Abstract

The extension of the Jones matrix formalism to higher-order transverse modes using N x N matrices presented in a previous paper [8] is applied to laser resonators. The resonator discussed in detail has a TEM01* Hermite-Gaussian mode, an axially symmetric polarizer combined with an axially symmetric phase shifter as a rear mirror and a folding mirror with conventional polarization dependent reflectivity and phase shift. The analysis reveals some useful regimes, where the output polarization is close to radial or azimuthal and the sensitivity to variations in the phase shift of the folding mirror is minimized.

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References

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  1. S. Quabis, R. Dorn, O. Gloeckl, M. Reichle, and M. Eberler, “Reduction of the spot size by using a radially polarized laser beam,” in Proceedings of IEEE International Seminar on Novel Trends in Nonlinear Laser Spectroscopy and High-Precision Measurements in Optics, eds.: S. N. Bagaev, V. N. Zadkov, and S. M. Arakelian, Proc. SPIE 4429, 105–111 (2001).
  2. V. G. Niziev and A. V. Nesterov, “„Influence of beam polarization on laser cutting efficiency,” J. Appl. Phys. D 32(13), 1455–1461 (1999).
    [CrossRef]
  3. M. Abdou Ahmed, A. Voss, M. Vogel, A. Austerschulte, J. Schulz, V. Metsch, T. Moser, and Th. Graf, “Radially polarized high-power lasers,” in Proceedings of IEEE XVII International Symposium on Gas Flow, Chemical Lasers, and High-Power Lasers (GCL-HPL), 15.-19., September 2008, Lisbon, Portugal, Proc. SPIE 7131, 71311I (2008).
    [CrossRef]
  4. I. Moshe, S. Jackel, and A. Meir, “Production of radially or azimuthally polarized beams in solid-state lasers and the elimination of thermally induced birefringence effects,” Opt. Lett. 28(10), 807–809 (2003).
    [CrossRef] [PubMed]
  5. M. A. Ahmed, J. Schulz, A. Voss, O. Parriaux, J. C. Pommier, and Th. Graf, “Radially polarized 3 kW beam from a CO2 laser with an intracavity resonant grating mirror,” Opt. Lett. 32(13), 1824–1826 (2007).
    [CrossRef] [PubMed]
  6. M. A. Ahmed, A. Voss, M. M. Vogel, and T. Graf, “Multilayer polarizing grating mirror used for the generation of radial polarization in Yb:YAG thin-disk lasers,” Opt. Lett. 32(22), 3272–3274 (2007).
    [CrossRef] [PubMed]
  7. T. Moser, J. Balmer, D. Delbeke, P. Muys, S. Verstuyft, and R. Baets, “Intracavity generation of radially polarized CO2 laser beams based on a simple binary dielectric diffraction grating,” Appl. Opt. 45(33), 8517–8522 (2006).
    [CrossRef] [PubMed]
  8. A. Voss, M. A. Ahmed, and Th. Graf, “Extension of the Jones matrix formalism to higher-order transverse modes,” Opt. Lett. 32(1), 83–85 (2007).
    [CrossRef]

2008 (1)

M. Abdou Ahmed, A. Voss, M. Vogel, A. Austerschulte, J. Schulz, V. Metsch, T. Moser, and Th. Graf, “Radially polarized high-power lasers,” in Proceedings of IEEE XVII International Symposium on Gas Flow, Chemical Lasers, and High-Power Lasers (GCL-HPL), 15.-19., September 2008, Lisbon, Portugal, Proc. SPIE 7131, 71311I (2008).
[CrossRef]

2007 (3)

2006 (1)

2003 (1)

1999 (1)

V. G. Niziev and A. V. Nesterov, “„Influence of beam polarization on laser cutting efficiency,” J. Appl. Phys. D 32(13), 1455–1461 (1999).
[CrossRef]

Abdou Ahmed, M.

M. Abdou Ahmed, A. Voss, M. Vogel, A. Austerschulte, J. Schulz, V. Metsch, T. Moser, and Th. Graf, “Radially polarized high-power lasers,” in Proceedings of IEEE XVII International Symposium on Gas Flow, Chemical Lasers, and High-Power Lasers (GCL-HPL), 15.-19., September 2008, Lisbon, Portugal, Proc. SPIE 7131, 71311I (2008).
[CrossRef]

Ahmed, M. A.

Austerschulte, A.

M. Abdou Ahmed, A. Voss, M. Vogel, A. Austerschulte, J. Schulz, V. Metsch, T. Moser, and Th. Graf, “Radially polarized high-power lasers,” in Proceedings of IEEE XVII International Symposium on Gas Flow, Chemical Lasers, and High-Power Lasers (GCL-HPL), 15.-19., September 2008, Lisbon, Portugal, Proc. SPIE 7131, 71311I (2008).
[CrossRef]

Baets, R.

Balmer, J.

Delbeke, D.

Graf, T.

Graf, Th.

