- Email: [email protected]
Numerical analysis of non-confocal configurations of a hybrid stable–unstable resonator
Optics Communications 284 (2011) 999–1003
Contents lists available at ScienceDirect
Optics Communications j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / o p t c o m
Numerical analysis of non-confocal configurations of a hybrid stable–unstable resonator Antonio Lapucci ⁎, Marco Ciofini Istituto Nazionale di Ottica, C.N.R., Largo Enrico Fermi 6, I-50125 Firenze, Italy
a r t i c l e
i n f o
Article history: Received 24 August 2010 Received in revised form 5 October 2010 Accepted 5 October 2010 Keywords: Laser optics Laser resonators Unstable resonators Slab lasers
a b s t r a c t The results of numerical simulations are reported modeling the behavior of a unidimensional Hybrid Stable– Unstable Resonator (HSUR) in strongly non-confocal configurations. Our analysis shows that such a resonator setup can produce cavity losses consistent with our design requirements and extract a Beam with Propagation Parameter better than 3 mm mrad. © 2010 Elsevier B.V. All rights reserved.
1. Introduction Hybrid Stable–Unstable Resonators (HSUR) have been largely used for thin and wide (slab-type) laser active media. Bourne and Dyer [1], in 1979, firstly adopted a spherical concave mirror and a cylindrical convex one, in combination with a rare gas halide slab active medium. Successively several gas [2] and solid state [3] lasers with slab shaped gain regions have been realized applying confocal HSURs. Confocal configurations either of the positive- or negative-branch types were experimentally tested as well as numerically analyzed in [4] and [5]. Nevertheless, when designing a laser, there are situations in which the use of a non-confocal configuration might be more desirable. This was for example the case of a recent work [6] of ours in which we experimentally investigated the adoption of a HSUR on a Nd:YAG slab with zig-zag internal propagation in the narrow cross-section transverse direction. In our case the stable transverse direction required a relatively short mirror radius of curvature (ROC). Indeed a zig-zag propagation in the active medium would have made the operation with longer mirror ROC too sensitive to misalignments in the stable transverse direction. The obvious advantage in the use of a simple concave spherical mirror and a cylindrical convex scraping one, brought us to the adoption of a definitely non-confocal configuration for the unstable transverse direction. Naturally this might be the case in many other situations, as for example when suitable optics are not available, or when thermal lens effects may bring out of the confocal configuration at certain power loading regimes. In 1995 Chandra et al. [7] adopted a non-confocal
⁎ Corresponding author. Tel.: +39 055 2308226; fax: +39 055 2337755. E-mail address: [email protected] (A. Lapucci). 0030-4018/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2010.10.021
unstable resonator in a solid-state dye laser based on a gradientreflectivity mirror (GRM). In their case, building a confocal resonator with the same cavity length and magnification, would have required a very short ROC GRM which is quite difficult to fabricate. It is worth noting, that there is very little information in the literature about the structure of “modes” extracted from unstable resonators that do not have a simple “canonical” configuration. In this paper we show the results of a numerical analysis based on diffractive propagation algorithms performed to determine the expected cavity losses and extracted beam properties when adopting a non-confocal HSUR setup. Our analysis demonstrates that strongly non-confocal configurations can produce suitable values of resonator out-coupling coefficient with a good quality beam extraction, generously widening the acceptable range of design parameters such as mirror ROCs, cavity length and mirror tilts. 2. Numerical analysis method Given the symmetry of the problem and the polarization selecting cavity we can simply consider the field distribution in our resonator to be scalar and separable into two x- and y-dependent complex amplitude functions. In ref. [6] we already demonstrated that the beam profile in the y-transverse direction, i.e. the direction parallel to the narrow cross-section of the YAG slab, is correctly described by a superposition of a limited (b8) number of low order Gaussian modes. This is a consequence of the HSUR producing a stable resonator configuration with a low Fresnel number in this transverse direction. The M2y values reported in [6] are consistent with this modal analysis. Our present study will be focused on the amplitude and phase xdependent distributions in the lateral direction parallel to the larger
1000
A. Lapucci, M. Ciofini / Optics Communications 284 (2011) 999–1003
cross-section of the YAG slab in which the HSUR determines an unstable resonator geometry. The analysis is performed by means of a numerical code based on Fresnel diffraction integral propagation algorithms in an iterative “Fox and Li” approach [4]. Referring to the experiments of ref. [6] we adopt in this numerical analysis a wavelength of 1.064 μm. With the purpose of understanding the viability of strongly non-confocal configurations we neglect the effects of the presence of a gain medium with the saturation and thermal lens effect that it generally produces. This assumption has various motivations. First of all it has been shown [8,9] that the general character of “loaded resonator” modes is very little changed from that of “bare resonator” ones, the main difference being the reduced amplitude of higher peaks. Secondly the cold cavity represents the simplest and most general model of the optical system, to which corrections such as thermal lens may be added as additional optical elements. Finally, our zig-zag face-pumped system proved to be scarcely affected by internal lens effects [10]. Our numerical algorithm reconstructs the resonator mode successively simulating the effect of mirror curvature, tilt (phase corrections) and apertures (amplitude truncations), and the effect of diffraction, applying an FFT based convolution with the impulse response function of e.m. propagation in the Fresnel approximation [4]: uðx; LÞ = hðx; LÞ⊗uðx; 0Þ
ð1Þ
