- Email: [email protected]
Three-dimensional geometrical analysis and the characteristics of laser generation in a multilobe mirror cavity
15 June 1999
Optics Communications 164 Ž1999. 297–305 www.elsevier.comrlocateroptcom
Full length article
Three-dimensional geometrical analysis and the characteristics of laser generation in a multilobe mirror cavity S.L. Popyrin, I.V. Sokolov, A.V. Yurkin
)
General Physics Institute, Russian Academy of Sciences, 38 VaÕiloÕ Street, 117942 Moscow, Russia Received 26 August 1998; received in revised form 10 February 1999; accepted 23 April 1999
Abstract The geometric optical properties of the plane–plane cavity and the multilobe mirror cavity are compared. The threshold pumping energy of the multilobe mirror cavity is experimentally measured. The relation between the ray geometry and the loss factor of the cavity is derived from the energy balance equations. It is shown that the three-dimensional system of rays in a multilobe mirror cavity contains not only the two-dimensional periodic trajectories but also three-dimensional helical trajectories similar to the off-axis Žskew. ray trajectories in spherical mirror Fabry–Perot type interferometers. Actually, the multilobe mirror cavity consists of many cavities of different types which are bound to each other. q 1999 Published by Elsevier Science B.V. All rights reserved. PACS: 42.60.Da; 42.15.Gs Keywords: Laser cavity; Multilobe mirror; Threshold energy of pumping; Ray geometry
1. Introduction The scheme of a modeless laser was proposed by Ewart w1x. There is no longitudinal mode structure in this optical device. The active medium of this laser consists of several active regions inside a dye cell. Four or six active regions w1x are sufficient for obtaining the low beam divergence and smooth beam profile w2x. In fact, the devise operates as a travelling wave amplifier of spontaneously emitted fluorescence. The modeless laser was used for diagnostic and spectral measurements w2,3x. )
Corresponding author. Fax: q7-95-135-8011; E-mail: [email protected]
In this work, we investigate the laser scheme similar in part to that mentioned above. To obtain a smooth laser beam, we apply multilobe mirrors proposed in Ref. w4x. The multilobe mirror consists of many non-parallel semi-transparent planes inclined relative to the axis so there is no longitudinal mode structure in this optical device. In the multilobe mirror cavity light rays are inclined at different angles in different areas of laser active medium. Because of this, light mixing and multiple scattering takes place in this cavity, improving the homogeneity and axial symmetry of the laser beam. It can be suggested that in the multilobe mirror cavity, feedback of all frequencies excited in the active medium occurs. The multilobe mirrors are used in high-power
0030-4018r99r$ - see front matter q 1999 Published by Elsevier Science B.V. All rights reserved. PII: S 0 0 3 0 - 4 0 1 8 Ž 9 9 . 0 0 2 1 3 - 8
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S.L. Popyrin et al.r Optics Communications 164 (1999) 297–305
solid state Žneodymium glass, ruby, and Nd:Yag. lasers. Experimental results obtained with the multilobe mirror cavity are described in detail in Refs. w4,5x. With this cavity, these results are promising and show that it is possible to obtain the smooth highpower axially symmetric focused Žas well as nonfocused. laser beam. Such beams could be used for plasma generation w6x, for making round holes with smooth edge in hard materials, etc. The calculations of the two-dimensional Ž2D. ray distribution for a multilobe mirror cavity are presented in Ref. w5x. This paper presents results of three-dimensional Ž3D. analysis which is more realistic than the previous 2D analysis. The general description of a laser equipped with the multilobe mirror cavity is presented in Section 2. In Section 3, we compare the geometric optics properties of the plane–plane cavity and the multilobe mirror cavity and explain the experimentally measured dependence of the threshold pumping energy of the multilobe mirror cavity on the angular off-set. In Section 4, the losses associated with ray leaving the cavity aperture are estimated on the basis of energy balance equations. It is also shown that the 2D analysis can be reduced to tracing a number of periodic trajectories. In Section 5, the 3D analysis of the multilobe mirror cavity with six reflected planes is performed. It is shown that the 3D system of rays in this multilobe mirror cavity contains 2D periodic trajectories and 3D helical trajectories similar to the off-axis Žskew. ray trajectories in spherical mirror Fabry–Perot type interferometers.
2. Multilobe mirrors for lasers The multilobe mirror consists of a set of glass, or quarts, wedges arranged successively in a pile and inclined at a small angle g relative to the axis and rotated about the axis at angles b s 2prk, where k is the number of the reflected planes w4,5x. Fig. 1a shows the general view of the multilobe mirror cavity. The cavity consists of a multilobe mirror Ž1. and a non-transmitting plane mirror Ž2.; an active medium Ž3. is placed between these mirrors. The larger the number k, the higher the degree of homogeneity of the beam. k s 8–12 is sufficient for obtaining the homogeneous energy distribution of laser beam in many practical cases. However, in Ref. w6x, in order to obtain a uniform laser plasma, k s 24 Ž12 wedges. was used. It is well known that high power solid state lasers emit a large number of modes with fluctuating phases and amplitudes w7x and have a broad emission spectrum. Usually, solid-state free-running lasers generate stochastic spikes w7x. The same phenomena are observed for the multilobe mirror neodymium-glass laser under study. From preliminary spectral measurements, its bandwidth is estimated to be around 30 nm. From a geometrical standpoint, the angular off-sets g of the multilobe mirror planes ŽFig. 1a. coincide with the inclination angles of tangential planes Ž4. to the cone Žaxicon. w8x surface Ž5. ŽFig. 1b..The thickness of the multilobe mirror pile is h - L, where L is the cavity length ŽFig. 1b.. However, the geometric optic properties of the set of tangential planes
Fig. 1. Ža. General view of the multilobe mirror cavity: Ž1. multilobe mirror, Ž2. non-transmitting mirror, Ž3. active element; Žb. arrangement of the set of tangential planes Ž4. to the cone Žaxicon. surface Ž5.. The angular off-sets g of the multilobe mirror planes Ž1. are the same as those of tangential planes Ž4..
