Kaleidoscope modes in large aperture Porro prism resonators

Transcription

1 Kaleidoscoe modes in large aerture Porro rism resonators Liesl Burger,2,* and Andrew Forbes,2 CSIR National Laser Centre, PO Box 395, Pretoria 000, South Africa 2 School of Physics, University of KwaZulu Natal, Private Bag X5400, Durban 4000, South Africa Corresonding authors: *lburger@csir.co.za, aforbes@csir.co.za Abstract: We aly a new method of modeling Porro rism resonators, using the concet of rotating loss screens, to study stable and unstable Porro rism resonator. We show that the reviously observed etal like modal outut is in fact only the lowest order mode, and reveal that a variety of kaleidoscoe beam modes will be roduced by these resonators when the intra cavity aertures are sufficiently large to allow higher order modes to oscillate. We also show that only stable resonators will roduce these modes Otical Society of America OCIS codes: ( ) Otical resonators; ( ) Physical otics; (40.340) Laser resonators; ( ) Prisms; (40.040) Lasers and laser otics. References and Links. B. A. See, K. Fuelo, and R. Seymour, An Assessment of the Crossed Porro Prism Resonator, Technical Memorandum ERL-062-TM, Electronics Research Lab Adelaide (Australia) (980). 2. M. Henriksson, L. Sjöqvista, and T. Uhrwing, Numerical simulation of a battlefield Nd:YAG laser, Proc. SPIE 5989, 59890I (2005). 3. M. Henriksson and L. Sjöqvist, Numerical simulation of a flashlam umed Nd:YAG laser, Reort # ISRN FOI-R-70-SE, Swedish Defence Research Agency, Linkoeing, Sensor Technology (2007). 4. M. Ishizu, Laser Oscillator, US Patent (2004). 5. A. Raaort and L. Weichman, Laser Resonator Design Using Otical Ray Tracing Software: Comarisons with Simle Analytical Models and Exerimental Results, IEEE J. Quantum Electron (200). 6. I. A. Litvin, L. Burger, and A. Forbes, Petal like modes in Porro rism resonators, Ot. Exress 5, (2007). 7. N. Hodgson and H. Weber, Laser Resonators and Beam Proagation: Fundamentals, Advanced Concets and Alications (Sringer, 2005), Cha W. Liu, Y. Huo, X. Yin, and D. Zhao, Modes of Multi-End-Pumed Nonlanar Ring Laser, IEEE Photonics Technol. Lett. 7, (2005). 9. C. Bollig, W. A. Clarkson, D. C. Hanna, D. S. Lovering, and G. C. W. Jones, Single-frequency oeration of a monolithic Nd:glass ring laser via the acousto-otics effect, Ot. Commun. 33, (997). 0. Z. Bouchal, Nondiffracting otical beams: hysical roerties, exeriments, and alications, Czech. J. Phys. 53, (2003).. M. Anguiano Morales, A. Martinez, M. D. Iturbe Castillo, and S. Chavez Cerda, Different field distributions obtained with an axicon and an amlitude mask, Ot. Commun. 28, (2008).. Introduction Porro risms have the useful roerty that all rays incident on the rism are reflected back arallel to the initial roagation direction, indeendent of the angle of incidence, thus making them insensitive to misalignment. This roerty is used in a Porro rism resonator, in which the end mirrors of a simle flat flat resonator have been relaced by Porro risms. Such resonators have been exloited for their ruggedness, and found use in alications where a laser beam is required at a large distance from the source, and where the source is not a stable latform; for examle, range finding and laser designators, mostly for the military [ 4]. Comrehensive models of these resonators exist [2,3,5], and have considered gain and outut couling loss, olarization, stability and temoral behavior, but desite the ubiquitous nature (C) 2008 OSA 8 August 2008 / Vol. 6, No. 7 / OPTICS EXPRESS 2707

