Abstract

An analysis of the increase in the beam-propagation factor, M2, of a higher order transverse mode beam caused by quartic phase aberration after transmission through a spherically aberrated lens is reported. The analysis shows that for a given beam size, the increase in the M2 parameter is less for a higher order transverse mode beam as compared to that for a diffraction limited beam. Experimental results using a multimode laser diode beam show good agreement with the theoretical results.

© 2009 Optical Society of America

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References

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  1. A. E. Siegman, “New developments in laser resonators,” Proc. SPIE 1224, 2-14 (1990).
    [CrossRef]
  2. H. H. Hopkins, Wave Theory of Aberrations (Oxford University, 1950).
  3. W. T. Welford, Aberrations of the Symmetric Optical System (Academic, 1974).
  4. A. E. Siegman, “Analysis of laser beam quality degradation caused by spherical aberration,” Appl. Opt. 32, 5893-5901(1993).
    [CrossRef] [PubMed]
  5. J. A. Ruff and A. E. Siegman, “Measurement of beam quality degradation due to spherical aberration in a simple lens,” Opt. Quantum Electron. 26, 629-632(1994).
    [CrossRef]
  6. R. L. Phillips and L. C. Andrews, “Spot size and divergence of Laguerre Gaussian beams of any order,” Appl. Opt. 22, 643-644 (1983).
    [CrossRef] [PubMed]
  7. W. Koechner, Solid State Laser Engineering, 6th ed. (Springer-Verlag, 2006), Chap. 5.1.1.
  8. R. Kapoor, P. K. Mukhopadhyay, J. George, and S. K. Sharma, “An alternative approach to determine the spot size of a multi-mode laser beam and its applications to diode laser beams,” Pramana J. Phys. 53, 307-319 (1999).
    [CrossRef]
  9. D. Wright, “Beamwidths of a diffracted laser using four proposed methods,” Opt. Quantum Electron. 24, S1129-S1135(1992).
    [CrossRef]
  10. A. E. Siegman, “Output beam propagation and beam quality from a multimode stable-cavity laser,” IEEE J. Quantum Electron. 29, 1212-1217 (1993).
    [CrossRef]

1999 (1)

R. Kapoor, P. K. Mukhopadhyay, J. George, and S. K. Sharma, “An alternative approach to determine the spot size of a multi-mode laser beam and its applications to diode laser beams,” Pramana J. Phys. 53, 307-319 (1999).
[CrossRef]

1994 (1)

J. A. Ruff and A. E. Siegman, “Measurement of beam quality degradation due to spherical aberration in a simple lens,” Opt. Quantum Electron. 26, 629-632(1994).
[CrossRef]

1993 (2)

A. E. Siegman, “Output beam propagation and beam quality from a multimode stable-cavity laser,” IEEE J. Quantum Electron. 29, 1212-1217 (1993).
[CrossRef]

A. E. Siegman, “Analysis of laser beam quality degradation caused by spherical aberration,” Appl. Opt. 32, 5893-5901(1993).
[CrossRef] [PubMed]

1992 (1)

D. Wright, “Beamwidths of a diffracted laser using four proposed methods,” Opt. Quantum Electron. 24, S1129-S1135(1992).
[CrossRef]

1990 (1)

A. E. Siegman, “New developments in laser resonators,” Proc. SPIE 1224, 2-14 (1990).
[CrossRef]

1983 (1)

Andrews, L. C.

George, J.

R. Kapoor, P. K. Mukhopadhyay, J. George, and S. K. Sharma, “An alternative approach to determine the spot size of a multi-mode laser beam and its applications to diode laser beams,” Pramana J. Phys. 53, 307-319 (1999).
[CrossRef]

Hopkins, H. H.

H. H. Hopkins, Wave Theory of Aberrations (Oxford University, 1950).

Kapoor, R.

R. Kapoor, P. K. Mukhopadhyay, J. George, and S. K. Sharma, “An alternative approach to determine the spot size of a multi-mode laser beam and its applications to diode laser beams,” Pramana J. Phys. 53, 307-319 (1999).
[CrossRef]

Koechner, W.

W. Koechner, Solid State Laser Engineering, 6th ed. (Springer-Verlag, 2006), Chap. 5.1.1.

Mukhopadhyay, P. K.

R. Kapoor, P. K. Mukhopadhyay, J. George, and S. K. Sharma, “An alternative approach to determine the spot size of a multi-mode laser beam and its applications to diode laser beams,” Pramana J. Phys. 53, 307-319 (1999).
[CrossRef]

Phillips, R. L.

Ruff, J. A.

J. A. Ruff and A. E. Siegman, “Measurement of beam quality degradation due to spherical aberration in a simple lens,” Opt. Quantum Electron. 26, 629-632(1994).
[CrossRef]

Sharma, S. K.

R. Kapoor, P. K. Mukhopadhyay, J. George, and S. K. Sharma, “An alternative approach to determine the spot size of a multi-mode laser beam and its applications to diode laser beams,” Pramana J. Phys. 53, 307-319 (1999).
[CrossRef]

Siegman, A. E.

