- Email: [email protected]
Generation of spiral-type laser beams
I5 August
1997
OPTICS COMMUNICATIONS ELSEVIER
Optics Communications
141 (1997) 59-64
Generation of spiral-type laser beams E. Abramochkin, Lebeder Physics Institute, Russian Academy Received 27 November
N. Losevsky, V. Volostnikov of Sciences. Nom-Sadoxzp
Street 221, Sanxzra 443011. Russia
1996; revised IO March 1997; accepted 22 April 1997
Abstract It has been shown theoretically that spiral-type beams are modes of a ring resonator with a beam rotator. The generation of spiral-type beams by this technique is demonstrated experimentally. 0 1997 Published by Elsevier Science B.V.
1. Introduction
It is known [ 11, that light beams in lasers with stable resonators are well described by Hermite-Gauss and Laguerre-Gauss modes. One of the main characteristics of these beams is the conservation of the intensity distribution under propagation and focusing if its scale is neglected. This property is usually called structural stability. In Ref. [2] the existence problem for a more general family of light beams that keep their intensity structure under propagation and focusing if scale and rotation are neglected has been investigated by theoretical and experimental means. Light fields of this kind were found, described completely and named spiral-type beams. These beams are a topic of considerable interest in various fields of optics (see for example Ref. [3]). Spiral-type beams provide the possibility of constructing structurally stable beams with a variety of spatial intensity distributions. In Ref. [4] beams with intensity distribution shaped as an arbitrary planar curve were found theoretically and a sample beam for equilateral triangle boundary was realized in an experiment. However, usually spiral-type beams are obtained through transformation of an ordinary laser beam by means of various amplitude-phase actions (see also Ref. [5]). The purpose of this work is to consider the possibility of generation of spiral-type beams directly inside a laser resonator. Experimental results of beam generation are presented.
2. Spiral-type beams in a laser resonator
In Ref. [2] it was found that in general an arbitrary spiral-type beam with the rotation parameter terms of Laguerre-Gauss modes Y,,,,(r,(p) as follows:
B0 may be described
in
where k is the wave number. l/Q = l/R - Zi/Xu * is a complex-valued parameter of curvature, Lrl(r) are Laguerre polynomials. and N(0,) is a set of integer pairs (n,m) such that 2n + Irnl + H,tn + 1 = y0 = const. Using the well-known 0030.4018/97/$17.00 0 1997 Published PII SOO30-4018(97)00215-O
by Elsevier Science B.V. All rights reserved
60
E. Abramochkin
et al. / Optics Communications
I41
f I9971 59-64
transformation of Lagueme-Gauss beams by an optical system with the ABCD matrix (see for example Ref. [6]) we obtain the same transformation of the field (1)
(2) where L, is the optical length along the system axis, W: = ~“1 A + B/Ql’. @ = arg(A + B/Q), Q, = ( AQ + B)/(CQ + D) and AD - BC = 1. Let now ABCD be a round-trip matrix of a laser with the stable resonator. The self-consistency condition for each term of the field (1) under one complete resonator transit Q, = Q gives (Ref. [6]) WI=&‘,
B
kd -= 2
,
Q, = arccos
/I - +( A + D)’
A-I-D 2 ( 1
Then, using the identity 2n + Irnl + 1 = y0 - @,m, we obtain from Eq. (2) that the evolution of spiral-type follows:
beam (1) is as
(3) Here f., is the optical length of one complete resonator transit. From Eq. (3) it is seen that the beam under one complete resonator transit acquires the additional phase (P,,,= XL, - y,,arccos(i( A + D)) and rotates through the angle ON= 0,arccos(f(A + D)). Thus, it is seen from the self-consistency condition that the beam must be rotated through the angle of either - 8, or 27r - 0,,,. It is known, that the beam rotation can be made in a ring resonator by introducing the Dove prism (so called “resonator with a field rotation” [7]). Resonators of this kind were used in order to improve the homogeneity of the transverse distribution of laser irradiance and the angle of the beam rotation (usually 90” or 180”) was not connected with the resonator parameter A + D. If in the resonator a beam rotation through an angle of either - 0, or 2~- 0, is realized and kLL, - y,arccos
A+D 2 i
=2%-q !
