Abstract

We present a consistent multimode theory that describes the coupling of single photons generated by collinear Type-I parametric down-conversion into single-mode optical fibers. We have calculated an analytic expression for the fiber diameter which maximizes the pair photon count rate. For a given focal length and wavelength, a lower limit of the fiber diameter for satisfactory coupling is obtained.

©2004 Optical Society of America

1. Introduction

The optical process of spontaneous parametric down-conversion (SPDC) involves the virtual absorption and spontaneous splitting of an incident (pump) photon in a transparent nonlinear crystal producing two lower-frequency (signal and idler) photons [13]. The pairs of photons can be entangled in a multi-parameter space of frequency, momentum and polarization. In type-I SPDC the photons are frequency-entangled and the signal and idler photons have parallel polarizations orthogonal to the pump polarization. Entangled photons have been used to demonstrate quantum nonlocality [4,5], quantum teleportation [68] and, more recently, quantum information processing [911] and quantum cryptography [12,13]. One of the challenges in recent practical schemes such as quantum cryptography and quantum communication is to maximize the efficiency of coupling of single photons into single-mode optical fibers. Several models and experiments have been developed to predict and measure the coupling efficiencies of down-converted light into single-mode fibers [14,15,16]. In the model and experiment of Kurtsiefer et. al. [14], the angular distribution of non-collinear type-II down-converted light for a given spectral bandwidth is calculated. Their idea is to match the angular distribution of the photon pairs to the angular width of the fiber mode. Bovino et. al. [16] produced a more rigorous model in which the dependence of the coupling efficiency on crystal length and walk-off were investigated. The importance of hyper-entanglement in type-I down-conversion and its possible use as a resource in the field quantum information has been recently discussed [22]. Single photon on-demand sources have also been designed using an array of type-I down-converters [23]. In contrast to previous work, we present a detailed theoretical model describing single-photon mode coupling in the simple situation of collinear type-I down-conversion. The pair photon count rate is calculated and an analytic expression which determines the condition for optimum single-photon coupling in single-mode optical fibers is obtained in terms of experimental parameters.

2. Amplitude for pair detection in single-mode fibers

The amplitude for detecting photon pairs at conjugate space-time points (r1,t 1) and (r2,t 2) is defined by

A(2)=EH+(r1,t1)EH+(r2,t2)

where t 1 and t 2 are detection times of signal and idler photons and EH(ri, t) are the Heisenberg electric field operators [18]. In the steady state the right-hand-side of Eq. (1) can be expressed as

d3r3d3k1d3k2Uk1λ1*(r3)Uk2λ2*(r3)Uk0λ0(r3)fp(r3)(ωk02ε0)12(ωk12ε0)12(ωk22ε0)12
×αk0,0EI(+)(r2,t2)EI(+)(r1,t1)ωk0,k1,k2δ(ωk1ωk2ωk0)

EI (ri ,t) are the interaction-picture electric field operators at some arbitrary but specified detection points r1 and r2 and fp (r3) is a function which describes the shape of the pump in the transverse direction; Ukiλi(r3) are plane-wave modes describing the electromagnetic field in the crystal with k1,k2 as the wave vectors of the signal and idler photons, k0 is the wave vector of the incident pump photon and λi are polarization indices. (As in our previous studies we have not introduced the effects of a change in the linear refractive index between the crystal and its surroundings. The incorporation of this difference does not qualitatively change our conclusions). We have therefore assumed that the nonlinear crystal is embedded in a medium whose linear refractive is the same as that of the crystal. Optical fibers used in the analysis are assumed to have the same refractive index as the crystal.

The initial state of the electromagnetic field |0,αk0, consists of a coherent state with wave-vector k 0 (the monochromatic pump beam) with other modes in the vacuum state |0〉. We take the quantized electric field in the fiber as

E+(r,t)=iλλd3kd3k(ωk2ε0)12εkλβkλ,kλUkλf(r)eiωtakλ

where the coupling coefficient β k⃗λ,kλ is a projection onto the appropriate fiber mode and is defined as [19]

βkλkλ=dxdyUkλin(x,y)Ukλf*(x,y)

Ukλin(r⃗) are spatial modes incident on the fiber and Ukλf(r⃗) are the fiber modes. For single-mode fibers we assume a spatial mode profile that is independent of k⃗ and λ. We may therefore suppress the fiber mode indices in Eq. (4) to simplify the notation. After performing the integration over the volume of the crystal and evaluating the expectation value, Eq. (2) simplifies to

d3k1d3k2f˜p(k1t+k2t)sinc(Δk1zk2zd2)βk1λ1βk2λ2Uf(r1)Uf(r2)
×(ωk12ε0)12(ωk22ε0)12eiωk1t1eiωk2t2δ(ωk0ωk1ωk2)

where d is the length of the crystal, Δk1zk2z=k0k1zk2z and the signal/idler wave vectors are defined as ki =k ix +ŷk iy +zk iz where i=1,2 denote signal, idler modes and kix ,kiy ,kiz are components of the wave vector along the x,y,z directions respectively. p (k⃗) is the two-dimensional Fourier transform of fp (r⃗). For degenerate signal and idler photons emitted at a cone angle θ* to the pump and for frequencies close to the degenerate frequency, the sinc function in Eq. (5) can be approximated to unity in a first-order analysis. If we assume that the divergence of the pump is negligible over the length of the crystal, i.e., the crystal is sufficiently thin, and that the transverse shape of the pump is gaussian, we can take the shape function as

f(rt)exp(x2+y2wp)

Thus, f˜p(k)exp(14wp2k2), where wp is the effective width of the pump beam. In terms of photon frequencies, a first-order analysis gives

f˜p(k1t+k2t)exp(14wp2v2(ωk1ωk2)2sin2θ*)

where v is the first-order dispersion coefficient of the nonlinear crystal. The collinear situation is obtained for θ*=0.