J. A. Ruff and A. E. Siegman, “Measurement of beam quality degradation due to spherical aberration in a simple lens,” Opt. Quantum Electron. 26, 629-632(1994).
[CrossRef]

A. E. Siegman, “Analysis of laser beam quality degradation caused by spherical aberration,” Appl. Opt. 32, 5893-5901(1993).
[CrossRef] [PubMed]

A. E. Siegman, “Output beam propagation and beam quality from a multimode stable-cavity laser,” IEEE J. Quantum Electron. 29, 1212-1217 (1993).
[CrossRef]

A. E. Siegman, “New developments in laser resonators,” Proc. SPIE 1224, 2-14 (1990).
[CrossRef]

Welford, W. T.

W. T. Welford, Aberrations of the Symmetric Optical System (Academic, 1974).

Wright, D.

D. Wright, “Beamwidths of a diffracted laser using four proposed methods,” Opt. Quantum Electron. 24, S1129-S1135(1992).
[CrossRef]

Appl. Opt. (2)

IEEE J. Quantum Electron. (1)

A. E. Siegman, “Output beam propagation and beam quality from a multimode stable-cavity laser,” IEEE J. Quantum Electron. 29, 1212-1217 (1993).
[CrossRef]

Opt. Quantum Electron. (2)

J. A. Ruff and A. E. Siegman, “Measurement of beam quality degradation due to spherical aberration in a simple lens,” Opt. Quantum Electron. 26, 629-632(1994).
[CrossRef]

D. Wright, “Beamwidths of a diffracted laser using four proposed methods,” Opt. Quantum Electron. 24, S1129-S1135(1992).
[CrossRef]

Pramana J. Phys. (1)

R. Kapoor, P. K. Mukhopadhyay, J. George, and S. K. Sharma, “An alternative approach to determine the spot size of a multi-mode laser beam and its applications to diode laser beams,” Pramana J. Phys. 53, 307-319 (1999).
[CrossRef]

Proc. SPIE (1)

A. E. Siegman, “New developments in laser resonators,” Proc. SPIE 1224, 2-14 (1990).
[CrossRef]

Other (3)

H. H. Hopkins, Wave Theory of Aberrations (Oxford University, 1950).

W. T. Welford, Aberrations of the Symmetric Optical System (Academic, 1974).

W. Koechner, Solid State Laser Engineering, 6th ed. (Springer-Verlag, 2006), Chap. 5.1.1.

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Figures (4)

Fig. 1
Fig. 1

Variation of the correction factor F p , l for higher order transverse mode LG beam TEM p , l as a function the M 2 parameter. The value of the order of the LG beam p was varied from 0 to 21 in the simulation, and the value of index l was taken to be either “0” or “1”. The M 2 was estimated using ( 2 p + l + 1 ) . The last data point with M 2 = 100 was estimated for p = 49 and l = 1 .

Fig. 2
Fig. 2

Schematic of the experimental setup to study the effect of quartic phase aberration introduced due to spherical aberration effect of a plano-convex lens on a fiber coupled laser diode beam at 808 nm .

Fig. 3
Fig. 3

Propagation method to measure M 2 parameter for the case when the separation of the lens L 1 from the tip of the fiber coupled laser diode F20518 was 50 mm . The scattered points are the data points and the solid line represents the best numeri cal fit obtained with M 2 = 46.7 ± 2.5 , z 0 = 11.63 ± 0.7 , and W 0 = 0.145 ± 0.008 mm .

Fig. 4
Fig. 4

Variation of the M 2 parameter with normalized spot size at the lens location W / W q for the laser diodes F20514 and F20518. The measured data, theoretical fit, and expected variation with F p , l = 1.00 are shown for both the cases.

Equations (12)

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( M f 2 ) 2 = ( M i 2 ) 2 + ( 2 π ) 2 16 λ 2 β r 2 ( C 4 f r 4 ¯ f 3 ) 2 ,
β r = ( r 2 ¯ r 6 ¯ r 4 ¯ 2 r 4 ¯ 2 ) 1 / 2 .
M f 2 = 1 + ( w W q ) 8 ,
W q = [ f 3 λ 2 2 π C 4 f ] 1 / 4 .
W = 2 p + l + 1 w = M w .
M 2 = 2 p + l + 1.
M f 2 = ( M i 2 ) 2 + ( W W q p , l ) 8 ,
W q p , l W q = ( W 4 2 β r 2 r 4 ¯ ) 1 / 4 = F p , l .
M f 2 = ( M i 2 ) 2 + ( W F p , l W q ) 8 .
I [ p , l , r , φ ] = 1 N ( 2 π w 2 ) ρ [ r ] l L p l [ ρ [ r ] ] 2 c os 2 [ l φ ] e ρ [ r ] ,
N = r = 0 φ = 0 2 π ( 2 π w 2 ) ρ [ r ] l L p l [ ρ [ r ] ] 2 c os 2 [ l φ ] e ρ [ r ] r d r d φ .
r 2 m ¯ = 1 N r = 0 φ = 0 2 π I [ p , l , r , φ ] r 2 m + 1 d r d φ ,

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