Abstract

We present a new scheme for an optical resonator for production of Bessel and Bessel–Gauss light beams. The resonator with Bessel modes is composed of two plane mirrors with an axicon placed close to one of them. If this mirror is concave, the modes are Bessel–Gauss light beams. Analytical expressions relating parameters of the resonator and characteristics of its modes are obtained and analyzed. The results are verified with the Fox–Li algorithm. The resonator scheme was implemented in an experiment to confirm the possibility of the generation of zero-order Bessel beams. It was found that multipass modes can also oscillate in the resonator if its apertures are large enough.

© 2001 Optical Society of America

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  1. J. Durnin, “Exact solutions for nondiffracting beams,” J. Opt. Soc. Am. A 4, 651–654 (1987).
    [CrossRef]
  2. J. Durnin, J. J. Miceli, J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
    [CrossRef] [PubMed]
  3. Z. Bouchal, J. Wagner, M. Chlup, “Self-reconstruction of a distorted nondiffracting beam,” Opt. Commun. 151, 207–211 (1998).
    [CrossRef]
  4. T. Wulle, S. Herminghaus, “Nonlinear optics of Bessel beams,” Phys. Rev. Lett. 58, 1401–1404 (1987).
  5. J. Arlt, K. Dholakia, L. Allen, M. J. Padgett, “Efficiency of second-harmonic generation with Bessel beams,” Phys. Rev. A 60, 2438–2441 (1999).
    [CrossRef]
  6. C. F. R. Caron, R. M. Potvliege, “Optimum conical angle of a Bessel–Gauss beam for low-order harmonic generation in gases,” J. Opt. Soc. Am. B 16, 1377–1384 (1999).
    [CrossRef]
  7. R. Gadonas, A. Marcinkevicius, A. Piskarskas, V. Smilgevicius, A. Stabinis, “Travelling wave optical parametric generator pumped by a conical beam,” Opt. Commun. 146, 253–256 (1997).
    [CrossRef]
  8. V. N. Belyi, N. S. Kazak, N. V. Kondratyuk, N. A. Khilo, A. A. Shagov, “Generation of the second harmonics of Bessel light beams in a KTP crystal,” Quantum Electron. 28, 1011–1016 (1998).
    [CrossRef]
  9. M. Florjanczyk, R. Tremblay, “Guiding of atoms in a travelling-wave laser trap formed by the axicon,” Opt. Commun. 73, 448–450 (1989).
    [CrossRef]
  10. I. Manek, Yu. B. Ovchinnikov, R. Grimm, “Generation of a hollow laser beam for atom trapping using an axicon,” Opt. Commun. 147, 67–70 (1998).
    [CrossRef]
  11. F. Gori, G. Guattari, C. Padovani, “Bessel–Gauss beams,” Opt. Commun. 64, 491–495 (1987).
    [CrossRef]
  12. J. Arlt, K. Dholakia, “Generation of high-order Bessel beams by use of an axicon,” Opt. Commun. 177, 297–301 (2000).
    [CrossRef]
  13. C. Paterson, R. Smith, “Higher-order Bessel waves produced by axicon-type computer-generated holograms,” Opt. Commun. 124, 121–130 (1996).
    [CrossRef]
  14. J. Durnin, J. H. Eberly, “Diffraction free arrangement,” U.S. Patent4887885, December19, 1989.
  15. K. Uehara, H. Kikuchi, “Generation of nearly diffraction-free laser beams,” Appl. Phys. B 48, 125–129 (1989).
    [CrossRef]
  16. J. K. Jabczyňski, “A ‘diffraction-free’ resonator,” Opt. Commun. 77, 292–294 (1990).
    [CrossRef]
  17. P. Pääkkönen, J. Turunen, “Resonators with Bessel–Gauss modes,” Opt. Commun. 156, 359–366 (1998).
    [CrossRef]
  18. A. Khilo, E. Katranji, A. Ryzhevich, “Resonators with Bessel and Bessel–Gauss modes,” presented at the Conference on Laser Optics, St. Petersburg, Russia, June 26–30, 2000, paper TuA3.
  19. J. R. Leger, D. Chen, Zh. Wang, “Diffractive optical element for mode shaping of a Nd:YAG laser,” Opt. Lett. 19, 108–110 (1994).
    [CrossRef] [PubMed]
  20. I. A. Ramsay, J. J. Degnan, “A ray analysis of optical resonators formed by two spherical mirrors,” Appl. Opt. 9, 385–398 (1970).
    [CrossRef] [PubMed]