M. Abdou Ahmed, A. Voss, M. Vogel, A. Austerschulte, J. Schulz, V. Metsch, T. Moser, and Th. Graf, “Radially polarized high-power lasers,” in Proceedings of IEEE XVII International Symposium on Gas Flow, Chemical Lasers, and High-Power Lasers (GCL-HPL), 15.-19., September 2008, Lisbon, Portugal, Proc. SPIE 7131, 71311I (2008).
[CrossRef]

A. Voss, M. A. Ahmed, and Th. Graf, “Extension of the Jones matrix formalism to higher-order transverse modes,” Opt. Lett. 32(1), 83–85 (2007).
[CrossRef]

M. A. Ahmed, J. Schulz, A. Voss, O. Parriaux, J. C. Pommier, and Th. Graf, “Radially polarized 3 kW beam from a CO2 laser with an intracavity resonant grating mirror,” Opt. Lett. 32(13), 1824–1826 (2007).
[CrossRef] [PubMed]

Jackel, S.

Meir, A.

Metsch, V.

M. Abdou Ahmed, A. Voss, M. Vogel, A. Austerschulte, J. Schulz, V. Metsch, T. Moser, and Th. Graf, “Radially polarized high-power lasers,” in Proceedings of IEEE XVII International Symposium on Gas Flow, Chemical Lasers, and High-Power Lasers (GCL-HPL), 15.-19., September 2008, Lisbon, Portugal, Proc. SPIE 7131, 71311I (2008).
[CrossRef]

Moser, T.

M. Abdou Ahmed, A. Voss, M. Vogel, A. Austerschulte, J. Schulz, V. Metsch, T. Moser, and Th. Graf, “Radially polarized high-power lasers,” in Proceedings of IEEE XVII International Symposium on Gas Flow, Chemical Lasers, and High-Power Lasers (GCL-HPL), 15.-19., September 2008, Lisbon, Portugal, Proc. SPIE 7131, 71311I (2008).
[CrossRef]

T. Moser, J. Balmer, D. Delbeke, P. Muys, S. Verstuyft, and R. Baets, “Intracavity generation of radially polarized CO2 laser beams based on a simple binary dielectric diffraction grating,” Appl. Opt. 45(33), 8517–8522 (2006).
[CrossRef] [PubMed]

Moshe, I.

Muys, P.

Nesterov, A. V.

V. G. Niziev and A. V. Nesterov, “„Influence of beam polarization on laser cutting efficiency,” J. Appl. Phys. D 32(13), 1455–1461 (1999).
[CrossRef]

Niziev, V. G.

V. G. Niziev and A. V. Nesterov, “„Influence of beam polarization on laser cutting efficiency,” J. Appl. Phys. D 32(13), 1455–1461 (1999).
[CrossRef]

Parriaux, O.

Pommier, J. C.

Schulz, J.

M. Abdou Ahmed, A. Voss, M. Vogel, A. Austerschulte, J. Schulz, V. Metsch, T. Moser, and Th. Graf, “Radially polarized high-power lasers,” in Proceedings of IEEE XVII International Symposium on Gas Flow, Chemical Lasers, and High-Power Lasers (GCL-HPL), 15.-19., September 2008, Lisbon, Portugal, Proc. SPIE 7131, 71311I (2008).
[CrossRef]

M. A. Ahmed, J. Schulz, A. Voss, O. Parriaux, J. C. Pommier, and Th. Graf, “Radially polarized 3 kW beam from a CO2 laser with an intracavity resonant grating mirror,” Opt. Lett. 32(13), 1824–1826 (2007).
[CrossRef] [PubMed]

Verstuyft, S.

Vogel, M.

M. Abdou Ahmed, A. Voss, M. Vogel, A. Austerschulte, J. Schulz, V. Metsch, T. Moser, and Th. Graf, “Radially polarized high-power lasers,” in Proceedings of IEEE XVII International Symposium on Gas Flow, Chemical Lasers, and High-Power Lasers (GCL-HPL), 15.-19., September 2008, Lisbon, Portugal, Proc. SPIE 7131, 71311I (2008).
[CrossRef]

Vogel, M. M.

Voss, A.

Appl. Opt. (1)

J. Appl. Phys. D (1)

V. G. Niziev and A. V. Nesterov, “„Influence of beam polarization on laser cutting efficiency,” J. Appl. Phys. D 32(13), 1455–1461 (1999).
[CrossRef]

Opt. Lett. (4)

Proc. SPIE (1)

M. Abdou Ahmed, A. Voss, M. Vogel, A. Austerschulte, J. Schulz, V. Metsch, T. Moser, and Th. Graf, “Radially polarized high-power lasers,” in Proceedings of IEEE XVII International Symposium on Gas Flow, Chemical Lasers, and High-Power Lasers (GCL-HPL), 15.-19., September 2008, Lisbon, Portugal, Proc. SPIE 7131, 71311I (2008).
[CrossRef]

Other (1)

S. Quabis, R. Dorn, O. Gloeckl, M. Reichle, and M. Eberler, “Reduction of the spot size by using a radially polarized laser beam,” in Proceedings of IEEE International Seminar on Novel Trends in Nonlinear Laser Spectroscopy and High-Precision Measurements in Optics, eds.: S. N. Bagaev, V. N. Zadkov, and S. M. Arakelian, Proc. SPIE 4429, 105–111 (2001).

