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Misalignment sensitivity of Nd:YAG unstable resonator with self-filtering aperture
Volume 76, number 5,6
OPTICS COMMUNICATIONS
15 May 1990
MISALIGNMENT SENSITIVITY OF Nd:YAG UNSTABLE RESONATOR WITH S E L F - F I L T E R I N G A P E R T U R E Li H o M I N t and K. V O G L E R Friedrich-Schiller-University, Jena, Department of Physics, Max- Wien-Platz 1, GDR 6900 Jena, GDR Received 20 November 1989
The influence of aperture and mirror misalignment on the output energy and mode profile of an unstable Nd:YAG laser resonator using a self-filteringaperture has been investigated experimentally. The results are discussed within the framework of theoretical descriptions of misalignment sensitivity of spherical resonators. Our positive branch unstable resonator proves to be rather insensitive against misalignment concerning the energy and even the mode profile of the Nd:YAG laser output. Beam steering has been found to be the most pronounced effect.
1. Introduction In most cases unstable resonators ( U R s ) provide the best way to extract diffraction-limited beams efficiently from a large-volume, high-gain laser medium with a fundamental mode well adapted to the active media cross-section. From the view-point of reliability and handing in real practical application the stability of the laser output parameters are o f great importance. Misalignment o f a laser resonator increases the cavity losses and influences both the laser output power (or energy) and the laser mode pattern. C o m m o n l y URs are regarded:to be generally more sensitive to misalignment than stable resonators (SRs). This may result from the traditional somewhat misleading names. On the contrary, however, URs attain a lower misalignment sensitivity because in operation they already involve high resonator losses. Thus some additional extra losses caused by mirror misalignment will not strongly influence the output energy, at least, if the laser is driven well above threshold. The influence of mirror tilting on mode losses and output power has been discussed theoretically from the early days o f laser development [ 1-4], considering even higher aberration effects [ 5 ]. But there are only a few experimental works concerning real ' On leave from Phjongiang Technical University.
measurements o f misalignment sensitivity of U R s of CO2 (e.g. ref. [ 6 ] ) and N d : Y A G lasers (e.g. refs. [7,8]). In discussion o f our results we will refer mainly to these two last mentioned very detailed papers, which are well adapted to be compared with experimental findings. We present here some experimental results o f misalignment sensitivity derived from a new type of positive branch U R with self-filtering (SF) aperture described elsewhere [ 9 ].
2. Brief theoretical compendium Following the considerations of Hauck, Kortz, and Weber [ 7 ] the loss factor V per resonator bounce due to misalignment by a tilting angle a o f a mirror decreases by an approximate square dependence, V(ai) -- Vo[ 1
-
( a i / Oloi) 2 ] ,
( 1)
and therefore the losses o f the resonator increase according to AV(oli) = l -- V(cti). Here V0 denotes the loss factor of the o p t i m u m adjusted resonator. The misalignment sensitivity of the resonator is characterized by ao~ denoting a tilt angle which introduces additional losses of 10% to the cavity. If mirror M~ is tilted by an angle a~ this causes a displacement of the intensity pattern at mirror M~ or mirror M~ itself according to [ 7,8 ]
0030-4018/90/$03.50 © Elsevier Science Publishers B.V. (North-Holland)
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OPTICS C O M M U N I C A T I O N S
Aj,=a,L/(1-glg2),
i#j
(2)
( i , j = 1, 2 referring to mirror M~ or M2), and A,=o~igjL/ (1 - g i g 2 ) , respectively (small angle approximation [10]). Moreover, these simple geometric considerations deliver the so-called beam steering angle 0~, i.e. the rotation of the optical axis of the laser resonator [ 6,7 ],
O, =oe~ ( l - g j ) / ( 1 - g , g2) .