S.L. Popyrin et al.r Optics Communications 164 (1999) 297–305
appear to be significantly different from those of the axicon. As is known, the geometric-optics approach is used to describe the processes in laser cavities with high Fresnel number: F s d 2r4 l L 4 1, where d is the aperture and l is the wavelength. The multilobe mirror cavities are characterized by the dimensionless number m s dr4g L by analogy with the Fresnel number F. In our analysis we also use the geometric-optics approach based on the use of periodic trajectories in the multilobe mirror cavity. Each periodical trajectory is formed by a group of rays rather than by a single ray. Since angular off-sets of mirrors are small, we assume that the rays in multilobe mirror cavity model are inclined at small angles ´ with respect to the axis:
´ s Ž "2 n q l . g ,
Ž 1.
where n s 0, 1, 2, . . . and 0 F l F 1 w5x. The value of l is different for each group of rays. The rays of each group can be arranged sequentially from a single ray w5x.
3. Trajectories and the threshold pumping energy for the multilobe mirror cavity In this study, we restrict ourselves to analysis of trajectories formed by a single group of rays whose inclination angles ´ are given by Eq. Ž1., where n s 0, 1 and l s 0. Thus, we only use the rays inclined to the axis by the angles ´ s 0, "2g . At first we consider the 2D case. Fig. 2 shows the simplest case for the group of rays inclined at small angles 0 and "2g relative to the axis. Fig. 2a shows the general view of rays in the multilobe mirror cavity for m s 1 and k s 2. There exists only one elementary periodic M-shaped trajectory CB2 EA1 D within the cavity aperture. To constructed trajectories we apply the method of images which is used in geometric optics w8x. Fig. 2b shows the images constructed for the rays inclined at 0 and "2g . For the planes A1 B1 , A 2 B2 , we construct their images AX1 B1X , AX2 B2X which are mirror-symmetric about the plane CD as shown in Fig. 2b. The broken-line trajectory A1 EB2 in Fig. 2a is imaged by the straight-line trajectory AX1 B2 in Fig. 2b. Then, for the plane CD, we construct its images CY DY and CZ DZ which are
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mirror-symmetric about the planes A 2 B2 , AX1 B1X respectively as shown in Fig. 2c. As a result, the M-shaped trajectory CB2 EA1 D in Fig. 2a is imaged by the straight line CY DZ in Fig. 2c. One can see that the system of the periodic trajectories in the multilobe mirror cavity reduces to periodic trajectories in the plane–parallel cavity with the plane mirrors CY DY , CZ DZ . Hence, like in the plane–parallel cavity, plane waves can exist in the multilobe cavity. However, the degree of coherence of this cavity is lower than that in the ideal plane– parallel cavity w5x. This is explained by the fact that the plane mirrors of the multilobe cavity are inclined to the cavity axis, there are trajectories non-parallel to the axis. Fig. 3 shows the typical experimental dependence of the pumping threshold energy E upon the angle of inclination Žmisalignment. d between the non-transmitting and exit mirrors of the laser cavity. A GOS1001 free-running neodymium glass laser Ž d s 4.5 cm; L s 1 m; the output energy up to 1000 J., is equipped either with the plane–plane cavity Ždotted curve 1. or with the multilobe mirror cavity Ž k s 12, multilobe mirror consists of 6 identical glass wedges, the reflection coefficient of whole multilobe pile is 33%. with various angles g Žsolid curves 2, 3, and 4.. In the plane–plane cavity, the periodic ray trajectories take place only in the case of very precise alignment Ž d s 0.. If the cavity planes are not parallel to each other Ž d ) 0., then the ray trajectories are non-periodic and leave the cavity, so threshold pumping energy E Žcurve 1. increases by several fold w9x. In the multilobe mirror cavity, the periodic ray trajectories take place if the angle of inclination d between the planes of the non-transmitting mirror and multilobe mirror is not more than g Ži.e., d F g .. The variation in E does not exceed several percent if d F "g 1 , "g 2 , "g 3 Žcurves 2, 3, and 4 correspondingly.. No periodic trajectories can be obtained for d ) g . That is why the threshold pumping energy increases sharply for d ) g Žcurves 2, 3, and 4.. Such a dependence is observed, if the number of the multilobe mirror planes equal to k s 6 or more. A similar dependence is observed for the Q-switch laser using the passive switch ŽLiF crystal.. The duration of giant pulse is t f 30–40 ns. It corre-
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Fig. 2. Group of rays in the multilobe mirror cavity with m s 1 and k s 2: Ža. periodic ray trajectories Žsolid lines. and ray trajectories leaving the cavity Ždotted lines.; Žb. images of rays reflected from the plane CD of non-transmitting mirror; Žc. images of rays reflected X X from the planes A 2 B2 and A1 B1. The scheme of the multilobe mirror cavity with trajectories is shown by bold lines. Indices Ž1., Ž2. refer to X X the multilobe and non-transmitting mirrors; Ž1 ., Ž2 . refer to their images, respectively.
sponds to more than 10 passes between the cavity mirrors. We can suggest that the quasi-steady field distribution is reached during the giant pulse, periodic trajectories take place in this case. We can assume that quasi-steady field distribution can be established even before the laser giant pulse builds up, because the time interval between the beginning of passive Q-switching and beginning of the giant pulse is equal to several microseconds, and thus is much longer than the duration t of the giant pulse w7x. For the more steady field distribution to be reached during short giant pulse, the length L of the
multilobe mirror cavity should be chosen as short as possible.