then the self-consistency condition with the generation frequency
F,(r,cp) = F(r,p)
takes place and the field (1) will be an eigenmode
of this resonator
(4) Here q is an axial index, c is the speed of light. Let us discuss Eq. (4). It is seen that. in contrast with an ordinary resonator without a beam rotator, Lagueme-Gauss modes _E”,.,( Y,cp) are frequency degenerated when 2 n + 1ml + 0,m = const and the mode composition depends on the rotation which we introduce into the resonator. It is easy to understand, that if one takes into account that the rotation of the distribution of the complex amplitude of Laguerre-Gauss beam P,!“,.,( Y,cp) through an angle 0 is equal to acquiring an additional phase me: Y&(r.cp + 0) =~~,,(r,p)e’““. Thus, spiral-type beams are modes of specific ring resonators containing a beam rotator.
3. Experimental
setup
The experimental setup is presented in Fig. 1. An argon ion ring laser operating at 488 nm contains a plane mirror M, (reflectivity 0.94) and spherical mirrors M?, M, (R, = R, = R = 3 m, reflectivity 0.995, 0.98). The distances between
Fig. 1. Experimental setup
E. Abramochkin et al. / Optics Communications 141 I1997159-64
61
mirrors are M, MZ = M, M, = I = 1.27 m, M,M, = I, = 2.4 m. The resonator scheme shaped as an obtuse-angled triangle has been selected in order to reduce the influence of astigmatism of the mirrors M, and M,. The beam rotation is performed by Dove prism P (under rotation of the prism through angle (Y the beam rotates through angle 2a). It should be noted that the prism P rotates the complex amplitude of a beam distribution, but the influence of the prism on the beam polarization is weak. The prism action is that the beam polarization becomes slightly elliptical. Intensity of the beam component that is normal to the resonator plane depends on the angle of rotation of the prism P and is O-5% of the intensity of the beam component that lies in the resonator plane. Elliptical polarization of the beam takes place in the resonator section P - M, - MZ - until active element. On the rest of the resonator the beam polarization is determined by orientation of Brewster windows of the active element and lies in the resonator plane. The beam generated by the laser has been observed and registered after partially passing mirror M, with the help of objective 0 and microscope M in the plane of screen S. The matrix of complete round-trip of the resonator starting at M, is as follows: I
41 l-
(
R
A c
21” x+R’
4 -:+R?
41’
411” 2’+‘o-
4’”
--
-E-
l-;-
\
41’1, \
411, R
210 y+R”
+-
R’
=
4 11,
-0.939 -0.267
/
After complete round-trip the phase shift for the lowest-order mode in an empty resonator is @ = arccos( - 0.939) - 160” = 0.89~, and therefore, the longitudinal- and transverse-mode frequency separation at zero angle of the prism rotation are selection has been realized by inserting a thin wire W into the _ 60 MHz and - 27 MHz, respectively. Transverse-mode beam zone (the wire diameter is 15 mkm).
4. Experiment The angle of rotation of the prism P for self-consistency 00
cy = - -
2
arccos i
AfD 2
of a spiral-type beam with the rotation parameter
0, is
= - B, x 80”. i
During the experiments it was found that spiral-type beams with various rotation parameters are obtained for some different values of the angle of the prism rotation, in particular, (Y= - 13~X 78”. The deviation has been typical for all our experiments and, as we see it, is the result of the presence of the active medium. 4.1. Spiral-type beam for the prism rotation angle (Y = - 26” The experiment results are presented in Fig. 2. (At the screen spiral-type beams with positive 0, look like clockwise rotated ones.> Numerical simulation for the field F(r,cp) =_Yo,_2(r,(p) + 2_5& ,(r,cp> that corresponds to the rotation parameter B. = { is presented in Fig. 3. The beam phase contains 4 dislocations df the wave front, the signs of the central dislocation and peripheral ones are opposite.
a
b
c
Fig. 2. Experimentally registered intensities of the spiral-type beam before (a), at (b), and after (c) the waist plane. The prism rotation angle is LY= -26”.
