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Optics Express

  • Editor: Michael Duncan
  • Vol. 13, Iss. 13 — Jun. 27, 2005
  • pp: 4952–4962
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Very high-order pure Laguerre-Gaussian mode selection in a passive Q-switched Nd:YAG laser

Amiel A. Ishaaya, Nir Davidson, and Asher A. Friesem  »View Author Affiliations


Optics Express, Vol. 13, Issue 13, pp. 4952-4962 (2005)
http://dx.doi.org/10.1364/OPEX.13.004952


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Abstract

An intra-cavity phase element, combined with a passive Q-switch saturable absorber and a suitable intra-cavity aperture, can provide extremely high mode discrimination, so as to obtain laser operation with single, pure, very high order Laguerre-Gaussian mode. With a Nd:YAG laser setup, well controlled and extremely stable Q-switched operation in the degenerate Laguerre-Gaussian TEM04, TEM14, TEM24, TEM34 , and TEM44 modes was obtained. The measured output energy per pulse for each of these modes was 5.2mJ, 7.5mJ, 10mJ, 12.5mJ, and 13.7mJ respectively, compared to 2.5mJ for the Gaussian mode without the phase element (more than a five fold increase in output energy). Correcting the phase for these modes, so that all transverse lobes have uniform phase, results in a very bright and narrow central lobe in the far field intensity distribution that can theoretically contain more than 90% of the output energy.

© 2005 Optical Society of America

1. Introduction

Pure high order transverse mode operation in laser cavities has been originally suggested in the 1960’s, and since then experimentally investigated in various types of laser configurations [1

1. M. Rioux, P. A. Belanger, and M. Cormier, “High-order circular-mode selection in a conical resonator,” Appl. Opt. 16, 1791–1792 (1977). [CrossRef] [PubMed]

] -[7

7. Y. Yadin, J. Scheuer, and M. Orenstein, “Light flowers from coherent vertical-cavity surface-emitting laser arrays,” Opt. Lett. 27, 1908–1910 (2002). [CrossRef]

]. In recent years rather efficient and robust high-order transverse mode selection in laser resonators was achieved by resorting to intra-cavity phase elements [8

8. R. Oron, N. Davidson, E. Hasman, and A. A. Friesem, “Transverse mode shaping and selection in laser resonators,” Progress in Optics 42, 325–386 (2001). [CrossRef]

, 9

9. A. A. Ishaaya, N. Davidson, G. Machavariani, E. Hasman, and A.A. Friesem, “Efficient selection of high-order Laguerre-Gaussian modes in a Q-switched Nd:YAG laser,” IEEE J. Quantum Electron. 39, 74–82 (2003). [CrossRef]

]. When placed in the resonator, the phase element introduces high losses to undesired modes and low losses to the selected mode. A limiting aperture is also used in the conventional manner to further discriminate against higher transverse modes. With such phase elements, efficient selection of pure TEM 01, TEM 02, TEM 03 and helical TEM 01* Laguerre-Gaussian (LG) modes was experimentally demonstrated, leading to an increase up to a factor of 2 in output power with respect to the fundamental TEM 00 mode. This was shown for CW and pulsed operation, and for different gain mediums (CO2, Nd:YAG). Since the entropy of a single high-order mode is equal to that of the fundamental mode [10

10. T. Graf and J. E. Balmer, “Laser beam quality, entropy and the limits of beam shaping,” Opt. Commun. 131, 77–83 (1996). [CrossRef]

], it is thermodynamically possible to efficiently transform a single high order mode into a Gaussian beam, thereby achieving excellent beam quality as well as high output power. Such a transformation can be performed externally by means of two specially designed phase elements [11

11. N. Davidson, A. A. Friesem, and E. Hasman, “Diffractive elements for annular laser beam transformation,” Appl. Phys. Lett. 61, 381–383 (1992). [CrossRef]

], or, as demonstrated recently, by coherently adding various transverse parts of the mode [12

12. A. A. Ishaaya, G. Machavariani, N. Davidson, E. Hasman, and A. A. Friesem, “Conversion of a high-order mode beam into a nearly Gaussian beam using a single interferometric element,” Opt. Lett. 28, 504–506 (2003). [CrossRef] [PubMed]

].