3. Calculation of the coupling coefficient

The coupling coefficient is defined in Eq.. (4) as the overlap integral of the input modes and the fiber mode. For our calculations in the collinear geometry the fibers are positioned along the z-axis. In experiments one needs to use a beam splitter to separate the beams as is shown in Fig. 1. A lens of focal length f is used to focus collinear signal and idler beams onto identical single-mode fibers through a 50/50 beam splitter. The pair count rate is then measured by single-photon detectors D1 and D2 connected to a correlator.

Since the beam splitter introduces constant phase factors into the pair detection amplitude, our calculations will also be valid for this experiment.

 

Fig. 1. Schematic of the experimental setup to determine pair photon coupling in fibers

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Assuming that the input face of the fibers is at the image plane of the lens and that plane-wave modes, described by the free-field interaction-picture electric field operators in Eq. (2), are incident on the lens, we can employ diffraction theory to obtain the electric field modes at the image plane of the lens [20]. We obtain

Ukλin(x,y)=1ikdeik(d+d)eik2d(x2+y2)02π0Rrdθdr(ei(k2dk2f)r2eik(xcosθd+ysinθd)r)

where d is the distance from the center of the crystal to the lens and d is the distance from the lens to the image plane. We are also assuming that a circular region of the lens of radius R is being irradiated by the down-converted light. We take the fiber modes, Uf (x, y), as Gaussian modes [21] defined by

Uf(x,y)=1w0πexp(x2+y22w02)

where w0 is the radius of the fiber mode. Substituting Eq. (8) and Eq. (9) in Eq. (4) we obtain the coupling coefficient β as

βkλw02d2+ikw04dkdw0(d2+k2w04)(1e(Ak+iBk)R2)(Ak+iBk)

where Ak and Bk are defined as

Ak=k2w022(d2+k2w04)
Bk=(12d12f)kk3w042d(d2+k2w04)

4. Two-photon count rate in single-mode fibers

Consider the situation in which the fiber is in the focal plane of the lens, i.e., d =f. We assume that Gaussian filters are used to select frequencies close to the degenerate frequency. Then for k 2w04f 2 and to first-order in the dispersion of the crystal, the two-photon amplitude in Eq. (5) approximates to

A(2)eiτxx2Δ2(1e(a+bx))2dx

where

a=(k*2w022f2ik*3w042f3)R2
b=(k*w02ngcf2i32ngk*2w042cf3)R2

τ is the time difference between the detection of signal and idler photons, k* is the degenerate phase-matched wave number, ng is the group refractive index and Δ is the bandwidth of the down-converted light reaching the fibers. To obtain the count rate we take the modulus square of the integral in Eq. (12). Typical photon detectors have resolving times much larger than the coherence time of the down-converted photons. The limits in the integration with respect to τ may then be taken as effectively from τ=-∞ to τ=∞.

The τ integration gives a delta function which reduces a double integral to a single x-integral. After performing the x-integral and using the approximation k 2w04f 2 for a typical bandwidth Δ≈1012 rads-1, we obtain, to a good approximation,

C(14ey+6e2y4e3y+e4y)

where C is the pair count rate and y=k*2w02R22f2. This approaches a maximum when y≥4.4. This gives the fiber radius w01.5fλπR for optimal coupling where λ is the wavelength in the fiber. For a fiber diameter equal to the Rayleigh width of the diffraction pattern of the aperture, i.e., w0=fλπR, the two-photon count rate in the fiber is only 56% of the maximum.

5. Conclusion

We have presented, from a first-principles calculation, a fully quantized multimode theory to describe the coupling of single photons generated by collinear Type-I SPDC into single-mode optical fibers. We have thus produced a practical recipe to maximize the coupling of single photons into optical fibers. As pointed out earlier this is important for quantum cryptography and quantum communication applications. For the type-I collinear down-conversion studied here, we find no significant dependence of the coupling on the transverse width of the pump beam and crystal length which have been obtained for non-collinear geometries. The important parameters for the collinear case are simply the photon wavelength, the focal length of the lens and the fiber diameter.

References and links

1. D. C. Burnham and D. L. Weinberg, “Observation of simultaneity in parametric production of optical photon pairs,” Phys. Rev. Lett. 25, 84–87 (1970). [CrossRef]  

2. Z. Y. Ou, X. Y. Zou, L. J. Wang, and L. Mandel, “Experiment on nonclassical fourth-order interference,” Phys. Rev. A 42, 2957–2965 (1990). [CrossRef]   [PubMed]  

3. C. K. Hong and L. Mandel, “Theory of parametric frequency down-conversion of light,” Phys. Rev. A 31, 2409–2418 (1985). [CrossRef]   [PubMed]  

4. Z. Y. Ou and L. Mandel, “Violation of Bell’s inequality and classical probability in a two-photon correlation experiment,” Phys. Rev. Lett. 61, 50–53 (1988). [CrossRef]   [PubMed]  

5. P. G. Kwiat, K. Mattle, H. Weinfurter, A. Zeilinger, A. V. Sergienko, and Y. H. Shih, Phys. Rev. Lett. 75, 4337–4341 (1995). [CrossRef]   [PubMed]  

6. C. H. Bennett, G. Brassard, C. Crepeau, R. Jozsa, A. Peres, and W. K. Wootters, “Teleporting an unknown quantum state via classical and Einstein-Podolsky-Rosen channels,” Phys. Rev. Lett. 70, 1895–1899 (1993). [CrossRef]   [PubMed]  

7. S. L. Braunstein and H. J. Kimble, “Teleportation of continuous quantum variables,” Phys. Rev. Lett. 80, 869–872 (1998). [CrossRef]  

8. L. Vaidman, “Teleportation of quantum states,” Phys. Rev. A 49, 1473–1476 (1994). [CrossRef]   [PubMed]  

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

10. J-W Pan, D. Bouwmeester, H. Weinfurter, and A. Zeilinger, “Experimental entanglement swapping : Entangling photons that never interacted,” Phys. Rev. Lett. 80, 3891–3894 (1998). [CrossRef]  

