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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 20, Iss. 22 — Oct. 22, 2012
  • pp: 25022–25029
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Pulsed Sagnac source of polarization entangled photon pairs

Ana Predojević, Stephanie Grabher, and Gregor Weihs  »View Author Affiliations


Optics Express, Vol. 20, Issue 22, pp. 25022-25029 (2012)
http://dx.doi.org/10.1364/OE.20.025022


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Abstract

Photonic quantum information experiments demand bright and highly entangled photon pair sources. The combination of periodic poling and collinear excitation geometry allows the use of considerably longer crystals for parametric down-conversion. We demonstrate a picosecond-pulsed laser pumped source of high quality polarization entangled photon pairs. The phase of the output biphoton state is affected by the relative phase of the two-color interferometer and the phase of the nonlinearly interacting Gaussian beams. We measure the influence of these onto the phase of the output state. The presented source is a promising candidate for a compact, semiconductor laser driven source of entangled photon pairs.

© 2012 OSA

1. Introduction

Experiments with multiple entangled photon pairs have enabled the exploration of new quantum communication protocols [1

1. D. Bouwmeester, J.-W. Pan, K. Mattle, M. Eibl, H. Weinfurter, and A. Zeilinger, “Experimental quantum teleportation,” Nature 390, 575–579 (1997). [CrossRef]

3

3. J.-W. Pan, S. Gasparoni, R. Ursin, G. Weihs, and A. Zeilinger, “Experimental entanglement purification of arbitrary unknown states,” Nature 423, 417–422 (2003). [CrossRef] [PubMed]

], the generation and characterization of multipartite entangled states [4

4. J.-W. Pan, M. Daniell, S. Gasparoni, G. Weihs, and A. Zeilinger “Experimental demonstration of four-photon entanglement and high-fidelity teleportation,” Phys. Rev. Lett. 86, 4435–4438 (2001). [CrossRef] [PubMed]

, 5

5. M. Eibl, N. Kiesel, M. Bourennane, C. Kurtsiefer, and H. Weinfurter “Experimental realization of a three-qubit entangled W state,” Phys. Rev. Lett. 92, 077901 (2004). [CrossRef] [PubMed]

], and implementations of few-qubit quantum circuits and quantum simulations. Most of these experiments employ traditional spontaneous parametric down-conversion sources driven by femtosecond pulsed lasers. The types of phase-matching together with the large spectral bandwidth of a femtosecond pulsed laser demand considerable experimental effort in order to reach high quality of the output entangled state [6

6. M. Hendrych, M. Mičuda, and J. P. Torres “Tunable control of the frequency correlations of entangled photons,” Opt. Lett. 32(16,) 2339–2341 (2007). [CrossRef] [PubMed]

8

8. R. Rangarajan, M. Goggin, and P. Kwiat “Optimizing type-I polarization-entangleg photons,” Opt. Express 17, 18920–18933 (2009). [CrossRef]

]. These phase-matching schemes further limit the useful conversion crystal length [9

9. P. S. K. Lee, M. P. van Exter, and J. P. Woerdman“Increased polarization-entangled photon flux via thinner crystals,” Phys. Rev. A 70, 043818 (2004). [CrossRef]

] and thus the lack of efficiency has to be compensated for by using very intense pump pulses with all the associated experimental difficulties.

Our source is a picosecond pulsed collinear source of entangled photon pairs with high quality performance. We used this source to experimentally demonstrate the effect of the Gaussian beam phase on the phase of the entangled output state. The experimental apparatus we use, including laser, doubling, down-conversion, and detection are techniques which could be applied to other wavelengths. Furthermore, due to the very relaxed requirements on the power and pulse length of the pump laser this source could be driven by a mode-locked semiconductor laser, thus dramatically reducing the cost and complexity of the experimental set-up.

2. Experimental set-up and source characterisation

Our source consists of a periodically poled KTiOPO4 (PPKTP) crystal embedded in a triangular two-color Sagnac interferometer [10

10. B.-S. Shi and A. Tomita “Generation of a pulsed polarization entangled photon pair using a Sagnac interferometer,” Phys. Rev. A 69, 013803 (2004). [CrossRef]

14

14. M. Hentschel, H. Hübel, A. Poppe, and A. Zeilinger “Pulsed Sagnac source of narrow-band polarization entangled photons,” Opt. Express 15, 15377–15386 (2007).