When trying to select very high-order modes, it is necessary to discriminate against many lower order modes. Such discrimination is difficult to achieve with a single intra-cavity phase element. Thus, the ability to select a pure transverse mode in a laser resonator seemed to be practically limited only to relatively low order modes, and hence resulted only in a modest increase in output powers compared to TEM 00 operation.

A passive Q-switch saturable absorber, where the transmission is intensity dependent, can provide additional discrimination between laser modes [13

13. W. R. Sooy, “The natural selection of modes in a passive q-switched laser,” Appl. Phys. Lett. 7, 36–37 (1965). [CrossRef]

, 14

14. A. Chandonnet, M. Piche, and N. McCarthy, “Beam Narrowing by saturable absorber in a Nd:YAG Laser,” Opt. Commun. 75, 123–128 (1990). [CrossRef]

]. Indeed, in some cases, a passive Q-switch saturable absorber can serve as a dynamic spatial filter [15

15. A. E. Siegman, Lasers, p. 1038 (University Science Books, Sausalito, California, 1986).

] and force TEM 00 operation even with intra-cavity apertures that are larger than needed for such operation. Further increase of the aperture size typically results in extremely erratic high-order or multimode lasing. This inability to obtain large mode volume in passive Q-switched lasers, either in a single high-order mode or multimode operation, poses a practical limit to the effective power extraction from such lasers.

In this article we show how an intra-cavity phase element, combined with the properties of a passive Q-switch saturable absorber and with a suitable intra-cavity aperture, can provide extremely high modal discrimination, enabling extremely stable and controlled operation of very high order LG modes (with as much as 40 lobes). These modes have a significantly larger mode volume than that of the Gaussian mode, resulting in more than 5-fold increase in output power. We further show that correcting the phase of these modes, so that they have transverse uniform phase, results in a bright central lobe in the far field.

2. Basic principles

A basic configuration for high-order mode selection with a binary phase element and a saturable absorber is schematically shown in Fig. 1. Similar to the case of non Q-switched operation (without a saturable absorber), the phase element, positioned near one of the cavity mirrors, introduces high diffraction losses to all the the undesired modes and very low losses to the desired mode, whereas the aperture provides for discrimination against modes of higher order than desired [8

8. R. Oron, N. Davidson, E. Hasman, and A. A. Friesem, “Transverse mode shaping and selection in laser resonators,” Progress in Optics 42, 325–386 (2001). [CrossRef]

, 9

9. A. A. Ishaaya, N. Davidson, G. Machavariani, E. Hasman, and A.A. Friesem, “Efficient selection of high-order Laguerre-Gaussian modes in a Q-switched Nd:YAG laser,” IEEE J. Quantum Electron. 39, 74–82 (2003). [CrossRef]

]. The passive Q-switch saturable absorber introduces additional transverse mode discrimination. The additional discrimination due to the passive Q-switch saturable absorber can be attributed to two factors. One is the rather slow opening of the switch (many round trips within the cavity), during which the mode competition introduces transverse discrimination. The other, more dominant factor, results from the fact that the transmission of the absorber depends on the intensity, which varies spatially, forming a dynamic spatial filter. We will now consider the later factor in some detail.

Fig. 1. A basic configuration for high-order mode selection in a laser resonator with an intra-cavity phase element and a passive Q-switch saturable absorber.
Fig. 2. Transmission of a typical saturable absorber (see ref. [16]).