11. D. Bouwmeester, J-W Pan, M. Daniell, H. Weinfurter, and A. Zeilinger, Phys. Rev. Lett. 82, 1345–1349 (1999). [CrossRef]  

12. A. K. Ekert, J. G. Rarity, P. R. Tapster, and G. M. Palma, “Practical quantum cryptography based on two-photon interferometry,” Phys. Rev. Lett. 69 (9), 1293–1295 (1992). [CrossRef]   [PubMed]  

13. W. Tittel, J. Brendel, H. Zbinden, and N. Gisin, “Quantum cryptography using entangled photons in energy-time Bell states,” Phys. Rev. Lett. 84 (20), 4737–4740, 2000. [CrossRef]   [PubMed]  

14. C. Kurtsiefer, M. Oberparleiter, and H. Weinfurter, “High-efficiency entangled photon pair collection in type-II parametric fluorescence,” Phys. Rev. A 64, 023802 (2001). [CrossRef]  

15. S. Castelletto, I. P. Degiovanni, M. Ware, and A. Migdall, “Coupling efficiencies in single photon on-demand sources,” ArXiv:quant-ph/0311099 (2003).

16. F. A. Bovino, P. Varisco, A. M. Colla, G. Castagnoli, G. D. Giuseppe, and A. V. Sergienko, “Effective fiber coupling of entangled photons for quantum communication,” Opt. Commun. 227, 343–348 (2003). [CrossRef]  

17. E. R. Pike and S. Sarkar, The Quantum Theory of Radiation, Oxford University Press (1995).

18. R. Andrews, E. R. Pike, and Sarben Sarkar, “Photon correlations and interference in type-I optical parametric down-conversion,” J. Opt. B: Quantum semiclass. Opt. 1, 588–597 (1999). [CrossRef]  

19. A. Yariv, Optical Electronics in Modern Communications (5th Edition, Oxford Series in Electrical and Computer Engineering).

20. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), chaps. 7 and 8.

21. Q. Cao and S. Chi, “Approximate Analytical Description for Fundamental-Mode Fields of Graded-Index Fibers: Beyond the Gaussian Approximation,” J. Lightwave Technol. 19, 54- (2001). [CrossRef]  

22. A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature 412, 313–316 (2001). [CrossRef]   [PubMed]  

23. A. L. Migdall, D. Branning, and S. Castelleto, “Tailoring single-photon and multiphoton probabilities of a single-photon on-demand source,” Phys. Rev. A , 66, 053805-1–053805-4 (2002). [CrossRef]  

References

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  1. D. C. Burnham and D. L. Weinberg, “Observation of simultaneity in parametric production of optical photon pairs,” Phys. Rev. Lett. 25, 84–87 (1970).
    [Crossref]
  2. Z. Y. Ou, X. Y. Zou, L. J. Wang, and L. Mandel, “Experiment on nonclassical fourth-order interference,” Phys. Rev. A 42, 2957–2965 (1990).
    [Crossref] [PubMed]
  3. C. K. Hong and L. Mandel, “Theory of parametric frequency down-conversion of light,” Phys. Rev. A 31, 2409–2418 (1985).
    [Crossref] [PubMed]
  4. Z. Y. Ou and L. Mandel, “Violation of Bell’s inequality and classical probability in a two-photon correlation experiment,” Phys. Rev. Lett. 61, 50–53 (1988).
    [Crossref] [PubMed]
  5. P. G. Kwiat, K. Mattle, H. Weinfurter, A. Zeilinger, A. V. Sergienko, and Y. H. Shih, Phys. Rev. Lett. 75, 4337–4341 (1995).
    [Crossref] [PubMed]
  6. C. H. Bennett, G. Brassard, C. Crepeau, R. Jozsa, A. Peres, and W. K. Wootters, “Teleporting an unknown quantum state via classical and Einstein-Podolsky-Rosen channels,” Phys. Rev. Lett. 70, 1895–1899 (1993).
    [Crossref] [PubMed]
  7. S. L. Braunstein and H. J. Kimble, “Teleportation of continuous quantum variables,” Phys. Rev. Lett. 80, 869–872 (1998).
    [Crossref]
  8. L. Vaidman, “Teleportation of quantum states,” Phys. Rev. A 49, 1473–1476 (1994).
    [Crossref] [PubMed]
  9. D. Bouwmeester, J-W Pan, K. Mattle, M. Eibl, H. Weinfurter, and A. Zeilinger, “Experimental quantum teleportation,” Nature 390, 575–579 (1997).
    [Crossref]
  10. J-W Pan, D. Bouwmeester, H. Weinfurter, and A. Zeilinger, “Experimental entanglement swapping : Entangling photons that never interacted,” Phys. Rev. Lett. 80, 3891–3894 (1998).
    [Crossref]
  11. D. Bouwmeester, J-W Pan, M. Daniell, H. Weinfurter, and A. Zeilinger, Phys. Rev. Lett. 82, 1345–1349 (1999).
    [Crossref]
  12. A. K. Ekert, J. G. Rarity, P. R. Tapster, and G. M. Palma, “Practical quantum cryptography based on two-photon interferometry,” Phys. Rev. Lett. 69 (9), 1293–1295 (1992).
    [Crossref] [PubMed]
  13. W. Tittel, J. Brendel, H. Zbinden, and N. Gisin, “Quantum cryptography using entangled photons in energy-time Bell states,” Phys. Rev. Lett. 84 (20), 4737–4740, 2000.
    [Crossref] [PubMed]
  14. C. Kurtsiefer, M. Oberparleiter, and H. Weinfurter, “High-efficiency entangled photon pair collection in type-II parametric fluorescence,” Phys. Rev. A 64, 023802 (2001).
    [Crossref]
  15. S. Castelletto, I. P. Degiovanni, M. Ware, and A. Migdall, “Coupling efficiencies in single photon on-demand sources,” ArXiv:quant-ph/0311099 (2003).
  16. F. A. Bovino, P. Varisco, A. M. Colla, G. Castagnoli, G. D. Giuseppe, and A. V. Sergienko, “Effective fiber coupling of entangled photons for quantum communication,” Opt. Commun. 227, 343–348 (2003).
    [Crossref]
  17. E. R. Pike and S. Sarkar, The Quantum Theory of Radiation, Oxford University Press (1995).
  18. R. Andrews, E. R. Pike, and Sarben Sarkar, “Photon correlations and interference in type-I optical parametric down-conversion,” J. Opt. B: Quantum semiclass. Opt. 1, 588–597 (1999).
    [Crossref]
  19. A. Yariv, Optical Electronics in Modern Communications (5th Edition, Oxford Series in Electrical and Computer Engineering).
  20. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), chaps. 7 and 8.
  21. Q. Cao and S. Chi, “Approximate Analytical Description for Fundamental-Mode Fields of Graded-Index Fibers: Beyond the Gaussian Approximation,” J. Lightwave Technol. 19, 54- (2001).
    [Crossref]
  22. A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature 412, 313–316 (2001).
    [Crossref] [PubMed]
  23. A. L. Migdall, D. Branning, and S. Castelleto, “Tailoring single-photon and multiphoton probabilities of a single-photon on-demand source,” Phys. Rev. A,  66, 053805-1–053805-4 (2002).
    [Crossref]

