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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 16, Iss. 17 — Aug. 18, 2008
  • pp: 12707–12714
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Kaleidoscope modes in large aperture Porro prism resonators

Liesl Burger and Andrew Forbes  »View Author Affiliations


Optics Express, Vol. 16, Issue 17, pp. 12707-12714 (2008)
http://dx.doi.org/10.1364/OE.16.012707


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Abstract

We apply a new method of modeling Porro prism resonators, using the concept of rotating loss screens, to study stable and unstable Porro prism resonator. We show that the previously observed petal–like modal output is in fact only the lowest order mode, and reveal that a variety of kaleidoscope beam modes will be produced by these resonators when the intra–cavity apertures are sufficiently large to allow higher order modes to oscillate. We also show that only stable resonators will produce these modes.

© 2008 Optical Society of America

1. Introduction

In this paper we apply the model in [6

6. I. A. Litvin, L. Burger, and A. Forbes, “Petal-like modes in Porro prism resonators,” Opt. Express 15, 14065–14077 (2007). [CrossRef] [PubMed]

] to stable and unstable Porro prism resonators with large intracavity apertures. In section (2) we briefly review the conditions under which petal–like modes are found. In section (3) we consider large aperture stable resonators, and show that higher order modes exist and can be made to resonate if the intracavity apertures are sufficiently large. Further we make use of non–planar, unidirectional resonance analysis [7–9

7. N. Hodgson and H. Weber, Laser Resonators and Beam Propagation: Fundamentals, Advanced Concepts and Applications (Springer, 2005), Chap. 20.

] to understand the oscillating modes supported in these resonators (section (4)). These higher order modes bear close resemblance to recently reported kaleidoscope modes [10

10. Z. Bouchal, “Nondiffracting optical beams: physical properties, experiments, and applications,” Czech. J. Phys. 53, 537–624 (2003). [CrossRef]

,11

11. M. Anguiano-Morales, A. Martinez, M. D. Iturbe-Castillo, and S. Chavez-Cerda, “Different field distributions obtained with an axicon and an amplitude mask,” Opt. Commun. 281, 401–407 (2008). [CrossRef]

]. In [10

10. Z. Bouchal, “Nondiffracting optical beams: physical properties, experiments, and applications,” Czech. J. Phys. 53, 537–624 (2003). [CrossRef]

] the kaleidoscope modes were generated external to a laser cavity using a loss screen with remarkable similarity to that found inside stable Porro prism resonators. These similarities and the implications thereof are discussed in section (5). This leads to the conclusion, in section (6), that the petal–like modes hitherto reported are in fact only the lowest order modes, while higher order kaleidoscope modes are possible given sufficient transverse spatial extent to oscillate.

2. Porro prism resonator model

We followed the approach to modeling Porro prism resonators detailed in [6

6. I. A. Litvin, L. Burger, and A. Forbes, “Petal-like modes in Porro prism resonators,” Opt. Express 15, 14065–14077 (2007). [CrossRef] [PubMed]

], but briefly review the salient facts here to enhance clarity and readability of the paper. To build a physical optics model of a typical Porro prism resonator (see Fig. 1), the prisms are modeled as mirrors with rotating loss screens.

Fig. 1. A typical Porro prism based Nd:YAG laser with passive Q–switch, showing the following optical elements: Porro prisms (elements a and h); intra–cavity lenses (elements b and g); a beamsplitter cube (element c); a quarter wave plate (element d), and a passive Q–switch (element e). The prism apexes are shown in red.

This leads to a discrete set of Porro angles – the angle between the apexes of the opposite prisms when the prisms are viewed along the resonator length (direction of blue arrow in Fig. 1) – that allow the rotating loss screens to repeat on themselves, a necessary condition to generate the petal–like modes. At these discrete angles, α, the oscillating field is sub–divided into a finite number (N) of petals, given by:

N=j2πα
(1)

where

α=iπm,
(2)

and i, j, and m are integers, with certain constraints placed on j [6

6. I. A. Litvin, L. Burger, and A. Forbes, “Petal-like modes in Porro prism resonators,” Opt. Express 15, 14065–14077 (2007). [CrossRef] [PubMed]

]. The combination of allowed Porro angles and associated number of petals is illustrated in Fig. 2(a) together with examples of the petal–like modes (Fig. 2(b)) commonly observed from Porro resonators.

