Abstract

A detailed study of the axicon-based Bessel–Gauss resonator with concave output coupler is presented. We employ a technique to convert the Huygens–Fresnel integral self-consistency equation into a matrix equation and then find the eigenvalues and the eigenfields of the resonator at one time. A paraxial ray analysis is performed to find the self-consistency condition to have stable periodic ray trajectories after one or two round trips. The fast-Fourier-transform-based Fox and Li algorithm is applied to describe the three-dimensional intracavity field distribution. Special attention was directed to the dependence of the output transverse profiles, the losses, and the modal-frequency changes on the curvature of the output coupler and the cavity length. The propagation of the output beam is discussed.

© 2003 Optical Society of America

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References

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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
  4. E. Abramochkin, V. Volostnikov, “Spiral-type beams,” Opt. Commun. 102, 336–350 (1993).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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2001 (2)

J. Rogel-Salazar, G. H. C. New, S. Chávez-Cerda, “Bessel–Gauss beam optical resonator,” Opt. Commun. 190, 117–122 (2001).
[CrossRef]

A. N. Khilo, E. G. Katranji, A. A. Ryzhevich, “Axicon-based Bessel resonator: analytical description and experiment,” J. Opt. Soc. Am. A 18, 1986–1992 (2001).
[CrossRef]

2000 (2)

1998 (3)

1997 (1)

1996 (2)

S. Chávez-Cerda, G. S. McDonald, G. H. C. New, “Nondiffracting beams: travelling, standing, rotating and spiral waves,” Opt. Commun. 123, 225–233 (1996).
[CrossRef]

C. Paterson, R. Smith, “Helicon waves: propagation-invariant waves in a rotating coordinate system,” Opt. Commun. 124, 131–140 (1996).
[CrossRef]

1993 (1)

E. Abramochkin, V. Volostnikov, “Spiral-type beams,” Opt. Commun. 102, 336–350 (1993).
[CrossRef]

1992 (2)

G. Scott, N. McArdle, “Efficient generation of nearly diffraction-free beams using an axicon,” Opt. Eng. 31, 2640–2643 (1992).
[CrossRef]

P. A. Bélanger, R. L. Lachance, C. Paré, “Super-Gaussian output from a CO2 laser by using a graded-phase mirror resonator,” Opt. Lett. 17, 739–741 (1992).
[CrossRef]

1991 (1)

1989 (3)

1988 (1)

1987 (3)

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

J. Durnin, J. J. Micely, 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]

1980 (1)

1966 (1)

H. Kogelnik, T. Li, “Laser beams and resonators,” Proc. IEEE 54, 1312–1329 (1966).
[CrossRef]

1954 (1)

Abramochkin, E.

E. Abramochkin, V. Volostnikov, “Spiral-type beams,” Opt. Commun. 102, 336–350 (1993).
[CrossRef]

Bélanger, P. A.

Borghi, R.

Chávez-Cerda, S.

J. Rogel-Salazar, G. H. C. New, S. Chávez-Cerda, “Bessel–Gauss beam optical resonator,” Opt. Commun. 190, 117–122 (2001).
[CrossRef]

S. Chávez-Cerda, G. H. C. New, “Evolution of focused Hankel waves and Bessel beams,” Opt. Commun. 181, 369–378 (2000).
[CrossRef]

J. C. Gutiérrez-Vega, M. D. Iturbe-Castillo, S. Chávez-Cerda, “Alternative formulation for invariant optical fields: Mathieu beams,” Opt. Lett. 25, 1493–1495 (2000).
[CrossRef]

S. Chávez-Cerda, G. S. McDonald, G. H. C. New, “Nondiffracting beams: travelling, standing, rotating and spiral waves,” Opt. Commun. 123, 225–233 (1996).
[CrossRef]

J. Rogel-Salazar, G. H. C. New, P. Muys, J. C. Gutiérrez-Vega, S. Chávez-Cerda, “Bessel–Gauss resonators,” in Laser Resonators IV, A. V. Kudryashov, A. H. Paxton, eds., Proc. SPIE4270, 52–63 (2001).

Chen, Nan-Xian

Cong, Wen-Xiang

Dente, G. C.

Durnin, J.

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

J. Durnin, J. J. Micely, 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. patent4,887,885, December19, 1989.

Eberly, J. H.

J. Durnin, J. J. Micely, 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. patent4,887,885, December19, 1989.

Flannery, B. P.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes (Cambridge U. Press, Cambridge, UK, 1986), Chap. 11.

Friberg, A. T.

Gori, F.

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

Gu, Ben-Yuan

Guattari, G.

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

Gutiérrez-Vega, J. C.

J. C. Gutiérrez-Vega, M. D. Iturbe-Castillo, S. Chávez-Cerda, “Alternative formulation for invariant optical fields: Mathieu beams,” Opt. Lett. 25, 1493–1495 (2000).
[CrossRef]

J. Rogel-Salazar, G. H. C. New, P. Muys, J. C. Gutiérrez-Vega, S. Chávez-Cerda, “Bessel–Gauss resonators,” in Laser Resonators IV, A. V. Kudryashov, A. H. Paxton, eds., Proc. SPIE4270, 52–63 (2001).