(3)
Here L and gj mean the effective resonator length L = L - l/n and the resonator g parameter, g~= 1 - L~ r~, where/S is the mirror distance, l the length of the laser rod with a refractive index n, r~ the radius of curvature of mirror M,). The quantity
(4)
S~ = O,/o~g
is the angular sensitivity of the mirror M~ [6 ]. These formulas obviously demonstrate the more serious influence of misalignment of the mirror possessing the larger radius of curvature, i.e. M2 in fig. 1. For instance the magnitude of beam steering of mirror M2 in fig. 1 is expected to be 4 times that of M~ for the same tilting angle o~: IS2/S~I=M. Expressing the angular sensitivity of the more sensitive mirror for confocal resonators by the magnificiation M yields
SpB=2M/(M-1),
SNB=2M/(M+I),
(5)
for positive branch (PB) and negative branch (NB) UR. Thus, concerning beam steering, the sensitivity
"l 20 2o
MI
;I
Q
2A
~
P
'
'
Nd :YAG
fit,
M2
Fig. 1. Positive branch unstable resonator with self-filtering aperture ( P B / S F U R ) . Mirrors M~: r = - 0 . 5 m; M2:r2=2.0 m, both with 100% reflectivity. Magnification M = I r2/r~l = 4. Resonator length L = 0 . 6 7 m; SF-aperture 2A= 1.0 mm; distance & = 6 . 5 cm; dielectric polarizing beam splitter P; quarter wave plate 2/4; (Q: Q-switch, not used here).
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15 May 1990
of the positive branch UR (PBUR) proves to be somewhat larger than that of the NBUR, at least at low magnification factors [ 3 ]. On the other hand the NBUR is shown to be more sensitive with respect to variations of resonator length [ 6 ]. Taking into account a limiting aperture within the laser resonator and equal tilting angles a of PBUR and NBUR, this will cause smaller perturbations of the laser output at PBUR, because of its smaller resonator length L (eq. ( 2 ) ) . This has been also verified experimentally [ 11 ]. Considering such a limiting aperture one can define a common misalignment sensitivity D, for the mirror Mi in stable resonators [7],
ztL(g2~ '/2 l + g , ge D2= ~ - k ~ . / ( 1 - g i g 2 ) 3/2 "
(6)
which can be related to O~o,in eq. ( 1 ) by C~o,= 1/D,. Beside C~oione can also use a so-called misalignment parameter Z as defined in ref. [8] to compare tolerable tilt angles,
Z, = Laoi/ a ,
(7 )
where a is the limiting aperture. All these forementioned expressions can be in principle applied to URs too, delivering the general theoretical result that large magnification factors M reduce the misalignment sensitivity for the more sensitive mirror (at constant resonator length L). It is relatively straightforward to calculate from the rised losses (including a, ao) the reduction of laser output energy [7,10] but is rather complicated to deduce from such values the changes in mode pattern. This is done for URs in ref. [8] by solving numerically the Kirchhoff's integral equation for certain tilt angles. The utilization of UR with a self-filtering (SF) aperture in the positive [9] or negative [12] branch ( P B / S F U R and N B / S F U R ) considerably improves the near-field intensity distribution of the laser beam, because of the strong diffraction influence of this SFaperture and simultaneous image relaying [13]. A lower misalignment sensitivity of such resonators could be expected theoretically, but corresponding calculations are not announced to the authors knowledge.
Volume 76, number 5,6
OPTICS COMMUNICATIONS
3. Experimental results
3.1. UR Nd: YAG laser We have investigated a PB/SFUR consisting of the following resonator elements fig. l: a convex mirror M I and a concave mirror ME with mirror radii of curvature of r~ = - 0 . 5 m and r2 = 2.0 m, respectively and a 100% reflectivity. The mirror distance was L = 0 . 6 7 m guaranteeing with the internal laser rod the confocal condition of a collimated telescopic output beam at an UR-magnification of M = 4. The laser radiation was coupled out by a dielectric polarizing beam splitter P, adjusted for optimum output coupling of s-polarized light by rotation of the quarter wave plate 2/4 [ 14]. The SF-condition was realized by insertion of a well dimensioned aperture 2A = 1.0 m m at a certain
15 May 1990
distance 1A =6.5 cm from mirror M~ according to the demands given previously [9 ]. The single flashlamp laser head contains a Nd:YAG rod of a dimension of 6 × 90 m m 2 with antireflection coated front faces (Monocrystaly Turnov, CSSR). The overall resonator losses have been estimated to be A V= 25% and the UR pumping threshold was about Ev(th) = 8 J. Normally the laser was driven well above laser threshold at Ep = 50... 100 J.
