PubMed]

*H*⊗

*H*setting provided around 20 cps while the

*V*⊗

*V*setting yielded about 135 cps. At 80.0° angle of incidence, these count rates increased to around 120 cps and 260 cps, respectively. In view of these low count rates, 400 s counting times were used for these setting of the polarisers. The other 8 polariser settings required to determine the phase used 200 s counting times at 70.0° and 75.0°, but 100 s counting times at 80.0°. 100 s counting times were used during the calibration, except the

*H*⊗

*H*and

*V*⊗

*V*settings for which 200 s was used. The experimental values given in the Table are one standard deviation errors based on the Poisson statistics of the count rates and are seen to be in good agreement with their expected values.

*ψ*as we expect this to be 1. Some photon loss is inevitable on entering and leaving the prism and this reduced the observed count rate slightly but had no other effect on the measurements.

## 6. Conclusion

## Acknowledgements

## References and links

1. | R. M. A. Azzam and N. M. Bashara, |

2. | D. E. Aspnes, “Expanding horizons: new developments in ellipsometry and polarimetry,” Thin Solid Films |

3. | K. Vedam, “Spectroscopic ellipsometry: a historical overview,” Thin Solid Films |

4. | T. E. Jenkins, “Multiple-angle-of-incidence ellipsometry,” J. Phys. D Appl. Phys. |

5. | P. G. Kwiat, E. Waks, A. G. White, I. Appelbaum, and P. H. Eberhard, “Ultrabright source of polarization-entangled photons,” Phys. Rev. A |

6. | D. F. V. James, P. G. Kwiat, W. J. Munro, and A. G. White, “Measurement of qubits,” Phys. Rev. A |

7. | A. Aspect, “Bell’s inequality test: more ideal than ever,” Nature (London) |

8. | J. L. O’Brien, G. J. Pryde, A. G. White, T. C. Ralph, and D. Branning, “Demonstration of an all-optical quantum Controlled-NOT Gate,” Nature (London) |

9. | N. Gisin, G. Ribordy, W. Tittel, and H. Zbinden, “Quantum cryptography,” Rev. Mod. Phys. |

10. | A. F. Abouraddy, K. C. Toussaint, A. V. Sergienko, B. E. A. Saleh, and M. C. Teich, “Ellipsometric measurements by use of photon pairs generated by spontaneous parametric downconversion,” Opt. Lett. |

11. | A. F. Abouraddy, K. C. Toussaint, A. V. Sergienko, B. E. A. Saleh, and M. C. Teich, “Entangled-Photon Ellip-sometry,” J. Opt. Soc. Am. B |

12. | A. V. Sergienko and G. S. Jaeger, “Quantum information processing and precise optical measurement with entangled-photon pairs,” Contemp. Phys. |

13. | K. C. Toussaint, G. D. Giuseppe, K. J. Bycenski, A. V. Sergienko, B. E. A. Saleh, and M. C. Teich, “Quantum ellipsometry using correlated-photon beams,” Phys. Rev. A |

14. | A. F. Abouraddy, A. V. Sergienko, B. E. A. Saleh, and M. C. Teich, “Quantum entanglement and the two-photon Stokes parameters,” Opt. Commun. |

15. | A. G. White, D. F. V. James, P. H. Eberhard, and P. G. Kwiat, “Nonmaximally Entangled States: Production, Characterization and Utilization,” Phys. Rev. Lett. |

16. | T. B. Pittman, D. V. Strekalov, D. N. Klyshko, M. H. Rubin, A. V. Sergienko, and Y. H. Shih, “Two-photon geometric optics,” Phys. Rev. A |

17. | D. Dehlinger and M. W. Mitchell, “Entangled photon apparatus for the undergraduate laboratory,” Am. J. Phys. |

18. | D. Dehlinger and M. W. Mitchell, “Entangled photons, nonlocality, and Bell inequalities in the under graduate laboratory,” Am. J. Phys. |

19. | W. J. Munro, K. Nemoto, and A. G. White, “The Bell inequality: a measure of entanglement?” J. Mod. Opt. |

20. | T. S. Larchuk, M. C. Teich, and B. E. A. Saleh, “Statistics of Entangled-Photon Coincidences in Parametric Downconversion,” Ann. NY Acad. Sci. |

21. | E. D. Palik, ed., |

22. | J. Altepeter, E. Jeffrey, and P. Kwiat, “Phase-compensated ultra-bright source of entangled photons,” Opt. Express |

**OCIS Codes**

(120.2130) Instrumentation, measurement, and metrology : Ellipsometry and polarimetry