4. The energy balance equations The equivalent schemes are widely used for calculations of light or electron beams as is shown in Ref. w10x. Taking this approach, we used the equivalent light-guide scheme to obtain the equations of light energy balance after the great number of ray passes between the mirrors ŽFig. 4.. Here, we examined the
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shows five double passes of the light ray through the cavity. After each double pass, the light rays BB, BA, AA and AB ŽFig. 4b. split into two rays; one of those rays is the part of the periodic trajectory, whereas the other ray leaves the cavity. Let y be the energy loss factor of the multilobe mirror cavity for one double pass of rays in the cavity. Let w 1 , w 2 , w 3 , and w4 be the light energies propagated along the rays BB, BA, AA, and AB respectively after a great number of passes ŽFig. 4b.. Then, the energy balance equations are written in the following form:
y w 1 s Rw 2 , y w 2 s Rw 3 , y w 3 s Rw4 , y w4 s Rw1 . Fig. 3. Dependence of the pumping threshold energy E upon the angle of inclination d between the non-transmitting and exit mirrors of the laser cavity: for the plane–plane cavity Ždotted curve 1.; and for the multilobe mirror cavity with k s12 and different values of g Žsolid curves.: Ž2. g 1 s1 mrad, Ž3. g 2 s1.5 mrad, Ž4. g 3 s 2 mrad. The curves Ž2–4. are fitted to the experimental points shown in the figure.
empty cavity without active medium. Our consideration is based on the following assumptions: the angles "2g are small; the thickness h of the multilobe mirror pile is small, compared with the cavity length h < L; the length of the rays within each double pass is f 2 L; the reflection coefficient of each plane of a multilobe mirror is R < 1. While assuming the mirror-pile to be thin, the cavity scheme for m s 1 and k s 2 is simplified as shown in Fig. 4a Žcompare with Fig. 2a.. Fig. 4b
Ž 2. In this simplest case, we obtain y s R. The examination of the more complicated 2D case with m s 2 can be reduced to the examination of two M-shaped trajectories shown in Fig. 5a. This scheme consists of two symmetric parts. Each part is equivalent to the elementary M-shaped trajectory shown in Fig. 4a. The method of images developed in Section 3 can be applied for each of these Mshaped trajectories. In Fig. 5b each of the rays BB, BA, and AB splits into two rays: one ray is the part of a periodic trajectory, and the other leaves the cavity whereas the ray AA splits into two rays AB, both being parts of the periodic trajectories. It is evident that the energy propagated along two rays BA is transferred to the energy along the ray
Fig. 4. 2D simplified scheme of ray trajectories as in Fig. 2: Ža. periodic M-shaped trajectory within the aperture d; Žb. equivalent light-guide system of rays. The periodic trajectories are shown by solid lines, trajectories leaving the cavity are shown by dotted lines. The thickness of the multilobe mirror pile is h < L and m s 1. Indices Ž1., Ž2. refer to the multilobe and non-transmitting mirrors, respectively.
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Fig. 5. 2D scheme of trajectories for m s 2: Ža. two bound elementary M-shaped trajectories; Žb. the equivalent light-guide system of rays. Indices Ž1., Ž2. refer to the multilobe and non-transmitting mirrors, respectively.
AA. The energy balance equations are written in the following form: y w 1 s Rw4 , y w 2 s Rw1 , y w 3 s 2 Rw 2 , y w4 s Rw 3 . Ž 3. It is easy to find y by solving Eq. Ž3.: y s 2 1r4 R. If we take w 1 s 1, then from Eq. Ž3. we can calculate w 2 s 2 3r4r2, w 3 s 2 1r2 , and w4 s 2 1r4 . We can write the expression for the energy propagated along light rays inclined at angles 0 and "2g Žby using the subscript indices Ž0. and Ž2., respectively.: PŽ0. s 2 w 1 q w 3 and PŽ2. s 2Ž w 2 q w4 ., respectively. The whole energy is P s PŽ0. q PŽ2. . The relative angular energy distribution after a great number of passes in the far field zone can be estimated as QŽ0. s PŽ0.rP s Ž2 1r4 y 1.Ž2 1r2 q 1. f 0.457 for rays at angles 0 and QŽ2. s PŽ2.rP s 2 1r4 Ž2 1r4 y 1.Ž2 1r2 q 1. f 0.543 for rays at angles "2g . From comparison of Fig. 4b and Fig. 5b, we see that the total number of rays allowed in the multilobe cavity increases as m increases and n increases. This effect explains the decrease in the threshold pumping energy for larger m and smaller g Žcurves 2–4 in Fig. 3.. 5. 3D ray distribution in the multilobe mirror cavity The 3D distribution of off-axis ray trajectories in a spherical mirror Fabry–Perot type interferometer was calculated in Refs. w11,12x.
