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Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 21, Iss. 18 — Sep. 9, 2013
  • pp: 21076–21086
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Selection and generation of multipass modes in an open resonator

V. G. Niziev  »View Author Affiliations


Optics Express, Vol. 21, Issue 18, pp. 21076-21086 (2013)
http://dx.doi.org/10.1364/OE.21.021076


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Abstract

The original numerical wave model of the open resonator has been employed in the investigation of conditions for multipass mode generation. It is shown that for Fresnel numbers larger than unity, multiple reflections of radiation from the stable resonator mirrors lead to sustained quasi-stationary oscillations which are indicative of multipass mode generation. Various types of ray trajectories have been considered at the paraxial resonance conditions. Trajectory selecting techniques are suggested to provide the high quality output beams at large Fresnel numbers. The results of numerical experiments on amplitude-phase distribution of output radiation are presented for the suggested schemes.

© 2013 OSA

1. Introduction

Despite the numerous works devoted to physics of open resonators, from pioneer research [1

1. A. G. Fox and T. Li, “Resonant modes in a maser interferometer,” Bell Syst. Tech. J. 40(2), 453–488 (1961). [CrossRef]

, 2

2. H. Kogelnik and T. Li, “Laser beams and resonators,” Appl. Opt. 5(10), 1550–1567 (1966). [CrossRef] [PubMed]

] including the well-known monographs [3

3. A. E. Siegman, Lasers (University Science, 1986).

6

6. L. A. Weinstein, Open resonators and open waveguides (Golem Press, Boulder, 1969).

] to current publications, a number of principal issues still remain open. The evidential scientific clarification of these problems can favor the optimization of designing open laser resonators that offers high-quality beam generation at the Fresnel numbers (F) larger than unity.

The classical solutions for the resonator with round mirrors, Laguerre-Gaussian modes (for example [2

2. H. Kogelnik and T. Li, “Laser beams and resonators,” Appl. Opt. 5(10), 1550–1567 (1966). [CrossRef] [PubMed]

, 7

7. S. Solimeno, B. Crosignani, and P. DiPorto, Guiding, Diffraction and Confinement of Optical Radiation (Academic Press, 1986).

],), have been obtained with a number of severe limitations. One of them is the assumption that modes forming inside a resonator are single-pass ones. It is a widely held opinion in the classical works on numerical simulation of resonators that, after multiple reflections from the resonator mirrors, any initial field distribution is transformed to a static condition with a given amplitude-phase distribution. However, this opinion was only supported by the calculations made for resonators at the stability boundary [1

1. A. G. Fox and T. Li, “Resonant modes in a maser interferometer,” Bell Syst. Tech. J. 40(2), 453–488 (1961). [CrossRef]

].

In addition to the universally recognized approach to the description of the transverse modes in the stable resonator, works exist which deal with the peculiarities of the ray traces in the resonator and have been done in the context of geometric optics, for instance [8

8. I. A. Ramsay and J. J. Degnan, “A Ray Analysis of Optical Resonators Formed by Two Spherical Mirrors,” Appl. Opt. 9(2), 385–398 (1970). [CrossRef] [PubMed]

]. This poses the question of a possible relationship between the wave and geometric optics in the description of stable resonators, and of “single-pass” and “multipass” modes.

It is worth mentioning a number of experimental and theoretical works devoted to “multipass modes” [9

9. C.-H. Chen, P.-T. Tai, W.-F. Hsieh, and M.-D. Wei, “Multibeam-waist modes in an end-pumped Nd-YVO4 laser,” J. Opt. Soc. Am. B 20(6), 1220–1226 (2003). [CrossRef]

13

13. J. Dingjan, M. P. van Exter, and J. P. Woerdman, “Geometric modes in a single-frequency Nd:YVO4 laser,” Opt. Commun. 188(5-6), 345–351 (2001). [CrossRef]