62
Fig. 3. Intensity (a). intensity level contours (b), and phase Cc) of the spiral-type
4.2. Spiral-type
beam with the rotation parameter
H, = {
beam for the prism rotation angle (Y = - 15.5”
The experimental results are that corresponds to the rotation of the wave front: the sign of broken) is opposite to the signs
presented in Fig. 4. Numerical simulation for the field F( r. cp) =_2’& &r. cp) + 2 LL?~,~( r, cp) parameter O,, = f is presented in Fig. 5. In this case the beam phase contains 7 dislocations the second order dislocation in the center (in the experiment this degeneration is slightly of the other 5 dislocations.
4.3. Spirul-type beam for the prism rotation angle cy = - 12” The experiment results are presented in Fig. 6. Numerical simulation for the field F(r,ip) =9’&r,q) + 2Y0,z(r,~P) is presented in Fig. 7. This field corresponds to the rotation parameter O,, = - 1 and contains 2 dislocations of the wave front with the same sign. In this case it was found that under the prism rotation through the angle (Y= - 12” the beam changes the rotation direction and, therefore, the excitable spiral-type beam after complete round-trip of the resonator is rotated through
Fig. 4. Experimentally is a = - 15.5”.
registered
intensities of the spiral-type
beam before (a), at (b), and after (c) the waist plane. The prism rotation angle
Fig. 5. Intensity (a), intensity level contours (b). and phase (c) of the spiral-type
beam with the rotation parameter
0, = f.
63
a
b
C
Fig. 6. Experimentally registered intensities of the spiral-type beam before (a), at (b), and after Cc)the waist plane. The prism rotation angle is cy= - 12”.
a
b
Fig. 7. Intensity (a), intensity level contours (b), and phase Cc) of the spiral-type
c
beam with the rotation parameter
0” = -
1.
the angle ON= - 156”, and the prism gives an additional rotation for it in the same direction until - 180”. As a result. the beam is symmetrical to the 180” rotation. 4.4. Spiral-type beam for the prism rotation angle CY= - 6” This case is similar to the previous one, but with the rotation parameter B0 = - i: the excitable spiral-type beam after complete round-trip of the resonator rotates through the angle ON= 8, X 156” = -78”, and the prism gives an additional rotation for it in the same direction until -90”. As a result, the beam is symmetrical to the 90” rotation. The experiment results are presented in Fig. 8. Numerical simulation for the field F(r,cp) =~‘&(Y,(P) + 5P0,_ ,(r.cp) is presented in Fig. 9. This field contains 5 dislocations of the wave front: the central dislocation sign is opposite to the signs of the other 4 dislocations. The possibility of formation of various spiral-type beams in this experiment is restricted by reflective and diffractive losses. Firstly, when the angle of the prism rotation is large enough (for example, (Y= -78” for a spiral-type beam with 0, = l), the angle of incidence of the beam on the Dove prism is far from the Brewster angle and the losses in the resonator
Fig. 8. Experimentally registered intensities of the spiral-type beam before (a). at (b), and after (c) the waist plane. The prism rotation angle is cy = - 6”.