where expð jkzÞ kx2 hðx; zÞ = exp j 1=2 ð jkzÞ 2z
! ð2Þ
is the propagation impulse response function, and u(x,z) is the xdependent complex field distribution after a z distance propagation. With a plane wave as the initial condition, resonator round trips are repeated up to the convergence of the “eigen-value” defined as the ratio between the amplitudes of two successive self-similar distributions. An eigen-value is considered to have reached convergence when its recursive calculation determines relative variations of less than 10−4. A set of more than one hundred different initial conditions was actually tested in a number of resonator configurations in order to be sure that the procedure converges towards the lowest loss spatial distribution. In Fig. 1 the typical outputs of our code are shown proving the correct reconstruction of “geometrical” and “diffractive”
properties of the intracavity field distributions. These outputs refer to the confocal resonator configuration, schematically depicted on the upper right corner. The mirror ROCs adopted in all the numerics presented in this paper are those of the optical setup of ref. [6]. Namely R1 = −1000 mm (diverging cylindrical) and R2 = 1250 mm (converging spherical). As a consequence the confocal cavity length is 125 mm. Our active medium slab has a 10 mm wide larger crosssection and the output aperture turns out to be 2 mm in the one-sided HSUR confocal configuration. In the resonator scheme two planes, P1 and P2, are evidenced on which we characterize four different intracavity field distributions. Intensity and phase distributions plotted in bold and blue, are related to the field number one, that is coupled out of the resonator. The phase distributions plotted in the lower part of the figure clearly show how the recursive code correctly reconstructs the wavefront curvature. Left-propagating wavefronts (with respect to the resonator scheme in the figure) are flat, indicating a collimated beam. As in previous numerical analysis [4,5] the intensity distributions present deep ripples caused by the diffraction from mirror hard edges. A similar analysis has been performed for various resonator configurations. Fig. 2 shows an example of non-confocal resonator distributions. These distributions correspond to a cavity length of 225 mm, close to the value of the resonator adopted in [6]. As in Fig. 1 the bold and blue profiles refer to the field from which the resonator extracts the output beam. Different from the case of the confocal resonator, the left-propagating fields (1 and 4) do not have a flat wavefront but a concave one indicating a converging beam. Correspondingly, the intensity distribution on the out-coupling mirror (topleft corner of the figure) doesn't extend in the whole 10 mm wide resonator cross-section, unless some tilt is given to the mirror alignment. Thus the extracted beam turns out to have a much thinner cross-section. In the following paragraph we present the results of a detailed analysis of non-confocal configurations performed varying two resonator parameters: the cavity length and the tilt of the rear mirror.
3. Field distributions and cavity losses in non-confocal configurations In Fig. 3 we show some typical intensity and phase distributions on the resonator output plane for strongly non-confocal configurations.
Fig. 1. Top-left: Intensity distribution of the left-propagating field on plane P1. Top-right: Confocal HSUR scheme. Bottom: Wavefront (field phase distribution) of the 1-2-3-4 beams propagating in the resonator as specified by the arrows in the scheme. The output wavefront (1) is plotted in bold and blue.
A. Lapucci, M. Ciofini / Optics Communications 284 (2011) 999–1003
1001
Fig. 2. Top-left: Intensity distribution of the left-propagating field on plane P1. Top-right: Non-confocal HSUR scheme. Bottom: Wavefront (field phase distribution) of the 1-2-3-4 beams propagating in the non-confocal resonator. The output wavefront (1) is plotted in bold and blue.
The figure refers to a 225 mm cavity length and the three different columns are obtained with different resonator alignments (−100 μrad; 0 μrad; +100 μrad). A positive tilt angle corresponds to a rotation of the rear mirror towards the resonator aperture. This turns out into “opening” the resonator, in the sense that a larger fraction of the propagating field will be extracted from the out-coupling side of the cylindrical convex mirror. A negative tilt angle, on the contrary, means rotating the rear mirror towards the other side of the cylindrical mirror. From the intensity distributions shown in the upper row of Fig. 3 one can clearly see that in negative tilt configurations the geometrical wavefront together with diffraction from the mirror edges act to confine the field distribution on the mirror surface, thus strongly reducing the loss coefficient. Whereas positive mirror tilts tend to bring a larger part of the intensity distribution into the out-coupling port of the resonator. This mechanism is the base of the general structure of the cavity loss curves
tilt = - 100 μrad
that we calculated with our diffractive numerical simulation code, and that are shown in Fig. 4. In the bottom left panel of Fig. 3 we also evidence how the wavefront of the beam extracted from the strongly non-confocal resonator is affected only by a significant tilt and a weak defocusing. These are both phase variations that do not modify the extracted beam focusability and thus beam propagation factors such as M2, as we will discuss in the following paragraph. In Fig. 4 the cavity loss curves are shown obtained with different cavity lengths and mirror tilts. Referring to an experimental setup such as that of ref. [6] we analyze cavity lengths starting from the confocal situation (125 mm) and growing towards the stability boundary (250 mm). As previously anticipated, total cavity losses with positive mirror tilts are higher with respect to those of the properly aligned HSUR, and lower with negative mirror tilts. Actually the total losses start
tilt = 100 μrad
Intensity (arb.un.)
tilt = 0 μrad
Phase (λ units)
40 20 0 -20 -40 -5
0
5
-5
0
5
-5
0
5
Transverse Position (mm) Fig. 3. Intensity and phase distributions for different tilt angles in strongly non-confocal configurations (Lc = 225 mm). Rectangular windows in the upper row show the resonator extraction port.