Indebetouw, G.

Iturbe-Castillo, M. D.

Katranji, E. G.

Khilo, A. N.

Kikuchi, H.

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

Kogelnik, H.

H. Kogelnik, T. Li, “Laser beams and resonators,” Proc. IEEE 54, 1312–1329 (1966).
[CrossRef]

Lachance, R. L.

Latham, W. P.

Li, T.

H. Kogelnik, T. Li, “Laser beams and resonators,” Proc. IEEE 54, 1312–1329 (1966).
[CrossRef]

McArdle, N.

G. Scott, N. McArdle, “Efficient generation of nearly diffraction-free beams using an axicon,” Opt. Eng. 31, 2640–2643 (1992).
[CrossRef]

McDonald, G. S.

S. Chávez-Cerda, G. S. McDonald, G. H. C. New, “Nondiffracting beams: travelling, standing, rotating and spiral waves,” Opt. Commun. 123, 225–233 (1996).
[CrossRef]

McLeod, J. H.

Micely, J. J.

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

Muys, P.

J. Rogel-Salazar, G. H. C. New, P. Muys, J. C. Gutiérrez-Vega, S. Chávez-Cerda, “Bessel–Gauss resonators,” in Laser Resonators IV, A. V. Kudryashov, A. H. Paxton, eds., Proc. SPIE4270, 52–63 (2001).

New, G. H. C.

J. Rogel-Salazar, G. H. C. New, S. Chávez-Cerda, “Bessel–Gauss beam optical resonator,” Opt. Commun. 190, 117–122 (2001).
[CrossRef]

S. Chávez-Cerda, G. H. C. New, “Evolution of focused Hankel waves and Bessel beams,” Opt. Commun. 181, 369–378 (2000).
[CrossRef]

S. Chávez-Cerda, G. S. McDonald, G. H. C. New, “Nondiffracting beams: travelling, standing, rotating and spiral waves,” Opt. Commun. 123, 225–233 (1996).
[CrossRef]

J. Rogel-Salazar, G. H. C. New, P. Muys, J. C. Gutiérrez-Vega, S. Chávez-Cerda, “Bessel–Gauss resonators,” in Laser Resonators IV, A. V. Kudryashov, A. H. Paxton, eds., Proc. SPIE4270, 52–63 (2001).

Pääkkönen, P.

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

Padovani, C.

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

Paré, C.

Paterson, C.

C. Paterson, R. Smith, “Helicon waves: propagation-invariant waves in a rotating coordinate system,” Opt. Commun. 124, 131–140 (1996).
[CrossRef]

Piestun, R.

Press, W. H.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes (Cambridge U. Press, Cambridge, UK, 1986), Chap. 11.

Rogel-Salazar, J.

J. Rogel-Salazar, G. H. C. New, S. Chávez-Cerda, “Bessel–Gauss beam optical resonator,” Opt. Commun. 190, 117–122 (2001).
[CrossRef]

J. Rogel-Salazar, G. H. C. New, P. Muys, J. C. Gutiérrez-Vega, S. Chávez-Cerda, “Bessel–Gauss resonators,” in Laser Resonators IV, A. V. Kudryashov, A. H. Paxton, eds., Proc. SPIE4270, 52–63 (2001).

Ryzhevich, A. A.

Santarsiero, M.

Scott, G.

G. Scott, N. McArdle, “Efficient generation of nearly diffraction-free beams using an axicon,” Opt. Eng. 31, 2640–2643 (1992).
[CrossRef]

Shamir, J.

Siegman, A. E.

A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986).

Smith, R.

C. Paterson, R. Smith, “Helicon waves: propagation-invariant waves in a rotating coordinate system,” Opt. Commun. 124, 131–140 (1996).
[CrossRef]

Teukolsky, S. A.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes (Cambridge U. Press, Cambridge, UK, 1986), Chap. 11.

Turunen, J.

Uehara, K.

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

Vasara, A.

Vetterling, W. T.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes (Cambridge U. Press, Cambridge, UK, 1986), Chap. 11.

Volostnikov, V.

E. Abramochkin, V. Volostnikov, “Spiral-type beams,” Opt. Commun. 102, 336–350 (1993).
[CrossRef]

Appl. Opt. (2)

Appl. Phys. B (1)

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

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (6)

Opt. Commun. (7)

J. Rogel-Salazar, G. H. C. New, S. Chávez-Cerda, “Bessel–Gauss beam optical resonator,” Opt. Commun. 190, 117–122 (2001).
[CrossRef]

S. Chávez-Cerda, G. H. C. New, “Evolution of focused Hankel waves and Bessel beams,” Opt. Commun. 181, 369–378 (2000).
[CrossRef]

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

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

E. Abramochkin, V. Volostnikov, “Spiral-type beams,” Opt. Commun. 102, 336–350 (1993).
[CrossRef]

S. Chávez-Cerda, G. S. McDonald, G. H. C. New, “Nondiffracting beams: travelling, standing, rotating and spiral waves,” Opt. Commun. 123, 225–233 (1996).
[CrossRef]

C. Paterson, R. Smith, “Helicon waves: propagation-invariant waves in a rotating coordinate system,” Opt. Commun. 124, 131–140 (1996).
[CrossRef]

Opt. Eng. (1)

G. Scott, N. McArdle, “Efficient generation of nearly diffraction-free beams using an axicon,” Opt. Eng. 31, 2640–2643 (1992).
[CrossRef]

Opt. Lett. (4)

Phys. Rev. Lett. (1)

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

Proc. IEEE (1)

H. Kogelnik, T. Li, “Laser beams and resonators,” Proc. IEEE 54, 1312–1329 (1966).
[CrossRef]

Other (4)

A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986).