The Porro prism resonator investigated in this study was based on the system shown in Fig. 1, with an optical path length from prism to prism of L=10 cm, and lasing at λ=1064 nm. Two Porro prisms at either end of the laser formed the resonator, replacing traditional mirrors. The stability of the resonators was determined by the two identical intra–cavity lenses (of focal length f) each placed adjacent to a prism. The resonator was confined in the transverse direction by circular apertures placed immediately in front of each prism, and with radius a.

Fig. 2. (a) Plot of the discrete set of angles α that give rise to a petal pattern, with the corresponding number of petals to be observed; (b) example of petal–like modes for α=60° (top) and α=45° (bottom).

3. Generalized modal patterns

In this section we vary both the lens focal lengths f, and the aperture radius a in order to investigate the impact of resonator stability and effective Fresnel number on the oscillating modes. Because of the symmetry of the lens-aperture configuration, any chosen stable resonator can be described in terms of just two parameters:

G=g1=g2
(3)

and

NF=a2λL.
(4)

Here G is the equivalent G–parameter of the resonator, and NF is the effective Fresnel number; λ denotes the wavelength of the laser light in vacuum and L is the total optical path length inside the resonator.

The first observation is that unstable resonators do not generate repeating petal–like patterns, while stable resonators do. We preempt our discussion to follow later with the following geometrical optics argument: a ray traversing the resonator must return to a loss–free sub–division in order to create the complete petal pattern. The lack of ray repeatability and confinement in an unstable resonator precludes this from happening, and hence only stable resonators exhibit the petal–like modes. We can further eliminate loss as a mechanism to explain this observation in that the loss for both stable and unstable resonators was set arbitrarily in this study and yet did not influence the observation of petals, or the lack thereof. The discussion to follow will therefore concentrate on stable resonators only. Without any loss of generality, all spatial modes to follow are calculated at the face of one of the Porro prisms, and may be propagated to any other plane if so desired.

Consider by way of example three stable resonators chosen so that G=0.75 and with Porro angles (α) of 60°, 45° and 30° respectively. When the intracavity aperture is very small (NF ~1.5), no mode is able to resonate. At intermediate aperture sizes (NF ~3.5) the conventional petal–like modes are observed, with 6, 8 and 12 petals for α=60°, 45° and 30° respectively. At large aperture sizes (NF>6) the petal–like modes give way to more complex mode patterns. This increase in mode complexity as the aperture size increases suggests that the petal–like modes are in fact the lowest order modes of Porro prism resonators, while previously unreported higher order modes also exist, and can be made to resonate if given sufficiently large transverse freedom. These results are shown in Fig. 3.

Fig. 3. Modal patterns for the three Porro angles with increasing effective Fresnel number to the right in each row. As NF is increased (through an increase in aperture size), the modes become more complex, departing from the petal–like standard. (Media 1), (Media 2), (Media 3).

While the results are illustrated for α=45°, similar results are found at other Porro angles.

Fig. 4. The output modes of a number of Porro prism resonators arranged as a function of G (rows) and NF (columns). Note that in the petal–like cases the single repeating mode is shown, while in the higher order mode cases, only one of the oscillating modes is shown (Media 4).

4. Mode periodicity

Fig. 5. A multi-pass beam pass is possible for a given resonator configuration. If the gain region is small and central then a Gaussian mode is expected. The resonator can be forced into a higher multi-pass mode by off-centre pumping.

Such a configuration leads to a complex output beam pattern based on the possible beam paths through the resonator, which we can refer to as beam loops. Since each beam loop has a particular output pattern, it is convenient to refer to these patterns as modes of the resonator. Thus the modes and they periodicity are linked by the choice of resonator parameters.