3.2. Influence of SF-aperture Fig. 2 shows the influence of a misadjustment of the SF-aperture measured by a beam profile imaging system [ 15 ] across the beam diameter. Fig. 2a gives the intensity distribution of the P B / S F U R output in the near-field which obviously meets the theoreti-
Fig. 2. Near-field beam profile of the PB/SFUR and PBUR as measured with the beam profile diagnostic system [ 17 ]. (a) Well adjusted SF-aperture, 2A= 1.0 ram, Ep=90 J; (b) misadjusted SF-aperture, ~0.5 mm displacement; (c) misadjusted SF-aperture, .~0.12 mm displacement; (d) PBUR without SF-aperture (random intensity distribution ).
359
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OPTICS COMMUNICATIONS
cally expected Airy-function-like intensity distribution. If the aperture is misadjusted, this will result in a clear deterioration of the beam profile (fig. 2b,c), whereas the output energy only little decreases. In fig. 2c the output energy has reduced to about 60% only. These distortions of the mode pattern are very similar to those caused by mirror misalignment (fig. 3), showing clearly the equivalence of aperture displacement and mirror tilting [6]. In fig. 2d the near-field intensity distribution of the equivalent P B U R without SF-aperture is given for comparison showing the typical r a n d o m Fresnel modulation across beam diameter [ 16 ].
15 May 1990
3. 3. Mirror misalignment In fig. 3 the influence of P B / S F U R mirror misalignment on near-field intensity distribution is demonstrated. This was measured by a diode line combined with a plotter. The misalignment is provoked by a definite tilt angle of mirrors M~ or M2 (fig. 3a,b respectively). The mode patterns observed experimentally are very similar to those derived numerically from the solution of Kirchhoff's integral equations in ref. [ 8 ]. The intensity distribution is quite asymmetric as in the case of misadjustment of the aperture but still remains relatively smooth. The mirror M~ with small radius of curvature proved to
I a.u. 1,0
05 !(a)'~7 ,
o
i
i
rB1_
I ~u. 1.0-
0.5 ¸
(b) 2
I
0
I
_ _
2
Fig. 3. Intensity distribution across the beam diameter in the near-field of the PB/SFUR given above. (a) Tilting of mirror M~ in one direction a~; (b) tilting of mirror M2 to both sides -+a2. 0, 1, 2, 3, 4 denote tilting angles of ai=0, 150, 300, 450, and 600 l~rad. The beam radius rB= 1.8 mm. 360
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OPTICS COMMUNICATIONS
be considerable less sensitive to misalignment. Theoretically one would aspect a sensitivity of M~ reduced by a factor of 4 (eq. ( 3 ) ) as compared with mirror M2, while we found only a factor of about 3 (oq =450 prad can be related to a 2 = 150 Ixrad). As compared with a SR running in the TEMoo mode the tolerable misadjustment of M~ is obviously larger (ct(SR) < 50 /~rad whereas ct2 ( P B / S F U R ) ~ 150 prad). Better misalignment sensitivity of U R in contrast to SR has also been observed in resonators using tapered mirrors [ 17 ]. Fig. 4 shows the far-field beam steering and amount of output energy caused by tilting one of the mirrors M~ o r M 2 in a distance of 6 m. The output energy slopes down only to 50% for considerable large tilt angles of mirror misalignment as or=800 prad for M2. The misalignment sensitivity of M~ is still lower. Thus, as seen from fig. 4a, the influence of mirror misalignment on the output energy is small, espe-
15 May 1990
cially with respect to M~, of course the large beam steering evoked simultaneously (fig. 4b) may be still very unconvenient for most applications. For a 10% decrease in output energy we have estimated a tilt angle ao~ ~ 270 prad and ao2 ~ 150 prad of mirror M~ and M 2 , respectively. This corresponds with an experimental misalignment sensitivity of D1=3.70×103 rad -~ for M~ and D2=6.67×103 rad-1 for M2, i.e. misadjustment of mirror M~ is less critical for reduction of output power by a factor of 1.80 as compared with mirror M E. Theoretically one would expect a ratio of D 2 / D I = ( g l / g 2 ) 1/2 = M l / 2 = 2 ,
as derived from eq. (6). The tolerable tilt angles in P B / S F U R are considerable larger than obtained for large volume TEMoo stable resonator [7] where experimental values of ao = 20 p.rad had been determined. Compared with
EL
[ao.] 10
05"
®
(a) -8oo -oo - oo -ioo
2oo
6'oo 8oo
As
2 1 0 -1 -2
(b) - oo -6 o
- 'oo -2'00
G
2'oo
goo
8oo ~roO]
Fig. 4. Decrease of output energy (a) and displacement of the beam (b) in the far-field of the PB/SFUR Versus mirror misalignment (tilt angle or). l: Tilting of mirror M~; 2: tilting of mirror M2. (a) Output energy; (b) beam displacement (beam steering), far-field distance about 6 m.