2003 (1)

F. A. Bovino, P. Varisco, A. M. Colla, G. Castagnoli, G. D. Giuseppe, and A. V. Sergienko, “Effective fiber coupling of entangled photons for quantum communication,” Opt. Commun. 227, 343–348 (2003).
[Crossref]

2002 (1)

A. L. Migdall, D. Branning, and S. Castelleto, “Tailoring single-photon and multiphoton probabilities of a single-photon on-demand source,” Phys. Rev. A,  66, 053805-1–053805-4 (2002).
[Crossref]

2001 (3)

Q. Cao and S. Chi, “Approximate Analytical Description for Fundamental-Mode Fields of Graded-Index Fibers: Beyond the Gaussian Approximation,” J. Lightwave Technol. 19, 54- (2001).
[Crossref]

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature 412, 313–316 (2001).
[Crossref] [PubMed]

C. Kurtsiefer, M. Oberparleiter, and H. Weinfurter, “High-efficiency entangled photon pair collection in type-II parametric fluorescence,” Phys. Rev. A 64, 023802 (2001).
[Crossref]

2000 (1)

W. Tittel, J. Brendel, H. Zbinden, and N. Gisin, “Quantum cryptography using entangled photons in energy-time Bell states,” Phys. Rev. Lett. 84 (20), 4737–4740, 2000.
[Crossref] [PubMed]

1999 (2)

R. Andrews, E. R. Pike, and Sarben Sarkar, “Photon correlations and interference in type-I optical parametric down-conversion,” J. Opt. B: Quantum semiclass. Opt. 1, 588–597 (1999).
[Crossref]

D. Bouwmeester, J-W Pan, M. Daniell, H. Weinfurter, and A. Zeilinger, Phys. Rev. Lett. 82, 1345–1349 (1999).
[Crossref]

1998 (2)

J-W Pan, D. Bouwmeester, H. Weinfurter, and A. Zeilinger, “Experimental entanglement swapping : Entangling photons that never interacted,” Phys. Rev. Lett. 80, 3891–3894 (1998).
[Crossref]

S. L. Braunstein and H. J. Kimble, “Teleportation of continuous quantum variables,” Phys. Rev. Lett. 80, 869–872 (1998).
[Crossref]

1997 (1)

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

1995 (1)

P. G. Kwiat, K. Mattle, H. Weinfurter, A. Zeilinger, A. V. Sergienko, and Y. H. Shih, Phys. Rev. Lett. 75, 4337–4341 (1995).
[Crossref] [PubMed]

1994 (1)

L. Vaidman, “Teleportation of quantum states,” Phys. Rev. A 49, 1473–1476 (1994).
[Crossref] [PubMed]

1993 (1)

C. H. Bennett, G. Brassard, C. Crepeau, R. Jozsa, A. Peres, and W. K. Wootters, “Teleporting an unknown quantum state via classical and Einstein-Podolsky-Rosen channels,” Phys. Rev. Lett. 70, 1895–1899 (1993).
[Crossref] [PubMed]

1992 (1)

A. K. Ekert, J. G. Rarity, P. R. Tapster, and G. M. Palma, “Practical quantum cryptography based on two-photon interferometry,” Phys. Rev. Lett. 69 (9), 1293–1295 (1992).
[Crossref] [PubMed]

1990 (1)

Z. Y. Ou, X. Y. Zou, L. J. Wang, and L. Mandel, “Experiment on nonclassical fourth-order interference,” Phys. Rev. A 42, 2957–2965 (1990).
[Crossref] [PubMed]

1988 (1)

Z. Y. Ou and L. Mandel, “Violation of Bell’s inequality and classical probability in a two-photon correlation experiment,” Phys. Rev. Lett. 61, 50–53 (1988).
[Crossref] [PubMed]

1985 (1)

C. K. Hong and L. Mandel, “Theory of parametric frequency down-conversion of light,” Phys. Rev. A 31, 2409–2418 (1985).
[Crossref] [PubMed]

1970 (1)

D. C. Burnham and D. L. Weinberg, “Observation of simultaneity in parametric production of optical photon pairs,” Phys. Rev. Lett. 25, 84–87 (1970).
[Crossref]

Andrews, R.

R. Andrews, E. R. Pike, and Sarben Sarkar, “Photon correlations and interference in type-I optical parametric down-conversion,” J. Opt. B: Quantum semiclass. Opt. 1, 588–597 (1999).
[Crossref]

Bennett, C. H.

C. H. Bennett, G. Brassard, C. Crepeau, R. Jozsa, A. Peres, and W. K. Wootters, “Teleporting an unknown quantum state via classical and Einstein-Podolsky-Rosen channels,” Phys. Rev. Lett. 70, 1895–1899 (1993).
[Crossref] [PubMed]

Bouwmeester, D.