We calculated the 3D ray distribution in the multilobe mirror cavity for m s 2 and k s 6. Fig. 6 shows the group of rays for m s 2. The periodic trajectories projected onto the plane YZ are shown in Fig. 6a. Fig. 6b illustrates the equivalent scheme which allows us to trace the periodic trajectories among the ray projections onto the plane YZ. These periodic trajectories projected onto the plane XY are shown in Fig. 6c. The general view of the group of rays is shown in Fig. 6d. To simplify analysis of the 3D distribution, we use, as above, the subscript index Ž0. to denote the rays parallel to the axis Žthe inclination angle is 0., and the subscript index Ž2. to denote the rays inclined at the angle 2g to the axis. In the 3D case, each ray splits after each double path through the cavity into six rays. Among these rays we find periodic trajectories. The number of the rays of periodic trajectories within each double ray path is: 1 ray AAŽ0. , 6 rays BBŽ0. , BAŽ2. , ABŽ2. , and 12 rays BBŽ2. — 31 rays total. We consider the process of the energy redistribution between the rays of periodic trajectories after each double path through the cavity. As follows from the equivalent scheme, the energy is redistributed between the directions along the rays as follows: from 6 rays BAŽ2. to ray AAŽ0. , from 2 rays BBŽ2. and ray ABŽ2. to ray BBŽ0. , from ray BBŽ0. and ray BBŽ2. to ray BBŽ2. , from ray BBŽ0. to ray BAŽ2. , from ray AAŽ0. and 2 rays BAŽ2. to ray ABŽ2. . Let w 1 , w 2 , w 3 , w4 , and w5 be the ray energies propagated along the rays AAŽ0. , BBŽ0. , BBŽ2. , BAŽ2. ,
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Fig. 6. 3D scheme of trajectories for k s 6 and m s 2: Ža. periodic trajectories projected onto the plane YZ; Žb. equivalent light guide system for the periodic trajectories projected onto the plane YZ; Žc. periodic trajectories projected onto the plane XY; Žd. general view of the 3D scheme of periodic trajectories in the multilobe mirror cavity. Also shown are the examples of trajectories: the 2D M-shaped trajectories consisting of rays BBŽ2. –BBŽ0. –BBŽ2. –BBŽ0. and AAŽ0. –ABŽ2. –BBŽ0. –BAŽ2. are shown by the bold solid lines, the 3D helical trajectory consisting of rays BBŽ2. –BBŽ2. –BBŽ2. –BBŽ2. is shown by the bold dotted lines. Indices Ž1., Ž2. refer to the multilobe and non-transmitting mirrors, respectively.
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and ABŽ2. respectively. As in Section 4, n is the energy loss factor for one double pass of rays in the cavity, and R is the reflection coefficient of each plane of a multilobe mirror. Then, the energy balance equations take the form:
y w 1 s 6 Rw4 , y w 2 s R Ž 2 w 3 q w5 . , y w 3 s R Ž w 2 q w 3 . , y w4 s Rw 2 , y w5 s R Ž w1 q 2 w4 . .
Ž 4.
By introducing the m s nrR from Eq. Ž4. we derive a single equation of power five:
m5 y m4 y 2 m3 y 2 m2 y 4m q 6 s 0.
Ž 5.
By numerical solving this equation, we obtain m f 2.3354. If we take w 1 s 1, then from Eq. Ž4. we can calculate w 2 f 0.9090, w 3 f 0.6807, w4 f 0.3892, and w5 f 0.7615. We can write the expression for the energy propagated at angles 0 and 2g : PŽ0. s w1 q 6w 2 and PŽ2. s 6Ž2 w 3 q w4 q w5 ., respectively. The whole energy is P s PŽ0. q PŽ2. . The relative angular energy distribution in the far field zone can be estimated as QŽ0. s PŽ0.rP f 0.2998 and QŽ2. s PŽ2.rP f 0.7002. The energy redistribution between the directions along the rays takes place after great number of ray passes within the aperture in this case. In Fig. 6 the 2D periodic M-shaped trajectories BBŽ2. –BBŽ0. –BBŽ2. –BBŽ0. and AA Ž0. –ABŽ2. –BBŽ0. – BAŽ0. , which are similar to those presented in Fig. 4, are shown by the bold solid lines and the 3D helical trajectory BBŽ2. –BBŽ2. –BBŽ2. –BBŽ2. which is similar to those exiting in spherical mirror interferometers w11,12x, is shown by the dashed line. It should be noted that all the rays AAŽ0. , BBŽ0. , BBŽ2. , BAŽ2. , ABŽ2. in one double path are parts of the M-shaped 2D trajectories. However, the rays BBŽ2. are parts of the helical 3D trajectories too. The projections of the helical 3D trajectories onto the plane YZ ŽFig. 6b. and XY ŽFig. 6c. can be reduced to periodic trajectories in a plane–parallel cavity. The method of images developed in Section 3 can be applied for each of these 2D M-shaped trajectories and for the projections of the helical 3D trajectories. 3D analysis shows that in fact, the multilobe mirror cavity is equivalent to a large number of cavities of different types which are bound with each other.
6. Summary Our analysis shows that, in contrast to a plane– plane cavity, the threshold pumping energy in the multilobe mirror cavity is almost independent of the angle of misalignment d for d F g . The threshold pumping energy of the multilobe mirror cavity depends on the inclination angle g of a multilobe mirror; its value decreases as g decreases, and consequently, the dimensionless m number increases. The use of the energy balance equations makes it possible to relate the geometry of rays to the loss factor of the cavity and to calculate light energy distribution in the cavity not only for the 2D case but also for the 3D case. By applying the method of images, we reduce the examination of 2D periodic M-shaped trajectories in a multilobe mirror cavity to the simple case of the ray propagation in a plane– parallel cavity. The 3D system of rays in a multilobe mirror cavity contains not only 2D periodic trajectories but also 3D helical trajectories similar to the off-axis Žskew. ray trajectories in spherical mirror Fabry–Perot type interferometers. Like in the plane–parallel cavity, plane waves exist in the multilobe cavity. 3D analysis shows that in fact, the multilobe mirror cavity is equivalent to a large number of cavities of different types which are bound with each other.
Acknowledgements The authors thank N.F. Larionova for help and useful discussions.
References w1x P. Ewart, Opt. Commun. 55 Ž1985. 124. w2x P. Snowdon, S.M. Skippon, P. Ewart, Appl. Opt. 30 Ž1991. 1008. w3x P. Ewart, G.M. Lloyd, R. Bratfalean, D.J. Walker, R.B. Williams, 16th International Conference on Coherent and Nonlinear Optics ’1998, Moscow, Russia, URSS Publishers, 1998, p. 96. w4x A.V. Yurkin, Sov. J. Quantum Electron. 21 Ž1991. 447. w5x A.V. Yurkin, Opt. Commun. 114 Ž1995. 393.