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e x p e r i m e n t a l results o f a n o r m a l U R the m i s a l i g n m e n t p a r a m e t e r Z (eq. ( 7 ) ), in o u r r e s o n a t o r is also larger: Z t ~ 4 0 0 m r a d a n d Z 2 ~ 2 2 5 m r a d (as c o m pared with Z = 150 m r a d in ref. [8] ), s h o w i n g the s t a b i l i z a t i o n o f laser o u t p u t by the SF-aperture.
4. Conclusion It has b e e n shown experimentally that o u r n e w P B / S F U R has a r a t h e r low sensitivity to m i r r o r misal i g n m e n t at least c o n c e r n i n g the o u t p u t energy. T h e changes in the near-field m o d e p a t t e r n caused by m i s a d j u s t m e n t o f the S F - a p e r t u r e or r e s o n a t o r mirrors a p p e a r to be m o r e serious. T h e i n t e n s i t y distrib u t i o n r e m a i n s h o w e v e r relatively s m o o t h . Beam steering with only little energy r e d u c t i o n has b e e n f o u n d to be the m o s t p r o n o u n c e d effect o f m i r r o r m i s a l i g n m e n t in the far-field. T h e d e m a n d s for mec h a n i c a l stability are m o r e s t r i n g e n t for the m i r r o r with larger r a d i u s o f c u r v a t u r e . T h e e x p e r i m e n t a l results can be well u n d e r s t o o d w i t h i n the f r a m e w o r k o f existing theoretical c o n s i d e r a t i o n from literature. Investigations o f the influence o f t h e r m a l lensing [ 4 ] o n P B / S F U R o u t p u t , o p e r a t i n g at high r e p e t i t i o n rates are in progress.
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15 May 1990
References [ 1 ] R.L. Sanderson and W. Streifer, Appl. Optics 8 (1969) 2241. [2] C. Santana and L.B. Felsen, Appl. Optics 17 (1978) 2352. [ 3 ] A.N. Chester, IEEE J. Quantum Electron. QE-9 ( 1973 ) 209. [4] J. Steffen, J.P. L6rtsch+r and G. Herziger, IEEE J. Quantum Electron. QE-8 (1972) 239. [5] K.E. Oughstun, J. Opt. Soc. Am. A 3 (1986) 1114, 1585. [6] W.F. Krupke and W.R. Sooy, IEEE J. Quantum Electron. QE-5 (1969) 575. [7] R. Hauck, H.D. Kortz and H. Weber, Appl. Optics 19 (1980) 598. [8] R. Hauck, N. Hodgson and H. Weber, J. Mod. Optics 35 (1988) 165. [9] Li Ho Min and K. Vogler, Optics Comm. 74 (1989) 79. [ 10] W. Koechner, Solid state laser engineering (Springer, Berlin, 1976) p. 190. [ 11 ] K. Vogler, Progress in the field of unstable laser resonators for pulsed Nd: YAG lasers, Laser Jahrbuch 2 (Vulkan, Essen, 1990) to be published. [ 12 ] P.G. Gobbi, S. Morosi, G.C. Reali and A.S. Zarkasi, Appl. Optics 24 ( 1985 ) 26. [ 13 ] A.H. Paxton and T.C. Salvi, Optics Comm. 26 ( 1978 ) 305. [14] Li Ho Min and K. Vogler, submitted to J. Mod. Optics (1989). [15] C. Resag and V. Schellenberger, 3. WGB Conference (Mitweida, GDR, 1988), data sheet of the laser beam analysis system of the TH Ilmenau, GDR. [ 16 ] D.B. Rensch and A.N. Chester, Appl. Optics 12 ( 1973 ) 997. [17] S. De Silvestri, P. Laporta, V. Magni, O. Svelto and B. Majocchi, Optics Lett. 13 (1988) 201.