64
E. Abramochkirl et al. / Optics Communicatioru
141 (1997) 59-64
Fig. 9. Intensity (a), intensity level contours (b), and phase (c) of the spiral-type beam with the rotation parameter 00 = - f.
become excessive. Secondly, for example, the construction of the beam with 0, = $ requires the presence of Laguerre-Gauss modes -Z,.,, and X2,,,> for those lm, - m,l is 8 at least, and diffractive losses in the resonator are excessively large for these modes. From Eq. (3) it can be seen that the angle of the beam rotation under complete round-trip in the resonator may be decreased by the change of its configuration, i.e. the parameter A + D. However, in the experiment we have no possibility to do this because of scheme construction restrictions. It is interesting to note that each realized beam prefers to conserve its structure when the prism slightly rotates near cy= - BOX 156”. Under the prism rotation the change of the beam looks like if the beam slightly rotates around its center of symmetry. Moreover, the beams with negative 8, value rotate in the direction of the prism rotation and the beams with positive 0, value rotate in the opposite direction. In our view, this property requires more detailed investigation and is the result of nonlinear effects within the active medium.
5. Conclusions It was shown that spiral-type beams are modes of a specific ring laser containing a beam rotator. For some values of the rotation parameter 0,, inside-resonator generation of spiral-type beams has been obtained experimentally and is in good accordance with the numerical simulation. In our opinion, these results show the possibility of construction of spiral-type beams with given spatial characteristics in a laser with appropriate resonator and amplification coefficients of the active medium.
Acknowledgements This work is partially supported by the Russian Fund for Fundamental
Research (grant 96-02-16430).
References [I] H. Kogelnik, T. Li, Proc. IEEE 54 (1966) 1312. [2] [3] [4] [5] [6] [7]
E. Abramochkin, V. Volostnikov. Optics Comm. 102 (1993) 336. V. Tikhonenko, J. Christou, B. Luther-Daves, J. Opt. Sot. Am. B 11 (1995) 2046. E. Abramochkin. V. Volostnikov, Optics Comm. 125 (1996) 302. E. Abramochkin. V. Volostnikov, Optics Comm. 83 (1991) 123. A. Gerard, J.M. Burch, Introduction to matrix methods in optics (Wiley, New York, 1975). Yu.A. Anan’ev, Optical resonators and laser beams (Nauka, Moscow, 1989).
1997
OPTICS COMMUNICATIONS ELSEVIER
Optics Communications
141 (1997) 59-64
Generation of spiral-type laser beams E. Abramochkin, Lebeder Physics Institute, Russian Academy Received 27 November
N. Losevsky, V. Volostnikov of Sciences. Nom-Sadoxzp
Street 221, Sanxzra 443011. Russia
1996; revised IO March 1997; accepted 22 April 1997
Abstract It has been shown theoretically that spiral-type beams are modes of a ring resonator with a beam rotator. The generation of spiral-type beams by this technique is demonstrated experimentally. 0 1997 Published by Elsevier Science B.V.
1. Introduction
It is known [ 11, that light beams in lasers with stable resonators are well described by Hermite-Gauss and Laguerre-Gauss modes. One of the main characteristics of these beams is the conservation of the intensity distribution under propagation and focusing if its scale is neglected. This property is usually called structural stability. In Ref. [2] the existence problem for a more general family of light beams that keep their intensity structure under propagation and focusing if scale and rotation are neglected has been investigated by theoretical and experimental means. Light fields of this kind were found, described completely and named spiral-type beams. These beams are a topic of considerable interest in various fields of optics (see for example Ref. [3]). Spiral-type beams provide the possibility of constructing structurally stable beams with a variety of spatial intensity distributions. In Ref. [4] beams with intensity distribution shaped as an arbitrary planar curve were found theoretically and a sample beam for equilateral triangle boundary was realized in an experiment. However, usually spiral-type beams are obtained through transformation of an ordinary laser beam by means of various amplitude-phase actions (see also Ref. [5]). The purpose of this work is to consider the possibility of generation of spiral-type beams directly inside a laser resonator. Experimental results of beam generation are presented.