3. Experimental procedure and results

To experimentally investigate mode selection in the presence of a saturable absorber, we used a pulsed Nd:YAG laser arrangement shown in Fig. 3. It includes a 70 cm long plano-concave stable resonator, with a concave (R=1.5 m) output coupler of 40% reflectivity at 1064 nm and a high-reflective flat mirror (the equivalent stability parameters [16

16. W. Koechner, Solid-state laser engineering (Springer-Verlag, 5th ed., Germany, 1999, p. 509).

] are g1=1 and g2=0.53). A flash lamp pumped Nd:YAG rod of 5 mm diameter and 10 cm length (1.1% doping) served as the gain medium. Throughout the experiments, the rod was pumped with a pulse rate of 0.5 Hz at a power level suitable for single pulse operation. Under these pumping conditions, the focal length of the thermal lens in the rod was measured to be more than 20 m. A thin film polarizer was inserted in order to obtain P-polarization operation. The intra-cavity aperture diameter was varied between 1 mm and 5 mm in order to allow selection of various transverse modes. A binary phase element with 8 azimuthal sections (see insert in Fig. 3), corresponding to the 0 and π phase regions of the TEM 04 degenerate LG mode, was positioned 4.5 cm from the rear mirror. This element was made of fused silica and was fabricated using photolithographic and reactive ion etching technologies to form the specific accurate depth profiles. These were subsequently coated with anti-reflection layers for 1064 nm. A plastic dye saturable absorber (BDM, made by Kodak), positioned 6.5 cm from the output coupler, was used for passive Q-switching. The near and far field intensity distributions were detected with CCD cameras, and a fast photodiode connected to an oscilloscope measured the temporal pulse shape and ensured single pulse operation (within one pump pulse).

Initially we operated the laser with the passive Q-switch saturable absorber but without the intra-cavity phase element. Single TEM 00 mode operation was obtained with intra-cavity aperture diameters as large as 1.8 mm. These values are larger than the 1.4 mm diameter required for selection of the TEM00 mode in free-running (non Q-switched) operation. The measured pulse energy was 2.5 mJ. The near and far field intensity distributions of the TEM 00 mode are shown in Fig. 4. The M 2 values calculated from these distributions were 1.03 and 1.08 for the X and Y directions respectively. Increasing the aperture diameter above 1.8 mm and below 2.3 mm resulted in TEM 01 operation, but this was very sensitive to alignment of the resonator mirrors. Increasing further the aperture diameter resulted in uncontrolled high-order mode lasing, where in each successive pulse a different mode was observed.

We then inserted the binary phase element for selecting the TEM 04 mode into the resonator,

and increased the aperture diameter to 3.5 mm. As expected, the laser output beam had a TEM 04 mode distribution. Increasing further the aperture diameter to 4.2 mm, 4.3 mm, 4.5 mm, and 4.8 mm, resulted in laser output beams with the TEM 14, TEM 24, TEM 34, and TEM 44 degenerate LG mode distributions, respectively. Lasing with these pure high order modes was found to be stable, and rather insensitive to alignment. The near and far field intensity distributions of these output beams are shown in Fig. 5. The measured output energy per pulse for the TEM 04, TEM 14, TEM 24, TEM 34, and TEM 44 mode distributions, were 5.2mJ, 7.5mJ, 10mJ, 12.5mJ, and 13.7mJ respectively. These results indicate that more than a five fold increase in output energy can be obtained as compared to that from a conventional passive Q-switch laser resonator with the Gaussian mode distribution (without the intra-cavity phase element).

Fig. 3. Experimental passive Q-switched Nd:YAG laser arrangement for intra-cavity highorder transverse mode selection.

Fig. 4. Experimental near and far field intensity distributions of the Gaussian TEM 00 mode in passive Q-switched operation. These were obtained with an intra-cavity aperture diameter of 1.8 mm and without an intra-cavity phase element.
Fig. 5. Experimental near and far field intensity distributions of high order modes (degenerate Laguerre-Gaussian TEM 04, TEM 14, TEM 24, TEM 34, and TEM 44 modes).