This periodicity can be defined as the number of conventional round trips (double passes) required for any ray to return to an initial position and orientation, and can be found using geometrical ray analysis. If the Porro resonator matrix M is given by:

M=(112L01)(10f11)(10f11)(1L01)(10f11)(10f11)(112L01),
(5)

then an initial ray, which can be thought of as any element of a mode pattern, can be written as a two–row vector ν 0 describing both the position and angular deviation of the ray. After p round trips through the resonator, ν 0 will be transformed into a new vector νp according to:

vp=Mpv0=iλipαivi,
(6)

where λi and ν⃗i are the eigenvalues and eigenvectors of the matrix M respectively and αi are the coefficients required for the expansion of ν 0 in terms of the eigenvectors. For repeatability of the mode we require νp=ν 0, found from the solution to the simultaneous equations (for each i) λpi=1.

This approach allows the periodicity of the cycling modes to be determined analytically, and compared to the periodic pattern observed in the spot size data from the numerical model. The results are illustrated graphically in Fig. 6, as well as in Table 1.

Fig. 6. Plot of spot size for 12 000 round trips (double passes) through a resonator with Porro angle 30° for G=0.9, NF=9.4, illustrating the periodic nature of the spot size and showing eventual convergence. The sequence of modes through one period is also show.

Table 1. Periodicity comparison

table-icon
View This Table

Table 1 shows the agreement in periodicity predicted by geometric resonator theory compared to that observed in the numerical model for the beam loop modes. The numerical model also correctly predicts the higher order beam modes to have higher losses than the lower order petal–like mode. Because our numerical model allows the modes to oscillate indefinitely, loss selection ultimately results in the convergence of all starting fields to the petal–like patterns, as shown in Fig. 6 (see also Fig. 3 ‘large aperture mode’ movie). In the presence of gain and hence a limited build–up time, such a convergence would not necessarily take place.

It is pertinent at this point to discuss the possibility of the experimental observation of these complex beam patterns. Their losses are such that in a mode competing environment they are distinct from the petal–like patterns for a time period in the order of 1–2 µs, which is comparable to the mode build–up time of a typical actively Q–switched Porro prism laser (see Fig. 1). Thus while we cannot prove analytically that these complex beams are transverse modes of the resonator, their lifetime is such that it is very likely they are transverse modes, and there should be the possibility of observing them experimentally. There are however some limitations and technical challenges to such an experiment. It is likely that in a conventional linear standing wave resonator some combination of these modes might appear, and with a time averaged measurement a multi–mode pattern would be observed. We believe that we have already observed this.

Fig. 7. (a) Petal mode, (b) Experimental beam pattern, (c) Average of 5 cycles of higher-order modes at 1000 round trips.

Figures 7 (a)–(b) shows the comparison of a previously calculated petal pattern together with experimental verification [6

6. I. A. Litvin, L. Burger, and A. Forbes, “Petal-like modes in Porro prism resonators,” Opt. Express 15, 14065–14077 (2007). [CrossRef] [PubMed]

]. A time averaged output in the time period of the complex modes is shown in Fig. 7(c). Two observations can be made: firstly, the resulting pattern is again similar to a petal–like pattern, despite no petal–like mode component in the sequence, and secondly, the pattern shows an elongation of the energy distribution, and a departure from the compact petals seen in Fig. 7(a). The latter is more consistent with the experimentally observed pattern, which was measured on a stable resonator with large apertures. This suggests (but does not prove) that the complex modes we predict do indeed exist, and are stable enough with low enough losses to be resonant in the cavity. In this sense they are likely to be viewed as higher order modes of the resonator.

So how to measure these modes? It is possible that the approach of others in selecting multi–pass modes might be employed, together with knowledge of our particular field distributions, as predicted in this work. It has been shown that either preferentially increasing the gain [8

8. W. Liu, Y. Huo, X. Yin, and D. Zhao, “Modes of Multi-End-Pumped Nonplanar Ring Laser,” IEEE Photonics Technol. Lett. 17, 1776–1778 (2005). [CrossRef]

] or the loss [9

9. C. Bollig, W. A. Clarkson, D. C. Hanna, D. S. Lovering, and G. C. W. Jones, “Single-frequency operation of a monolithic Nd:glass ring laser via the acousto-optics effect,” Opt. Commun. 133, 221–224 (1997). [CrossRef]

] for a particular path can force oscillation of a particular multi–pass beam mode. The challenge is to adapt such approaches to mode selection in Porro prism resonators.