S.L. Popyrin et al.r Optics Communications 164 (1999) 297–305 w6x G.A. Askar’yan, M.G. Korolev, A.V. Yurkin, K.L. Yakushkin, Quantum Electron. 25 Ž1995. 289. w7x O. Svelto, Principles of Lasers, Plenum, New York, 1989, sects. 5.35, 5.42, 5.43. w8x L.M. Soroko, Axicons and meso-optical imaging devices, in: E. Wolf ŽEd.., Progress in Optics XXVII, Elsevier, Amsterdam, 1989, pp. 109–160. w9x B.R. Belostotskii, Yu.V. Lubavskii, V.M. Ovchinnikov, Prin-
w10x w11x w12x
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ciples of Laser Engineering, Sovetskoe Radio, Moskow, 1972, p. 207, Fig. 2.43 Žin Russian.. J.R. Pierce, Theory and Design of Electron Beams, Van Nostrand, New York, 1954, pp. 194–213. D. Herriot, H. Kogelnik, R. Kompfer, Appl. Opt. 3 Ž1964. 523. D. Herriot, H. Shulte, Appl. Opt. 4 Ž1965. 883.
Optics Communications 164 Ž1999. 297–305 www.elsevier.comrlocateroptcom
Full length article
Three-dimensional geometrical analysis and the characteristics of laser generation in a multilobe mirror cavity S.L. Popyrin, I.V. Sokolov, A.V. Yurkin
)
General Physics Institute, Russian Academy of Sciences, 38 VaÕiloÕ Street, 117942 Moscow, Russia Received 26 August 1998; received in revised form 10 February 1999; accepted 23 April 1999
Abstract The geometric optical properties of the plane–plane cavity and the multilobe mirror cavity are compared. The threshold pumping energy of the multilobe mirror cavity is experimentally measured. The relation between the ray geometry and the loss factor of the cavity is derived from the energy balance equations. It is shown that the three-dimensional system of rays in a multilobe mirror cavity contains not only the two-dimensional periodic trajectories but also three-dimensional helical trajectories similar to the off-axis Žskew. ray trajectories in spherical mirror Fabry–Perot type interferometers. Actually, the multilobe mirror cavity consists of many cavities of different types which are bound to each other. q 1999 Published by Elsevier Science B.V. All rights reserved. PACS: 42.60.Da; 42.15.Gs Keywords: Laser cavity; Multilobe mirror; Threshold energy of pumping; Ray geometry
1. Introduction The scheme of a modeless laser was proposed by Ewart w1x. There is no longitudinal mode structure in this optical device. The active medium of this laser consists of several active regions inside a dye cell. Four or six active regions w1x are sufficient for obtaining the low beam divergence and smooth beam profile w2x. In fact, the devise operates as a travelling wave amplifier of spontaneously emitted fluorescence. The modeless laser was used for diagnostic and spectral measurements w2,3x. )
Corresponding author. Fax: q7-95-135-8011; E-mail: [email protected]
In this work, we investigate the laser scheme similar in part to that mentioned above. To obtain a smooth laser beam, we apply multilobe mirrors proposed in Ref. w4x. The multilobe mirror consists of many non-parallel semi-transparent planes inclined relative to the axis so there is no longitudinal mode structure in this optical device. In the multilobe mirror cavity light rays are inclined at different angles in different areas of laser active medium. Because of this, light mixing and multiple scattering takes place in this cavity, improving the homogeneity and axial symmetry of the laser beam. It can be suggested that in the multilobe mirror cavity, feedback of all frequencies excited in the active medium occurs. The multilobe mirrors are used in high-power
0030-4018r99r$ - see front matter q 1999 Published by Elsevier Science B.V. All rights reserved. PII: S 0 0 3 0 - 4 0 1 8 Ž 9 9 . 0 0 2 1 3 - 8
298
S.L. Popyrin et al.r Optics Communications 164 (1999) 297–305
solid state Žneodymium glass, ruby, and Nd:Yag. lasers. Experimental results obtained with the multilobe mirror cavity are described in detail in Refs. w4,5x. With this cavity, these results are promising and show that it is possible to obtain the smooth highpower axially symmetric focused Žas well as nonfocused. laser beam. Such beams could be used for plasma generation w6x, for making round holes with smooth edge in hard materials, etc. The calculations of the two-dimensional Ž2D. ray distribution for a multilobe mirror cavity are presented in Ref. w5x. This paper presents results of three-dimensional Ž3D. analysis which is more realistic than the previous 2D analysis. The general description of a laser equipped with the multilobe mirror cavity is presented in Section 2. In Section 3, we compare the geometric optics properties of the plane–plane cavity and the multilobe mirror cavity and explain the experimentally measured dependence of the threshold pumping energy of the multilobe mirror cavity on the angular off-set. In Section 4, the losses associated with ray leaving the cavity aperture are estimated on the basis of energy balance equations. It is also shown that the 2D analysis can be reduced to tracing a number of periodic trajectories. In Section 5, the 3D analysis of the multilobe mirror cavity with six reflected planes is performed. It is shown that the 3D system of rays in this multilobe mirror cavity contains 2D periodic trajectories and 3D helical trajectories similar to the off-axis Žskew. ray trajectories in spherical mirror Fabry–Perot type interferometers.