]. In the experiments, these modes were excited under special conditions. A disk from Nd3+:YVO4 was used as an active medium. A beam of the pump laser moved relative to the disk axis, and the area of pumping was less than 1% of the disk area. The authors attribute the “multipass modes” to superposition of usual higher-order modes. Works [14

14. B. Sterman, A. Gabay, S. Yatsiv, and E. Dagan, “Off-axis folded laser beam trajectories in a strip-line CO2 laser,” Opt. Lett. 14(23), 1309–1311 (1989). [CrossRef] [PubMed]

, 15

15. D. Dick and F. Hanson, “M modes in a diode side-pumped Nd:glass slab laser,” Opt. Lett. 16(7), 476–477 (1991). [CrossRef] [PubMed]

] made an attempt to employ multipass modes in the hemiconfocal resonator in the lasers having the slit geometry of an active medium.

The present work is aimed at the integrated description of single-pass and multipass modes, which will permit to define their mutual place in the practice of open resonators application. The description of properties of the multipass modes will help in working out the recommendations for resonator designing to provide high quality of laser radiation at large F.

2. Numerical model of an open resonator

3. Paraxial resonances

For a Fresnel number F=rm2/λL (rm is a mirror radius, λ is wavelength, L is a resonator length) on the order of unity, the result of the numerical experiment for establishing the field distribution in the stable resonator was as expected. After multiple reflections, the distribution starts to resemble the principal Laguerre-Gaussian TEM00 mode of the stable resonator. However, for F considerably larger than unity, the picture is drastically changed. With all the examined parameters of the stable resonators g1 = g2 = g, 0<g<1, the field does not display a stationary distribution. The steady-state mode reveals the quasi-stationary oscillations of the field, Fig. 1
Fig. 1 The illustration of stationary (or quasi-stationary) generation established after multiple bounces in the stable resonators at different stability parameters g1 = g2. |E| is electric field amplitude, the Fresnel number is 4.5.
.

The analysis of these results and others obtained in our work unambiguously points to excitation of multipass modes. The classical conceptions of mode competition are, in principle, valid, though they are usually applied in the treatment of single-pass modes competition with each other, neglecting the possibilities of generating not clearly understood multipass modes. In the stable resonator with F>1, the multipass pattern of beam propagation has a competitive advantage over the classical single-pass modes. With a typical field radius, the single-pass Laguerre-Gaussian modes show higher diffraction loss at the mirror edges than the multipass modes. The loss of single-pass modes at the mirror edges is the same for each pass, while the field radius of a multipass mode is changed on each pass and the averaged loss of the multipass mode is lower.

In [8

8. I. A. Ramsay and J. J. Degnan, “A Ray Analysis of Optical Resonators Formed by Two Spherical Mirrors,” Appl. Opt. 9(2), 385–398 (1970). [CrossRef] [PubMed]

], a paraxial resonance equation is derived. It gives the mirror separation as a function of the radii of curvature of the mirrors and an integer N which is the number of return transits necessary to form a closed ray path.
g1g2=1+cosθ2;θ=2πKN;0KN/2
(1)
gi=1L/Ri, i = 1,2 are the parameters of open resonator stability diagram.

4. Ray trajectories of multipass modes

The ray traces in the resonator are geometrically described by the well-known ray matrix technique [8

8. I. A. Ramsay and J. J. Degnan, “A Ray Analysis of Optical Resonators Formed by Two Spherical Mirrors,” Appl. Opt. 9(2), 385–398 (1970). [CrossRef] [PubMed]

]. The trajectories presented below are not merely illustrative, but the relative position of their vertexes on the mirrors has been calculated. The terminology introduced in [8

8. I. A. Ramsay and J. J. Degnan, “A Ray Analysis of Optical Resonators Formed by Two Spherical Mirrors,” Appl. Opt. 9(2), 385–398 (1970). [CrossRef] [PubMed]

] will be used. In this work, the possible closed trajectories under paraxial resonance are divided into two categories called ecliptic and non-ecliptic (Fig. 2
Fig. 2 The typical ray trajectories at different N of paraxial resonance. The parameters of resonator stability are gi = 1-L/Ri.
).