5. Kaleidoscope modes

In [10

10. Z. Bouchal, “Nondiffracting optical beams: physical properties, experiments, and applications,” Czech. J. Phys. 53, 537–624 (2003). [CrossRef]

] these field distributions were proposed as a result of the coherent superposition of n cosine gratings, each rotated with angular increments of ψ=π/n. The similarity between this and a rotating loss on the field at angles α=π/n (Eq. (1) with i=1) in Porro prism resonators probably accounts for the likeness in output modes. In [11

11. M. Anguiano-Morales, A. Martinez, M. D. Iturbe-Castillo, and S. Chavez-Cerda, “Different field distributions obtained with an axicon and an amplitude mask,” Opt. Commun. 281, 401–407 (2008). [CrossRef]

] kaleidoscope modes were generated using crossed apertures to sub–divide the input field to an axicon. This type of obstruction pattern is identical to the final loss field observed in Porro prism resonators (see Fig. 8 of [6

6. I. A. Litvin, L. Burger, and A. Forbes, “Petal-like modes in Porro prism resonators,” Opt. Express 15, 14065–14077 (2007). [CrossRef] [PubMed]

]). While such fields were previously created external to the laser cavity, we have shown that the fundamental property of field sub–division in Porro prisms can produce similar fields directly from the laser cavity.

The ubiquitous nature of Porro prism resonators makes a study of such modes necessary in its own right, but there also exists the possibility of using such complex modes to excite complex photonic crystal structures, and so further study is required.

6. Conclusion

We have applied a previously developed mathematical model of intracavity Porro prisms to stable and unstable Porro prism resonators with large intracavity apertures. We have shown that higher order modes exist only if NF is sufficiently large, and that these higher order modes closely resemble recently reported kaleidoscope modes due to the fundamental property of field sub–division in Porro prism resonators. The appearance of first the petal mode and then increasingly complex kaleidoscope modes with increasing aperture size leads to the conclusion that the petal–like modes are the lowest order modes of Porro prism resonators, while higher order modes exist in the form of kaleidoscope–like fields. We also predict that the standard petal mode is only observable from stable Porro prism resonators, and indicate how the stability criteria (G parameter) impacts on the cyclical nature of the higher order modes. We believe it is possible to observe these modes experimentally, but acknowledge that there are some technical challenges to overcome before doing so.

Acknowledgment

We would like to gratefully acknowledge the useful discussions and advice from Dr Christoph Bollig.

References and Links

1.

B. A. See, K. Fuelop, and R. Seymour, “An Assessment of the Crossed Porro Prism Resonator,” Technical Memorandum ERL-0162-TM, Electronics Research Lab Adelaide (Australia) (1980).

2.

M. Henriksson, L. Sjöqvista, and T. Uhrwing, “Numerical simulation of a battlefield Nd:YAG laser,” Proc. SPIE 5989, 59890I (2005). [CrossRef]

3.

M. Henriksson and L. Sjöqvist, “Numerical simulation of a flashlamp pumped Nd:YAG laser,” Report # ISRN FOI-R-1710-SE, Swedish Defence Research Agency, Linkoeping, Sensor Technology (2007).

4.

M. Ishizu, “Laser Oscillator,” US Patent 6816533 (2004).

5.

A. Rapaport and L. Weichman, “Laser Resonator Design Using Optical Ray Tracing Software: Comparisons with Simple Analytical Models and Experimental Results,” IEEE J. Quantum Electron. 371401–1408 (2001). [CrossRef]

6.

I. A. Litvin, L. Burger, and A. Forbes, “Petal-like modes in Porro prism resonators,” Opt. Express 15, 14065–14077 (2007). [CrossRef] [PubMed]

7.

N. Hodgson and H. Weber, Laser Resonators and Beam Propagation: Fundamentals, Advanced Concepts and Applications (Springer, 2005), Chap. 20.