D. Bouwmeester, J-W Pan, M. Daniell, H. Weinfurter, and A. Zeilinger, Phys. Rev. Lett. 82, 1345–1349 (1999).
[Crossref]

J-W Pan, D. Bouwmeester, H. Weinfurter, and A. Zeilinger, “Experimental entanglement swapping : Entangling photons that never interacted,” Phys. Rev. Lett. 80, 3891–3894 (1998).
[Crossref]

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

Bovino, F. A.

F. A. Bovino, P. Varisco, A. M. Colla, G. Castagnoli, G. D. Giuseppe, and A. V. Sergienko, “Effective fiber coupling of entangled photons for quantum communication,” Opt. Commun. 227, 343–348 (2003).
[Crossref]

Branning, D.

A. L. Migdall, D. Branning, and S. Castelleto, “Tailoring single-photon and multiphoton probabilities of a single-photon on-demand source,” Phys. Rev. A,  66, 053805-1–053805-4 (2002).
[Crossref]

Brassard, G.

C. H. Bennett, G. Brassard, C. Crepeau, R. Jozsa, A. Peres, and W. K. Wootters, “Teleporting an unknown quantum state via classical and Einstein-Podolsky-Rosen channels,” Phys. Rev. Lett. 70, 1895–1899 (1993).
[Crossref] [PubMed]

Braunstein, S. L.

S. L. Braunstein and H. J. Kimble, “Teleportation of continuous quantum variables,” Phys. Rev. Lett. 80, 869–872 (1998).
[Crossref]

Brendel, J.

W. Tittel, J. Brendel, H. Zbinden, and N. Gisin, “Quantum cryptography using entangled photons in energy-time Bell states,” Phys. Rev. Lett. 84 (20), 4737–4740, 2000.
[Crossref] [PubMed]

Burnham, D. C.

D. C. Burnham and D. L. Weinberg, “Observation of simultaneity in parametric production of optical photon pairs,” Phys. Rev. Lett. 25, 84–87 (1970).
[Crossref]

Cao, Q.

Q. Cao and S. Chi, “Approximate Analytical Description for Fundamental-Mode Fields of Graded-Index Fibers: Beyond the Gaussian Approximation,” J. Lightwave Technol. 19, 54- (2001).
[Crossref]

Castagnoli, G.

F. A. Bovino, P. Varisco, A. M. Colla, G. Castagnoli, G. D. Giuseppe, and A. V. Sergienko, “Effective fiber coupling of entangled photons for quantum communication,” Opt. Commun. 227, 343–348 (2003).
[Crossref]

Castelleto, S.

A. L. Migdall, D. Branning, and S. Castelleto, “Tailoring single-photon and multiphoton probabilities of a single-photon on-demand source,” Phys. Rev. A,  66, 053805-1–053805-4 (2002).
[Crossref]

Castelletto, S.

S. Castelletto, I. P. Degiovanni, M. Ware, and A. Migdall, “Coupling efficiencies in single photon on-demand sources,” ArXiv:quant-ph/0311099 (2003).

Chi, S.

Q. Cao and S. Chi, “Approximate Analytical Description for Fundamental-Mode Fields of Graded-Index Fibers: Beyond the Gaussian Approximation,” J. Lightwave Technol. 19, 54- (2001).
[Crossref]

Colla, A. M.

F. A. Bovino, P. Varisco, A. M. Colla, G. Castagnoli, G. D. Giuseppe, and A. V. Sergienko, “Effective fiber coupling of entangled photons for quantum communication,” Opt. Commun. 227, 343–348 (2003).
[Crossref]

Crepeau, C.

C. H. Bennett, G. Brassard, C. Crepeau, R. Jozsa, A. Peres, and W. K. Wootters, “Teleporting an unknown quantum state via classical and Einstein-Podolsky-Rosen channels,” Phys. Rev. Lett. 70, 1895–1899 (1993).
[Crossref] [PubMed]

Daniell, M.

D. Bouwmeester, J-W Pan, M. Daniell, H. Weinfurter, and A. Zeilinger, Phys. Rev. Lett. 82, 1345–1349 (1999).
[Crossref]

Degiovanni, I. P.

S. Castelletto, I. P. Degiovanni, M. Ware, and A. Migdall, “Coupling efficiencies in single photon on-demand sources,” ArXiv:quant-ph/0311099 (2003).

Eibl, M.

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

Ekert, A. K.

A. K. Ekert, J. G. Rarity, P. R. Tapster, and G. M. Palma, “Practical quantum cryptography based on two-photon interferometry,” Phys. Rev. Lett. 69 (9), 1293–1295 (1992).
[Crossref] [PubMed]

Gisin, N.

W. Tittel, J. Brendel, H. Zbinden, and N. Gisin, “Quantum cryptography using entangled photons in energy-time Bell states,” Phys. Rev. Lett. 84 (20), 4737–4740, 2000.
[Crossref] [PubMed]

Giuseppe, G. D.

F. A. Bovino, P. Varisco, A. M. Colla, G. Castagnoli, G. D. Giuseppe, and A. V. Sergienko, “Effective fiber coupling of entangled photons for quantum communication,” Opt. Commun. 227, 343–348 (2003).
[Crossref]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), chaps. 7 and 8.

Hong, C. K.

C. K. Hong and L. Mandel, “Theory of parametric frequency down-conversion of light,” Phys. Rev. A 31, 2409–2418 (1985).
[Crossref] [PubMed]

Jozsa, R.

C. H. Bennett, G. Brassard, C. Crepeau, R. Jozsa, A. Peres, and W. K. Wootters, “Teleporting an unknown quantum state via classical and Einstein-Podolsky-Rosen channels,” Phys. Rev. Lett. 70, 1895–1899 (1993).
[Crossref] [PubMed]

Kimble, H. J.

S. L. Braunstein and H. J. Kimble, “Teleportation of continuous quantum variables,” Phys. Rev. Lett. 80, 869–872 (1998).
[Crossref]

Kurtsiefer, C.