]. Here, two counter-propagating beams simultaneously pump the crystal and thus, the down-converted photon pairs can be created in either direction. Unlike a traditional Sagnac interferometer this geometry uses a polarizing beamsplitter as entrance and output port. A schematic of the Sagnac source is given in Fig. 1. A half-waveplate placed after the polarizing beamsplitter rotates the polarization of one of the pump beams. This way both pump beams have the correct polarization needed for the down-conversion process. Since the down-conversion process is of type II, the created photon pairs suffer a temporal walk-off caused by the birefringence of the crystal. This can be compensated for using the half-waveplate inside the loop, which therefore has a second function: It erases the information about the direction in which the down-conversion took place by making photons with the same time shift leave the interferometer beamsplitter through the same output port. An additional advantage of this geometry is the intrinsic phase stability of the Sagnac interferometer.

Fig. 1 Experimental set-up. The frequency doubled Ti:Sapphire picosecond laser (blue) is directed via a single mode fiber (SMF) onto the Sagnac loop. The pump light is split at the polarizing beamsplitter (PBS). A half-waveplate (HWP) placed inside the loop rotates the polarization of one of the pump directions in front of the crystal (PPKTP). The photon pairs produced in the counter-clockwise direction pass through the same waveplate on their way out of the loop. The photons leave the loop through the polarizing beamsplitter (PBS) and are coupled into single mode optical fibers (output 1 and output 2). The remaining pump light is eliminated by two color glass filters (CG). The phase φp of the pump beam is used to adjust the phase between the two conversion processes and thus to set the phase of the output entangled state.

The source is pumped by 2 ps long laser pulses. The light derived from a 76 MHz repetition rate Ti:Sapphire laser is frequency doubled in an Bismuth Triborate (BIBO) nonlinear crystal. The generated 404 nm light is passed through a short single-mode fiber for spatial filtering and focused into the down-conversion crystal. The power after spatial filtering can be raised to up to several milliwatts without affecting the pump spectrum by fiber nonlinearity.

The nonlinear crystal is a 15 mm long, anti-reflection coated type-II PPKTP placed in the middle of the Sagnac-loop, which also contains a dual-wavelength polarizing beamsplitter and a dual-wavelength waveplate. The produced photon pairs enter single mode fibers that act as spatial filters. Broadband color glass filters are the only other filtering used. The photons are detected by silicon avalanche photodiodes, and the photon statistics are recorded by a multi-channel event timer.

The possible range of target output states is
|Ψ=|HV+eiφ|VH
(1)
where the phase φ can be adjusted by various means as detailed below. The phase settings φ = 0, π correspond to the Bell states Ψ+ and Ψ, respectively. The Ψ+ -state we created in the experiment yields a correlation visibility of 98.70(9)% in the A/D (±45°) basis and 99.88(3)% in the H/V (horizontal/vertical) basis, where accidental coincidence counts of 2 counts per second have been subtracted. Figure 2 shows the results of the visibility measurements. In addition, we performed state tomography of the output state [15

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

]. The fidelity of the reconstructed two-photon density matrix with the maximally entangled state was found to be F=0.982(7) while the tangle was T=0.965(8). Here also accidental coincidence counts of 2 counts per second have been subtracted. The raw data yield following results: 99.5(4)% and 98.40(9)% for the visibilities in the H/V and the A/D basis, respectively, and F=0.981(7) and T=0.964(8) for the state fidelity and the tangle. These measurements were performed with a pump power of 300μW and each measurement point was averaged over 10 s. The density matrix was reconstructed using the maximum-likelihood estimation method. In order to obtain the measurement errors we performed a 100 run Monte Carlo simulation of the data with a Poissonian noise model applied to the measured values. We note that the measurements of the achieved tangle and state fidelity were performed via long single mode optical fibers which could have led to some averaging of the measurement results over different bases due to polarization fluctuations in the fibers. We measured the total brightness per pump power of our source to be 39 700 pairs/s per mW. Spectral widths (FWHM) of the produced protons were measured to be 1.79 nm and 2.67 nm.

Fig. 2 Coincidence counts from the picosecond pulsed Sagnac source as a function of the polarizer angle in one output port. For the circles / squares the polarizer in the other output port was set to horizontal (H) / −45° (A), respectively. The solid lines are sinusoidal fits to the data. The Poissonian error bars are smaller than the symbols. The lower count rate of this measurement compared with the reported brightness originates from linear polarizers used in the measurement which exhibit low transmission (≈62%).