2000

J. Arlt, K. Dholakia, “Generation of high-order Bessel beams by use of an axicon,” Opt. Commun. 177, 297–301 (2000).
[CrossRef]

1999

J. Arlt, K. Dholakia, L. Allen, M. J. Padgett, “Efficiency of second-harmonic generation with Bessel beams,” Phys. Rev. A 60, 2438–2441 (1999).
[CrossRef]

C. F. R. Caron, R. M. Potvliege, “Optimum conical angle of a Bessel–Gauss beam for low-order harmonic generation in gases,” J. Opt. Soc. Am. B 16, 1377–1384 (1999).
[CrossRef]

1998

V. N. Belyi, N. S. Kazak, N. V. Kondratyuk, N. A. Khilo, A. A. Shagov, “Generation of the second harmonics of Bessel light beams in a KTP crystal,” Quantum Electron. 28, 1011–1016 (1998).
[CrossRef]

Z. Bouchal, J. Wagner, M. Chlup, “Self-reconstruction of a distorted nondiffracting beam,” Opt. Commun. 151, 207–211 (1998).
[CrossRef]

I. Manek, Yu. B. Ovchinnikov, R. Grimm, “Generation of a hollow laser beam for atom trapping using an axicon,” Opt. Commun. 147, 67–70 (1998).
[CrossRef]

P. Pääkkönen, J. Turunen, “Resonators with Bessel–Gauss modes,” Opt. Commun. 156, 359–366 (1998).
[CrossRef]

1997

R. Gadonas, A. Marcinkevicius, A. Piskarskas, V. Smilgevicius, A. Stabinis, “Travelling wave optical parametric generator pumped by a conical beam,” Opt. Commun. 146, 253–256 (1997).
[CrossRef]

1996

C. Paterson, R. Smith, “Higher-order Bessel waves produced by axicon-type computer-generated holograms,” Opt. Commun. 124, 121–130 (1996).
[CrossRef]

1994

1990

J. K. Jabczyňski, “A ‘diffraction-free’ resonator,” Opt. Commun. 77, 292–294 (1990).
[CrossRef]

1989

K. Uehara, H. Kikuchi, “Generation of nearly diffraction-free laser beams,” Appl. Phys. B 48, 125–129 (1989).
[CrossRef]

M. Florjanczyk, R. Tremblay, “Guiding of atoms in a travelling-wave laser trap formed by the axicon,” Opt. Commun. 73, 448–450 (1989).
[CrossRef]

1987

T. Wulle, S. Herminghaus, “Nonlinear optics of Bessel beams,” Phys. Rev. Lett. 58, 1401–1404 (1987).

J. Durnin, “Exact solutions for nondiffracting beams,” J. Opt. Soc. Am. A 4, 651–654 (1987).
[CrossRef]

J. Durnin, J. J. Miceli, J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[CrossRef] [PubMed]

F. Gori, G. Guattari, C. Padovani, “Bessel–Gauss beams,” Opt. Commun. 64, 491–495 (1987).
[CrossRef]

1970

Allen, L.

J. Arlt, K. Dholakia, L. Allen, M. J. Padgett, “Efficiency of second-harmonic generation with Bessel beams,” Phys. Rev. A 60, 2438–2441 (1999).
[CrossRef]

Arlt, J.

J. Arlt, K. Dholakia, “Generation of high-order Bessel beams by use of an axicon,” Opt. Commun. 177, 297–301 (2000).
[CrossRef]

J. Arlt, K. Dholakia, L. Allen, M. J. Padgett, “Efficiency of second-harmonic generation with Bessel beams,” Phys. Rev. A 60, 2438–2441 (1999).
[CrossRef]

Belyi, V. N.