OPTICS COMMUNICATIONS
15 May 1990
MISALIGNMENT SENSITIVITY OF Nd:YAG UNSTABLE RESONATOR WITH S E L F - F I L T E R I N G A P E R T U R E Li H o M I N t and K. V O G L E R Friedrich-Schiller-University, Jena, Department of Physics, Max- Wien-Platz 1, GDR 6900 Jena, GDR Received 20 November 1989
The influence of aperture and mirror misalignment on the output energy and mode profile of an unstable Nd:YAG laser resonator using a self-filteringaperture has been investigated experimentally. The results are discussed within the framework of theoretical descriptions of misalignment sensitivity of spherical resonators. Our positive branch unstable resonator proves to be rather insensitive against misalignment concerning the energy and even the mode profile of the Nd:YAG laser output. Beam steering has been found to be the most pronounced effect.
1. Introduction In most cases unstable resonators ( U R s ) provide the best way to extract diffraction-limited beams efficiently from a large-volume, high-gain laser medium with a fundamental mode well adapted to the active media cross-section. From the view-point of reliability and handing in real practical application the stability of the laser output parameters are o f great importance. Misalignment o f a laser resonator increases the cavity losses and influences both the laser output power (or energy) and the laser mode pattern. C o m m o n l y URs are regarded:to be generally more sensitive to misalignment than stable resonators (SRs). This may result from the traditional somewhat misleading names. On the contrary, however, URs attain a lower misalignment sensitivity because in operation they already involve high resonator losses. Thus some additional extra losses caused by mirror misalignment will not strongly influence the output energy, at least, if the laser is driven well above threshold. The influence of mirror tilting on mode losses and output power has been discussed theoretically from the early days o f laser development [ 1-4], considering even higher aberration effects [ 5 ]. But there are only a few experimental works concerning real ' On leave from Phjongiang Technical University.
measurements o f misalignment sensitivity of U R s of CO2 (e.g. ref. [ 6 ] ) and N d : Y A G lasers (e.g. refs. [7,8]). In discussion o f our results we will refer mainly to these two last mentioned very detailed papers, which are well adapted to be compared with experimental findings. We present here some experimental results o f misalignment sensitivity derived from a new type of positive branch U R with self-filtering (SF) aperture described elsewhere [ 9 ].
2. Brief theoretical compendium Following the considerations of Hauck, Kortz, and Weber [ 7 ] the loss factor V per resonator bounce due to misalignment by a tilting angle a o f a mirror decreases by an approximate square dependence, V(ai) -- Vo[ 1
-
( a i / Oloi) 2 ] ,
( 1)
and therefore the losses o f the resonator increase according to AV(oli) = l -- V(cti). Here V0 denotes the loss factor of the o p t i m u m adjusted resonator. The misalignment sensitivity of the resonator is characterized by ao~ denoting a tilt angle which introduces additional losses of 10% to the cavity. If mirror M~ is tilted by an angle a~ this causes a displacement of the intensity pattern at mirror M~ or mirror M~ itself according to [ 7,8 ]
0030-4018/90/$03.50 © Elsevier Science Publishers B.V. (North-Holland)
357
Volume 76, number 5,6
OPTICS C O M M U N I C A T I O N S
Aj,=a,L/(1-glg2),
i#j
(2)
( i , j = 1, 2 referring to mirror M~ or M2), and A,=o~igjL/ (1 - g i g 2 ) , respectively (small angle approximation [10]). Moreover, these simple geometric considerations deliver the so-called beam steering angle 0~, i.e. the rotation of the optical axis of the laser resonator [ 6,7 ],
O, =oe~ ( l - g j ) / ( 1 - g , g2) .