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Figures (10)

Fig. 1
Fig. 1

Schematic setup of the laser resonator with folding mirror and radially symmetric polarizing rear mirror (a), the reflection properties of the mirrors are described by complex amplitude reflectivities as indicated; orthogonal set of polarization states for the TEM01* Hermite-Gaussian mode in RAH representation (b), the donut-shaped amplitude distribution of the electrical field is shown in gray scale, arrows indicate the local orientation of the field vector.

Fig. 2
Fig. 2

Round-trip loss factors ηi of the thin-disk laser resonator (details in the text) in dependence of the phase shift δ in the folding mirror for three different values ψ of the phase shift in the polarizing rear mirror.

Fig. 3
Fig. 3

Round-trip loss factor η3 of the thin-disk laser resonator (details in the text) in dependence of the phase shift δ in the folding mirror for different values ψ of the phase shift in the polarizing rear mirror.

Fig. 4
Fig. 4

Round-trip loss factor η3 of the thin-disk laser resonator (details in the text) in dependence of the phase shift ψ in the polarizing rear mirror for different values δ of the phase shift in the folding mirror.

Fig. 5
Fig. 5

Radiality R3 of the thin-disk laser resonator (details in the text) in dependence of the phase shift δ in the folding mirror for different values of the phase shift ψ in the polarizing rear mirror.

Fig. 6
Fig. 6

Radiality R3 of the thin-disk laser resonator (details in the text) in dependence of the phase shift ψ in the polarizing rear mirror for different values δ of the phase shift in the folding mirror.

Fig. 7
Fig. 7

Normalized round-trip loss factor η3 of the thin-disk laser resonator (details in the text) in dependence of the phase shifts in the polarizing rear mirror and in the folding mirror.

Fig. 8
Fig. 8

Radiality R3 of the thin-disk laser resonator (details in the text) in dependence of the phase shifts in the polarizing rear mirror and in the folding mirror.

Fig. 9
Fig. 9

Round-trip loss factor η3 of the thin-disk laser resonator (details in the text) in dependence of the reflectivity difference |s|2 -|t|2 in the polarizing rear mirror for different phase shifts ψ in the polarizing rear mirror and a fixed phase shift δ = 1° in the folding mirror.

Fig. 10
Fig. 10

Radiality R3 of the thin-disk laser resonator (details in the text) in dependence of the reflectivity difference |s|2 -|t|2 in the polarizing rear mirror for different phase shifts ψ in the polarizing rear mirror and a fixed phase shift δ = 1° in the folding mirror.

Equations (13)

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M p o l r o t ( ϕ ) = D ( ϕ ) ( s 0 0 t ) D ( ϕ ) , D ( ϕ ) = ( cos ( ϕ ) sin ( ϕ ) sin ( ϕ ) cos ( ϕ ) )
M p o l r o t     m i r r o r , R A H = M p o l r o t , R A H M m i r r o r , R A H = 1 2 ( 2 s 0 0 0 0 2 t 0 0 0 0 s + t 0 0 0 0 ( s + t ) )
λ 1 = r s , λ 2 = r t , λ 3 = r s + t 2 ,   and   λ 4 = r s + t 2
| λ 3 , 4 | = r 2 | s + t | = r 2 | s | 2 + | t | 2 + 2 | s | | t | cos ( arg ( s ) arg ( t ) ) < r 2 ( | s | + | t | )
M f o l d , R A H = M m i r r o r , R A H M A B C D , R A H ( A = u ; D = v ; B = C = 0 ) =
1 2 ( u + v 0 u v 0 0 ( u + v ) 0 v u u v 0 u + v 0 0 v u 0 ( u + v ) )
M r e s o , R A H = r M m i r r o r , R A H M f o l d , R A H M p o l r o t m i r r o r , R A H M f o l d , R A H =
r 8 ( A + B 0 D 0 0 C B 0 E D 0 A B 0 0 E 0 C + B )
with   A = ( 3 s + t ) ( u 2 + v 2 )     ,     C = ( 3 t + s ) ( u 2 + v 2 )     ,     D = ( 3 s + t ) ( u 2 v 2 ) ,         E = ( 3 t + s ) ( u 2 v 2 )     ,     a n d     B = 2 ( s t ) u v
λ 1 , 2 = r 8 [ C ± B 2 + E 2 ]   and   λ 3 , 4 = r 8 [ A ± B 2 + D 2 ]
V 1 , 2 = ( 0 ± B 2 + E 2 B E 0 1 )   and   V 3 , 4 = ( 1 0 ± B 2 + D 2 B D 0 )
R i = | V i ( 1 0 0 0 ) T | 2 / | V i | 2   and   A i = | V i ( 0 1 0 0 ) T | 2 / | V i | 2 ,
H 1 i = | V i ( 0 0 1 0 ) T | 2 / | V i | 2   and   H 2 i = | V i ( 0 0 0 1 ) T | 2 / | V i | 2 ,

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