2. Multilobe mirrors for lasers The multilobe mirror consists of a set of glass, or quarts, wedges arranged successively in a pile and inclined at a small angle g relative to the axis and rotated about the axis at angles b s 2prk, where k is the number of the reflected planes w4,5x. Fig. 1a shows the general view of the multilobe mirror cavity. The cavity consists of a multilobe mirror Ž1. and a non-transmitting plane mirror Ž2.; an active medium Ž3. is placed between these mirrors. The larger the number k, the higher the degree of homogeneity of the beam. k s 8–12 is sufficient for obtaining the homogeneous energy distribution of laser beam in many practical cases. However, in Ref. w6x, in order to obtain a uniform laser plasma, k s 24 Ž12 wedges. was used. It is well known that high power solid state lasers emit a large number of modes with fluctuating phases and amplitudes w7x and have a broad emission spectrum. Usually, solid-state free-running lasers generate stochastic spikes w7x. The same phenomena are observed for the multilobe mirror neodymium-glass laser under study. From preliminary spectral measurements, its bandwidth is estimated to be around 30 nm. From a geometrical standpoint, the angular off-sets g of the multilobe mirror planes ŽFig. 1a. coincide with the inclination angles of tangential planes Ž4. to the cone Žaxicon. w8x surface Ž5. ŽFig. 1b..The thickness of the multilobe mirror pile is h - L, where L is the cavity length ŽFig. 1b.. However, the geometric optic properties of the set of tangential planes
Fig. 1. Ža. General view of the multilobe mirror cavity: Ž1. multilobe mirror, Ž2. non-transmitting mirror, Ž3. active element; Žb. arrangement of the set of tangential planes Ž4. to the cone Žaxicon. surface Ž5.. The angular off-sets g of the multilobe mirror planes Ž1. are the same as those of tangential planes Ž4..
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appear to be significantly different from those of the axicon. As is known, the geometric-optics approach is used to describe the processes in laser cavities with high Fresnel number: F s d 2r4 l L 4 1, where d is the aperture and l is the wavelength. The multilobe mirror cavities are characterized by the dimensionless number m s dr4g L by analogy with the Fresnel number F. In our analysis we also use the geometric-optics approach based on the use of periodic trajectories in the multilobe mirror cavity. Each periodical trajectory is formed by a group of rays rather than by a single ray. Since angular off-sets of mirrors are small, we assume that the rays in multilobe mirror cavity model are inclined at small angles ´ with respect to the axis:
´ s Ž "2 n q l . g ,
Ž 1.
where n s 0, 1, 2, . . . and 0 F l F 1 w5x. The value of l is different for each group of rays. The rays of each group can be arranged sequentially from a single ray w5x.
3. Trajectories and the threshold pumping energy for the multilobe mirror cavity In this study, we restrict ourselves to analysis of trajectories formed by a single group of rays whose inclination angles ´ are given by Eq. Ž1., where n s 0, 1 and l s 0. Thus, we only use the rays inclined to the axis by the angles ´ s 0, "2g . At first we consider the 2D case. Fig. 2 shows the simplest case for the group of rays inclined at small angles 0 and "2g relative to the axis. Fig. 2a shows the general view of rays in the multilobe mirror cavity for m s 1 and k s 2. There exists only one elementary periodic M-shaped trajectory CB2 EA1 D within the cavity aperture. To constructed trajectories we apply the method of images which is used in geometric optics w8x. Fig. 2b shows the images constructed for the rays inclined at 0 and "2g . For the planes A1 B1 , A 2 B2 , we construct their images AX1 B1X , AX2 B2X which are mirror-symmetric about the plane CD as shown in Fig. 2b. The broken-line trajectory A1 EB2 in Fig. 2a is imaged by the straight-line trajectory AX1 B2 in Fig. 2b. Then, for the plane CD, we construct its images CY DY and CZ DZ which are
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mirror-symmetric about the planes A 2 B2 , AX1 B1X respectively as shown in Fig. 2c. As a result, the M-shaped trajectory CB2 EA1 D in Fig. 2a is imaged by the straight line CY DZ in Fig. 2c. One can see that the system of the periodic trajectories in the multilobe mirror cavity reduces to periodic trajectories in the plane–parallel cavity with the plane mirrors CY DY , CZ DZ . Hence, like in the plane–parallel cavity, plane waves can exist in the multilobe cavity. However, the degree of coherence of this cavity is lower than that in the ideal plane– parallel cavity w5x. This is explained by the fact that the plane mirrors of the multilobe cavity are inclined to the cavity axis, there are trajectories non-parallel to the axis. Fig. 3 shows the typical experimental dependence of the pumping threshold energy E upon the angle of inclination Žmisalignment. d between the non-transmitting and exit mirrors of the laser cavity. A GOS1001 free-running neodymium glass laser Ž d s 4.5 cm; L s 1 m; the output energy up to 1000 J., is equipped either with the plane–plane cavity Ždotted curve 1. or with the multilobe mirror cavity Ž k s 12, multilobe mirror consists of 6 identical glass wedges, the reflection coefficient of whole multilobe pile is 33%. with various angles g Žsolid curves 2, 3, and 4.. In the plane–plane cavity, the periodic ray trajectories take place only in the case of very precise alignment Ž d s 0.. If the cavity planes are not parallel to each other Ž d ) 0., then the ray trajectories are non-periodic and leave the cavity, so threshold pumping energy E Žcurve 1. increases by several fold w9x. In the multilobe mirror cavity, the periodic ray trajectories take place if the angle of inclination d between the planes of the non-transmitting mirror and multilobe mirror is not more than g Ži.e., d F g .. The variation in E does not exceed several percent if d F "g 1 , "g 2 , "g 3 Žcurves 2, 3, and 4 correspondingly.. No periodic trajectories can be obtained for d ) g . That is why the threshold pumping energy increases sharply for d ) g Žcurves 2, 3, and 4.. Such a dependence is observed, if the number of the multilobe mirror planes equal to k s 6 or more. A similar dependence is observed for the Q-switch laser using the passive switch ŽLiF crystal.. The duration of giant pulse is t f 30–40 ns. It corre-
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Fig. 2. Group of rays in the multilobe mirror cavity with m s 1 and k s 2: Ža. periodic ray trajectories Žsolid lines. and ray trajectories leaving the cavity Ždotted lines.; Žb. images of rays reflected from the plane CD of non-transmitting mirror; Žc. images of rays reflected X X from the planes A 2 B2 and A1 B1. The scheme of the multilobe mirror cavity with trajectories is shown by bold lines. Indices Ž1., Ž2. refer to X X the multilobe and non-transmitting mirrors; Ž1 ., Ž2 . refer to their images, respectively.
sponds to more than 10 passes between the cavity mirrors. We can suggest that the quasi-steady field distribution is reached during the giant pulse, periodic trajectories take place in this case. We can assume that quasi-steady field distribution can be established even before the laser giant pulse builds up, because the time interval between the beginning of passive Q-switching and beginning of the giant pulse is equal to several microseconds, and thus is much longer than the duration t of the giant pulse w7x. For the more steady field distribution to be reached during short giant pulse, the length L of the
multilobe mirror cavity should be chosen as short as possible.