A. Lapucci, M. Ciofini / Optics Communications 284 (2011) 999–1003
Near Field Intensity (arb. un.)
1002
We now analyze in detail the propagation properties of the beams extracted from the resonator configurations described in the previous paragraph. In Fig. 5 we compare the near-field aspect of the beam scraped out from the intracavity propagating field by the hard edged side of the cylindrical mirror. Intensity distribution plots in the upper windows (red lines) are those obtained in the confocal resonator condition (Lc = 125 mm) for two different mirror alignments, namely the correctly aligned (0 μrad tilt — on the left) and a slightly opened one (+100 μrad tilt — on the right). Intensity profiles plotted in the lower windows (blue lines) are the extracted intensity distributions obtained with a longer cavity (Lc = 225 mm) and the same mirror tilts. It is clear that in this latter case the extracted beam is thinner as a consequence of the converging character of the intracavity field propagating towards the out-coupling mirror. The positive mirror tilt,
Lc = 225 mm - Tilt 0 μrad
Lc = 225 mm - Tilt 100 μrad
2
3
4
5
6
7
1
2
3
4
5
6
7
Transverse Position (mm) Fig. 5. Near-field intensity distributions for different tilt angles and cavity lengths.
in this cavity configuration, more sensibly acts to increase the extracted beam width. The beams emerging from the confocal resonator always present a squarish shape with different ripple profiles for different tilts superimposed on it. On the contrary the beam produced by the strongly non-confocal resonator may show a truncated bell-like or multi-peaked shape. This last fact is related to the lower equivalent Fresnel Number [8] of the longer cavity determining a lower number of intensity oscillations on the output aperture. As a consequence the far-field intensity distributions result weakly affected by the mirror tilt, in the case of the confocal resonator, and more sensibly in the case of the longer cavity. In Fig. 6 we show the far-field intensity distributions of the four sample beams of Fig. 5 obtained from confocal (upper plots — red line) and non-confocal (lower plots — blue line) resonator configurations. One can see that, in the case of a confocal resonator, different mirror tilts produce only tiny modifications of the far-field side lobes. Whereas, in the case of longer non-confocal cavities the far-field main lobe changes with tilt as a consequence of a different extracted beam width and side lobes significantly change as consequence of a larger variation in the nearfield intensity profile. In Fig. 7 we characterize these features of the beams extracted from the different resonator configurations in terms of the most used beam propagation quality factor, i.e. M2 [13]. As it is well known [13] M2 is not well indicated to characterize beams produced by hard edged unstable resonators, but it remains the most widely used quality factor parameter and the only one with definitely established measurement procedures [6]. In the case of the beams
Far Field Intensity (arb. un.)
4. Extracted beam properties
Lc = 125 mm - Tilt 100 μrad
1
Fig. 4. Cavity losses resulting from the numerical simulations based on the diffractive propagation algorithm, for different cavity lengths and mirror tilt angles. Large symbols denote the geometrical loss estimations for 0-angle tilt and different cavity lengths.
to grow again for larger negative tilt angle values (angles larger than 175 μrad in our present design), as a consequence of an increasing field spill out from the wrong side of the out-coupling mirror. This effect is more pronounced in the case of cavities with longer mirror radii of curvature as we detail in a different work [11]. Here we focus our attention on the results that our numerical analysis gives in the range of parameters of interest for our resonator configuration [6]. The large symbols plotted at the center of Fig. 4 show the geometrical cavity loss values for the aligned resonators of different lengths that have been calculated by means of a classical ray tracing optical design software package [12]. The main results of our simulations are thus the following. Losses calculated with the diffractive algorithm result generally higher than those geometrically estimated. This difference reduces when we move from the confocal resonator length towards longer cavities. Losses calculated taking diffraction effects into account present, as it is well known [4,8], a “swinging” dependence from cavity parameters, such as mirror tilt in our figure. This feature turns out to be well pronounced for lengths close to the confocal configuration and sensibly smeared out for larger length values. This same swinging behavior appears in the dependence from other parameters such as the length itself [11]. Of course total cavity losses smaller than the geometrically calculated value can be found “closing” the cavity with a rear mirror rotation towards negative tilt values, for all the investigated cavity lengths. Finally, and most interesting from the resonator designer viewpoint, cavity loss values equivalent to the aligned confocal resonator are easily obtained in longer non-confocal cavities by means of a “slightly opened” mirror tilt. Note that with the usual aspect-ratio of slab lasers adopting this kind of resonator [6], the needed mirror tilt leaves the intracavity field distribution to gain-medium overlap factor almost unaffected.
Lc = 125 mm - Tilt 0 μrad
Lc = 125 mm - Tilt 0 μrad
Lc = 125 mm - Tilt 100 μrad
Lc = 225 mm - Tilt 0 μrad
Lc = 225 mm - Tilt 100 μrad
-5
0
5
-5
0
5
Diffraction Angle (mrad) Fig. 6. Far-field intensity distributions for different tilt angles and cavity lengths.