As expected, significant correlation was found between the multimode intensity distributions in non Q-switched (and in actively Q-switched) operation and the modes selected in passive Q-switched operation, in the same resonator configuration. This is shown in Fig. 7. Figures 7(a) and 7(b) show the experimentally detected far field multimode intensity distributions in active Q-switched operation, with the TEM 04 phase element and aperture diameters of 3.5 mm and 4.8 mm respectively. As evident, these two multimode distributions differ significantly. Figures 7(c) and 7(d) show only the peaks of the distributions of Figs. 7(a) and 7(b) respectively (this was obtained by discarding all intensity values below 85% of the maximum intensity). Now, assuming that the initial spatial transmission distribution of the saturable absorber, T(x,y), is proportional to peaks in these multimode distributions, we calculated for each of the two distributions shown in Figs. 7(c) and 7(d) the effective transmission for the LG TEM ρ4 modes, with ρ=0,1,2,3, 4. The effective transmission, Teff, for each mode was calculated by multiplying the mode intensity distribution with the spatial transmission T(x,y), and normalizing the result with respect to the mode distribution (see equation in Table I).The waist parameter of the TEM ρ4 modes was extracted from the experimental data in passive Q-switched operation. The calculated effective transmission values for each case are shown in Table I. As evident, the mode with the highest effective transmission in the case of a 3.5 mm aperture diameter is the TEM 04 mode, and in the case of a 4.8 mm aperture diameter is the TEM 44 mode; the corresponding calculated intensity distributions of these modes are shown in Figs. 7(e) and 7(f)), and they are in excellent agreement with the experimental results for passive Q-switched operation shown in Fig. 5.

Fig. 6. Far field intensity distribution of the laser output beam when operated with an active Q-switch, a TEM 04 phase element, and a 4.8 mm intra-cavity aperture diameter.
Fig. 7. Experimental multimode intensity distributions in active Q-switched operation and the corresponding selected modes in passive Q-switched operation. (a) and (b) experimental far field intensity distributions in active Q-switched operation for aperture diameters of 3.5 mm and 4.8 mm; (c) and (d) intensity distributions of (a) and (b) above a threshold of 85% of the maximum intensity; (e) and (f) corresponding calculated LG TEM 04 TEM 44 mode distributions that fit (c) and (d).

Table 1. Calculated effective transmission through the saturable absorber for each LG mode distribution, assuming the induced transmission patterns shown in Figs. 7(c) and 7(d). With Teff=T(x,y)·Imode·(x,y)·dxdyImode(x,y)·dxdy..

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Temporal pulse shape measurements for the high order mode passive Q-switched pulses were also performed. The results, shown in Fig. 8, reveal how the temporal pulse shape depends on the purity of the selected transverse modes. Figures 8(a) and 8(b) show a representative far field intensity distribution for a pure mode and the corresponding pulse shape. The measured pulse width (FWHM) was 44nsec. Figures 8(c) and 8(d) show what happens when a pure mode is not discriminated and selected, i.e. when the aperture diameter is such that two or more modes lase simultaneously. As evident, the far field intensity distribution is not that of a pure mode, and the corresponding temporal pulse shape has a double-hump shape, probably corresponding to the lasing of two (or more) participating modes. When the pump power is increased, two discrete laser pulses can be obtained within one pump pulse. In this case two high-order transverse modes lase, as shown in Figs. 8(e) and 8(f), where each of the modes corresponds to one of the pulses. Here a combination of TEM 04 and TEM 0,16 was observed in a single CCD camera frame (the time delay between the two pulses was about 24 µsec). We believe that the TEM 04 mode with the lowest losses lases with the first pulse and depletes the energy in the gain regions that correspond to this mode. Then the TEM 0,16 mode lases with the second pulse by exploiting energy that is stored in the outer part of the gain region.

Fig. 8. Experimental far field intensity distributions and corresponding temporal pulse shapes in passive Q-switched operation. (a) and (b) far field intensity distribution and corresponding pulse shape for a pure high-order mode selection; (c) and (d) far field intensity distribution and the corresponding pulse shape, for an impure mode selection; (e) and (f) far field intensity distribution and timing sequence for a double-pulse operation.