4. The energy balance equations The equivalent schemes are widely used for calculations of light or electron beams as is shown in Ref. w10x. Taking this approach, we used the equivalent light-guide scheme to obtain the equations of light energy balance after the great number of ray passes between the mirrors ŽFig. 4.. Here, we examined the
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shows five double passes of the light ray through the cavity. After each double pass, the light rays BB, BA, AA and AB ŽFig. 4b. split into two rays; one of those rays is the part of the periodic trajectory, whereas the other ray leaves the cavity. Let y be the energy loss factor of the multilobe mirror cavity for one double pass of rays in the cavity. Let w 1 , w 2 , w 3 , and w4 be the light energies propagated along the rays BB, BA, AA, and AB respectively after a great number of passes ŽFig. 4b.. Then, the energy balance equations are written in the following form:
y w 1 s Rw 2 , y w 2 s Rw 3 , y w 3 s Rw4 , y w4 s Rw1 . Fig. 3. Dependence of the pumping threshold energy E upon the angle of inclination d between the non-transmitting and exit mirrors of the laser cavity: for the plane–plane cavity Ždotted curve 1.; and for the multilobe mirror cavity with k s12 and different values of g Žsolid curves.: Ž2. g 1 s1 mrad, Ž3. g 2 s1.5 mrad, Ž4. g 3 s 2 mrad. The curves Ž2–4. are fitted to the experimental points shown in the figure.
empty cavity without active medium. Our consideration is based on the following assumptions: the angles "2g are small; the thickness h of the multilobe mirror pile is small, compared with the cavity length h < L; the length of the rays within each double pass is f 2 L; the reflection coefficient of each plane of a multilobe mirror is R < 1. While assuming the mirror-pile to be thin, the cavity scheme for m s 1 and k s 2 is simplified as shown in Fig. 4a Žcompare with Fig. 2a.. Fig. 4b
Ž 2. In this simplest case, we obtain y s R. The examination of the more complicated 2D case with m s 2 can be reduced to the examination of two M-shaped trajectories shown in Fig. 5a. This scheme consists of two symmetric parts. Each part is equivalent to the elementary M-shaped trajectory shown in Fig. 4a. The method of images developed in Section 3 can be applied for each of these Mshaped trajectories. In Fig. 5b each of the rays BB, BA, and AB splits into two rays: one ray is the part of a periodic trajectory, and the other leaves the cavity whereas the ray AA splits into two rays AB, both being parts of the periodic trajectories. It is evident that the energy propagated along two rays BA is transferred to the energy along the ray
Fig. 4. 2D simplified scheme of ray trajectories as in Fig. 2: Ža. periodic M-shaped trajectory within the aperture d; Žb. equivalent light-guide system of rays. The periodic trajectories are shown by solid lines, trajectories leaving the cavity are shown by dotted lines. The thickness of the multilobe mirror pile is h < L and m s 1. Indices Ž1., Ž2. refer to the multilobe and non-transmitting mirrors, respectively.
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Fig. 5. 2D scheme of trajectories for m s 2: Ža. two bound elementary M-shaped trajectories; Žb. the equivalent light-guide system of rays. Indices Ž1., Ž2. refer to the multilobe and non-transmitting mirrors, respectively.
AA. The energy balance equations are written in the following form: y w 1 s Rw4 , y w 2 s Rw1 , y w 3 s 2 Rw 2 , y w4 s Rw 3 . Ž 3. It is easy to find y by solving Eq. Ž3.: y s 2 1r4 R. If we take w 1 s 1, then from Eq. Ž3. we can calculate w 2 s 2 3r4r2, w 3 s 2 1r2 , and w4 s 2 1r4 . We can write the expression for the energy propagated along light rays inclined at angles 0 and "2g Žby using the subscript indices Ž0. and Ž2., respectively.: PŽ0. s 2 w 1 q w 3 and PŽ2. s 2Ž w 2 q w4 ., respectively. The whole energy is P s PŽ0. q PŽ2. . The relative angular energy distribution after a great number of passes in the far field zone can be estimated as QŽ0. s PŽ0.rP s Ž2 1r4 y 1.Ž2 1r2 q 1. f 0.457 for rays at angles 0 and QŽ2. s PŽ2.rP s 2 1r4 Ž2 1r4 y 1.Ž2 1r2 q 1. f 0.543 for rays at angles "2g . From comparison of Fig. 4b and Fig. 5b, we see that the total number of rays allowed in the multilobe cavity increases as m increases and n increases. This effect explains the decrease in the threshold pumping energy for larger m and smaller g Žcurves 2–4 in Fig. 3.. 5. 3D ray distribution in the multilobe mirror cavity The 3D distribution of off-axis ray trajectories in a spherical mirror Fabry–Perot type interferometer was calculated in Refs. w11,12x.