2. Spiral-type beams in a laser resonator
In Ref. [2] it was found that in general an arbitrary spiral-type beam with the rotation parameter terms of Laguerre-Gauss modes Y,,,,(r,(p) as follows:
B0 may be described
in
where k is the wave number. l/Q = l/R - Zi/Xu * is a complex-valued parameter of curvature, Lrl(r) are Laguerre polynomials. and N(0,) is a set of integer pairs (n,m) such that 2n + Irnl + H,tn + 1 = y0 = const. Using the well-known 0030.4018/97/$17.00 0 1997 Published PII SOO30-4018(97)00215-O
by Elsevier Science B.V. All rights reserved
60
E. Abramochkin
et al. / Optics Communications
I41
f I9971 59-64
transformation of Lagueme-Gauss beams by an optical system with the ABCD matrix (see for example Ref. [6]) we obtain the same transformation of the field (1)
(2) where L, is the optical length along the system axis, W: = ~“1 A + B/Ql’. @ = arg(A + B/Q), Q, = ( AQ + B)/(CQ + D) and AD - BC = 1. Let now ABCD be a round-trip matrix of a laser with the stable resonator. The self-consistency condition for each term of the field (1) under one complete resonator transit Q, = Q gives (Ref. [6]) WI=&‘,
B
kd -= 2
,
Q, = arccos
/I - +( A + D)’
A-I-D 2 ( 1
Then, using the identity 2n + Irnl + 1 = y0 - @,m, we obtain from Eq. (2) that the evolution of spiral-type follows:
beam (1) is as
(3) Here f., is the optical length of one complete resonator transit. From Eq. (3) it is seen that the beam under one complete resonator transit acquires the additional phase (P,,,= XL, - y,,arccos(i( A + D)) and rotates through the angle ON= 0,arccos(f(A + D)). Thus, it is seen from the self-consistency condition that the beam must be rotated through the angle of either - 8, or 27r - 0,,,. It is known, that the beam rotation can be made in a ring resonator by introducing the Dove prism (so called “resonator with a field rotation” [7]). Resonators of this kind were used in order to improve the homogeneity of the transverse distribution of laser irradiance and the angle of the beam rotation (usually 90” or 180”) was not connected with the resonator parameter A + D. If in the resonator a beam rotation through an angle of either - 0, or 2~- 0, is realized and kLL, - y,arccos
A+D 2 i
=2%-q !
then the self-consistency condition with the generation frequency
F,(r,cp) = F(r,p)
takes place and the field (1) will be an eigenmode
of this resonator
(4) Here q is an axial index, c is the speed of light. Let us discuss Eq. (4). It is seen that. in contrast with an ordinary resonator without a beam rotator, Lagueme-Gauss modes _E”,.,( Y,cp) are frequency degenerated when 2 n + 1ml + 0,m = const and the mode composition depends on the rotation which we introduce into the resonator. It is easy to understand, that if one takes into account that the rotation of the distribution of the complex amplitude of Laguerre-Gauss beam P,!“,.,( Y,cp) through an angle 0 is equal to acquiring an additional phase me: Y&(r.cp + 0) =~~,,(r,p)e’““. Thus, spiral-type beams are modes of specific ring resonators containing a beam rotator.
3. Experimental
setup
The experimental setup is presented in Fig. 1. An argon ion ring laser operating at 488 nm contains a plane mirror M, (reflectivity 0.94) and spherical mirrors M?, M, (R, = R, = R = 3 m, reflectivity 0.995, 0.98). The distances between
Fig. 1. Experimental setup
E. Abramochkin et al. / Optics Communications 141 I1997159-64
61
mirrors are M, MZ = M, M, = I = 1.27 m, M,M, = I, = 2.4 m. The resonator scheme shaped as an obtuse-angled triangle has been selected in order to reduce the influence of astigmatism of the mirrors M, and M,. The beam rotation is performed by Dove prism P (under rotation of the prism through angle (Y the beam rotates through angle 2a). It should be noted that the prism P rotates the complex amplitude of a beam distribution, but the influence of the prism on the beam polarization is weak. The prism action is that the beam polarization becomes slightly elliptical. Intensity of the beam component that is normal to the resonator plane depends on the angle of rotation of the prism P and is O-5% of the intensity of the beam component that lies in the resonator plane. Elliptical polarization of the beam takes place in the resonator section P - M, - MZ - until active element. On the rest of the resonator the beam polarization is determined by orientation of Brewster windows of the active element and lies in the resonator plane. The beam generated by the laser has been observed and registered after partially passing mirror M, with the help of objective 0 and microscope M in the plane of screen S. The matrix of complete round-trip of the resonator starting at M, is as follows: I
41 l-
(
R
A c
21” x+R’
4 -:+R?