A. Lapucci, M. Ciofini / Optics Communications 284 (2011) 999–1003
1003
this way we can see that a typical M2 measure would produce a value in the range 1.5–2.5 for the confocal resonator and in the range 2–5 for the non-confocal resonator, as long as one can expect experimental measurements to reveal intensity levels that are 2–5% of the peak value. This is perfectly consistent with what we found in our beam characterization in [6], adopting the ISO standard rule to determine the beam M2 [14]. We also show in the inset of Fig. 7 the encircled energy plots, that is the fraction of the beam energy that propagates within a certain angle from the beam axis as a function of the diffraction angle. Angles are normalized to the diffraction limit value, and thus the abscissa of this plot is a Times Diffraction Limit (TDL) factor. In such a manner one can see that the fraction of energy contained in the main lobe may vary from 70 to 90%. Confocal configurations have higher percentages, but there are also non-confocal setups with more than 80% of the power contained in the far-field principal lobe. This kind of information is of particular significance when the use of a space filter in the beam-conditioning chain is envisaged to increase the propagation quality of the beam emitted by a slab laser with this kind of an unstable resonator. 5. Conclusion
Fig. 7. Beam quality estimations for beams extracted from resonators with different configurations. (A) refers to confocal configurations, and (B) to non-confocal ones.
Intensity (arb.un.)
produced by our resonator the presence of several low power side lobes diffracted at very large propagation angles would produce rather large M2 value estimations unless a lower threshold is imposed to the field intensity values considered in the variance-based M2 calculations [13]. For this reason we plot in Fig. 7 the M2 values calculated adopting different lower threshold levels in the amplitude of far-field intensity distributions. M2 values are evaluated for several beams obtained with our simulations for different cavity lengths and alignments. In
In Fig. 8 we show a comparison of intensity distributions obtained with our numerical analysis (left column — blue lines) and in the experiment of ref. [6] (right column — red lines) both in the near- and far-field. It evidently shows that our simulations correctly reconstruct the spatial structure of the extracted beams. Only minor differences can be evidenced in the intensity levels of the different peaks from those of the experimental profiles. These differences are to be attributed to the presence, in the experimental case, of active medium saturation and detector non-linear response. This represents a validation of our numerical code in the ability to simulate the diffractive propagation of our HSUR configurations. The analysis of cavity losses and extracted beam properties performed with the numerical code enabled us to explain the results obtained in our past experiments [6] carried out with a strongly non-confocal resonator configuration and to prove that out-coupling coefficients compatible with the design requirements and good quality extracted beams can be obtained also in this non-canonical cavity setup. These results allow to largely widen the range of acceptable parameters for the unstable direction when designing a hybrid resonator (HSUR). In the case of our slab laser they enabled us to adopt mirror ROCs much shorter than those prescribed for the confocal cavity, given our active medium length. This choice produced a significant improvement in terms of alignment tolerance in the stable transverse direction with a zig-zag propagation.
References
2
3
4
5
6
10
9
8
7
6
5
4
Intensity (arb.un.)
Transverse position (mm)
[1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13]
-10
-5
0
5
10
-10
-5
0
5
10
Diffraction Angle (mrad) Fig. 8. Comparison of some sample field distributions both in the near- (upper row) and far-field (lower row) obtained by our numerical simulations (left column plots — blue line) with the beam profiles measured in the experimental setup of ref. [6] (right column plots — red line).
[14]
L. Bourn, P.E. Dyer, Opt. Commun. 31 (1979) 193. P.E. Jackson, H.J. Baker, D.R. Hall, Appl. Phys. Lett. 54 (1989) 1950. K. Du, N. Wu, J. Xu, J. Giesekus, P. Loosen, R. Poprawe, Opt. Lett. 23 (1998) 370. A. Lapucci, A. Labate, F. Rossetti, S. Mascalchi, Appl. Opt. 35 (1996) 3185. A. Wasilewski, H.J. Baker, D.R. Hall, Appl. Opt. 39 (2000) 6174. M. Ciofini, E. Favilla, A. Lapucci, E. Sani, Opt. Laser Technol. 39 (2007) 1380. S. Chandra, T.H. Allik, J.A. Hutchinson, Opt. Lett. 20 (1995) 2387. A.E. Siegman, Lasers, University Science Books, Mill Valley Calif, 1986 Chap. 22. Z. Wang, X. Ye, T. Fang, J. Xu, Opt. Rev. 12 (2005) 391. A. Lapucci, M. Ciofini, E. Favilla, E. Sani, M. De Rosa, G. Mincuzzi, A.S.D. Rocco, J. Eur. Opt. Soc. Rapid Publ. (2007) 2,07020. A. Lapucci, M. Ciofini, “Non-confocal hybrid stable–unstable resonators: parametric study of the optical losses”, in press. Trace-Pro 3.3. Lambda-Research Corp. www.lambdares.com . M.W. Sasnett, Propagation of multimode laser beams—the M2 factor, in: D.R. Hall, P.E. Jackson (Eds.), The Physics and Technology of Laser Resonators, Hilger, London, 1989, p. 132. ISO standard 11146-1 “Lasers and laser-related equipment — test methods for laser beam widths, divergence angle and beam propagation factor — Part 1: stigmatic and simple astigmatic beams” and 11146-3 “Lasers and laser-related equipment — Test methods for laser beam widths, divergence angle and beam propagation factor — Part 3: alternative test methods and geometrical laser beam classification and propagation.”