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes (Cambridge U. Press, Cambridge, UK, 1986), Chap. 11.

J. Durnin, J. H. Eberly, “Diffraction free arrangement,” U.S. patent4,887,885, December19, 1989.

J. Rogel-Salazar, G. H. C. New, P. Muys, J. C. Gutiérrez-Vega, S. Chávez-Cerda, “Bessel–Gauss resonators,” in Laser Resonators IV, A. V. Kudryashov, A. H. Paxton, eds., Proc. SPIE4270, 52–63 (2001).

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Figures (11)

Fig. 1
Fig. 1

(a), (b) An axicon transforms an incident plane wave into a converging conical wave. (c) The constituent plane waves of the conical waves superpose in the interference region, building up a Bessel beam.

Fig. 2
Fig. 2

Design of the resonator with (a) a refractive axicon or (b) a reflective axicon. The spherical mirror with radius of curvature R is placed a distance L from the axicon plane. (c) Lens-guide equivalent resonator. Not that light crosses twice through the refractive axicon.

Fig. 3
Fig. 3

Self-reproducibility condition for stable ray trajectories after one and two round trips.

Fig. 4
Fig. 4

Profiles of the magnitude and the phase of the eigenfield at (a), (b) the axicon plane and (c), (d) the output plane for an output flat mirror (R). The thin curve in (c) represents the theoretical output Bessel beam J0(ktr).

Fig. 5
Fig. 5

Profiles of the magnitude and the phase of the eigenfield at (a), (b) the axicon plane and (c), (d) the output plane for an output spherical mirror (R=10L). The plus signs represent the theoretical zero-order Bessel–Gauss beam J0(ktr)exp(-r2/w2).

Fig. 6
Fig. 6

Same as Fig. 5, except that the plus signs represent the theoretical second-order Bessel–Gauss beam J2(ktr)exp(-r2/w2).

Fig. 7
Fig. 7

Loss (Γ) and relative phase shift (Δβ/π) behavior as a function of the normalized radius of curvature R/L for the first lower orders l=0,,3. The dashed lines in (a) represent the losses in the limit when R tends to infinity.

Fig. 8
Fig. 8

Passive three-dimensional intracavity field distributions in the resonator with spherical output mirror (R=10 L): (a) dominant J0 Bessel–Gauss beam, (b) second-order J2 Bessel–Gauss beam.

Fig. 9
Fig. 9

Transverse field pattern at the output and axicon planes corresponding to the length factors μ=0.8, 1, and 1.2 for both plane and spherical output mirrors.

Fig. 10
Fig. 10

Loss and resonant frequency shift behavior as a function of length factor μ for both plane and spherical output mirrors.

Fig. 11
Fig. 11

Numerical propagation of the output beam of the ABGR for (a) plane and (b) spherical R=10L output couplers.

Tables (2)

Tables Icon

Table 1 Geometrical Stability Conditions

Tables Icon

Table 2 Diffractive Losses for the First Ten Modes

Equations (23)

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

θ0=arcsin(n sin α)-α(n-1)α,
T(r)=exp(-ikθ0r),
L=a2 tan θ0a2θ0=a2(n-1)α.
U(r)=J0(ktr)exp(-r2/w2)exp(iΦ),
r2=r1,
θ2=θ1-2θ0=-2θ0r1r1+θ1,
r2θ2=10-2θ0/r11r1θ1.
ABCD=1L0110-2/R11L0110-2θ0/r11
=1-2LR+-2θ0r12L1-LR2L1-LR-2θ0r11-2LR-2R1-2LR
=D+-2θ0r1BBCD.
ABCDrθ=γrθ.
raθa=γa-DC,rbθb=γb-DC.
A-γBCD-γr1θ1=0.
r=4γθ0L(1-L/R)2γ(1-2L/R)-γ2-1.
roner-t=(L-R)θ0,rtwor-t=Lθ0=a2.
ulp(r2)=0Kl(r1, r2)ulp(r1)dr1,
Kl(r1, r2)=(-i)l+1kBr1JlkBr1r2×expik2B(Ara2+Dr22),
ABCD=1-2L/R2L(1-L/R)-2/R1-2L/R.
γlpulp(r2)=0Hl(r1, r2)ulp(r1)dr1,
γlp=|γlp|exp(iβlp),
γulp=(H*W)*ulp,
νlp=ν0q+βlpπ,
Δν=ν0(Δβ/π).

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