3. Phase of the output state

It had been pointed out earlier [16

16. D. R. Hamel “Realization of novel entangled photon sources using periodically poled materials,” Master’s thesis, University of Waterloo (2010).

] that geometrically centering the nonlinear crystal in the loop of a Sagnac source results in an optimum output tangle of the generated state and that displacing the crystal from the loop center results in a decrease of entanglement. To investigate this phenomenon we made a series of measurements where we performed state tomography of the generated entangled state for different positions of the crystal in the loop. From the reconstructed density matrices we calculated the fidelity of the measured state with respect to the states |Ψ+〉 and |Ψ〉, the phase between the state components |HV〉 and |VH〉, and tangle as shown in Fig. 3(a, b and c). At the nominal center position we set the pump phase to produce the |Ψ+〉 state. Figure 3(b) shows the real and imaginary part of an off-diagonal element (i.e. the coherence between |HV〉 and |VH〉) of the reconstructed density matrix. The tangle of the state maintained its high value (around 90%) throughout the measurement, but decreased when moving away from the central position, as expected due to the imbalance between the clockwise and counter-clockwise amplitudes.

Fig. 3 The error bars are smaller than the symbols. (a) The fidelity of the generated entangled state to the states Ψ+ and Ψ changes anti-symmetrically as the crystal is displaced from the initial position (near 4th data point). (b) The real and the imaginary parts of the off-diagonal density matrix element indicate the changing phase among the |HV〉 and |VH〉 components. (c) The tangle for all positions of the crystal.

We found that the quality of the entanglement itself only decreases very slowly away from the center (see Fig. 3(c)) As detailed below the major effect observed is simply a crystal position-dependent phase shift which changes the phase of the output state. We identified two effects that contribute to this shift, the larger contribution coming from the dispersion of air and the smaller one from the Gouy phase of the interacting beams.

The refractive index difference between the blue and red wavelength in air is approximately 7 × 10−6. This amounts to a phase shift between the clockwise and counter-clockwise beams of φair(x) = 0.07πx where x is the shift of the crystal away from the loop center in mm. Consequently, this shift is transferred to the phase of the entangled state. However, it does not yet fully account for the observed effect and we further consider the exact phase acquired in the nonlinear interaction.

Due to the changing wave-front curvature a propagating Gaussian beam acquires an additional phase compared to a plane wave, which is called the Gouy phase. In the complex field amplitude of the Gaussian beam this derives from the expression 11+iτ=exp(iϕGouy)/1+τ2 with ϕGouy = − arctanτ, where τ = z/z0 is the scaled position z in the propagation direction and z0 is the Rayleigh range of the beam. In particular, passing through a focus a Gaussian beam receives a π phase shift.

The efficiency of nonlinear up-conversion processes between Gaussian beams was calculated in detail by Boyd and Kleinman [17

17. G. D. Boyd and D. A. Kleiman “Parametric interaction of focused Gaussian light beams,” J. Appl. Phys. 39, 3597–3640 (1968). [CrossRef]

]. For simplicity we adopt their formalism for the case where all the involved beams have the same Rayleigh range z0 and where there is no birefringent walk-off, which is appropriate for the crystal we use. For a nonlinear crystal of geometrical length L or, equivalently, focusing parameter ξ = L/(2z0), the converted complex on-axis field amplitude for a beam centered at position f within the crystal is then proportional to
H(σ,ξ,f)=f/z0ξf/z0exp(iστ)1+iτdτ.
(2)

At f = L/2 the focus is centered in the crystal. The highest conversion efficiency is obtained at a nominal zero phase difference between the pump and down-conversion beams. The phase mismatch σ = z0 Δk is set by the linear dispersive properties of the crystal (Δk = kpumpksignalkidler) and the denominator in Eq. (2) comes from the Gouy phases of the three interacting waves [18

18. R. S. Bennink“Optimal collinear Gaussian beams for spontaneous parametric down-conversion,” Phys. Rev. A 81, 053805 (2010). [CrossRef]

20

20. N. Lastzka and R. Schnabel “The Gouy phase shift in nonlinear interactions of waves,” Opt. Express 15, 7211–7217 (2007). [CrossRef] [PubMed]

]. When the crystal is moved from the loop center, if we have focus position f in the clockwise direction, we will have Lf in the counter-clockwise position. The resultant phase shift is then φGouy(σ, f) = arg{H(σ, ξ, f)} − arg{H(σ, ξ, Lf)} = 2arg{H(σ, ξ, f)}.