V. N. Belyi, N. S. Kazak, N. V. Kondratyuk, N. A. Khilo, A. A. Shagov, “Generation of the second harmonics of Bessel light beams in a KTP crystal,” Quantum Electron. 28, 1011–1016 (1998).
[CrossRef]

Bouchal, Z.

Z. Bouchal, J. Wagner, M. Chlup, “Self-reconstruction of a distorted nondiffracting beam,” Opt. Commun. 151, 207–211 (1998).
[CrossRef]

Caron, C. F. R.

Chen, D.

Chlup, M.

Z. Bouchal, J. Wagner, M. Chlup, “Self-reconstruction of a distorted nondiffracting beam,” Opt. Commun. 151, 207–211 (1998).
[CrossRef]

Degnan, J. J.

Dholakia, K.

J. Arlt, K. Dholakia, “Generation of high-order Bessel beams by use of an axicon,” Opt. Commun. 177, 297–301 (2000).
[CrossRef]

J. Arlt, K. Dholakia, L. Allen, M. J. Padgett, “Efficiency of second-harmonic generation with Bessel beams,” Phys. Rev. A 60, 2438–2441 (1999).
[CrossRef]

Durnin, J.

J. Durnin, “Exact solutions for nondiffracting beams,” J. Opt. Soc. Am. A 4, 651–654 (1987).
[CrossRef]

J. Durnin, J. J. Miceli, J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[CrossRef] [PubMed]

J. Durnin, J. H. Eberly, “Diffraction free arrangement,” U.S. Patent4887885, December19, 1989.

Eberly, J. H.

J. Durnin, J. J. Miceli, J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[CrossRef] [PubMed]

J. Durnin, J. H. Eberly, “Diffraction free arrangement,” U.S. Patent4887885, December19, 1989.

Florjanczyk, M.

M. Florjanczyk, R. Tremblay, “Guiding of atoms in a travelling-wave laser trap formed by the axicon,” Opt. Commun. 73, 448–450 (1989).
[CrossRef]

Gadonas, R.

R. Gadonas, A. Marcinkevicius, A. Piskarskas, V. Smilgevicius, A. Stabinis, “Travelling wave optical parametric generator pumped by a conical beam,” Opt. Commun. 146, 253–256 (1997).
[CrossRef]

Gori, F.

F. Gori, G. Guattari, C. Padovani, “Bessel–Gauss beams,” Opt. Commun. 64, 491–495 (1987).
[CrossRef]

Grimm, R.

I. Manek, Yu. B. Ovchinnikov, R. Grimm, “Generation of a hollow laser beam for atom trapping using an axicon,” Opt. Commun. 147, 67–70 (1998).
[CrossRef]

Guattari, G.

F. Gori, G. Guattari, C. Padovani, “Bessel–Gauss beams,” Opt. Commun. 64, 491–495 (1987).
[CrossRef]

Herminghaus, S.

T. Wulle, S. Herminghaus, “Nonlinear optics of Bessel beams,” Phys. Rev. Lett. 58, 1401–1404 (1987).

Jabczynski, J. K.

J. K. Jabczyňski, “A ‘diffraction-free’ resonator,” Opt. Commun. 77, 292–294 (1990).
[CrossRef]

Katranji, E.

A. Khilo, E. Katranji, A. Ryzhevich, “Resonators with Bessel and Bessel–Gauss modes,” presented at the Conference on Laser Optics, St. Petersburg, Russia, June 26–30, 2000, paper TuA3.

Kazak, N. S.

V. N. Belyi, N. S. Kazak, N. V. Kondratyuk, N. A. Khilo, A. A. Shagov, “Generation of the second harmonics of Bessel light beams in a KTP crystal,” Quantum Electron. 28, 1011–1016 (1998).
[CrossRef]

Khilo, A.

A. Khilo, E. Katranji, A. Ryzhevich, “Resonators with Bessel and Bessel–Gauss modes,” presented at the Conference on Laser Optics, St. Petersburg, Russia, June 26–30, 2000, paper TuA3.

Khilo, N. A.

V. N. Belyi, N. S. Kazak, N. V. Kondratyuk, N. A. Khilo, A. A. Shagov, “Generation of the second harmonics of Bessel light beams in a KTP crystal,” Quantum Electron. 28, 1011–1016 (1998).
[CrossRef]

Kikuchi, H.