Contents lists available at ScienceDirect
Optics Communications j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / o p t c o m
Numerical analysis of non-confocal configurations of a hybrid stable–unstable resonator Antonio Lapucci ⁎, Marco Ciofini Istituto Nazionale di Ottica, C.N.R., Largo Enrico Fermi 6, I-50125 Firenze, Italy
a r t i c l e
i n f o
Article history: Received 24 August 2010 Received in revised form 5 October 2010 Accepted 5 October 2010 Keywords: Laser optics Laser resonators Unstable resonators Slab lasers
a b s t r a c t The results of numerical simulations are reported modeling the behavior of a unidimensional Hybrid Stable– Unstable Resonator (HSUR) in strongly non-confocal configurations. Our analysis shows that such a resonator setup can produce cavity losses consistent with our design requirements and extract a Beam with Propagation Parameter better than 3 mm mrad. © 2010 Elsevier B.V. All rights reserved.
1. Introduction Hybrid Stable–Unstable Resonators (HSUR) have been largely used for thin and wide (slab-type) laser active media. Bourne and Dyer [1], in 1979, firstly adopted a spherical concave mirror and a cylindrical convex one, in combination with a rare gas halide slab active medium. Successively several gas [2] and solid state [3] lasers with slab shaped gain regions have been realized applying confocal HSURs. Confocal configurations either of the positive- or negative-branch types were experimentally tested as well as numerically analyzed in [4] and [5]. Nevertheless, when designing a laser, there are situations in which the use of a non-confocal configuration might be more desirable. This was for example the case of a recent work [6] of ours in which we experimentally investigated the adoption of a HSUR on a Nd:YAG slab with zig-zag internal propagation in the narrow cross-section transverse direction. In our case the stable transverse direction required a relatively short mirror radius of curvature (ROC). Indeed a zig-zag propagation in the active medium would have made the operation with longer mirror ROC too sensitive to misalignments in the stable transverse direction. The obvious advantage in the use of a simple concave spherical mirror and a cylindrical convex scraping one, brought us to the adoption of a definitely non-confocal configuration for the unstable transverse direction. Naturally this might be the case in many other situations, as for example when suitable optics are not available, or when thermal lens effects may bring out of the confocal configuration at certain power loading regimes. In 1995 Chandra et al. [7] adopted a non-confocal
⁎ Corresponding author. Tel.: +39 055 2308226; fax: +39 055 2337755. E-mail address: [email protected] (A. Lapucci). 0030-4018/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2010.10.021
unstable resonator in a solid-state dye laser based on a gradientreflectivity mirror (GRM). In their case, building a confocal resonator with the same cavity length and magnification, would have required a very short ROC GRM which is quite difficult to fabricate. It is worth noting, that there is very little information in the literature about the structure of “modes” extracted from unstable resonators that do not have a simple “canonical” configuration. In this paper we show the results of a numerical analysis based on diffractive propagation algorithms performed to determine the expected cavity losses and extracted beam properties when adopting a non-confocal HSUR setup. Our analysis demonstrates that strongly non-confocal configurations can produce suitable values of resonator out-coupling coefficient with a good quality beam extraction, generously widening the acceptable range of design parameters such as mirror ROCs, cavity length and mirror tilts. 2. Numerical analysis method Given the symmetry of the problem and the polarization selecting cavity we can simply consider the field distribution in our resonator to be scalar and separable into two x- and y-dependent complex amplitude functions. In ref. [6] we already demonstrated that the beam profile in the y-transverse direction, i.e. the direction parallel to the narrow cross-section of the YAG slab, is correctly described by a superposition of a limited (b8) number of low order Gaussian modes. This is a consequence of the HSUR producing a stable resonator configuration with a low Fresnel number in this transverse direction. The M2y values reported in [6] are consistent with this modal analysis. Our present study will be focused on the amplitude and phase xdependent distributions in the lateral direction parallel to the larger
1000
A. Lapucci, M. Ciofini / Optics Communications 284 (2011) 999–1003
cross-section of the YAG slab in which the HSUR determines an unstable resonator geometry. The analysis is performed by means of a numerical code based on Fresnel diffraction integral propagation algorithms in an iterative “Fox and Li” approach [4]. Referring to the experiments of ref. [6] we adopt in this numerical analysis a wavelength of 1.064 μm. With the purpose of understanding the viability of strongly non-confocal configurations we neglect the effects of the presence of a gain medium with the saturation and thermal lens effect that it generally produces. This assumption has various motivations. First of all it has been shown [8,9] that the general character of “loaded resonator” modes is very little changed from that of “bare resonator” ones, the main difference being the reduced amplitude of higher peaks. Secondly the cold cavity represents the simplest and most general model of the optical system, to which corrections such as thermal lens may be added as additional optical elements. Finally, our zig-zag face-pumped system proved to be scarcely affected by internal lens effects [10]. Our numerical algorithm reconstructs the resonator mode successively simulating the effect of mirror curvature, tilt (phase corrections) and apertures (amplitude truncations), and the effect of diffraction, applying an FFT based convolution with the impulse response function of e.m. propagation in the Fresnel approximation [4]: uðx; LÞ = hðx; LÞ⊗uðx; 0Þ
ð1Þ
where expð jkzÞ kx2 hðx; zÞ = exp j 1=2 ð jkzÞ 2z
! ð2Þ