(3)
Here L and gj mean the effective resonator length L = L - l/n and the resonator g parameter, g~= 1 - L~ r~, where/S is the mirror distance, l the length of the laser rod with a refractive index n, r~ the radius of curvature of mirror M,). The quantity
(4)
S~ = O,/o~g
is the angular sensitivity of the mirror M~ [6 ]. These formulas obviously demonstrate the more serious influence of misalignment of the mirror possessing the larger radius of curvature, i.e. M2 in fig. 1. For instance the magnitude of beam steering of mirror M2 in fig. 1 is expected to be 4 times that of M~ for the same tilting angle o~: IS2/S~I=M. Expressing the angular sensitivity of the more sensitive mirror for confocal resonators by the magnificiation M yields
SpB=2M/(M-1),
SNB=2M/(M+I),
(5)
for positive branch (PB) and negative branch (NB) UR. Thus, concerning beam steering, the sensitivity
"l 20 2o
MI
;I
Q
2A
~
P
'
'
Nd :YAG
fit,
M2
Fig. 1. Positive branch unstable resonator with self-filtering aperture ( P B / S F U R ) . Mirrors M~: r = - 0 . 5 m; M2:r2=2.0 m, both with 100% reflectivity. Magnification M = I r2/r~l = 4. Resonator length L = 0 . 6 7 m; SF-aperture 2A= 1.0 mm; distance & = 6 . 5 cm; dielectric polarizing beam splitter P; quarter wave plate 2/4; (Q: Q-switch, not used here).
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15 May 1990
of the positive branch UR (PBUR) proves to be somewhat larger than that of the NBUR, at least at low magnification factors [ 3 ]. On the other hand the NBUR is shown to be more sensitive with respect to variations of resonator length [ 6 ]. Taking into account a limiting aperture within the laser resonator and equal tilting angles a of PBUR and NBUR, this will cause smaller perturbations of the laser output at PBUR, because of its smaller resonator length L (eq. ( 2 ) ) . This has been also verified experimentally [ 11 ]. Considering such a limiting aperture one can define a common misalignment sensitivity D, for the mirror Mi in stable resonators [7],
ztL(g2~ '/2 l + g , ge D2= ~ - k ~ . / ( 1 - g i g 2 ) 3/2 "
(6)
which can be related to O~o,in eq. ( 1 ) by C~o,= 1/D,. Beside C~oione can also use a so-called misalignment parameter Z as defined in ref. [8] to compare tolerable tilt angles,
Z, = Laoi/ a ,
(7 )
where a is the limiting aperture. All these forementioned expressions can be in principle applied to URs too, delivering the general theoretical result that large magnification factors M reduce the misalignment sensitivity for the more sensitive mirror (at constant resonator length L). It is relatively straightforward to calculate from the rised losses (including a, ao) the reduction of laser output energy [7,10] but is rather complicated to deduce from such values the changes in mode pattern. This is done for URs in ref. [8] by solving numerically the Kirchhoff's integral equation for certain tilt angles. The utilization of UR with a self-filtering (SF) aperture in the positive [9] or negative [12] branch ( P B / S F U R and N B / S F U R ) considerably improves the near-field intensity distribution of the laser beam, because of the strong diffraction influence of this SFaperture and simultaneous image relaying [13]. A lower misalignment sensitivity of such resonators could be expected theoretically, but corresponding calculations are not announced to the authors knowledge.
Volume 76, number 5,6
OPTICS COMMUNICATIONS
3. Experimental results
3.1. UR Nd: YAG laser We have investigated a PB/SFUR consisting of the following resonator elements fig. l: a convex mirror M I and a concave mirror ME with mirror radii of curvature of r~ = - 0 . 5 m and r2 = 2.0 m, respectively and a 100% reflectivity. The mirror distance was L = 0 . 6 7 m guaranteeing with the internal laser rod the confocal condition of a collimated telescopic output beam at an UR-magnification of M = 4. The laser radiation was coupled out by a dielectric polarizing beam splitter P, adjusted for optimum output coupling of s-polarized light by rotation of the quarter wave plate 2/4 [ 14]. The SF-condition was realized by insertion of a well dimensioned aperture 2A = 1.0 m m at a certain
15 May 1990
distance 1A =6.5 cm from mirror M~ according to the demands given previously [9 ]. The single flashlamp laser head contains a Nd:YAG rod of a dimension of 6 × 90 m m 2 with antireflection coated front faces (Monocrystaly Turnov, CSSR). The overall resonator losses have been estimated to be A V= 25% and the UR pumping threshold was about Ev(th) = 8 J. Normally the laser was driven well above laser threshold at Ep = 50... 100 J.