K. Uehara, H. Kikuchi, “Generation of nearly diffraction-free laser beams,” Appl. Phys. B 48, 125–129 (1989).
[CrossRef]

Kondratyuk, N. V.

V. N. Belyi, N. S. Kazak, N. V. Kondratyuk, N. A. Khilo, A. A. Shagov, “Generation of the second harmonics of Bessel light beams in a KTP crystal,” Quantum Electron. 28, 1011–1016 (1998).
[CrossRef]

Leger, J. R.

Manek, I.

I. Manek, Yu. B. Ovchinnikov, R. Grimm, “Generation of a hollow laser beam for atom trapping using an axicon,” Opt. Commun. 147, 67–70 (1998).
[CrossRef]

Marcinkevicius, A.

R. Gadonas, A. Marcinkevicius, A. Piskarskas, V. Smilgevicius, A. Stabinis, “Travelling wave optical parametric generator pumped by a conical beam,” Opt. Commun. 146, 253–256 (1997).
[CrossRef]

Miceli, J. J.

J. Durnin, J. J. Miceli, J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[CrossRef] [PubMed]

Ovchinnikov, Yu. B.

I. Manek, Yu. B. Ovchinnikov, R. Grimm, “Generation of a hollow laser beam for atom trapping using an axicon,” Opt. Commun. 147, 67–70 (1998).
[CrossRef]

Pääkkönen, P.

P. Pääkkönen, J. Turunen, “Resonators with Bessel–Gauss modes,” Opt. Commun. 156, 359–366 (1998).
[CrossRef]

Padgett, M. J.

J. Arlt, K. Dholakia, L. Allen, M. J. Padgett, “Efficiency of second-harmonic generation with Bessel beams,” Phys. Rev. A 60, 2438–2441 (1999).
[CrossRef]

Padovani, C.

F. Gori, G. Guattari, C. Padovani, “Bessel–Gauss beams,” Opt. Commun. 64, 491–495 (1987).
[CrossRef]

Paterson, C.

C. Paterson, R. Smith, “Higher-order Bessel waves produced by axicon-type computer-generated holograms,” Opt. Commun. 124, 121–130 (1996).
[CrossRef]

Piskarskas, A.

R. Gadonas, A. Marcinkevicius, A. Piskarskas, V. Smilgevicius, A. Stabinis, “Travelling wave optical parametric generator pumped by a conical beam,” Opt. Commun. 146, 253–256 (1997).
[CrossRef]

Potvliege, R. M.

Ramsay, I. A.

Ryzhevich, A.

A. Khilo, E. Katranji, A. Ryzhevich, “Resonators with Bessel and Bessel–Gauss modes,” presented at the Conference on Laser Optics, St. Petersburg, Russia, June 26–30, 2000, paper TuA3.

Shagov, A. A.

V. N. Belyi, N. S. Kazak, N. V. Kondratyuk, N. A. Khilo, A. A. Shagov, “Generation of the second harmonics of Bessel light beams in a KTP crystal,” Quantum Electron. 28, 1011–1016 (1998).
[CrossRef]

Smilgevicius, V.

R. Gadonas, A. Marcinkevicius, A. Piskarskas, V. Smilgevicius, A. Stabinis, “Travelling wave optical parametric generator pumped by a conical beam,” Opt. Commun. 146, 253–256 (1997).
[CrossRef]

Smith, R.

C. Paterson, R. Smith, “Higher-order Bessel waves produced by axicon-type computer-generated holograms,” Opt. Commun. 124, 121–130 (1996).
[CrossRef]

Stabinis, A.

R. Gadonas, A. Marcinkevicius, A. Piskarskas, V. Smilgevicius, A. Stabinis, “Travelling wave optical parametric generator pumped by a conical beam,” Opt. Commun. 146, 253–256 (1997).
[CrossRef]

Tremblay, R.

M. Florjanczyk, R. Tremblay, “Guiding of atoms in a travelling-wave laser trap formed by the axicon,” Opt. Commun. 73, 448–450 (1989).
[CrossRef]

Turunen, J.