2 of these lasers in the field, for a long time the outut satial modes from such lasers were not fully understood. Recently a new aroach to modeling Porro rism resonators has been outlined, making use of rotating loss screens to mimic intracavity Porro risms, which correctly redicts all the salient features of the observed etal like satial modal atterns [6]. In this aer we aly the model in [6] to stable and unstable Porro rism resonators with large intracavity aertures. In section (2) we briefly review the conditions under which etal like modes are found. In section (3) we consider large aerture stable resonators, and show that higher order modes exist and can be made to resonate if the intracavity aertures are sufficiently large. Further we make use of non lanar, unidirectional resonance analysis [7 9] to understand the oscillating modes suorted in these resonators (section (4)). These higher order modes bear close resemblance to recently reorted kaleidoscoe modes [0,]. In [0] the kaleidoscoe modes were generated external to a laser cavity using a loss screen with remarkable similarity to that found inside stable Porro rism resonators. These similarities and the imlications thereof are discussed in section (5). This leads to the conclusion, in section (6), that the etal like modes hitherto reorted are in fact only the lowest order modes, while higher order kaleidoscoe modes are ossible given sufficient transverse satial extent to oscillate. 2. Porro rism resonator model We followed the aroach to modeling Porro rism resonators detailed in [6], but briefly review the salient facts here to enhance clarity and readability of the aer. To build a hysical otics model of a tyical Porro rism resonator (see Fig. ), the risms are modeled as mirrors with rotating loss screens. Fig.. A tyical Porro rism based Nd:YAG laser with assive Q switch, showing the following otical elements: Porro risms (elements a and h); intra cavity lenses (elements b and g); a beamslitter cube (element c); a quarter wave late (element d), and a assive Q switch (element e). The rism aexes are shown in red. This leads to a discrete set of Porro angles the angle between the aexes of the oosite risms when the risms are viewed along the resonator length (direction of blue arrow in Fig. ) that allow the rotating loss screens to reeat on themselves, a necessary condition to generate the etal like modes. At these discrete angles, α, the oscillating field is sub divided into a finite number (N) of etals, given by: where j2π N = () α (C) 2008 OSA 8 August 2008 / Vol. 6, No. 7 / OPTICS EXPRESS 2708

3 iπ α =, (2) m and i, j, and m are integers, with certain constraints laced on j [6]. The combination of allowed Porro angles and associated number of etals is illustrated in Fig. 2(a) together with examles of the etal like modes (Fig. 2(b)) commonly observed from Porro resonators. The Porro rism resonator investigated in this study was based on the system shown in Fig., with an otical ath length from rism to rism of L = 0 cm, and lasing at λ = 064 nm. Two Porro risms at either end of the laser formed the resonator, relacing traditional mirrors. The stability of the resonators was determined by the two identical intra cavity lenses (of focal length f) each laced adjacent to a rism. The resonator was confined in the transverse direction by circular aertures laced immediately in front of each rism, and with radius a Number of etals N Porro Angle (degrees) (b) (a) (b) Fig. 2. (a) Plot of the discrete set of angles α that give rise to a etal attern, with the corresonding number of etals to be observed; (b) examle of etal like modes for α =60 (to) and α =45 (bottom). The laser was modeled by successive asses through a folded out resonator without any gain, using an array size of to describe the oscillating field. The modal build u was started from a field comrising random noise, and continued until the round tri loss stabilized to within 0.5%. Each rism was assumed to be equivalent to a erfect mirror suerimosed on a rotating loss line as described in []. 3. Generalized modal atterns In this section we vary both the lens focal lengths f, and the aerture radius a in order to investigate the imact of resonator stability and effective Fresnel number on the oscillating modes. Because of the symmetry of the lens aerture configuration, any chosen stable resonator can be described in terms of just two arameters: and G = g = g 2 (3) N F 2 a = λl. (4) (C) 2008 OSA 8 August 2008 / Vol. 6, No. 7 / OPTICS EXPRESS 2709