C. Kurtsiefer, M. Oberparleiter, and H. Weinfurter, “High-efficiency entangled photon pair collection in type-II parametric fluorescence,” Phys. Rev. A 64, 023802 (2001).
[Crossref]

Kwiat, P. G.

P. G. Kwiat, K. Mattle, H. Weinfurter, A. Zeilinger, A. V. Sergienko, and Y. H. Shih, Phys. Rev. Lett. 75, 4337–4341 (1995).
[Crossref] [PubMed]

Mair, A.

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature 412, 313–316 (2001).
[Crossref] [PubMed]

Mandel, L.

Z. Y. Ou, X. Y. Zou, L. J. Wang, and L. Mandel, “Experiment on nonclassical fourth-order interference,” Phys. Rev. A 42, 2957–2965 (1990).
[Crossref] [PubMed]

Z. Y. Ou and L. Mandel, “Violation of Bell’s inequality and classical probability in a two-photon correlation experiment,” Phys. Rev. Lett. 61, 50–53 (1988).
[Crossref] [PubMed]

C. K. Hong and L. Mandel, “Theory of parametric frequency down-conversion of light,” Phys. Rev. A 31, 2409–2418 (1985).
[Crossref] [PubMed]

Mattle, K.

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

P. G. Kwiat, K. Mattle, H. Weinfurter, A. Zeilinger, A. V. Sergienko, and Y. H. Shih, Phys. Rev. Lett. 75, 4337–4341 (1995).
[Crossref] [PubMed]

Migdall, A.

S. Castelletto, I. P. Degiovanni, M. Ware, and A. Migdall, “Coupling efficiencies in single photon on-demand sources,” ArXiv:quant-ph/0311099 (2003).

Migdall, A. L.

A. L. Migdall, D. Branning, and S. Castelleto, “Tailoring single-photon and multiphoton probabilities of a single-photon on-demand source,” Phys. Rev. A,  66, 053805-1–053805-4 (2002).
[Crossref]

Oberparleiter, M.

C. Kurtsiefer, M. Oberparleiter, and H. Weinfurter, “High-efficiency entangled photon pair collection in type-II parametric fluorescence,” Phys. Rev. A 64, 023802 (2001).
[Crossref]

Ou, Z. Y.

Z. Y. Ou, X. Y. Zou, L. J. Wang, and L. Mandel, “Experiment on nonclassical fourth-order interference,” Phys. Rev. A 42, 2957–2965 (1990).
[Crossref] [PubMed]

Z. Y. Ou and L. Mandel, “Violation of Bell’s inequality and classical probability in a two-photon correlation experiment,” Phys. Rev. Lett. 61, 50–53 (1988).
[Crossref] [PubMed]

Palma, G. M.

A. K. Ekert, J. G. Rarity, P. R. Tapster, and G. M. Palma, “Practical quantum cryptography based on two-photon interferometry,” Phys. Rev. Lett. 69 (9), 1293–1295 (1992).
[Crossref] [PubMed]

Pan, J-W

D. Bouwmeester, J-W Pan, M. Daniell, H. Weinfurter, and A. Zeilinger, Phys. Rev. Lett. 82, 1345–1349 (1999).
[Crossref]

J-W Pan, D. Bouwmeester, H. Weinfurter, and A. Zeilinger, “Experimental entanglement swapping : Entangling photons that never interacted,” Phys. Rev. Lett. 80, 3891–3894 (1998).
[Crossref]

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

Peres, A.

C. H. Bennett, G. Brassard, C. Crepeau, R. Jozsa, A. Peres, and W. K. Wootters, “Teleporting an unknown quantum state via classical and Einstein-Podolsky-Rosen channels,” Phys. Rev. Lett. 70, 1895–1899 (1993).
[Crossref] [PubMed]

Pike, E. R.

R. Andrews, E. R. Pike, and Sarben Sarkar, “Photon correlations and interference in type-I optical parametric down-conversion,” J. Opt. B: Quantum semiclass. Opt. 1, 588–597 (1999).
[Crossref]

E. R. Pike and S. Sarkar, The Quantum Theory of Radiation, Oxford University Press (1995).

Rarity, J. G.

A. K. Ekert, J. G. Rarity, P. R. Tapster, and G. M. Palma, “Practical quantum cryptography based on two-photon interferometry,” Phys. Rev. Lett. 69 (9), 1293–1295 (1992).
[Crossref] [PubMed]

Sarkar, S.

E. R. Pike and S. Sarkar, The Quantum Theory of Radiation, Oxford University Press (1995).

Sarkar, Sarben

R. Andrews, E. R. Pike, and Sarben Sarkar, “Photon correlations and interference in type-I optical parametric down-conversion,” J. Opt. B: Quantum semiclass. Opt. 1, 588–597 (1999).
[Crossref]

Sergienko, A. V.

F. A. Bovino, P. Varisco, A. M. Colla, G. Castagnoli, G. D. Giuseppe, and A. V. Sergienko, “Effective fiber coupling of entangled photons for quantum communication,” Opt. Commun. 227, 343–348 (2003).
[Crossref]

P. G. Kwiat, K. Mattle, H. Weinfurter, A. Zeilinger, A. V. Sergienko, and Y. H. Shih, Phys. Rev. Lett. 75, 4337–4341 (1995).
[Crossref] [PubMed]

Shih, Y. H.

P. G. Kwiat, K. Mattle, H. Weinfurter, A. Zeilinger, A. V. Sergienko, and Y. H. Shih, Phys. Rev. Lett. 75, 4337–4341 (1995).
[Crossref] [PubMed]

Tapster, P. R.

A. K. Ekert, J. G. Rarity, P. R. Tapster, and G. M. Palma, “Practical quantum cryptography based on two-photon interferometry,” Phys. Rev. Lett. 69 (9), 1293–1295 (1992).
[Crossref] [PubMed]

Tittel, W.