The pure high-order LG mode beam distributions shown in Fig. 5, have a relatively high M 2 and large divergence, but in principle can be efficiently converted into a nearly Gaussian TEM 00 mode [10

10. T. Graf and J. E. Balmer, “Laser beam quality, entropy and the limits of beam shaping,” Opt. Commun. 131, 77–83 (1996). [CrossRef]

, 12

12. A. A. Ishaaya, G. Machavariani, N. Davidson, E. Hasman, and A. A. Friesem, “Conversion of a high-order mode beam into a nearly Gaussian beam using a single interferometric element,” Opt. Lett. 28, 504–506 (2003). [CrossRef] [PubMed]

]. Even without conversion, when these pure high-order modes are corrected so that they have a uniform transverse phase distribution in the near field, the far field intensity distribution has a central lobe with most of the energy contained within it, although M 2 remains the same [19

19. A. E. Siegman, “Binary phase plates cannot improve laser beam quality,” Opt. Lett. 18, 675–677 (1993). [CrossRef] [PubMed]

].

We calculated the relative energy as a function of the radius in the far field intensity distribution, for several degenerate LG modes with uniform phase. The results for TEM 04, TEM 14, TEM 24, TEM 34, and TEM 44 modes with uniform phase distributions, are shown in Fig. 9 (the result for the TEM 00 is also shown for comparison). As evident, in all cases, more than 85% of the total energy is contained in the far field central lobe. For the higher order modes, that have more lobes in the central area, the energy contained in the central lobe increases and the width of the lobe decreases. Specifically, for TEM 44 with uniform phase distribution the energy in the central lobe exceeds 90%, and it’s width decreased by about a factor of 2 when compared to that of the TEM 00. In addition, since the far field central lobe closely resembles a Gaussian distribution, using a single aperture in the far field to filter out the side lobes would result in nearly Gaussian beam quality with high peak intensity.

We performed an experiment to obtain a uniform phase distribution, and thereby a high narrow central lobe, for the far field intensity distribution of a laser operating with TEM 04 mode. In this experiment we placed a TEM 04 binary phase element in the optical path of the laser output. Figure 10 shows the calculated and experimental far field intensity distributions for two laser outputs. In one the near field distribution was the usual TEM 04 mode distribution, and in the other TEM 04 mode distribution with uniform phase. As expected, we obtained a central bright lobe for that with the TEM 04 with the uniform phase distribution. The measured energy percentage contained in the central lobe was 73%, somewhat lower than the calculated percentage of 85%. This reduction can be attributed to imperfections in the external phase element or to a slight impurity of the selected mode (in phase or amplitude). Thus, although it was shown that applying a uniform phase to laser modes by use of binary phase elements does not improve the M 2 [19

19. A. E. Siegman, “Binary phase plates cannot improve laser beam quality,” Opt. Lett. 18, 675–677 (1993). [CrossRef] [PubMed]

], far field distributions with most of the energy in the central lobe can be obtained. Accordingly, the energy-in-the-bucket criterion would be more suitable than the M 2 criterion for characterizing the beam quality. Finally, it should be noted that the need for an external phase element in this case could be eliminated by placing the intra-cavity phase element near the output coupler [9

9. A. A. Ishaaya, N. Davidson, G. Machavariani, E. Hasman, and A.A. Friesem, “Efficient selection of high-order Laguerre-Gaussian modes in a Q-switched Nd:YAG laser,” IEEE J. Quantum Electron. 39, 74–82 (2003). [CrossRef]

].

Fig. 9. Calculated energy percentage as a function of radius in the far field intensity distribution for the TEM 00 mode, and for several high-order LG modes with uniform phase.
Fig. 10. Calculated and experimental far field intensity distributions for a laser operating with a LG TEM 04 mode. (a) for typical TEM 04 with alternating 0 and π phases for adjacent lobes; (b) for TEM 04 with uniform phase.

4. Concluding remarks

Acknowledgments

The authors would like to thank E. Galun, V. Krupkin, E. Luria, I. Pe’er, I. Shoshan and M. Sirota, from Elop Electro-optics Industries, for the passive Q-switch and helpful discussions.

References and links

1.

M. Rioux, P. A. Belanger, and M. Cormier, “High-order circular-mode selection in a conical resonator,” Appl. Opt. 16, 1791–1792 (1977). [CrossRef] [PubMed]

2.

A. P. Kolchenko, A. G. Nikitenko, and Y. K. Troitskill, “Control of the structure of transverse laser modes by phase-shifting masks,” Sov. J. Quantum Electron. 10, 1013–1016 (1980). [CrossRef]

3.