4 Here G is the equivalent G arameter of the resonator, and N F is the effective Fresnel number; λ denotes the wavelength of the laser light in vacuum and L is the total otical ath length inside the resonator. The first observation is that unstable resonators do not generate reeating etal like atterns, while stable resonators do. We reemt our discussion to follow later with the following geometrical otics argument: a ray traversing the resonator must return to a loss free sub division in order to create the comlete etal attern. The lack of ray reeatability and confinement in an unstable resonator recludes this from haening, and hence only stable resonators exhibit the etal like modes. We can further eliminate loss as a mechanism to exlain 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 etals, or the lack thereof. The discussion to follow will therefore concentrate on stable resonators only. Without any loss of generality, all satial modes to follow are calculated at the face of one of the Porro risms, and may be roagated to any other lane if so desired. Consider by way of examle three stable resonators chosen so that G = 0.75 and with Porro angles (α ) of 60, 45 and 30 resectively. When the intracavity aerture is very small (N F ~.5), no mode is able to resonate. At intermediate aerture sizes (N F ~ 3.5) the conventional etal like modes are observed, with 6, 8 and 2 etals for α = 60, 45 and 30 resectively. At large aerture sizes (N F > 6) the etal like modes give way to more comlex mode atterns. This increase in mode comlexity as the aerture size increases suggests that the etal like modes are in fact the lowest order modes of Porro rism resonators, while reviously unreorted 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 atterns for the three Porro angles with increasing effective Fresnel number to the right in each row. As N F is increased (through an increase in aerture size), the modes become more comlex, dearting from the etal like standard. (Media ), (Media 2), (Media 3). Porro rism resonators aear to offer a rich landscae of ossible modes, many of which have not been associated with this tye of resonator reviously. While the revious discussion focused on one articular resonator for three Porro angles, we illustrate in Fig. 4 that the resonator arameter G also influences the oscillating mode, as one might exect. While the results are illustrated for α = 45, similar results are found at other Porro angles. (C) 2008 OSA 8 August 2008 / Vol. 6, No. 7 / OPTICS EXPRESS 270

5 Fig. 4. The outut modes of a number of Porro rism resonators arranged as a function of G (rows) and N F (columns). Note that in the etal like cases the single reeating mode is shown, while in the higher order mode cases, only one of the oscillating modes is shown (Media 4). 4. Mode eriodicity The higher order modes deicted in Fig. 3 and Fig. 4 exhibit an interesting feature: they reeat after a fixed number of asses through the resonator. This eriodicity is not a function of the Porro angle α but rather of G, and is the result of the resonator s comlex eigenvalues. One can understand this eriodicity if one considers the similarities to the well known Herriot cell resonator [7] and by following the ath of a ray through the resonator. Such resonators result in a eriodicity that is not a double ass through the resonator, as is the case in a standard Fabry Perot system, but rather is based on a uni directional analysis, where the number of asses can be made very large for a comlete round tri in this case round tri refers to the condition that the beam reeats a revious ath through the resonator. The number of reflections and the orientation of the beam, and hence the eriodicity of the resonator, can be controlled by judicious choice of the resonator arameters. This concet is illustrated in Fig. 5 where a standard resonator is oerated as a non lanar ring laser [8]. In this case the beam asses through the resonator six times (or reflects off each mirror three times) in a single round tri. Fig. 5. A multi ass beam ass is ossible for a given resonator configuration. If the gain region is small and central then a Gaussian mode is exected. The resonator can be forced into a higher multi ass mode by off centre uming. Such a configuration leads to a comlex outut beam attern based on the ossible beam aths through the resonator, which we can refer to as beam loos. Since each beam loo has a (C) 2008 OSA 8 August 2008 / Vol. 6, No. 7 / OPTICS EXPRESS 27

6 articular outut attern, it is convenient to refer to these atterns as modes of the resonator. Thus the modes and they eriodicity are linked by the choice of resonator arameters. This eriodicity can be defined as the number of conventional round tris (double asses) required for any ray to return to an initial osition and orientation, and can be found using geometrical ray analysis. If the Porro resonator matrix M is given by: = L 0 0 L 0 0 L M , (5) f f f f 0 then an initial ray, which can be thought of as any element of a mode attern, can be written as a two row vector v 0 describing both the osition and angular deviation of the ray. After round tris through the resonator, v 0 will be transformed into a new vector v according to: v i i = M v0 = λ α v, (6) where λ i and v i are the eigenvalues and eigenvectors of the matrix M resectively and α i are the coefficients required for the exansion of v 0 in terms of the eigenvectors. For reeatability of the mode we require v = v 0, found from the solution to the simultaneous equations (for each i) λ i =. This aroach allows the eriodicity of the cycling modes to be determined analytically, and comared to the eriodic attern observed in the sot size data from the numerical model. The results are illustrated grahically in Fig. 6, as well as in Table. i i Fig. 6. Plot of sot size for round tris (double asses) through a resonator with Porro angle 30 for G = 0.9, N F = 9.4, illustrating the eriodic nature of the sot size and showing eventual convergence. The sequence of modes through one eriod is also show. (C) 2008 OSA 8 August 2008 / Vol. 6, No. 7 / OPTICS EXPRESS 272