is the propagation impulse response function, and u(x,z) is the xdependent complex field distribution after a z distance propagation. With a plane wave as the initial condition, resonator round trips are repeated up to the convergence of the “eigen-value” defined as the ratio between the amplitudes of two successive self-similar distributions. An eigen-value is considered to have reached convergence when its recursive calculation determines relative variations of less than 10−4. A set of more than one hundred different initial conditions was actually tested in a number of resonator configurations in order to be sure that the procedure converges towards the lowest loss spatial distribution. In Fig. 1 the typical outputs of our code are shown proving the correct reconstruction of “geometrical” and “diffractive”
properties of the intracavity field distributions. These outputs refer to the confocal resonator configuration, schematically depicted on the upper right corner. The mirror ROCs adopted in all the numerics presented in this paper are those of the optical setup of ref. [6]. Namely R1 = −1000 mm (diverging cylindrical) and R2 = 1250 mm (converging spherical). As a consequence the confocal cavity length is 125 mm. Our active medium slab has a 10 mm wide larger crosssection and the output aperture turns out to be 2 mm in the one-sided HSUR confocal configuration. In the resonator scheme two planes, P1 and P2, are evidenced on which we characterize four different intracavity field distributions. Intensity and phase distributions plotted in bold and blue, are related to the field number one, that is coupled out of the resonator. The phase distributions plotted in the lower part of the figure clearly show how the recursive code correctly reconstructs the wavefront curvature. Left-propagating wavefronts (with respect to the resonator scheme in the figure) are flat, indicating a collimated beam. As in previous numerical analysis [4,5] the intensity distributions present deep ripples caused by the diffraction from mirror hard edges. A similar analysis has been performed for various resonator configurations. Fig. 2 shows an example of non-confocal resonator distributions. These distributions correspond to a cavity length of 225 mm, close to the value of the resonator adopted in [6]. As in Fig. 1 the bold and blue profiles refer to the field from which the resonator extracts the output beam. Different from the case of the confocal resonator, the left-propagating fields (1 and 4) do not have a flat wavefront but a concave one indicating a converging beam. Correspondingly, the intensity distribution on the out-coupling mirror (topleft corner of the figure) doesn't extend in the whole 10 mm wide resonator cross-section, unless some tilt is given to the mirror alignment. Thus the extracted beam turns out to have a much thinner cross-section. In the following paragraph we present the results of a detailed analysis of non-confocal configurations performed varying two resonator parameters: the cavity length and the tilt of the rear mirror.
3. Field distributions and cavity losses in non-confocal configurations In Fig. 3 we show some typical intensity and phase distributions on the resonator output plane for strongly non-confocal configurations.
Fig. 1. Top-left: Intensity distribution of the left-propagating field on plane P1. Top-right: Confocal HSUR scheme. Bottom: Wavefront (field phase distribution) of the 1-2-3-4 beams propagating in the resonator as specified by the arrows in the scheme. The output wavefront (1) is plotted in bold and blue.
A. Lapucci, M. Ciofini / Optics Communications 284 (2011) 999–1003
1001
Fig. 2. Top-left: Intensity distribution of the left-propagating field on plane P1. Top-right: Non-confocal HSUR scheme. Bottom: Wavefront (field phase distribution) of the 1-2-3-4 beams propagating in the non-confocal resonator. The output wavefront (1) is plotted in bold and blue.
The figure refers to a 225 mm cavity length and the three different columns are obtained with different resonator alignments (−100 μrad; 0 μrad; +100 μrad). A positive tilt angle corresponds to a rotation of the rear mirror towards the resonator aperture. This turns out into “opening” the resonator, in the sense that a larger fraction of the propagating field will be extracted from the out-coupling side of the cylindrical convex mirror. A negative tilt angle, on the contrary, means rotating the rear mirror towards the other side of the cylindrical mirror. From the intensity distributions shown in the upper row of Fig. 3 one can clearly see that in negative tilt configurations the geometrical wavefront together with diffraction from the mirror edges act to confine the field distribution on the mirror surface, thus strongly reducing the loss coefficient. Whereas positive mirror tilts tend to bring a larger part of the intensity distribution into the out-coupling port of the resonator. This mechanism is the base of the general structure of the cavity loss curves
tilt = - 100 μrad
that we calculated with our diffractive numerical simulation code, and that are shown in Fig. 4. In the bottom left panel of Fig. 3 we also evidence how the wavefront of the beam extracted from the strongly non-confocal resonator is affected only by a significant tilt and a weak defocusing. These are both phase variations that do not modify the extracted beam focusability and thus beam propagation factors such as M2, as we will discuss in the following paragraph. In Fig. 4 the cavity loss curves are shown obtained with different cavity lengths and mirror tilts. Referring to an experimental setup such as that of ref. [6] we analyze cavity lengths starting from the confocal situation (125 mm) and growing towards the stability boundary (250 mm). As previously anticipated, total cavity losses with positive mirror tilts are higher with respect to those of the properly aligned HSUR, and lower with negative mirror tilts. Actually the total losses start
tilt = 100 μrad
Intensity (arb.un.)
tilt = 0 μrad
Phase (λ units)
40 20 0 -20 -40 -5
0
5
-5
0
5
-5
0
5
Transverse Position (mm) Fig. 3. Intensity and phase distributions for different tilt angles in strongly non-confocal configurations (Lc = 225 mm). Rectangular windows in the upper row show the resonator extraction port.
