Abstract

We present the analytical and numerical analyses of two new resonator systems for generating flat–top–like beams. Both approaches lead to closed form expressions for the required cavity optics, but differ substantially in the design technique, with the first based on reverse propagation of a flattened Gaussian beam, and the second a metamorphosis of a Gaussian into a flat–top beam. We show that both have good convergence properties, and result in the desired stable mode.

© 2009 OSA

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References

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  1. Laser Beam Shaping, Theory and Techniques, ed Dickey/Holswade. (New York: Marcel Dekker, Inc.) (2000)
  2. P. A. Bélanger and C. Paré, “Optical resonators using graded-phase mirrors,” Opt. Lett. 16(14), 1057–1059 (1991).
    [CrossRef] [PubMed]
  3. C. Pare and P. A. Belanger, “Custom Laser Resonators Using Graded-Phase Mirror,” IEEE J. Quantum Electron. 28(1), 355–362 (1992).
    [CrossRef]
  4. P. A. Bélanger, R. L. Lachance, and C. Paré, “Super-Gaussian output from a CO(2) laser by using a graded-phase mirror resonator,” Opt. Lett. 17(10), 739–741 (1992).
    [CrossRef] [PubMed]
  5. J. R. Leger, D. Chen, and Z. Wang, “Diffractive optical element for mode shaping of a Nd:YAG laser,” Opt. Lett. 19(2), 108–110 (1994).
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  6. J. R. Leger, D. Chen, and K. Dai, “High modal discrimination in a Nd:YAG laser resonator with internal phase gratings,” Opt. Lett. 19(23), 1976–1978 (1994).
    [CrossRef] [PubMed]
  7. A. G. Fox and T. Li, “Resonant Modes in a Maser Interferometer,” Bell Syst. Tech. J. 40, 453–488 (1961).
  8. D. L. Shealy and J. A. Hoffnagle, “Laser beam shaping profiles and propagation,” Appl. Opt. 45(21), 5118–5131 (2006).
    [CrossRef] [PubMed]
  9. A. Forbes, H. J. Strydom, L. R. Botha, and E. Ronander, “Beam delivery for stable isotope separation,” Proc. SPIE 4770, 13–27 (2002).
    [CrossRef]
  10. F. Gori, “Flattened Gaussian beams,” Opt. Commun. 107(5-6), 335–341 (1994).
    [CrossRef]
  11. B. Lü and S. Luo, “General propagation equation of flattened Gaussian beams,” J. Opt. Soc. Am. A 17(11), 2001–2004 (2000).
    [CrossRef]
  12. L. A. Romero and F. M. Dickey, “Lossless laser beam shaping,” J. Opt. Soc. Am. A 13(4), 751–760 (1996).
    [CrossRef]
  13. F. M. Dickey and S. C. Holswade, “Gaussian laser beam profile shaping,” Opt. Eng. 35(11), 3285–3295 (1996).
    [CrossRef]
  14. I. A. Litvin and A. Forbes, “Gaussian mode selection with intracavity diffractive optics,” Opt. Lett. 34(19), 2991–2993 (2009).
    [CrossRef] [PubMed]
  15. L. Burger and A. Forbes, “Kaleidoscope modes in large aperture Porro prism resonators,” Opt. Express 16(17), 12707–12714 (2008).
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  16. S. A. Collins., “Lens-System Diffraction Integral Written in Terms of Matrix Optics,” J. Opt. Soc. Am. A 60(9), 1168–1177 (1970).
    [CrossRef]

2009 (1)

2008 (1)

2006 (1)

2002 (1)

A. Forbes, H. J. Strydom, L. R. Botha, and E. Ronander, “Beam delivery for stable isotope separation,” Proc. SPIE 4770, 13–27 (2002).
[CrossRef]

2000 (1)

1996 (2)

L. A. Romero and F. M. Dickey, “Lossless laser beam shaping,” J. Opt. Soc. Am. A 13(4), 751–760 (1996).
[CrossRef]

F. M. Dickey and S. C. Holswade, “Gaussian laser beam profile shaping,” Opt. Eng. 35(11), 3285–3295 (1996).
[CrossRef]

1994 (3)

1992 (2)

C. Pare and P. A. Belanger, “Custom Laser Resonators Using Graded-Phase Mirror,” IEEE J. Quantum Electron. 28(1), 355–362 (1992).
[CrossRef]

P. A. Bélanger, R. L. Lachance, and C. Paré, “Super-Gaussian output from a CO(2) laser by using a graded-phase mirror resonator,” Opt. Lett. 17(10), 739–741 (1992).
[CrossRef] [PubMed]

1991 (1)

1970 (1)

S. A. Collins., “Lens-System Diffraction Integral Written in Terms of Matrix Optics,” J. Opt. Soc. Am. A 60(9), 1168–1177 (1970).
[CrossRef]

1961 (1)

A. G. Fox and T. Li, “Resonant Modes in a Maser Interferometer,” Bell Syst. Tech. J. 40, 453–488 (1961).

Belanger, P. A.