3.2. Influence of SF-aperture Fig. 2 shows the influence of a misadjustment of the SF-aperture measured by a beam profile imaging system [ 15 ] across the beam diameter. Fig. 2a gives the intensity distribution of the P B / S F U R output in the near-field which obviously meets the theoreti-
Fig. 2. Near-field beam profile of the PB/SFUR and PBUR as measured with the beam profile diagnostic system [ 17 ]. (a) Well adjusted SF-aperture, 2A= 1.0 ram, Ep=90 J; (b) misadjusted SF-aperture, ~0.5 mm displacement; (c) misadjusted SF-aperture, .~0.12 mm displacement; (d) PBUR without SF-aperture (random intensity distribution ).
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OPTICS COMMUNICATIONS
cally expected Airy-function-like intensity distribution. If the aperture is misadjusted, this will result in a clear deterioration of the beam profile (fig. 2b,c), whereas the output energy only little decreases. In fig. 2c the output energy has reduced to about 60% only. These distortions of the mode pattern are very similar to those caused by mirror misalignment (fig. 3), showing clearly the equivalence of aperture displacement and mirror tilting [6]. In fig. 2d the near-field intensity distribution of the equivalent P B U R without SF-aperture is given for comparison showing the typical r a n d o m Fresnel modulation across beam diameter [ 16 ].
15 May 1990
3. 3. Mirror misalignment In fig. 3 the influence of P B / S F U R mirror misalignment on near-field intensity distribution is demonstrated. This was measured by a diode line combined with a plotter. The misalignment is provoked by a definite tilt angle of mirrors M~ or M2 (fig. 3a,b respectively). The mode patterns observed experimentally are very similar to those derived numerically from the solution of Kirchhoff's integral equations in ref. [ 8 ]. The intensity distribution is quite asymmetric as in the case of misadjustment of the aperture but still remains relatively smooth. The mirror M~ with small radius of curvature proved to
I a.u. 1,0
05 !(a)'~7 ,
o
i
i
rB1_
I ~u. 1.0-
0.5 ¸
(b) 2
I
0
I
_ _
2
Fig. 3. Intensity distribution across the beam diameter in the near-field of the PB/SFUR given above. (a) Tilting of mirror M~ in one direction a~; (b) tilting of mirror M2 to both sides -+a2. 0, 1, 2, 3, 4 denote tilting angles of ai=0, 150, 300, 450, and 600 l~rad. The beam radius rB= 1.8 mm. 360
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OPTICS COMMUNICATIONS
be considerable less sensitive to misalignment. Theoretically one would aspect a sensitivity of M~ reduced by a factor of 4 (eq. ( 3 ) ) as compared with mirror M2, while we found only a factor of about 3 (oq =450 prad can be related to a 2 = 150 Ixrad). As compared with a SR running in the TEMoo mode the tolerable misadjustment of M~ is obviously larger (ct(SR) < 50 /~rad whereas ct2 ( P B / S F U R ) ~ 150 prad). Better misalignment sensitivity of U R in contrast to SR has also been observed in resonators using tapered mirrors [ 17 ]. Fig. 4 shows the far-field beam steering and amount of output energy caused by tilting one of the mirrors M~ o r M 2 in a distance of 6 m. The output energy slopes down only to 50% for considerable large tilt angles of mirror misalignment as or=800 prad for M2. The misalignment sensitivity of M~ is still lower. Thus, as seen from fig. 4a, the influence of mirror misalignment on the output energy is small, espe-
15 May 1990
cially with respect to M~, of course the large beam steering evoked simultaneously (fig. 4b) may be still very unconvenient for most applications. For a 10% decrease in output energy we have estimated a tilt angle ao~ ~ 270 prad and ao2 ~ 150 prad of mirror M~ and M 2 , respectively. This corresponds with an experimental misalignment sensitivity of D1=3.70×103 rad -~ for M~ and D2=6.67×103 rad-1 for M2, i.e. misadjustment of mirror M~ is less critical for reduction of output power by a factor of 1.80 as compared with mirror M E. Theoretically one would expect a ratio of D 2 / D I = ( g l / g 2 ) 1/2 = M l / 2 = 2 ,
as derived from eq. (6). The tolerable tilt angles in P B / S F U R are considerable larger than obtained for large volume TEMoo stable resonator [7] where experimental values of ao = 20 p.rad had been determined. Compared with
EL
[ao.] 10
05"
®
(a) -8oo -oo - oo -ioo
2oo
6'oo 8oo
As
2 1 0 -1 -2
(b) - oo -6 o
- 'oo -2'00
G
2'oo
goo
8oo ~roO]
Fig. 4. Decrease of output energy (a) and displacement of the beam (b) in the far-field of the PB/SFUR Versus mirror misalignment (tilt angle or). l: Tilting of mirror M~; 2: tilting of mirror M2. (a) Output energy; (b) beam displacement (beam steering), far-field distance about 6 m.