P. Pääkkönen, J. Turunen, “Resonators with Bessel–Gauss modes,” Opt. Commun. 156, 359–366 (1998).
[CrossRef]

Uehara, K.

K. Uehara, H. Kikuchi, “Generation of nearly diffraction-free laser beams,” Appl. Phys. B 48, 125–129 (1989).
[CrossRef]

Wagner, J.

Z. Bouchal, J. Wagner, M. Chlup, “Self-reconstruction of a distorted nondiffracting beam,” Opt. Commun. 151, 207–211 (1998).
[CrossRef]

Wang, Zh.

Wulle, T.

T. Wulle, S. Herminghaus, “Nonlinear optics of Bessel beams,” Phys. Rev. Lett. 58, 1401–1404 (1987).

Appl. Opt.

Appl. Phys. B

K. Uehara, H. Kikuchi, “Generation of nearly diffraction-free laser beams,” Appl. Phys. B 48, 125–129 (1989).
[CrossRef]

J. Opt. Soc. Am. A

J. Opt. Soc. Am. B

Opt. Commun.

R. Gadonas, A. Marcinkevicius, A. Piskarskas, V. Smilgevicius, A. Stabinis, “Travelling wave optical parametric generator pumped by a conical beam,” Opt. Commun. 146, 253–256 (1997).
[CrossRef]

J. K. Jabczyňski, “A ‘diffraction-free’ resonator,” Opt. Commun. 77, 292–294 (1990).
[CrossRef]

P. Pääkkönen, J. Turunen, “Resonators with Bessel–Gauss modes,” Opt. Commun. 156, 359–366 (1998).
[CrossRef]

Z. Bouchal, J. Wagner, M. Chlup, “Self-reconstruction of a distorted nondiffracting beam,” Opt. Commun. 151, 207–211 (1998).
[CrossRef]

M. Florjanczyk, R. Tremblay, “Guiding of atoms in a travelling-wave laser trap formed by the axicon,” Opt. Commun. 73, 448–450 (1989).
[CrossRef]

I. Manek, Yu. B. Ovchinnikov, R. Grimm, “Generation of a hollow laser beam for atom trapping using an axicon,” Opt. Commun. 147, 67–70 (1998).
[CrossRef]

F. Gori, G. Guattari, C. Padovani, “Bessel–Gauss beams,” Opt. Commun. 64, 491–495 (1987).
[CrossRef]

J. Arlt, K. Dholakia, “Generation of high-order Bessel beams by use of an axicon,” Opt. Commun. 177, 297–301 (2000).
[CrossRef]

C. Paterson, R. Smith, “Higher-order Bessel waves produced by axicon-type computer-generated holograms,” Opt. Commun. 124, 121–130 (1996).
[CrossRef]

Opt. Lett.

Phys. Rev. A

J. Arlt, K. Dholakia, L. Allen, M. J. Padgett, “Efficiency of second-harmonic generation with Bessel beams,” Phys. Rev. A 60, 2438–2441 (1999).
[CrossRef]

Phys. Rev. Lett.

T. Wulle, S. Herminghaus, “Nonlinear optics of Bessel beams,” Phys. Rev. Lett. 58, 1401–1404 (1987).

J. Durnin, J. J. Miceli, J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[CrossRef] [PubMed]

Quantum Electron.

V. N. Belyi, N. S. Kazak, N. V. Kondratyuk, N. A. Khilo, A. A. Shagov, “Generation of the second harmonics of Bessel light beams in a KTP crystal,” Quantum Electron. 28, 1011–1016 (1998).
[CrossRef]

Other

A. Khilo, E. Katranji, A. Ryzhevich, “Resonators with Bessel and Bessel–Gauss modes,” presented at the Conference on Laser Optics, St. Petersburg, Russia, June 26–30, 2000, paper TuA3.

J. Durnin, J. H. Eberly, “Diffraction free arrangement,” U.S. Patent4887885, December19, 1989.

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Figures (8)

Fig. 1
Fig. 1

Profiles of amplitude (left-hand graphs) and phase (right-hand graphs) of (a) zero-order Bessel–Gauss beam at z=0.4w0/γ=4.6 mm (γ=1°, w0=0.2 mm, λ=0.633 μm); (b) the same beam at z=2w0/γ=23 mm; (c) Bessel beam limited with an aperture of radius Rmax=0.5 mm at a distance z=4Rmax/γ=115 mm from the aperture (γ=1°,λ=0.633 μm).