(270.0270) Quantum optics : Quantum optics

**ToC Category:**

Instrumentation, Measurement, and Metrology

**History**

Original Manuscript: May 9, 2006

Revised Manuscript: July 6, 2006

Manuscript Accepted: July 3, 2006

Published: August 7, 2006

**Citation**

David J. L. Graham, A. Scott Parkins, and Lionel R. Watkins, "Ellipsometry with polarisation-entangled photons," Opt. Express **14**, 7037-7045 (2006)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-16-7037

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### References

- R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, The Netherlands,1987).
- D. E. Aspnes, "Expanding horizons: new developments in ellipsometry and polarimetry," Thin Solid Films 455-456, 3 - 13 (2004). [CrossRef]
- K. Vedam, "Spectroscopic ellipsometry: a historical overview," Thin Solid Films 313-314, 1 - 9 (1998). [CrossRef]
- T. E. Jenkins, "Multiple-angle-of-incidence ellipsometry," J. Phys. D Appl. Phys. 32, R45 - R56 (1999). [CrossRef]
- P. G. Kwiat, E. Waks, A. G. White, I. Appelbaum, and P. H. Eberhard, "Ultrabright source of polarizationentangled photons," Phys. Rev. A 60, R773 - R776 (1999). [CrossRef]
- D. F. V. James, P. G. Kwiat, W. J. Munro, and A. G. White, "Measurement of qubits," Phys. Rev. A 64, 052312(2001). [CrossRef]
- A. Aspect, "Bell’s inequality test: more ideal than ever," Nature (London) 398, 189 - 190 (1999). [CrossRef]
- J. L. O’Brien, G. J. Pryde, A. G. White, T. C. Ralph, and D. Branning, "Demonstration of an all-optical quantum Controlled-NOT Gate," Nature (London) 426, 264 - 267 (2003). [CrossRef]
- N. Gisin, G. Ribordy, W. Tittel, and H. Zbinden, "Quantum cryptography," Rev. Mod. Phys. 74, 145 - 195 (2002). [CrossRef]
- A. F. Abouraddy, K. C. Toussaint, A. V. Sergienko, B. E. A. Saleh, andM. C. Teich, "Ellipsometric measurements by use of photon pairs generated by spontaneous parametric downconversion," Opt. Lett. 26, 1717 - 1719 (2001). [CrossRef]
- A. F. Abouraddy, K. C. Toussaint, A. V. Sergienko, B. E. A. Saleh, and M. C. Teich, "Entangled-Photon Ellipsometry," J. Opt. Soc. Am. B 19, 656 - 662 (2002). [CrossRef]
- A. V. Sergienko and G. S. Jaeger, "Quantum information processing and precise optical measurement with entangled-photon pairs," Contemp. Phys. 44, 341 - 356 (2003). [CrossRef]
- K. C. Toussaint, G. D. 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]
- A. F. Abouraddy, A. V. Sergienko, B. E. A. Saleh, and M. C. Teich, "Quantum entanglement and the two-photon Stokes parameters," Opt. Commun. 201, 93 - 98 (2002). [CrossRef]
- A. G. White, D. F. V. James, P. H. Eberhard, and P. G. Kwiat, "Nonmaximally Entangled States: Production, Characterization and Utilization," Phys. Rev. Lett. 83, 3103-3107 (1999). [CrossRef]
- T. B. Pittman, D. V. Strekalov, D. N. Klyshko, M. H. Rubin, A. V. Sergienko, and Y. H. Shih, "Two-photon geometric optics," Phys. Rev. A 53, R2804 - R2815 (1996). [CrossRef]
- D. Dehlinger and M. W. Mitchell, "Entangled photon apparatus for the undergraduate laboratory," Am. J. Phys. 70, 898 - 902 (2002). [CrossRef]
- D. Dehlinger and M. W. Mitchell, "Entangled photons, nonlocality, and Bell inequalities in the under graduate laboratory," Am. J. Phys. 70, 903 - 910 (2002). [CrossRef]
- W. J. Munro, K. Nemoto, and A. G. White, "The Bell inequality: a measure of entanglement?" J. Mod. Opt. 48, 1239 - 1246 (2001).
- T. S. Larchuk, M. C. Teich, and B. E. A. Saleh, "Statistics of Entangled-Photon Coincidences in Parametric Downconversion," Ann. NY Acad. Sci. 755, 680 - 686 (1995). [CrossRef]
- E. D. Palik, ed., Handbook of Optical Constants of Solids (Academic Press, San Diego, California, 1998).
- J. Altepeter, E. Jeffrey, and P. Kwiat, "Phase-compensated ultra-bright source of entangled photons," Opt. Express 13, 8951 - 8959 (2005). [CrossRef] [PubMed]

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