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e x p e r i m e n t a l results o f a n o r m a l U R the m i s a l i g n m e n t p a r a m e t e r Z (eq. ( 7 ) ), in o u r r e s o n a t o r is also larger: Z t ~ 4 0 0 m r a d a n d Z 2 ~ 2 2 5 m r a d (as c o m pared with Z = 150 m r a d in ref. [8] ), s h o w i n g the s t a b i l i z a t i o n o f laser o u t p u t by the SF-aperture.
4. Conclusion It has b e e n shown experimentally that o u r n e w P B / S F U R has a r a t h e r low sensitivity to m i r r o r misal i g n m e n t at least c o n c e r n i n g the o u t p u t energy. T h e changes in the near-field m o d e p a t t e r n caused by m i s a d j u s t m e n t o f the S F - a p e r t u r e or r e s o n a t o r mirrors a p p e a r to be m o r e serious. T h e i n t e n s i t y distrib u t i o n r e m a i n s h o w e v e r relatively s m o o t h . Beam steering with only little energy r e d u c t i o n has b e e n f o u n d to be the m o s t p r o n o u n c e d effect o f m i r r o r m i s a l i g n m e n t in the far-field. T h e d e m a n d s for mec h a n i c a l stability are m o r e s t r i n g e n t for the m i r r o r with larger r a d i u s o f c u r v a t u r e . T h e e x p e r i m e n t a l results can be well u n d e r s t o o d w i t h i n the f r a m e w o r k o f existing theoretical c o n s i d e r a t i o n from literature. Investigations o f the influence o f t h e r m a l lensing [ 4 ] o n P B / S F U R o u t p u t , o p e r a t i n g at high r e p e t i t i o n rates are in progress.
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References [ 1 ] R.L. Sanderson and W. Streifer, Appl. Optics 8 (1969) 2241. [2] C. Santana and L.B. Felsen, Appl. Optics 17 (1978) 2352. [ 3 ] A.N. Chester, IEEE J. Quantum Electron. QE-9 ( 1973 ) 209. [4] J. Steffen, J.P. L6rtsch+r and G. Herziger, IEEE J. Quantum Electron. QE-8 (1972) 239. [5] K.E. Oughstun, J. Opt. Soc. Am. A 3 (1986) 1114, 1585. [6] W.F. Krupke and W.R. Sooy, IEEE J. Quantum Electron. QE-5 (1969) 575. [7] R. Hauck, H.D. Kortz and H. Weber, Appl. Optics 19 (1980) 598. [8] R. Hauck, N. Hodgson and H. Weber, J. Mod. Optics 35 (1988) 165. [9] Li Ho Min and K. Vogler, Optics Comm. 74 (1989) 79. [ 10] W. Koechner, Solid state laser engineering (Springer, Berlin, 1976) p. 190. [ 11 ] K. Vogler, Progress in the field of unstable laser resonators for pulsed Nd: YAG lasers, Laser Jahrbuch 2 (Vulkan, Essen, 1990) to be published. [ 12 ] P.G. Gobbi, S. Morosi, G.C. Reali and A.S. Zarkasi, Appl. Optics 24 ( 1985 ) 26. [ 13 ] A.H. Paxton and T.C. Salvi, Optics Comm. 26 ( 1978 ) 305. [14] Li Ho Min and K. Vogler, submitted to J. Mod. Optics (1989). [15] C. Resag and V. Schellenberger, 3. WGB Conference (Mitweida, GDR, 1988), data sheet of the laser beam analysis system of the TH Ilmenau, GDR. [ 16 ] D.B. Rensch and A.N. Chester, Appl. Optics 12 ( 1973 ) 997. [17] S. De Silvestri, P. Laporta, V. Magni, O. Svelto and B. Majocchi, Optics Lett. 13 (1988) 201.