For a focusing parameter ξ and focus position f there is an optimum phase mismatch σopt(ξ, f), which best compensates the Gouy phase and achieves the highest conversion efficiency. Here we have to point out that we investigate this phase shift in spontaneous parametric down-conversion where the produced photon pair may be non-degenerate and σ depends very sensitively on all three wavelengths and the operating temperature. Because the Sagnac source works equally well for the degenerate and non-degenerate cases and we do not apply any narrow-band filtering we assume that at the chosen temperature the spectra of the two photons will only be determined by the function H and that the phase mismatch at the centers of these spectra is σopt.

In our experiment, the pump beam waist was measured to be w0 = 25(3)μm corresponding to z0 = 8.9(21) mm (inside the crystal) or ξ = 0.84(20). Using the values for atmospheric pressure, temperature, and humidity during the measurement, we modelled the air refractive index using Ciddor’s approach [21

21. Engineering metrology toolbox, http://emtoolbox.nist.gov/Wavelength/Ciddor.asp.

, 22

22. P. E. Ciddor “Refractive index of air: new equations for the visible and near infrared,” Appl. Opt. 35, 1566–1573 (1996). [CrossRef] [PubMed]

] resulting in a refractive index difference of 7.02(1)×10−6 for the wavelengths of 404 nm and 808 nm. Subtracting the resulting air phase shift (blue, dashed line in Fig 4(a)) from the measured total phase data yields the net (Gouy) phase.

Fig. 4 (a) The phase of the entangled state as a function of the crystal displacement from the Sagnac loop center shows a nearly linear dependence on the crystal displacement, where zero corresponds to the nominal loop center. The symbols represent the phases reconstructed from the tomography measurements (the statistical errors are smaller than the symbols). The solid line is a fit of the nonlinear Gouy phase model including the air dispersive shift. The contribution of the air at the mean pressure and humidity on the afternoon of the measurement (see the main text) is drawn as the blue dashed straight line. (b) In order to highlight the influence of the Gouy phase here we show data, where φair was subtracted. The fit to this data is based on Boyd and Kleinman’s theory and results in ξ = 0.706 ± 0.076 for the focusing parameter and 6.0±1.6mm for the initial displacement of the beam waists from the loop center. In addition we plot, for identical parameters, φGouy obtained from Boyd and Kleinman’s up-conversion theory, but for coincident clock- and counter-clockwise beam waist positions (black dashed line).

4. Conclusion

We have demonstrated a picosecond pulsed down-conversion source in collinear geometry. Our source exhibits very high visibility, a high level of entanglement and excellent brightness. Using picosecond pulses to create pulsed entangled photon pairs allows the use of semiconductor based pump sources. Due to their reduced cost and size, the use of mode-locked diode systems would make experiments using multiple entangled photon pairs more compact and affordable, and may enable us to perform more complex experiments than possible with the traditional femtosecond pump approach. Moreover, we have experimentally measured the phase shift in a nonlinear conversion process via the phase of the generated entangled state. We have characterized the magnitude of the effect this phase shift has on the phase of the bi-photon entangled state created in a Sagnac-geometry down-conversion source. This investigation can be applied to any source in this geometry, both pulsed and continuous wave.

Acknowledgments

References and links

1.

D. Bouwmeester, J.-W. Pan, K. Mattle, M. Eibl, H. Weinfurter, and A. Zeilinger, “Experimental quantum teleportation,” Nature 390, 575–579 (1997). [CrossRef]

2.

T. Jennewein, G. Weihs, J.-W. Pan, and A. Zeilinger “Experimental nonlocality proof of quantum teleportation and entanglement swapping,” Phys. Rev. Lett. 88, 017903 (2002). [CrossRef] [PubMed]

3.

J.-W. Pan, S. Gasparoni, R. Ursin, G. Weihs, and A. Zeilinger, “Experimental entanglement purification of arbitrary unknown states,” Nature 423, 417–422 (2003). [CrossRef] [PubMed]

4.

J.-W. Pan, M. Daniell, S. Gasparoni, G. Weihs, and A. Zeilinger “Experimental demonstration of four-photon entanglement and high-fidelity teleportation,” Phys. Rev. Lett. 86, 4435–4438 (2001). [CrossRef] [PubMed]

5.