Fig. 2
Fig. 2

Design of a resonator with Bessel–Gauss modes. 1, partially transparent plane mirror with radius of aperture R1 max; 2, spherical mirror with radius of curvature R placed at a distance L from the plane mirror, its annular aperture having radii R2 min and R2 max; 3, axicon with parameter α.

Fig. 3
Fig. 3

(a) Axicon parameter α and (b) mirror radius R versus resonator length L. Such α and R have to be taken for construction of a resonator whose Bessel–Gauss modes have γ=1° and beam radius w0, which is the curve parameter. λ=0.633 μm.

Fig. 4
Fig. 4

(a) Bessel–Gauss beam conicity angle γ and (b) width w0 versus resonator length L. The resonator comprises an axicon with α=1°, and a spherical mirror with curvature radius R, which is the curve parameter.

Fig. 5
Fig. 5

Transverse distribution of the absolute value of the amplitude of the zero resonator mode (top curve in each plot) and also of the Bessel–Gauss beam (second curve in each plot) on plane mirror 1. [The curves coincide almost completely in (b) and (c)]. The resonator parameters are L=0.1 m, λ=0.633 μm, γ=1°, and (a) w0= (Bessel beam); α=1°, R= (plane mirror) (R1 max=0.77 mm, R2 min=0.8 mm, R2 max=2.7 mm, diffraction energy loss per round trip=1%). (b) w0=0.2 mm; α=0.80°, R=0.49 m (R1 max=0.5 mm, R2 min=1.4 mm, R2 max=2.1 mm, diffraction loss=0.3%). (c) w0=0.1 mm; α=0.20°, R=0.125 m (R1 max=0.5 mm, R2 min=1.4 mm, R2 max=2.1 mm, diffraction loss=0.4%).

Fig. 6
Fig. 6

Schematic of the experimental dye laser that generates the Bessel light beam.

Fig. 7
Fig. 7

(a) CCD image and (c) radial intensity distribution on the output coupling mirror of the experimental dye laser; (b) CCD image and (d) radial intensity distribution of the spatial Fourier spectrum of the output field.

Fig. 8
Fig. 8

Some self-reproducing ray configurations: (a) M=1, (b) M=2, (c) M=3, (d) M=4.

Equations (17)

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

Um(ρ, ϕ, z)=aJm(kγρ)exp[i(kzz+mϕ)],
Um(ρ, ϕ, z=0)=aJm(kγρ)exp(-ρ2/w02)exp(imϕ),
Uexp(-ρ2/w02).
Am(ρ, ϕ, z)=a w0w(z)exp[iΦ(z)]Jmkγρ1+iz/zR×exp-1w2(z)-ik2Rg(z)×(ρ2+γ 2z2)exp(imϕ),
τ2(ρ)=exp[iφ2(ρ)].
φ2(ρ)=-2Φ(L)-kγ 2L2Rg(L)-kρ2Rg(L)
-2 argJmkγρ1+iL/zR.
φ2(ρ)=-kρ2Rg(L)-2 argJmkγρ1+iL/zR.
argJmkγρ1+iL/zRkγρ1+L2/zR2.
φ2(ρ)=-kρ2Rg(L)-2 kγρ1+L2/zR2.
α=γ1+4L2/(kw02)2,
R=L+(kw02)24L.
γ=α RR-L,
w02=2k [(R-L)L]1/2.
J0kγρ1+iL/zRcoskγρ1+iL/zR-π4=coskγρ1+L2/zR2-π4-i kγρ1+L2/zR2LzR=coskγρ1+L2/zR2-π4coshkγρ1+L2/zR2LzR+i sinkγρ1+L2/zR2-π4sinhkγρ1+L2/zR2LzR.
arctantankγρ1+L2/zR2-π4tanhkγρ1+L2/zR2LzR.
argJ0kγρ1+iL/zRarctantankγρ1+L2/zR2-π4=kγρ1+L2/zR2-π4.

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