If during the cycle the projected point reverses its motion and retraces the generated figure in the opposite direction, the ray path is ecliptic. In our case of axial symmetry the trajectory of ecliptic for three-pass resonance is shown in Fig. 2(a). It is the propagation of a thin ring inside the resonator. This trajectory resembles the case of a classical multipass resonator with the difference that the main resonator mirrors act as folding ones. Only two types of ecliptic trajectories (solid and dotted lines) depicted in Fig. 2(a) exist for the given resonator. The ray traces of the same ecliptic for other initial conditions of the ray can be obtained by compression (extension) of one of them symmetrically to the resonator axis.

If all portions of the projected figure are traced only once per cycle, the ray path is nonecliptic (Fig. 2(b)). This Fig. illustrates the symmetric type of ecliptic trajectories.

Under the chosen conditions of paraxial resonance different types of trajectories can be found. The rays propagating along these trajectories generate the transverse field structure of the multipass mode. The competition among the types of multipass modes offers a prevalence of the mode showing the lowest loss.

The cited above resonator model describes only axisymmetrical approaches to field distribution along the radius. Nevertheless, a particular opposite case can readily be imagined that involves the closed ray traces with respect to the azimuthal angle in the cylindrical coordinate system. It is so called non-planar versions of ray traces [8

8. I. A. Ramsay and J. J. Degnan, “A Ray Analysis of Optical Resonators Formed by Two Spherical Mirrors,” Appl. Opt. 9(2), 385–398 (1970). [CrossRef] [PubMed]

]. In the spherical-mirror resonators the closed ecliptic trajectories along the azimuthal coordinate are impossible. Ecliptic can only be found when the trajectory is located in the radial plane, while the variant of non-ecliptic for closed ray traces with respect to the azimuthal angle is well known (Fig. 3
Fig. 3 A six-pass closed ray trajectory with respect to the azimuthal angle.
). These modes were observed in particular in the coaxially pumped CO2 laser [18

18. V. I. Voronov, “Spatial characteristics of multipass modes in lasers with ring cross-section of active medium,” J. Tech. Phys. 65, 98–107 (1995).

].

5. Prospects for practical use of multipass modes

We now discuss the potential for high-quality laser radiation generation in the resonators with Fresnel number greater than unity, which make use of multipass modes. The foregoing data on generation of multipass modes allow one to understand the reasons for the low spatial quality of multimode generation for Fresnel numbers larger than unity. In this case, multipass modes are formed, and the conventional radiation coupling through a semitransparent mirror cannot provide an output beam with high spatial quality.. This conclusion is valid for any stable resonator having a high degeneracy (N small) paraxial resonance, since lower loss multipass volume modes can evolve. Under these conditions, the amplitude-phase distribution of the field at the output of the resonator is difficult to predict and it is therefore not productive to represent the radiation from such a resonator as a superposition of orthogonal Laguerre-Gaussian modes. Furthermore, this formal mathematical approach does not provide clear insight into the physics of multipass modes.

Given the diversity of manifestations of multipass transverse modes, it is difficult to provide general recommendations for upgrading beam quality. However, there are some specific cases which offer attractive prospects for generation of high-quality radiation at rather large Fresnel numbers. The general idea is as follows.

  • 1. By choosing the resonator parameters via Eq. (1), the conditions of paraxial resonance are achieved for the multipass mode with closed ray trajectories.
  • 2. The probable types of ray trajectories in the resonator are selected with subsequent generation of one of them and suppression of the others.
  • 3. Optimal coupling of radiation from the resonator is realized.