M. Piche and D. Cantin, “Enhancement of modal feedback in unstable resonators using mirrors with a phase step,” Opt. Lett. 16, 1135–1137 (1991). [CrossRef] [PubMed]

4.

K. M. Abramski, H. J. Baker, A. D. Colly, and D. R. Hall, “Single-mode selection using coherent imaging within a slab waveguide CO2 laser,” Appl. Phys. Lett. 60, 2469–2471 (1992). [CrossRef]

5.

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

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, 1976–1978 (1994). [CrossRef] [PubMed]

7.

Y. Yadin, J. Scheuer, and M. Orenstein, “Light flowers from coherent vertical-cavity surface-emitting laser arrays,” Opt. Lett. 27, 1908–1910 (2002). [CrossRef]

8.

R. Oron, N. Davidson, E. Hasman, and A. A. Friesem, “Transverse mode shaping and selection in laser resonators,” Progress in Optics 42, 325–386 (2001). [CrossRef]

9.

A. A. Ishaaya, N. Davidson, G. Machavariani, E. Hasman, and A.A. Friesem, “Efficient selection of high-order Laguerre-Gaussian modes in a Q-switched Nd:YAG laser,” IEEE J. Quantum Electron. 39, 74–82 (2003). [CrossRef]

10.

T. Graf and J. E. Balmer, “Laser beam quality, entropy and the limits of beam shaping,” Opt. Commun. 131, 77–83 (1996). [CrossRef]

11.

N. Davidson, A. A. Friesem, and E. Hasman, “Diffractive elements for annular laser beam transformation,” Appl. Phys. Lett. 61, 381–383 (1992). [CrossRef]

12.

A. A. Ishaaya, G. Machavariani, N. Davidson, E. Hasman, and A. A. Friesem, “Conversion of a high-order mode beam into a nearly Gaussian beam using a single interferometric element,” Opt. Lett. 28, 504–506 (2003). [CrossRef] [PubMed]

13.

W. R. Sooy, “The natural selection of modes in a passive q-switched laser,” Appl. Phys. Lett. 7, 36–37 (1965). [CrossRef]

14.

A. Chandonnet, M. Piche, and N. McCarthy, “Beam Narrowing by saturable absorber in a Nd:YAG Laser,” Opt. Commun. 75, 123–128 (1990). [CrossRef]

15.

A. E. Siegman, Lasers, p. 1038 (University Science Books, Sausalito, California, 1986).

16.

W. Koechner, Solid-state laser engineering (Springer-Verlag, 5th ed., Germany, 1999, p. 509).

17.

Y. F. Chen, T. M. Huang, C. F. Kao, C. L. Wang, and S. c. Wang, “Generation of Hermite-Gaussian modes in fiber-coupled laser-diode end-pumped lasers,” IEEE J. Quantum Electron. 33, 1025–1031 (1997). [CrossRef]

18.

Y. F. Chen, Y. P. Lan, and S. c. Wang, “Generation of Laguerre-Gaussian modes in fiber-coupled laser-diode end-pumped lasers,” Appl. Phys. B 72, 167–170 (2001). [CrossRef]

19.

A. E. Siegman, “Binary phase plates cannot improve laser beam quality,” Opt. Lett. 18, 675–677 (1993). [CrossRef] [PubMed]

OCIS Codes
(140.3300) Lasers and laser optics : Laser beam shaping
(140.3410) Lasers and laser optics : Laser resonators
(160.4670) Materials : Optical materials
(310.6860) Thin films : Thin films, optical properties
(340.7480) X-ray optics : X-rays, soft x-rays, extreme ultraviolet (EUV)

ToC Category:
Research Papers

History
Original Manuscript: April 22, 2005
Revised Manuscript: June 11, 2005
Published: June 27, 2005

Citation
Amiel Ishaaya, Nir Davidson, and Asher Friesem, "Very high-order pure Laguerre-Gaussian mode selection in a passive Q-switched Nd:YAG laser," Opt. Express 13, 4952-4962 (2005)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-13-13-4952