C. Pare and P. A. Belanger, “Custom Laser Resonators Using Graded-Phase Mirror,” IEEE J. Quantum Electron. 28(1), 355–362 (1992).
[CrossRef]

Bélanger, P. A.

Botha, L. R.

A. Forbes, H. J. Strydom, L. R. Botha, and E. Ronander, “Beam delivery for stable isotope separation,” Proc. SPIE 4770, 13–27 (2002).
[CrossRef]

Burger, L.

Chen, D.

Collins, S. A.

S. A. Collins., “Lens-System Diffraction Integral Written in Terms of Matrix Optics,” J. Opt. Soc. Am. A 60(9), 1168–1177 (1970).
[CrossRef]

Dai, K.

Dickey, F. M.

L. A. Romero and F. M. Dickey, “Lossless laser beam shaping,” J. Opt. Soc. Am. A 13(4), 751–760 (1996).
[CrossRef]

F. M. Dickey and S. C. Holswade, “Gaussian laser beam profile shaping,” Opt. Eng. 35(11), 3285–3295 (1996).
[CrossRef]

Forbes, A.

Fox, A. G.

A. G. Fox and T. Li, “Resonant Modes in a Maser Interferometer,” Bell Syst. Tech. J. 40, 453–488 (1961).

Gori, F.

F. Gori, “Flattened Gaussian beams,” Opt. Commun. 107(5-6), 335–341 (1994).
[CrossRef]

Hoffnagle, J. A.

Holswade, S. C.

F. M. Dickey and S. C. Holswade, “Gaussian laser beam profile shaping,” Opt. Eng. 35(11), 3285–3295 (1996).
[CrossRef]

Lachance, R. L.

Leger, J. R.

Li, T.

A. G. Fox and T. Li, “Resonant Modes in a Maser Interferometer,” Bell Syst. Tech. J. 40, 453–488 (1961).

Litvin, I. A.

Lü, B.

Luo, S.

Pare, C.

C. Pare and P. A. Belanger, “Custom Laser Resonators Using Graded-Phase Mirror,” IEEE J. Quantum Electron. 28(1), 355–362 (1992).
[CrossRef]

Paré, C.

Romero, L. A.

Ronander, E.

A. Forbes, H. J. Strydom, L. R. Botha, and E. Ronander, “Beam delivery for stable isotope separation,” Proc. SPIE 4770, 13–27 (2002).
[CrossRef]

Shealy, D. L.

Strydom, H. J.

A. Forbes, H. J. Strydom, L. R. Botha, and E. Ronander, “Beam delivery for stable isotope separation,” Proc. SPIE 4770, 13–27 (2002).
[CrossRef]

Wang, Z.

Appl. Opt. (1)

Bell Syst. Tech. J. (1)

A. G. Fox and T. Li, “Resonant Modes in a Maser Interferometer,” Bell Syst. Tech. J. 40, 453–488 (1961).

IEEE J. Quantum Electron. (1)

C. Pare and P. A. Belanger, “Custom Laser Resonators Using Graded-Phase Mirror,” IEEE J. Quantum Electron. 28(1), 355–362 (1992).
[CrossRef]

J. Opt. Soc. Am. A (3)

Opt. Commun. (1)

F. Gori, “Flattened Gaussian beams,” Opt. Commun. 107(5-6), 335–341 (1994).
[CrossRef]

Opt. Eng. (1)

F. M. Dickey and S. C. Holswade, “Gaussian laser beam profile shaping,” Opt. Eng. 35(11), 3285–3295 (1996).
[CrossRef]

Opt. Express (1)

Opt. Lett. (5)

Proc. SPIE (1)

A. Forbes, H. J. Strydom, L. R. Botha, and E. Ronander, “Beam delivery for stable isotope separation,” Proc. SPIE 4770, 13–27 (2002).
[CrossRef]

Other (1)

Laser Beam Shaping, Theory and Techniques, ed Dickey/Holswade. (New York: Marcel Dekker, Inc.) (2000)

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Figures (8)

Fig. 1
Fig. 1

A schematic of the resonator to be modeled: with output coupling at M 2. Mirrors M 1 and M 2 can either be considered as elements with non–spherical curvature, or as depicted above, as flat mirrors with an appropriate transmission DOE placed immediately in front of each.