As regards the selection of ray trajectories, generation of “azimuthal modes” has primarily to be suppressed. It has been demonstrated in [19

19. A. V. Nesterov and V. G. Niziev, “Laser beams with axially symmetric polarization,” J. Phys. D Appl. Phys. 33(15), 1817–1822 (2000). [CrossRef]

] that, if laser radiation exhibits an axisymmetrical polarization (radial or azimuthal), the field distribution automatically becomes uniform along the azimuthal angle. The employment of a intracavity grating mirror is, in its turn, an effective means of generating a beam with radial or azimuthal polarization [20

20. A. V. Nesterov, V. G. Niziev, and V. P. Yakunin, “Generation of high-power radially polarized beam,” J. Phys. D Appl. Phys. 32(22), 2871–2875 (1999). [CrossRef]

, 21

21. M. A. Ahmed, J. Schulz, A. Voss, O. Parriaux, J. C. Pommier, and T. 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]

]. Such mirrors are produced by various technologies including diamond turning. In terms of beam quality, it can be reasoned that the presence of such a mirror in the resonator will cause effective suppression of the “azimuthal modes”. This will lead to selection of symmetrical non-ecliptic and to suppression of other ecliptics lacking a similar degree of circular symmetry (Fig. 2(b)).

To make a correct selection of the kind of ray trajectory, it is necessary to determine the means by which radiation is coupled from the resonator. Examination of the trajectories for N = 3 in Fig. 2 suggests that the most promising is radiation output in the region of the vertex which is located in the resonator axis at the centre of the mirror. All the trajectories of this kind produced by symmetric compression (extension) relative to the resonator axis pass through this point. The radiation from this resonator will be coupled through the output region with partial reflection at the mirror centre, on the resonator axis, at the location of this vertex. Here, both types of trajectories (ecliptic and non-ecliptic) are possible (Fig. 2). Beam truncation at the edges of the output region can be avoided by proper choice of the diameter of the partially reflecting region, such that the field amplitude is minimal at the boundary.

To suppress the generation along ecliptic trajectories, a non-reflecting zone has to be created in the area of localization of the vertexes of these trajectories. In this instance, it must be a ring zone at one or both of the mirrors. The desirable location of non-reflecting zones on the mirrors is qualitatively clear from Fig. 4. The best values of the outer ring radius r/rm, and annular width Δr/rm were chosen at the numerical calculation. The selection criterion was the amplitude-phase distribution similar to Laguerre-Gaussian mode.

Figure 5
Fig. 5 Radial distribution of the field amplitude (upper curve) and phase (lower curve) on mirror 2 (Fig. 4) in the steady-state mode of quasi-stationary generation. The calculation parameters: g1 = g2 = 0.5, N = 3, non-ecliptic, azimuthal polarization. Two identical non-reflecting zones are located on the both mirrors. The outer ring radius is r/rm = 0.5, while the annular width Δr/rm = 0.03. The output zone radius (filled in the Fig.) is r/rm = 0.33, with a reflectance of 50% and a Fresnel number F = 4.5. The phase distribution is shown on the selected spherical surface; its radius is R = 1.43L.
illustrates the calculated amplitude-phase distribution of output radiation with the above construction of the resonator. It is similar to the distribution of Laguerre-Gaussian TEM11* mode with azimuthal (or radial) polarization. Owing to symmetry of the non-ecliptic trajectory used in generation (Fig. 2(b)), the beam coupling from the resonator can be accomplished through both the mirrors simultaneously.

The extent of active medium utilization inside the resonator in generation of the multipass mode can be estimated from Fig. 4. The filled zones illustrate the propagation of the beams of the extracted three-pass mode, relying on purely geometrical approach. Actually, in terms of the wave nature of beam propagation, filling is better, as the calculation results show (Fig. 5).

The second example concerns the widely used semiconfocal resonator (Fig. 2(c,d)). Its parameters conform to the four-pass paraxial-resonance mode. This resonator can traditionally be operated in two ways. The first technique uses an intracavity aperture at F~1 to provide generation of the fundamental mode TEM00. Its shortcoming is a small mode volume. Another method consists in “multimode generation” with radiation coupling through a semitransparent mirror at F>1. In this case, the beam quality is substantially degraded.