M. Eibl, N. Kiesel, M. Bourennane, C. Kurtsiefer, and H. Weinfurter “Experimental realization of a three-qubit entangled W state,” Phys. Rev. Lett. 92, 077901 (2004). [CrossRef] [PubMed]

6.

M. Hendrych, M. Mičuda, and J. P. Torres “Tunable control of the frequency correlations of entangled photons,” Opt. Lett. 32(16,) 2339–2341 (2007). [CrossRef] [PubMed]

7.

P. J. Mosley, J. S. Lundeen, B. J. Smith, P. Wasylczyk, A. B. U’Ren, C. Silberhorn, and I. A. Walmsley “Heralded generation of ultrafast single photons in pure quantum states,” Phys. Rev. Lett. 100, 133601 (2008). [CrossRef] [PubMed]

8.

R. Rangarajan, M. Goggin, and P. Kwiat “Optimizing type-I polarization-entangleg photons,” Opt. Express 17, 18920–18933 (2009). [CrossRef]

9.

P. S. K. Lee, M. P. van Exter, and J. P. Woerdman“Increased polarization-entangled photon flux via thinner crystals,” Phys. Rev. A 70, 043818 (2004). [CrossRef]

10.

B.-S. Shi and A. Tomita “Generation of a pulsed polarization entangled photon pair using a Sagnac interferometer,” Phys. Rev. A 69, 013803 (2004). [CrossRef]

11.

T. Kim, M. Fiorentino, and F. N. C. Wong “Phase-stable source of polarization-entangled photons using a polarization Sagnac interferometer,” Phys. Rev. A 73, 012316 (2006). [CrossRef]

12.

A. Fedrizzi, T. Herbst, A. Poppe, T. Jennewein, and A. Zeilinger “A wavelength-tunable fiber-coupled source of narrowband entangled photons,” Phys. Rev. A 77, 032314 (2008).

13.

O. Kuzucu and F. N. C. Wong “Pulsed Sagnac source of narrow-band polarization entangled photons,” Opt. Express 15, 15377–15386 (2007).

14.

M. Hentschel, H. Hübel, A. Poppe, and A. Zeilinger “Pulsed Sagnac source of narrow-band polarization entangled photons,” Opt. Express 15, 15377–15386 (2007).

15.

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

16.

D. R. Hamel “Realization of novel entangled photon sources using periodically poled materials,” Master’s thesis, University of Waterloo (2010).

17.

G. D. Boyd and D. A. Kleiman “Parametric interaction of focused Gaussian light beams,” J. Appl. Phys. 39, 3597–3640 (1968). [CrossRef]

18.

R. S. Bennink“Optimal collinear Gaussian beams for spontaneous parametric down-conversion,” Phys. Rev. A 81, 053805 (2010). [CrossRef]

19.

H. E. Major, C. B. E. Gawith, and P. G. R. Smith “Gouy phase compensation in quasi phase matching,” Opt. Commun. 281, 5036–5040 (2008). [CrossRef]

20.

N. Lastzka and R. Schnabel “The Gouy phase shift in nonlinear interactions of waves,” Opt. Express 15, 7211–7217 (2007). [CrossRef] [PubMed]

21.

Engineering metrology toolbox, http://emtoolbox.nist.gov/Wavelength/Ciddor.asp.

22.

P. E. Ciddor “Refractive index of air: new equations for the visible and near infrared,” Appl. Opt. 35, 1566–1573 (1996). [CrossRef] [PubMed]

OCIS Codes
(000.1600) General : Classical and quantum physics
(190.4410) Nonlinear optics : Nonlinear optics, parametric processes
(270.0270) Quantum optics : Quantum optics

ToC Category:
Quantum Optics

History
Original Manuscript: September 11, 2012
Revised Manuscript: October 4, 2012
Manuscript Accepted: October 4, 2012
Published: October 17, 2012

Citation
Ana Predojević, Stephanie Grabher, and Gregor Weihs, "Pulsed Sagnac source of polarization entangled photon pairs," Opt. Express 20, 25022-25029 (2012)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-22-25022