41’
411” 2’+‘o-
4’”
--
-E-
l-;-
\
41’1, \
411, R
210 y+R”
+-
R’
=
4 11,
-0.939 -0.267
/
After complete round-trip the phase shift for the lowest-order mode in an empty resonator is @ = arccos( - 0.939) - 160” = 0.89~, and therefore, the longitudinal- and transverse-mode frequency separation at zero angle of the prism rotation are selection has been realized by inserting a thin wire W into the _ 60 MHz and - 27 MHz, respectively. Transverse-mode beam zone (the wire diameter is 15 mkm).
4. Experiment The angle of rotation of the prism P for self-consistency 00
cy = - -
2
arccos i
AfD 2
of a spiral-type beam with the rotation parameter
0, is
= - B, x 80”. i
During the experiments it was found that spiral-type beams with various rotation parameters are obtained for some different values of the angle of the prism rotation, in particular, (Y= - 13~X 78”. The deviation has been typical for all our experiments and, as we see it, is the result of the presence of the active medium. 4.1. Spiral-type beam for the prism rotation angle (Y = - 26” The experiment results are presented in Fig. 2. (At the screen spiral-type beams with positive 0, look like clockwise rotated ones.> Numerical simulation for the field F(r,cp) =_Yo,_2(r,(p) + 2_5& ,(r,cp> that corresponds to the rotation parameter B. = { is presented in Fig. 3. The beam phase contains 4 dislocations df the wave front, the signs of the central dislocation and peripheral ones are opposite.
a
b
c
Fig. 2. Experimentally registered intensities of the spiral-type beam before (a), at (b), and after (c) the waist plane. The prism rotation angle is LY= -26”.
62
Fig. 3. Intensity (a). intensity level contours (b), and phase Cc) of the spiral-type
4.2. Spiral-type
beam with the rotation parameter
H, = {
beam for the prism rotation angle (Y = - 15.5”
The experimental results are that corresponds to the rotation of the wave front: the sign of broken) is opposite to the signs
presented in Fig. 4. Numerical simulation for the field F( r. cp) =_2’& &r. cp) + 2 LL?~,~( r, cp) parameter O,, = f is presented in Fig. 5. In this case the beam phase contains 7 dislocations the second order dislocation in the center (in the experiment this degeneration is slightly of the other 5 dislocations.
4.3. Spirul-type beam for the prism rotation angle cy = - 12” The experiment results are presented in Fig. 6. Numerical simulation for the field F(r,ip) =9’&r,q) + 2Y0,z(r,~P) is presented in Fig. 7. This field corresponds to the rotation parameter O,, = - 1 and contains 2 dislocations of the wave front with the same sign. In this case it was found that under the prism rotation through the angle (Y= - 12” the beam changes the rotation direction and, therefore, the excitable spiral-type beam after complete round-trip of the resonator is rotated through
Fig. 4. Experimentally is a = - 15.5”.
registered
intensities of the spiral-type
beam before (a), at (b), and after (c) the waist plane. The prism rotation angle
Fig. 5. Intensity (a), intensity level contours (b). and phase (c) of the spiral-type
beam with the rotation parameter
0, = f.