Choose an ecliptic with a vertex at the center of a flat mirror as the generation trajectory, Fig. 2(c,d). Couple the radiation from the resonator through the partially reflecting zone located in the centre of the flat mirror. The second ecliptic and the non-ecliptic are blocked by the non-reflecting zones at the mirrors. The results of calculation of the amplitude-phase distribution on the flat mirror obtained after multiple reflections are given in Fig. 6
Fig. 6 Radial distribution of the field amplitude (upper curve) and phase (lower curve) on the flat mirror in the steady-state mode of quasi-stationary generation. The calculation parameters: g1 = 1, g2 = 0.5, N = 4, ecliptic, azimuthal polarization. The resonator mirrors have two non-reflecting zones. On the flat mirror it has the form of a ring with a mean radius r/rm = 0.3 and a width Δr/rm = 0.1. On the concave mirror, the non-reflecting zone at the mirror centre has the radius r/rm = 0.25. The output zone radius (filled in the Fig.) is r/rm = 0.25, and has a reflectance of 50%, and a Fresnel number, F = 4.5. The phase distribution is shown on the selected spherical surface; its curvature radius is R = 3.3L.
. The distribution resembles that of Laguerre-Gaussian TEM01* mode with azimuthal (or radial) polarization.

In the third example, generation of an eight-pass mode is treated (Fig. 2(e,f)). Let us couple the radiation from the resonator through the partially reflecting zone that is at the centre of the right mirror. The ecliptic of Fig. 2(e) will be used as the selected trajectory. The second ecliptic that is symmetrical to the first one and has vertices on the left mirror and the non-ecliptic in Fig. 2(f) must be blocked. This is most conveniently done by placing a zero-reflection zone at the centre of the left mirror with a diameter large enough to introduce a substantial loss to the second ecliptic and non-ecliptic trajectories. This non-reflecting zone also suppresses the fundamental Laguerre-Gaussian mode.

Figure 7
Fig. 7 Radial distribution of the field amplitude (upper curve) and phase (lower curve) on the right mirror in the steady-state mode of quasi-stationary generation. The calculation parameters: g1 = g2 = 0.383, N = 8, ecliptic, azimuthal polarization. The radius of the non-reflecting circle zone on the left mirror is r/rm = 0.153, the radius of the output zone on the right mirror (filled in the Fig.) is r/rm = 0.21, the output zone reflectance is 50%, and the Fresnel number is F = 10.1. The phase distribution is shown on the mirror surface.
displays the results of calculating the radial distribution of the field amplitude and phase on the right mirror (Fig. 2(e)) in the steady-state mode of quasi-stationary generation.

As shown in the Fig., the amplitude-phase distribution of the field in the output zone is similar in its structure to that of TEM21* mode with azimuthal (or radial) polarization. In the discussed example high-quality radiation can be coupled in the form of one ring or two rings. This permits optimization of the amount of resonator feedback in terms of energy.

The suggested technology of generating high-quality laser radiation at large Fresnel numbers is based on employment of multipass modes under paraxial resonance. The sharpness of resonance under variation of the resonator length was studied in [8

8. I. A. Ramsay and J. J. Degnan, “A Ray Analysis of Optical Resonators Formed by Two Spherical Mirrors,” Appl. Opt. 9(2), 385–398 (1970). [CrossRef] [PubMed]

]. With the total resonator length of 125 cm, the typical width of resonance at half-height made 2 to 3 cm. This raises an additional problem of resonator realignment for resonance attainment. For this purpose, it would be useful to provide for a possibility of some variation in the resonator length as a trimming parameter.

A misalignment in one or both mirrors is known to result in the displacement and rotation of the resonator axis. This does not violate the paraxial resonance conditions and, in terms of multipass mode generation, the resonator resistance to misalignment remains the same as for a single-pass mode.