A. Lapucci, M. Ciofini / Optics Communications 284 (2011) 999–1003
Near Field Intensity (arb. un.)
1002
We now analyze in detail the propagation properties of the beams extracted from the resonator configurations described in the previous paragraph. In Fig. 5 we compare the near-field aspect of the beam scraped out from the intracavity propagating field by the hard edged side of the cylindrical mirror. Intensity distribution plots in the upper windows (red lines) are those obtained in the confocal resonator condition (Lc = 125 mm) for two different mirror alignments, namely the correctly aligned (0 μrad tilt — on the left) and a slightly opened one (+100 μrad tilt — on the right). Intensity profiles plotted in the lower windows (blue lines) are the extracted intensity distributions obtained with a longer cavity (Lc = 225 mm) and the same mirror tilts. It is clear that in this latter case the extracted beam is thinner as a consequence of the converging character of the intracavity field propagating towards the out-coupling mirror. The positive mirror tilt,
Lc = 225 mm - Tilt 0 μrad
Lc = 225 mm - Tilt 100 μrad
2
3
4
5
6
7
1
2
3
4
5
6
7
Transverse Position (mm) Fig. 5. Near-field intensity distributions for different tilt angles and cavity lengths.
in this cavity configuration, more sensibly acts to increase the extracted beam width. The beams emerging from the confocal resonator always present a squarish shape with different ripple profiles for different tilts superimposed on it. On the contrary the beam produced by the strongly non-confocal resonator may show a truncated bell-like or multi-peaked shape. This last fact is related to the lower equivalent Fresnel Number [8] of the longer cavity determining a lower number of intensity oscillations on the output aperture. As a consequence the far-field intensity distributions result weakly affected by the mirror tilt, in the case of the confocal resonator, and more sensibly in the case of the longer cavity. In Fig. 6 we show the far-field intensity distributions of the four sample beams of Fig. 5 obtained from confocal (upper plots — red line) and non-confocal (lower plots — blue line) resonator configurations. One can see that, in the case of a confocal resonator, different mirror tilts produce only tiny modifications of the far-field side lobes. Whereas, in the case of longer non-confocal cavities the far-field main lobe changes with tilt as a consequence of a different extracted beam width and side lobes significantly change as consequence of a larger variation in the nearfield intensity profile. In Fig. 7 we characterize these features of the beams extracted from the different resonator configurations in terms of the most used beam propagation quality factor, i.e. M2 [13]. As it is well known [13] M2 is not well indicated to characterize beams produced by hard edged unstable resonators, but it remains the most widely used quality factor parameter and the only one with definitely established measurement procedures [6]. In the case of the beams
Far Field Intensity (arb. un.)
4. Extracted beam properties
Lc = 125 mm - Tilt 100 μrad
1
Fig. 4. Cavity losses resulting from the numerical simulations based on the diffractive propagation algorithm, for different cavity lengths and mirror tilt angles. Large symbols denote the geometrical loss estimations for 0-angle tilt and different cavity lengths.
to grow again for larger negative tilt angle values (angles larger than 175 μrad in our present design), as a consequence of an increasing field spill out from the wrong side of the out-coupling mirror. This effect is more pronounced in the case of cavities with longer mirror radii of curvature as we detail in a different work [11]. Here we focus our attention on the results that our numerical analysis gives in the range of parameters of interest for our resonator configuration [6]. The large symbols plotted at the center of Fig. 4 show the geometrical cavity loss values for the aligned resonators of different lengths that have been calculated by means of a classical ray tracing optical design software package [12]. The main results of our simulations are thus the following. Losses calculated with the diffractive algorithm result generally higher than those geometrically estimated. This difference reduces when we move from the confocal resonator length towards longer cavities. Losses calculated taking diffraction effects into account present, as it is well known [4,8], a “swinging” dependence from cavity parameters, such as mirror tilt in our figure. This feature turns out to be well pronounced for lengths close to the confocal configuration and sensibly smeared out for larger length values. This same swinging behavior appears in the dependence from other parameters such as the length itself [11]. Of course total cavity losses smaller than the geometrically calculated value can be found “closing” the cavity with a rear mirror rotation towards negative tilt values, for all the investigated cavity lengths. Finally, and most interesting from the resonator designer viewpoint, cavity loss values equivalent to the aligned confocal resonator are easily obtained in longer non-confocal cavities by means of a “slightly opened” mirror tilt. Note that with the usual aspect-ratio of slab lasers adopting this kind of resonator [6], the needed mirror tilt leaves the intracavity field distribution to gain-medium overlap factor almost unaffected.
Lc = 125 mm - Tilt 0 μrad
Lc = 125 mm - Tilt 0 μrad
Lc = 125 mm - Tilt 100 μrad
Lc = 225 mm - Tilt 0 μrad
Lc = 225 mm - Tilt 100 μrad
-5
0
5
-5
0
5
Diffraction Angle (mrad) Fig. 6. Far-field intensity distributions for different tilt angles and cavity lengths.