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References

  1. M. Rioux, P. A. Belanger and M. Cormier, �??High-order circular-mode selection in a conical resonator,�?? Appl. Opt. 16, 1791�??1792 (1977). [CrossRef] [PubMed]
  2. A. P. Kolchenko, A. G. Nikitenko and Y. K. Troitskill, �??Control of the structure of transverse laser modes by phase-shifting masks,�?? Sov. J. Quantum Electron. 10, 1013�??1016 (1980). [CrossRef]
  3. M. Piche and D. Cantin, �??Enhancement of modal feedback in unstable resonators using mirrors with a phase step,�?? Opt. Lett. 16, 1135�??1137 (1991). [CrossRef] [PubMed]
  4. K. M. Abramski, H. J. Baker, A. D. Colly and D. R. Hall, �??Single-mode selection using coherent imaging within a slab waveguide CO2 laser,�?? Appl. Phys. Lett. 60, 2469�??2471 (1992). [CrossRef]
  5. P. A. Belanger, R. L. Lachance and C. Pare, �??Super-Gaussian output from a CO2 laser by using a graded-phase mirror resonator,�?? Opt. Lett. 17, 739�??741 (1992). [CrossRef] [PubMed]
  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, 1976�??1978 (1994). [CrossRef] [PubMed]
  7. Y. Yadin, J. Scheuer and M. Orenstein, �??Light flowers from coherent vertical-cavity surface-emitting laser arrays,�?? Opt. Lett. 27, 1908�??1910 (2002). [CrossRef]
  8. R. Oron, N. Davidson, E. Hasman and A. A. Friesem, �??Transverse mode shaping and selection in laser resonators,�?? Progress in Optics 42, 325�??386 (2001). [CrossRef]
  9. A. A. Ishaaya, N. Davidson, G. Machavariani, E. Hasman and A.A. Friesem, �??Efficient selection of high-order Laguerre-Gaussian modes in a Q-switched Nd:YAG laser,�?? IEEE J. Quantum Electron. 39, 74�??82 (2003). [CrossRef]
  10. T. Graf and J. E. Balmer, �??Laser beam quality, entropy and the limits of beam shaping,�?? Opt. Commun. 131, 77�??83 (1996). [CrossRef]
  11. N. Davidson, A. A. Friesem and E. Hasman, �??Diffractive elements for annular laser beam transformation,�?? Appl. Phys. Lett. 61, 381�??383 (1992). [CrossRef]
  12. A. A. Ishaaya, G. Machavariani, N. Davidson, E. Hasman and A. A. Friesem, �??Conversion of a high-order mode beam into a nearly Gaussian beam using a single interferometric element,�?? Opt. Lett. 28, 504�??506 (2003). [CrossRef] [PubMed]
  13. W. R. Sooy, �??The natural selection of modes in a passive q-switched laser,�?? Appl. Phys. Lett. 7, 36�??37 (1965). [CrossRef]
  14. A. Chandonnet, M. Piche and N. McCarthy, �??Beam Narrowing by saturable absorber in a Nd:YAG Laser,�?? Opt. Commun. 75, 123�??128 (1990). [CrossRef]
  15. A. E. Siegman, Lasers, p. 1038 (University Science Books, Sausalito, California, 1986).
  16. W. Koechner, Solid-state laser engineering (Springer-Verlag, 5th ed., Germany, 1999, p. 509).
  17. Y. F. Chen, T. M. Huang, C. F. Kao, C. L. Wang and S. c. Wang, �??Generation of Hermite-Gaussian modes in fiber-coupled laser-diode end-pumped lasers,�?? IEEE J. Quantum Electron. 33, 1025�??1031 (1997). [CrossRef]
  18. Y. F. Chen, Y. P. Lan and S. c. Wang, �??Generation of Laguerre-Gaussian modes in fiber-coupled laser-diode end-pumped lasers,�?? Appl. Phys. B 72, 167�??170 (2001). [CrossRef]
  19. A. E. Siegman, �??Binary phase plates cannot improve laser beam quality,�?? Opt. Lett. 18, 675�??677 (1993). [CrossRef] [PubMed]

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