As shown in [22

22. V. G. Niziev and D. Toebaert, “Formation of transverse mode in axially symmetric lasers,” Appl. Opt. 51(7), 954–962 (2012). [CrossRef] [PubMed]

], the structure of the active medium has a pronounced effect on the formation of the transverse structure of radiation. The above cited calculations assume a uniform distribution of the active medium gain. In the general case, calculations must allow for the geometry and properties of the active medium and would be similar to those made in [22

22. V. G. Niziev and D. Toebaert, “Formation of transverse mode in axially symmetric lasers,” Appl. Opt. 51(7), 954–962 (2012). [CrossRef] [PubMed]

].

6. Analysis of qualitative features of experimental results

The results of the present work permit the previously published experimental data to be critically analyzed. Thus, paper [8

8. I. A. Ramsay and J. J. Degnan, “A Ray Analysis of Optical Resonators Formed by Two Spherical Mirrors,” Appl. Opt. 9(2), 385–398 (1970). [CrossRef] [PubMed]

] reports the experiments with the CO2 laser. Radiation escaped through a hole in the mirror at the resonator axis. The resonator length could be varied without its misalignment. The main result of the experiment is the dependence of the power of radiation, emerging from the hole, on the resonator length. This dependence generally has a wide maximum with sharp narrow dips, which correspond to the conditions of paraxial resonances. At first glance, this does not in contrast with the results obtained (Figs. 5-7), which demonstrate effective generation at the selected conditions of paraxial resonances.

However, both these facts have a natural explanation. According to our research, practically always when they speak about “multimode” generation, we have some multipass modes possessing competitive advantages over the single pass LG modes. Two qualitatively different situations are now considered, depending on the resonator parameters.

  • a. The resonator parameters are out of the conditions of paraxial resonances. A transverse distribution is formed with the rays travelling along unclosed trajectories. In the experiment [8

    8. I. A. Ramsay and J. J. Degnan, “A Ray Analysis of Optical Resonators Formed by Two Spherical Mirrors,” Appl. Opt. 9(2), 385–398 (1970). [CrossRef] [PubMed]

    ], there is no competition between ray trajectories. All of them are very long and interact with the hole approximately in the same manner. It is a “usual situation”. A part of resonator energy exits from the hole. The comparatively high detector signal is obtained.
  • b. The conditions of paraxial resonances are satisfied. All ray trajectories are “well organized”, they are closed. For the given N, there are several types of closed trajectories. A competition between the multipass modes formed with different types of trajectories arises. For example, for N = 4 there are M- and W-type modes (the resonator axis is vertical). Making an outlet hole in one mirror, we insert the losses for the mode having a vertex in this very place (for example, for the M-mode). On the contrary, the W-mode practically does not interact with this very hole. The M-mode is depressed, the W-mode effectively generates. This results in the low detector signal of output power.

The experiment [8

8. I. A. Ramsay and J. J. Degnan, “A Ray Analysis of Optical Resonators Formed by Two Spherical Mirrors,” Appl. Opt. 9(2), 385–398 (1970). [CrossRef] [PubMed]

] was numerically simulated to illustrate this effect qualitatively, Fig. 8
Fig. 8 Radial distribution of the field amplitude on the mirror in the steady-state mode of quasi-stationary generation. The radius of the outlet hole (filled in the Fig.) is r/rm = 0.1. The Fresnel number F = 4.5. The polarization is plane. The calculation parameters are g1 = g2 = 0.5 (a) and g1 = g2 = 0.53 (b).
. A small hole was made in one mirror for detection of output power. There are no absorption zones for selection of trajectories. The output power at the paraxial resonance conditions, Fig. 8(a) is much lower, than in Fig. 8(b), when paraxial resonance is absent.

In our numerical experiments (Part 5), corresponding to paraxial resonance, we improved the quality of the useful mode by increasing the reflectivity of the hole from 0% to 50% and inserted the losses for the undesirable mode by setting an absorption zone on the second mirror. So we obtain high power at paraxial resonance conditions.