W. Tittel, J. Brendel, H. Zbinden, and N. Gisin, “Quantum cryptography using entangled photons in energy-time Bell states,” Phys. Rev. Lett. 84 (20), 4737–4740, 2000.
[Crossref] [PubMed]

Vaidman, L.

L. Vaidman, “Teleportation of quantum states,” Phys. Rev. A 49, 1473–1476 (1994).
[Crossref] [PubMed]

Varisco, P.

F. A. Bovino, P. Varisco, A. M. Colla, G. Castagnoli, G. D. Giuseppe, and A. V. Sergienko, “Effective fiber coupling of entangled photons for quantum communication,” Opt. Commun. 227, 343–348 (2003).
[Crossref]

Vaziri, A.

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature 412, 313–316 (2001).
[Crossref] [PubMed]

Wang, L. J.

Z. Y. Ou, X. Y. Zou, L. J. Wang, and L. Mandel, “Experiment on nonclassical fourth-order interference,” Phys. Rev. A 42, 2957–2965 (1990).
[Crossref] [PubMed]

Ware, M.

S. Castelletto, I. P. Degiovanni, M. Ware, and A. Migdall, “Coupling efficiencies in single photon on-demand sources,” ArXiv:quant-ph/0311099 (2003).

Weihs, G.

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature 412, 313–316 (2001).
[Crossref] [PubMed]

Weinberg, D. L.

D. C. Burnham and D. L. Weinberg, “Observation of simultaneity in parametric production of optical photon pairs,” Phys. Rev. Lett. 25, 84–87 (1970).
[Crossref]

Weinfurter, H.

C. Kurtsiefer, M. Oberparleiter, and H. Weinfurter, “High-efficiency entangled photon pair collection in type-II parametric fluorescence,” Phys. Rev. A 64, 023802 (2001).
[Crossref]

D. Bouwmeester, J-W Pan, M. Daniell, H. Weinfurter, and A. Zeilinger, Phys. Rev. Lett. 82, 1345–1349 (1999).
[Crossref]

J-W Pan, D. Bouwmeester, H. Weinfurter, and A. Zeilinger, “Experimental entanglement swapping : Entangling photons that never interacted,” Phys. Rev. Lett. 80, 3891–3894 (1998).
[Crossref]

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

P. G. Kwiat, K. Mattle, H. Weinfurter, A. Zeilinger, A. V. Sergienko, and Y. H. Shih, Phys. Rev. Lett. 75, 4337–4341 (1995).
[Crossref] [PubMed]

Wootters, W. K.

C. H. Bennett, G. Brassard, C. Crepeau, R. Jozsa, A. Peres, and W. K. Wootters, “Teleporting an unknown quantum state via classical and Einstein-Podolsky-Rosen channels,” Phys. Rev. Lett. 70, 1895–1899 (1993).
[Crossref] [PubMed]

Yariv, A.

A. Yariv, Optical Electronics in Modern Communications (5th Edition, Oxford Series in Electrical and Computer Engineering).

Zbinden, H.

W. Tittel, J. Brendel, H. Zbinden, and N. Gisin, “Quantum cryptography using entangled photons in energy-time Bell states,” Phys. Rev. Lett. 84 (20), 4737–4740, 2000.
[Crossref] [PubMed]

Zeilinger, A.

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature 412, 313–316 (2001).
[Crossref] [PubMed]

D. Bouwmeester, J-W Pan, M. Daniell, H. Weinfurter, and A. Zeilinger, Phys. Rev. Lett. 82, 1345–1349 (1999).
[Crossref]

J-W Pan, D. Bouwmeester, H. Weinfurter, and A. Zeilinger, “Experimental entanglement swapping : Entangling photons that never interacted,” Phys. Rev. Lett. 80, 3891–3894 (1998).
[Crossref]

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

P. G. Kwiat, K. Mattle, H. Weinfurter, A. Zeilinger, A. V. Sergienko, and Y. H. Shih, Phys. Rev. Lett. 75, 4337–4341 (1995).
[Crossref] [PubMed]

Zou, X. Y.

Z. Y. Ou, X. Y. Zou, L. J. Wang, and L. Mandel, “Experiment on nonclassical fourth-order interference,” Phys. Rev. A 42, 2957–2965 (1990).
[Crossref] [PubMed]

J. Lightwave Technol. (1)

Q. Cao and S. Chi, “Approximate Analytical Description for Fundamental-Mode Fields of Graded-Index Fibers: Beyond the Gaussian Approximation,” J. Lightwave Technol. 19, 54- (2001).
[Crossref]

J. Opt. B: Quantum semiclass. Opt. (1)

R. Andrews, E. R. Pike, and Sarben Sarkar, “Photon correlations and interference in type-I optical parametric down-conversion,” J. Opt. B: Quantum semiclass. Opt. 1, 588–597 (1999).
[Crossref]

Nature (2)

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

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature 412, 313–316 (2001).
[Crossref] [PubMed]

Opt. Commun. (1)

F. A. Bovino, P. Varisco, A. M. Colla, G. Castagnoli, G. D. Giuseppe, and A. V. Sergienko, “Effective fiber coupling of entangled photons for quantum communication,” Opt. Commun. 227, 343–348 (2003).
[Crossref]

Phys. Rev. A (5)

L. Vaidman, “Teleportation of quantum states,” Phys. Rev. A 49, 1473–1476 (1994).
[Crossref] [PubMed]

C. Kurtsiefer, M. Oberparleiter, and H. Weinfurter, “High-efficiency entangled photon pair collection in type-II parametric fluorescence,” Phys. Rev. A 64, 023802 (2001).
[Crossref]

Z. Y. Ou, X. Y. Zou, L. J. Wang, and L. Mandel, “Experiment on nonclassical fourth-order interference,” Phys. Rev. A 42, 2957–2965 (1990).
[Crossref] [PubMed]

C. K. Hong and L. Mandel, “Theory of parametric frequency down-conversion of light,” Phys. Rev. A 31, 2409–2418 (1985).
[Crossref] [PubMed]