7 Table. Periodicity comarison G Theory Model -.0 none none 0.0 none none Table shows the agreement in eriodicity redicted by geometric resonator theory comared to that observed in the numerical model for the beam loo modes. The numerical model also correctly redicts the higher order beam modes to have higher losses than the lower order etal 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 etal like atterns, as shown in Fig. 6 (see also Fig. 3 large aerture mode movie). In the resence of gain and hence a limited build u time, such a convergence would not necessarily take lace. It is ertinent at this oint to discuss the ossibility of the exerimental observation of these comlex beam atterns. Their losses are such that in a mode cometing environment they are distinct from the etal like atterns for a time eriod in the order of 2 μs, which is comarable to the mode build u time of a tyical actively Q switched Porro rism laser (see Fig. ). Thus while we cannot rove analytically that these comlex 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 ossibility of observing them exerimentally. There are however some limitations and technical challenges to such an exeriment. It is likely that in a conventional linear standing wave resonator some combination of these modes might aear, and with a time averaged measurement a multi mode attern would be observed. We believe that we have already observed this. (a) (b) (c) Fig. 7. (a) Petal mode, (b) Exerimental beam attern, (c) Average of 5 cycles of higher-order modes at 000 round tris. Figures 7 (a) (b) shows the comarison of a reviously calculated etal attern together with exerimental verification [6]. A time averaged outut in the time eriod of the comlex modes is shown in Fig. 7(c). Two observations can be made: firstly, the resulting attern is again similar to a etal like attern, desite no etal like mode comonent in the sequence, and secondly, the attern shows an elongation of the energy distribution, and a dearture from the comact etals seen in Fig. 7(a). The latter is more consistent with the exerimentally observed attern, which was measured on a stable resonator with large aertures. This suggests (but does not rove) that the comlex modes we redict do indeed exist, and are (C) 2008 OSA 8 August 2008 / Vol. 6, No. 7 / OPTICS EXPRESS 273

8 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 ossible that the aroach of others in selecting multi ass modes might be emloyed, together with knowledge of our articular field distributions, as redicted in this work. It has been shown that either referentially increasing the gain [8] or the loss [9] for a articular ath can force oscillation of a articular multi ass beam mode. The challenge is to adat such aroaches to mode selection in Porro rism resonators. 5. Kaleidoscoe modes The comlex higher order modes revealed in the revious section (see movie link in Fig. 4 for more examles) show distinct similarities to so called kaleidoscoe modes [0,]. The similarities are visual, which we acknowledge to be subjective given that such modes have not been ut on a firm mathematical basis, but more imortant similarities exist in the generating mechanisms. In [0] these field distributions were roosed as a result of the coherent suerosition 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. () with i = ) in Porro rism resonators robably accounts for the likeness in outut modes. In [] kaleidoscoe modes were generated using crossed aertures to sub divide the inut field to an axicon. This tye of obstruction attern is identical to the final loss field observed in Porro rism resonators (see Fig. 8 of [6]). While such fields were reviously created external to the laser cavity, we have shown that the fundamental roerty of field sub division in Porro risms can roduce similar fields directly from the laser cavity. The generating mechanisms in both [0] and [] have strong oints of commonality with how intracavity Porro risms are treated. However for comleteness we must oint out that the studies in question dealt with diffraction free beams created by lane waves traveling on cones, with no obvious link to our resonator. Desite this the outut modes bear very strong likeness in form, and erhas also in roerties. The ubiquitous nature of Porro rism resonators makes a study of such modes necessary in its own right, but there also exists the ossibility of using such comlex modes to excite comlex hotonic crystal structures, and so further study is required. 6. Conclusion We have alied a reviously develoed mathematical model of intracavity Porro risms to stable and unstable Porro rism resonators with large intracavity aertures. We have shown that higher order modes exist only if N F is sufficiently large, and that these higher order modes closely resemble recently reorted kaleidoscoe modes due to the fundamental roerty of field sub division in Porro rism resonators. The aearance of first the etal mode and then increasingly comlex kaleidoscoe modes with increasing aerture size leads to the conclusion that the etal like modes are the lowest order modes of Porro rism resonators, while higher order modes exist in the form of kaleidoscoe like fields. We also redict that the standard etal mode is only observable from stable Porro rism resonators, and indicate how the stability criteria (G arameter) imacts on the cyclical nature of the higher order modes. We believe it is ossible to observe these modes exerimentally, 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 Christoh Bollig. (C) 2008 OSA 8 August 2008 / Vol. 6, No. 7 / OPTICS EXPRESS 274