We calculated the 3D ray distribution in the multilobe mirror cavity for m s 2 and k s 6. Fig. 6 shows the group of rays for m s 2. The periodic trajectories projected onto the plane YZ are shown in Fig. 6a. Fig. 6b illustrates the equivalent scheme which allows us to trace the periodic trajectories among the ray projections onto the plane YZ. These periodic trajectories projected onto the plane XY are shown in Fig. 6c. The general view of the group of rays is shown in Fig. 6d. To simplify analysis of the 3D distribution, we use, as above, the subscript index Ž0. to denote the rays parallel to the axis Žthe inclination angle is 0., and the subscript index Ž2. to denote the rays inclined at the angle 2g to the axis. In the 3D case, each ray splits after each double path through the cavity into six rays. Among these rays we find periodic trajectories. The number of the rays of periodic trajectories within each double ray path is: 1 ray AAŽ0. , 6 rays BBŽ0. , BAŽ2. , ABŽ2. , and 12 rays BBŽ2. — 31 rays total. We consider the process of the energy redistribution between the rays of periodic trajectories after each double path through the cavity. As follows from the equivalent scheme, the energy is redistributed between the directions along the rays as follows: from 6 rays BAŽ2. to ray AAŽ0. , from 2 rays BBŽ2. and ray ABŽ2. to ray BBŽ0. , from ray BBŽ0. and ray BBŽ2. to ray BBŽ2. , from ray BBŽ0. to ray BAŽ2. , from ray AAŽ0. and 2 rays BAŽ2. to ray ABŽ2. . Let w 1 , w 2 , w 3 , w4 , and w5 be the ray energies propagated along the rays AAŽ0. , BBŽ0. , BBŽ2. , BAŽ2. ,
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Fig. 6. 3D scheme of trajectories for k s 6 and m s 2: Ža. periodic trajectories projected onto the plane YZ; Žb. equivalent light guide system for the periodic trajectories projected onto the plane YZ; Žc. periodic trajectories projected onto the plane XY; Žd. general view of the 3D scheme of periodic trajectories in the multilobe mirror cavity. Also shown are the examples of trajectories: the 2D M-shaped trajectories consisting of rays BBŽ2. –BBŽ0. –BBŽ2. –BBŽ0. and AAŽ0. –ABŽ2. –BBŽ0. –BAŽ2. are shown by the bold solid lines, the 3D helical trajectory consisting of rays BBŽ2. –BBŽ2. –BBŽ2. –BBŽ2. is shown by the bold dotted lines. Indices Ž1., Ž2. refer to the multilobe and non-transmitting mirrors, respectively.
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and ABŽ2. respectively. As in Section 4, n is the energy loss factor for one double pass of rays in the cavity, and R is the reflection coefficient of each plane of a multilobe mirror. Then, the energy balance equations take the form:
y w 1 s 6 Rw4 , y w 2 s R Ž 2 w 3 q w5 . , y w 3 s R Ž w 2 q w 3 . , y w4 s Rw 2 , y w5 s R Ž w1 q 2 w4 . .
Ž 4.
By introducing the m s nrR from Eq. Ž4. we derive a single equation of power five:
m5 y m4 y 2 m3 y 2 m2 y 4m q 6 s 0.
Ž 5.
By numerical solving this equation, we obtain m f 2.3354. If we take w 1 s 1, then from Eq. Ž4. we can calculate w 2 f 0.9090, w 3 f 0.6807, w4 f 0.3892, and w5 f 0.7615. We can write the expression for the energy propagated at angles 0 and 2g : PŽ0. s w1 q 6w 2 and PŽ2. s 6Ž2 w 3 q w4 q w5 ., respectively. The whole energy is P s PŽ0. q PŽ2. . The relative angular energy distribution in the far field zone can be estimated as QŽ0. s PŽ0.rP f 0.2998 and QŽ2. s PŽ2.rP f 0.7002. The energy redistribution between the directions along the rays takes place after great number of ray passes within the aperture in this case. In Fig. 6 the 2D periodic M-shaped trajectories BBŽ2. –BBŽ0. –BBŽ2. –BBŽ0. and AA Ž0. –ABŽ2. –BBŽ0. – BAŽ0. , which are similar to those presented in Fig. 4, are shown by the bold solid lines and the 3D helical trajectory BBŽ2. –BBŽ2. –BBŽ2. –BBŽ2. which is similar to those exiting in spherical mirror interferometers w11,12x, is shown by the dashed line. It should be noted that all the rays AAŽ0. , BBŽ0. , BBŽ2. , BAŽ2. , ABŽ2. in one double path are parts of the M-shaped 2D trajectories. However, the rays BBŽ2. are parts of the helical 3D trajectories too. The projections of the helical 3D trajectories onto the plane YZ ŽFig. 6b. and XY ŽFig. 6c. can be reduced to periodic trajectories in a plane–parallel cavity. The method of images developed in Section 3 can be applied for each of these 2D M-shaped trajectories and for the projections of the helical 3D trajectories. 3D analysis shows that in fact, the multilobe mirror cavity is equivalent to a large number of cavities of different types which are bound with each other.
6. Summary Our analysis shows that, in contrast to a plane– plane cavity, the threshold pumping energy in the multilobe mirror cavity is almost independent of the angle of misalignment d for d F g . The threshold pumping energy of the multilobe mirror cavity depends on the inclination angle g of a multilobe mirror; its value decreases as g decreases, and consequently, the dimensionless m number increases. The use of the energy balance equations makes it possible to relate the geometry of rays to the loss factor of the cavity and to calculate light energy distribution in the cavity not only for the 2D case but also for the 3D case. By applying the method of images, we reduce the examination of 2D periodic M-shaped trajectories in a multilobe mirror cavity to the simple case of the ray propagation in a plane– parallel cavity. The 3D system of rays in a multilobe mirror cavity contains not only 2D periodic trajectories but also 3D helical trajectories similar to the off-axis Žskew. ray trajectories in spherical mirror Fabry–Perot type interferometers. Like in the plane–parallel cavity, plane waves exist in the multilobe cavity. 3D analysis shows that in fact, the multilobe mirror cavity is equivalent to a large number of cavities of different types which are bound with each other.
Acknowledgements The authors thank N.F. Larionova for help and useful discussions.
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