A. L. Migdall, D. Branning, and S. Castelleto, “Tailoring single-photon and multiphoton probabilities of a single-photon on-demand source,” Phys. Rev. A,  66, 053805-1–053805-4 (2002).
[Crossref]

Phys. Rev. Lett. (9)

Z. Y. Ou and L. Mandel, “Violation of Bell’s inequality and classical probability in a two-photon correlation experiment,” Phys. Rev. Lett. 61, 50–53 (1988).
[Crossref] [PubMed]

P. G. Kwiat, K. Mattle, H. Weinfurter, A. Zeilinger, A. V. Sergienko, and Y. H. Shih, Phys. Rev. Lett. 75, 4337–4341 (1995).
[Crossref] [PubMed]

C. H. Bennett, G. Brassard, C. Crepeau, R. Jozsa, A. Peres, and W. K. Wootters, “Teleporting an unknown quantum state via classical and Einstein-Podolsky-Rosen channels,” Phys. Rev. Lett. 70, 1895–1899 (1993).
[Crossref] [PubMed]

S. L. Braunstein and H. J. Kimble, “Teleportation of continuous quantum variables,” Phys. Rev. Lett. 80, 869–872 (1998).
[Crossref]

J-W Pan, D. Bouwmeester, H. Weinfurter, and A. Zeilinger, “Experimental entanglement swapping : Entangling photons that never interacted,” Phys. Rev. Lett. 80, 3891–3894 (1998).
[Crossref]

D. Bouwmeester, J-W Pan, M. Daniell, H. Weinfurter, and A. Zeilinger, Phys. Rev. Lett. 82, 1345–1349 (1999).
[Crossref]

A. K. Ekert, J. G. Rarity, P. R. Tapster, and G. M. Palma, “Practical quantum cryptography based on two-photon interferometry,” Phys. Rev. Lett. 69 (9), 1293–1295 (1992).
[Crossref] [PubMed]

W. Tittel, J. Brendel, H. Zbinden, and N. Gisin, “Quantum cryptography using entangled photons in energy-time Bell states,” Phys. Rev. Lett. 84 (20), 4737–4740, 2000.
[Crossref] [PubMed]

D. C. Burnham and D. L. Weinberg, “Observation of simultaneity in parametric production of optical photon pairs,” Phys. Rev. Lett. 25, 84–87 (1970).
[Crossref]

Other (4)

S. Castelletto, I. P. Degiovanni, M. Ware, and A. Migdall, “Coupling efficiencies in single photon on-demand sources,” ArXiv:quant-ph/0311099 (2003).

E. R. Pike and S. Sarkar, The Quantum Theory of Radiation, Oxford University Press (1995).

A. Yariv, Optical Electronics in Modern Communications (5th Edition, Oxford Series in Electrical and Computer Engineering).

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), chaps. 7 and 8.

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Figures (1)

Fig. 1.
Fig. 1. Schematic of the experimental setup to determine pair photon coupling in fibers

Equations (18)

Equations on this page are rendered with MathJax. Learn more.

A ( 2 ) = E H + ( r 1 , t 1 ) E H + ( r 2 , t 2 )
d 3 r 3 d 3 k 1 d 3 k 2 U k 1 λ 1 * ( r 3 ) U k 2 λ 2 * ( r 3 ) U k 0 λ 0 ( r 3 ) f p ( r 3 ) ( ω k 0 2 ε 0 ) 1 2 ( ω k 1 2 ε 0 ) 1 2 ( ω k 2 2 ε 0 ) 1 2
× α k 0 , 0 E I ( + ) ( r 2 , t 2 ) E I ( + ) ( r 1 , t 1 ) ω k 0 , k 1 , k 2 δ ( ω k 1 ω k 2 ω k 0 )
E + ( r , t ) = i λ λ d 3 k d 3 k ( ω k 2 ε 0 ) 1 2 ε k λ β k λ , k λ U k λ f ( r ) e i ω t a k λ
β k λ k λ = d x d y U k λ in ( x , y ) U k λ f * ( x , y )
d 3 k 1 d 3 k 2 f ˜ p ( k 1 t + k 2 t ) sin c ( Δ k 1 z k 2 z d 2 ) β k 1 λ 1 β k 2 λ 2 U f ( r 1 ) U f ( r 2 )
× ( ω k 1 2 ε 0 ) 1 2 ( ω k 2 2 ε 0 ) 1 2 e i ω k 1 t 1 e i ω k 2 t 2 δ ( ω k 0 ω k 1 ω k 2 )
f ( r t ) exp ( x 2 + y 2 w p )
f ˜ p ( k 1 t + k 2 t ) exp ( 1 4 w p 2 v 2 ( ω k 1 ω k 2 ) 2 sin 2 θ * )
U k λ in ( x , y ) = 1 i k d e i k ( d + d ) e i k 2 d ( x 2 + y 2 ) 0 2 π 0 R r d θ d r ( e i ( k 2 d k 2 f ) r 2 e i k ( x cos θ d + y sin θ d ) r )
U f ( x , y ) = 1 w 0 π exp ( x 2 + y 2 2 w 0 2 )
β k λ w 0 2 d 2 + i k w 0 4 d k d w 0 ( d 2 + k 2 w 0 4 ) ( 1 e ( A k + i B k ) R 2 ) ( A k + i B k )
A k = k 2 w 0 2 2 ( d 2 + k 2 w 0 4 )
B k = ( 1 2 d 1 2 f ) k k 3 w 0 4 2 d ( d 2 + k 2 w 0 4 )
A ( 2 ) e i τ x x 2 Δ 2 ( 1 e ( a + b x ) ) 2 d x
a = ( k * 2 w 0 2 2 f 2 i k * 3 w 0 4 2 f 3 ) R 2
b = ( k * w 0 2 n g c f 2 i 3 2 n g k * 2 w 0 4 2 c f 3 ) R 2
C ( 1 4 e y + 6 e 2 y 4 e 3 y + e 4 y )

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