In [14

14. B. Sterman, A. Gabay, S. Yatsiv, and E. Dagan, “Off-axis folded laser beam trajectories in a strip-line CO2 laser,” Opt. Lett. 14(23), 1309–1311 (1989). [CrossRef] [PubMed]

, 15

15. D. Dick and F. Hanson, “M modes in a diode side-pumped Nd:glass slab laser,” Opt. Lett. 16(7), 476–477 (1991). [CrossRef] [PubMed]

] slab lasers are studied. The main disadvantage of these works is application of the coupled mirror that is semi-transparent over its whole surface. Therefore, they obtained the laser beam consisting of several spots travelling in space at different angles to each other.

The experiments with the multipass modes, cited in [9

9. C.-H. Chen, P.-T. Tai, W.-F. Hsieh, and M.-D. Wei, “Multibeam-waist modes in an end-pumped Nd-YVO4 laser,” J. Opt. Soc. Am. B 20(6), 1220–1226 (2003). [CrossRef]

13

13. J. Dingjan, M. P. van Exter, and J. P. Woerdman, “Geometric modes in a single-frequency Nd:YVO4 laser,” Opt. Commun. 188(5-6), 345–351 (2001). [CrossRef]

], are very special. They obtained generation just along a single ray trajectory created by the ray of a pumping diode laser. The conditions of our numerical simulation are opposite. We consider the spatially homogeneous active medium and study transverse mode formation by the multiple rays, travelling along the closed trajectories (at the chosen N).

Conclusion

The calculations were performed on the basis of the numerical wave model describing the axisymmetrical resonator [16

16. V. G. Niziev and R. V. Grishaev, “Dynamics of Mode Formation in an Open Resonator,” Appl. Opt. 49(34), 6582–6590 (2010). [CrossRef] [PubMed]

]. The model is based on the analytical description of radiation diffraction from a narrow ring. Reflection of an incident wave with the specified amplitude-phase distribution is referred to as the Green problem.

It was demonstrated that, in stable resonators with Fresnel numbers larger than unity, multiple reflections from the resonator mirrors lead to continuous quasi-stationary oscillations indicative of generation of the multipass modes. It has been shown that, in conditions of paraxial resonance, the multipass modes can provide high quality spatial modes. An approach to the design and construction of multipass resonators has been suggested and the methods of trajectory selection have been discussed for generating high-quality radiation at large Fresnel numbers. The results of numerical experiments on amplitude-phase distribution of output radiation are presented for the three- four- and eight-pass paraxial modes, and support the effectiveness of the proposed solutions for F = 4.5-10. The spatial quality of the output radiation is high in all the ecliptic and non-ecliptic trajectories at even and odd multiplicities of paraxial resonance.

References and links

1.

A. G. Fox and T. Li, “Resonant modes in a maser interferometer,” Bell Syst. Tech. J. 40(2), 453–488 (1961). [CrossRef]

2.

H. Kogelnik and T. Li, “Laser beams and resonators,” Appl. Opt. 5(10), 1550–1567 (1966). [CrossRef] [PubMed]

3.

A. E. Siegman, Lasers (University Science, 1986).

4.

N. Hodgson and H. Weber, Optical Resonators: Fundamentals, Advanced Concepts and Applications (Springer Verlag, Berlin, 1997).

5.

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6.

L. A. Weinstein, Open resonators and open waveguides (Golem Press, Boulder, 1969).

7.

S. Solimeno, B. Crosignani, and P. DiPorto, Guiding, Diffraction and Confinement of Optical Radiation (Academic Press, 1986).

8.

I. A. Ramsay and J. J. Degnan, “A Ray Analysis of Optical Resonators Formed by Two Spherical Mirrors,” Appl. Opt. 9(2), 385–398 (1970). [CrossRef] [PubMed]

9.

C.-H. Chen, P.-T. Tai, W.-F. Hsieh, and M.-D. Wei, “Multibeam-waist modes in an end-pumped Nd-YVO4 laser,” J. Opt. Soc. Am. B 20(6), 1220–1226 (2003). [CrossRef]

10.