A. Lapucci, M. Ciofini / Optics Communications 284 (2011) 999–1003
1003
this way we can see that a typical M2 measure would produce a value in the range 1.5–2.5 for the confocal resonator and in the range 2–5 for the non-confocal resonator, as long as one can expect experimental measurements to reveal intensity levels that are 2–5% of the peak value. This is perfectly consistent with what we found in our beam characterization in [6], adopting the ISO standard rule to determine the beam M2 [14]. We also show in the inset of Fig. 7 the encircled energy plots, that is the fraction of the beam energy that propagates within a certain angle from the beam axis as a function of the diffraction angle. Angles are normalized to the diffraction limit value, and thus the abscissa of this plot is a Times Diffraction Limit (TDL) factor. In such a manner one can see that the fraction of energy contained in the main lobe may vary from 70 to 90%. Confocal configurations have higher percentages, but there are also non-confocal setups with more than 80% of the power contained in the far-field principal lobe. This kind of information is of particular significance when the use of a space filter in the beam-conditioning chain is envisaged to increase the propagation quality of the beam emitted by a slab laser with this kind of an unstable resonator. 5. Conclusion
Fig. 7. Beam quality estimations for beams extracted from resonators with different configurations. (A) refers to confocal configurations, and (B) to non-confocal ones.
Intensity (arb.un.)
produced by our resonator the presence of several low power side lobes diffracted at very large propagation angles would produce rather large M2 value estimations unless a lower threshold is imposed to the field intensity values considered in the variance-based M2 calculations [13]. For this reason we plot in Fig. 7 the M2 values calculated adopting different lower threshold levels in the amplitude of far-field intensity distributions. M2 values are evaluated for several beams obtained with our simulations for different cavity lengths and alignments. In
In Fig. 8 we show a comparison of intensity distributions obtained with our numerical analysis (left column — blue lines) and in the experiment of ref. [6] (right column — red lines) both in the near- and far-field. It evidently shows that our simulations correctly reconstruct the spatial structure of the extracted beams. Only minor differences can be evidenced in the intensity levels of the different peaks from those of the experimental profiles. These differences are to be attributed to the presence, in the experimental case, of active medium saturation and detector non-linear response. This represents a validation of our numerical code in the ability to simulate the diffractive propagation of our HSUR configurations. The analysis of cavity losses and extracted beam properties performed with the numerical code enabled us to explain the results obtained in our past experiments [6] carried out with a strongly non-confocal resonator configuration and to prove that out-coupling coefficients compatible with the design requirements and good quality extracted beams can be obtained also in this non-canonical cavity setup. These results allow to largely widen the range of acceptable parameters for the unstable direction when designing a hybrid resonator (HSUR). In the case of our slab laser they enabled us to adopt mirror ROCs much shorter than those prescribed for the confocal cavity, given our active medium length. This choice produced a significant improvement in terms of alignment tolerance in the stable transverse direction with a zig-zag propagation.
References
2
3
4
5
6
10
9
8
7
6
5
4
Intensity (arb.un.)
Transverse position (mm)
[1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13]
-10
-5
0
5
10
-10
-5
0
5
10
Diffraction Angle (mrad) Fig. 8. Comparison of some sample field distributions both in the near- (upper row) and far-field (lower row) obtained by our numerical simulations (left column plots — blue line) with the beam profiles measured in the experimental setup of ref. [6] (right column plots — red line).
[14]
L. Bourn, P.E. Dyer, Opt. Commun. 31 (1979) 193. P.E. Jackson, H.J. Baker, D.R. Hall, Appl. Phys. Lett. 54 (1989) 1950. K. Du, N. Wu, J. Xu, J. Giesekus, P. Loosen, R. Poprawe, Opt. Lett. 23 (1998) 370. A. Lapucci, A. Labate, F. Rossetti, S. Mascalchi, Appl. Opt. 35 (1996) 3185. A. Wasilewski, H.J. Baker, D.R. Hall, Appl. Opt. 39 (2000) 6174. M. Ciofini, E. Favilla, A. Lapucci, E. Sani, Opt. Laser Technol. 39 (2007) 1380. S. Chandra, T.H. Allik, J.A. Hutchinson, Opt. Lett. 20 (1995) 2387. A.E. Siegman, Lasers, University Science Books, Mill Valley Calif, 1986 Chap. 22. Z. Wang, X. Ye, T. Fang, J. Xu, Opt. Rev. 12 (2005) 391. A. Lapucci, M. Ciofini, E. Favilla, E. Sani, M. De Rosa, G. Mincuzzi, A.S.D. Rocco, J. Eur. Opt. Soc. Rapid Publ. (2007) 2,07020. A. Lapucci, M. Ciofini, “Non-confocal hybrid stable–unstable resonators: parametric study of the optical losses”, in press. Trace-Pro 3.3. Lambda-Research Corp. www.lambdares.com . M.W. Sasnett, Propagation of multimode laser beams—the M2 factor, in: D.R. Hall, P.E. Jackson (Eds.), The Physics and Technology of Laser Resonators, Hilger, London, 1989, p. 132. ISO standard 11146-1 “Lasers and laser-related equipment — test methods for laser beam widths, divergence angle and beam propagation factor — Part 1: stigmatic and simple astigmatic beams” and 11146-3 “Lasers and laser-related equipment — Test methods for laser beam widths, divergence angle and beam propagation factor — Part 3: alternative test methods and geometrical laser beam classification and propagation.”