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References

  1. D. Bouwmeester, J.-W. Pan, K. Mattle, M. Eibl, H. Weinfurter, and A. Zeilinger, “Experimental quantum teleportation,” Nature390, 575–579 (1997). [CrossRef]
  2. T. Jennewein, G. Weihs, J.-W. Pan, and A. Zeilinger “Experimental nonlocality proof of quantum teleportation and entanglement swapping,” Phys. Rev. Lett.88, 017903 (2002). [CrossRef] [PubMed]
  3. J.-W. Pan, S. Gasparoni, R. Ursin, G. Weihs, and A. Zeilinger, “Experimental entanglement purification of arbitrary unknown states,” Nature423, 417–422 (2003). [CrossRef] [PubMed]
  4. J.-W. Pan, M. Daniell, S. Gasparoni, G. Weihs, and A. Zeilinger “Experimental demonstration of four-photon entanglement and high-fidelity teleportation,” Phys. Rev. Lett.86, 4435–4438 (2001). [CrossRef] [PubMed]
  5. M. Eibl, N. Kiesel, M. Bourennane, C. Kurtsiefer, and H. Weinfurter “Experimental realization of a three-qubit entangled W state,” Phys. Rev. Lett.92, 077901 (2004). [CrossRef] [PubMed]
  6. M. Hendrych, M. Mičuda, and J. P. Torres “Tunable control of the frequency correlations of entangled photons,” Opt. Lett.32(16,) 2339–2341 (2007). [CrossRef] [PubMed]
  7. P. J. Mosley, J. S. Lundeen, B. J. Smith, P. Wasylczyk, A. B. U’Ren, C. Silberhorn, and I. A. Walmsley “Heralded generation of ultrafast single photons in pure quantum states,” Phys. Rev. Lett.100, 133601 (2008). [CrossRef] [PubMed]
  8. R. Rangarajan, M. Goggin, and P. Kwiat “Optimizing type-I polarization-entangleg photons,” Opt. Express17, 18920–18933 (2009). [CrossRef]
  9. P. S. K. Lee, M. P. van Exter, and J. P. Woerdman“Increased polarization-entangled photon flux via thinner crystals,” Phys. Rev. A70, 043818 (2004). [CrossRef]
  10. B.-S. Shi and A. Tomita “Generation of a pulsed polarization entangled photon pair using a Sagnac interferometer,” Phys. Rev. A69, 013803 (2004). [CrossRef]
  11. T. Kim, M. Fiorentino, and F. N. C. Wong “Phase-stable source of polarization-entangled photons using a polarization Sagnac interferometer,” Phys. Rev. A73, 012316 (2006). [CrossRef]
  12. A. Fedrizzi, T. Herbst, A. Poppe, T. Jennewein, and A. Zeilinger “A wavelength-tunable fiber-coupled source of narrowband entangled photons,” Phys. Rev. A77, 032314 (2008).
  13. O. Kuzucu and F. N. C. Wong “Pulsed Sagnac source of narrow-band polarization entangled photons,” Opt. Express15, 15377–15386 (2007).
  14. M. Hentschel, H. Hübel, A. Poppe, and A. Zeilinger “Pulsed Sagnac source of narrow-band polarization entangled photons,” Opt. Express15, 15377–15386 (2007).
  15. D. F. V. James, P. Kwiat, W. Munro, and A. White“Measurement of qubits,” Phys. Rev. A64, 052312 (2001). [CrossRef]
  16. D. R. Hamel “Realization of novel entangled photon sources using periodically poled materials,” Master’s thesis, University of Waterloo (2010).
  17. G. D. Boyd and D. A. Kleiman “Parametric interaction of focused Gaussian light beams,” J. Appl. Phys.39, 3597–3640 (1968). [CrossRef]
  18. R. S. Bennink“Optimal collinear Gaussian beams for spontaneous parametric down-conversion,” Phys. Rev. A81, 053805 (2010). [CrossRef]
  19. H. E. Major, C. B. E. Gawith, and P. G. R. Smith “Gouy phase compensation in quasi phase matching,” Opt. Commun.281, 5036–5040 (2008). [CrossRef]
  20. N. Lastzka and R. Schnabel “The Gouy phase shift in nonlinear interactions of waves,” Opt. Express15, 7211–7217 (2007). [CrossRef] [PubMed]
  21. Engineering metrology toolbox, http://emtoolbox.nist.gov/Wavelength/Ciddor.asp .
  22. P. E. Ciddor “Refractive index of air: new equations for the visible and near infrared,” Appl. Opt.35, 1566–1573 (1996). [CrossRef] [PubMed]

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