H. H. Wu and W. F. Hsieh, “Observations of multipass transverse modes in an axially pumped solid-state laser with different fractionally degenerate resonator configurations,” J. Opt. Soc. Am. B 18(1), 7–12 (2001). [CrossRef]

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12.

C. H. Chen, P. Y. Huang, and C. W. Kuo, “Geometric modes outside the multi-bouncing fundamental Gaussian beam model,” J. Opt. 12(1), 015708 (2010). [CrossRef]

13.

J. Dingjan, M. P. van Exter, and J. P. Woerdman, “Geometric modes in a single-frequency Nd:YVO4 laser,” Opt. Commun. 188(5-6), 345–351 (2001). [CrossRef]

14.

B. Sterman, A. Gabay, S. Yatsiv, and E. Dagan, “Off-axis folded laser beam trajectories in a strip-line CO2 laser,” Opt. Lett. 14(23), 1309–1311 (1989). [CrossRef] [PubMed]

15.

D. Dick and F. Hanson, “M modes in a diode side-pumped Nd:glass slab laser,” Opt. Lett. 16(7), 476–477 (1991). [CrossRef] [PubMed]

16.

V. G. Niziev and R. V. Grishaev, “Dynamics of Mode Formation in an Open Resonator,” Appl. Opt. 49(34), 6582–6590 (2010). [CrossRef] [PubMed]

17.

A. V. Nesterov and V. G. Niziev, “Vector solution of the diffraction task using the Hertz vector,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 71(4), 046608 (2005). [CrossRef] [PubMed]

18.

V. I. Voronov, “Spatial characteristics of multipass modes in lasers with ring cross-section of active medium,” J. Tech. Phys. 65, 98–107 (1995).

19.

A. V. Nesterov and V. G. Niziev, “Laser beams with axially symmetric polarization,” J. Phys. D Appl. Phys. 33(15), 1817–1822 (2000). [CrossRef]

20.

A. V. Nesterov, V. G. Niziev, and V. P. Yakunin, “Generation of high-power radially polarized beam,” J. Phys. D Appl. Phys. 32(22), 2871–2875 (1999). [CrossRef]

21.

M. A. Ahmed, J. Schulz, A. Voss, O. Parriaux, J. C. Pommier, and T. 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]

22.

V. G. Niziev and D. Toebaert, “Formation of transverse mode in axially symmetric lasers,” Appl. Opt. 51(7), 954–962 (2012). [CrossRef] [PubMed]

OCIS Codes
(140.0140) Lasers and laser optics : Lasers and laser optics
(140.3410) Lasers and laser optics : Laser resonators

ToC Category:
Lasers and Laser Optics

History
Original Manuscript: June 27, 2013
Revised Manuscript: August 18, 2013
Manuscript Accepted: August 23, 2013
Published: September 3, 2013

Citation
V. G. Niziev, "Selection and generation of multipass modes in an open resonator," Opt. Express 21, 21076-21086 (2013)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-18-21076


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References

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  17. A. V. Nesterov and V. G. Niziev, “Vector solution of the diffraction task using the Hertz vector,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys.71(4), 046608 (2005). [CrossRef] [PubMed]
  18. V. I. Voronov, “Spatial characteristics of multipass modes in lasers with ring cross-section of active medium,” J. Tech. Phys.65, 98–107 (1995).
  19. A. V. Nesterov and V. G. Niziev, “Laser beams with axially symmetric polarization,” J. Phys. D Appl. Phys.33(15), 1817–1822 (2000). [CrossRef]
  20. A. V. Nesterov, V. G. Niziev, and V. P. Yakunin, “Generation of high-power radially polarized beam,” J. Phys. D Appl. Phys.32(22), 2871–2875 (1999). [CrossRef]
  21. M. A. Ahmed, J. Schulz, A. Voss, O. Parriaux, J. C. Pommier, and T. 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]
  22. V. G. Niziev and D. Toebaert, “Formation of transverse mode in axially symmetric lasers,” Appl. Opt.51(7), 954–962 (2012). [CrossRef] [PubMed]

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