8.

W. Liu, Y. Huo, X. Yin, and D. Zhao, “Modes of Multi-End-Pumped Nonplanar Ring Laser,” IEEE Photonics Technol. Lett. 17, 1776–1778 (2005). [CrossRef]

9.

C. Bollig, W. A. Clarkson, D. C. Hanna, D. S. Lovering, and G. C. W. Jones, “Single-frequency operation of a monolithic Nd:glass ring laser via the acousto-optics effect,” Opt. Commun. 133, 221–224 (1997). [CrossRef]

10.

Z. Bouchal, “Nondiffracting optical beams: physical properties, experiments, and applications,” Czech. J. Phys. 53, 537–624 (2003). [CrossRef]

11.

M. Anguiano-Morales, A. Martinez, M. D. Iturbe-Castillo, and S. Chavez-Cerda, “Different field distributions obtained with an axicon and an amplitude mask,” Opt. Commun. 281, 401–407 (2008). [CrossRef]

OCIS Codes
(140.0140) Lasers and laser optics : Lasers and laser optics
(140.3410) Lasers and laser optics : Laser resonators
(140.4780) Lasers and laser optics : Optical resonators
(230.5480) Optical devices : Prisms
(260.0260) Physical optics : Physical optics

ToC Category:
Lasers and Laser Optics

History
Original Manuscript: April 28, 2008
Revised Manuscript: June 28, 2008
Manuscript Accepted: August 1, 2008
Published: August 7, 2008

Citation
Liesl Burger and Andrew Forbes, "Kaleidoscope modes in large aperture Porro prism resonators," Opt. Express 16, 12707-12714 (2008)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-17-12707


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References

  1. B. A. See, K. Fuelop, and R. Seymour, "An Assessment of the Crossed Porro Prism Resonator," Technical Memorandum ERL-0162-TM, Electronics Research Lab Adelaide (Australia) (1980).
  2. M. Henriksson, L. Sjöqvista, and T. Uhrwing, "Numerical simulation of a battlefield Nd:YAG laser," Proc. SPIE 5989, 59890I (2005). [CrossRef]
  3. M. Henriksson and L. Sjöqvist, "Numerical simulation of a flashlamp pumped Nd:YAG laser," Report # ISRN FOI-R-1710-SE, Swedish Defence Research Agency, Linkoeping, Sensor Technology (2007).
  4. M. Ishizu, "Laser Oscillator," US Patent 6816533 (2004).
  5. A. Rapaport and L. Weichman, "Laser Resonator Design Using Optical Ray Tracing Software: Comparisons with Simple Analytical Models and Experimental Results," IEEE J. Quantum Electron. 37, 1401-1408 (2001). [CrossRef]
  6. I. A. Litvin, L. Burger, and A. Forbes, "Petal-like modes in Porro prism resonators," Opt. Express 15, 14065-14077 (2007). [CrossRef] [PubMed]
  7. N. Hodgson and H. Weber, Laser Resonators and Beam Propagation: Fundamentals, Advanced Concepts and Applications (Springer, 2005), Chap. 20.
  8. W. Liu, Y. Huo, X. Yin, and D. Zhao, "Modes of Multi-End-Pumped Nonplanar Ring Laser," IEEE Photon. Technol. Lett. 17, 1776-1778 (2005). [CrossRef]
  9. C. Bollig, W. A. Clarkson, D. C. Hanna, D. S. Lovering, and G. C. W. Jones, "Single-frequency operation of a monolithic Nd:glass ring laser via the acousto-optics effect," Opt. Commun. 133, 221-224 (1997). [CrossRef]
  10. Z. Bouchal, "Nondiffracting optical beams: physical properties, experiments, and applications," Czech. J. Phys. 53, 537-624 (2003). [CrossRef]
  11. M. Anguiano-Morales, A. Martinez, M. D. Iturbe-Castillo, and S. Chavez-Cerda, "Different field distributions obtained with an axicon and an amplitude mask," Opt. Commun. 281, 401-407 (2008). [CrossRef]

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