Fig. 2
Fig. 2

Calculated phase profile required for the DOE at mirror M 1. The requirement for the DOE at M 2 is that it is a planar surface.

Fig. 3
Fig. 3

The simulated field at mirror M 1 (red) and M 2 (blue): (a) intensity, showing a near perfect flat–top beam at M 2, with slight change in flatness after propagating across the resonator to M 1, (b) phase of the field, with a flat wavefront at M 2 as anticipated from the design.

Fig. 4
Fig. 4

The simulated field as it propagates across the resonator after stabilization, from M 1 (left) to M 2 (right). The perfect flat–top beam develops some intensity ‘structure’ as it propagates away from M 2. This is in accordance with the propagation properties of such fields, and may be minimized by suitable choice of Rayleigh range of the field.

Fig. 5
Fig. 5

The calculated required phases of the two DOEs, DOE1 in blue and DOE2 in red, to achieve the flat–top output mode.

Fig. 6
Fig. 6

The simulated field at mirror M 1 (red) and M 2 (blue): (a) intensity, showing a near perfect flat–top beam at M 2, changing into a perfect Gaussian after propagating across the resonator to M 1, (b) phase of the field, with a flat wavefront at M 1 as anticipated from the design.

Fig. 7
Fig. 7

The simulated field as it propagates across the resonator after stabilization, from M 1 (left) to M 2 (right). The perfect Gaussian beam (a) gradually changes into a perfect flat–top beam (e) on one pass through the resonator. In this design the field also decreases in size, as noted from the size of the grey scale images.

Fig. 8
Fig. 8

The simulated losses as a starting field of random noise is propagated through the resonator, shown as a function of the number of round trips taken, for: (a) resonator A and (b) resonator B. The losses stabilize in both resonators, and both show a characteristic oscillation in the losses as the field converges to the stable mode of lowest loss.

Equations (25)

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u(x)=ψ(x)exp[ikφ(x)],
φout(x)=φin(x)2φDOE(x).
x(φinx)ψin2(x)dx=x(φDOEx)ψin2(x)dx,
φDOE(x)=φin(x)φin(0).
φout(x)=φin(x).
u1(x1,L)=iλLu2(x2)exp(iπλL(x122x1x2+x22))dx2,
φDOE1=Arg[u1*(x,L)].
exp(iφout(x))=exp(iφin(x))=exp(i2φDOE(x))exp(iφin(x)),
φDOE(x)=φin(x),
uFTB(x)={u0,|x|<w0,|x|w,
uFGB(x,z)=ik2zexp[ikz2]exp[ikx22z]exp[(kxz)24(N+1w2+ik2z)]×m=0N(14)m1m!(N+1w2)m(N+1w2+ik2z)mH2m(kxz2N+1w2+ik2z)
u2(x2,f)=iλfu1(x1)exp[i(φSF(x1)ikx122f)]×exp(iπλf(x122x1x2+x22))dx1
φSF(x)=β{π22xwgerf(2xwg)+12exp([2xwg]2)12},
β=2πwgwFTBfλ.
φDOE1(x)=φSF(x)kx22f,
φDOE2(x)=[k2fx2+12βexpξ2(x)],
whereξ(x)=Inv{erf(2xwFTBπ)}.
u1(x1,L)=iλLX2X2u2(x2)exp(iπλLx122x1x2+x22)dx2=i=0Nu2(X2iΔx)X2iΔxX2(i+1)ΔxiλLexp(iπλL(x122x1x2+x22))dx2.
u1=Tu2
u1=(u1(X1).u1(X1iΔx).u1(X1))
u2=(u2(X2).u2(X2iΔx).u2(X2))
T=(T11T12..T1NT21T22.............TN1...TNN)
Tij=X2iΔxX2(i+1)ΔxiλLexp(iπλL(x222x2xj+xj2))dx2.
Tij=limΔx0Tij=iλLexp(iπλL(xi22xixj+xj2))Δx,
λu1=T1T2u1.

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