63
a
b
C
Fig. 6. Experimentally registered intensities of the spiral-type beam before (a), at (b), and after Cc)the waist plane. The prism rotation angle is cy= - 12”.
a
b
Fig. 7. Intensity (a), intensity level contours (b), and phase Cc) of the spiral-type
c
beam with the rotation parameter
0” = -
1.
the angle ON= - 156”, and the prism gives an additional rotation for it in the same direction until - 180”. As a result. the beam is symmetrical to the 180” rotation. 4.4. Spiral-type beam for the prism rotation angle CY= - 6” This case is similar to the previous one, but with the rotation parameter B0 = - i: the excitable spiral-type beam after complete round-trip of the resonator rotates through the angle ON= 8, X 156” = -78”, and the prism gives an additional rotation for it in the same direction until -90”. As a result, the beam is symmetrical to the 90” rotation. The experiment results are presented in Fig. 8. Numerical simulation for the field F(r,cp) =~‘&(Y,(P) + 5P0,_ ,(r.cp) is presented in Fig. 9. This field contains 5 dislocations of the wave front: the central dislocation sign is opposite to the signs of the other 4 dislocations. The possibility of formation of various spiral-type beams in this experiment is restricted by reflective and diffractive losses. Firstly, when the angle of the prism rotation is large enough (for example, (Y= -78” for a spiral-type beam with 0, = l), the angle of incidence of the beam on the Dove prism is far from the Brewster angle and the losses in the resonator
Fig. 8. Experimentally registered intensities of the spiral-type beam before (a). at (b), and after (c) the waist plane. The prism rotation angle is cy = - 6”.
64
E. Abramochkirl et al. / Optics Communicatioru
141 (1997) 59-64
Fig. 9. Intensity (a), intensity level contours (b), and phase (c) of the spiral-type beam with the rotation parameter 00 = - f.
become excessive. Secondly, for example, the construction of the beam with 0, = $ requires the presence of Laguerre-Gauss modes -Z,.,, and X2,,,> for those lm, - m,l is 8 at least, and diffractive losses in the resonator are excessively large for these modes. From Eq. (3) it can be seen that the angle of the beam rotation under complete round-trip in the resonator may be decreased by the change of its configuration, i.e. the parameter A + D. However, in the experiment we have no possibility to do this because of scheme construction restrictions. It is interesting to note that each realized beam prefers to conserve its structure when the prism slightly rotates near cy= - BOX 156”. Under the prism rotation the change of the beam looks like if the beam slightly rotates around its center of symmetry. Moreover, the beams with negative 8, value rotate in the direction of the prism rotation and the beams with positive 0, value rotate in the opposite direction. In our view, this property requires more detailed investigation and is the result of nonlinear effects within the active medium.
5. Conclusions It was shown that spiral-type beams are modes of a specific ring laser containing a beam rotator. For some values of the rotation parameter 0,, inside-resonator generation of spiral-type beams has been obtained experimentally and is in good accordance with the numerical simulation. In our opinion, these results show the possibility of construction of spiral-type beams with given spatial characteristics in a laser with appropriate resonator and amplification coefficients of the active medium.
Acknowledgements This work is partially supported by the Russian Fund for Fundamental
Research (grant 96-02-16430).
References [I] H. Kogelnik, T. Li, Proc. IEEE 54 (1966) 1312. [2] [3] [4] [5] [6] [7]
E. Abramochkin, V. Volostnikov. Optics Comm. 102 (1993) 336. V. Tikhonenko, J. Christou, B. Luther-Daves, J. Opt. Sot. Am. B 11 (1995) 2046. E. Abramochkin. V. Volostnikov, Optics Comm. 125 (1996) 302. E. Abramochkin. V. Volostnikov, Optics Comm. 83 (1991) 123. A. Gerard, J.M. Burch, Introduction to matrix methods in optics (Wiley, New York, 1975). Yu.A. Anan’ev, Optical resonators and laser beams (Nauka, Moscow, 1989).