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Diffraction-limited transverse-coherent radiation of pulsed capillary gas lasers with waveguide resonators
15 December 1995
OPTICS COMMUNICATIONS Optics Communications 122 ( 1995) 35-39
ECSEVIER
Diffraction-listed transverse-coherent radiation of pulsed capillary gas lasers with waveguide resonators S.V. Kukhlevsky, L. Kozma Janus Pannonius University, Department
of'Physics.Ifjusag u. 6. Pets H-7624. Hungary
Received 3 October 1994; revised version received 19 January 1995
Abstract The spatial d~st~bution and coherence of pulsed radiation generated in capillary gas lasers with optical feedback implemented by waveguide resonators with external mirrors have been analyzed. A waveguide resonator configuration has been developed which provided near diffraction-limited transverse-coherent radiation of a short-pulse waveguide N,-laser.
1. Introduction
Optical feedback in CW and pulsed capilary gas lasers with values of the Fresnel’s numbers of the discharge channel less than about 1 is usuaily implemented by waveguide Fabry-Perot resonators [ l--6]. Waveguide resonators are modeled by considering each end of the waveguide as a radiating source whose electric field distribution is a stationary solution of Maxwell’s equations for the waveguide boundary conditions, i.e., a linear combination of stationary waveguide modes. In the work [7] we have developed a method for the calculation of the spatial distribution and coherency of pulsed waveguide modes generated in capillary gas lasers with waveguide resonators. The results of the work show that the employment of waveguide FabryPerot resonators with internal mirrors for the formation of the spatially-coherent low-divergence radiation is reasonable only if a pulse duration is large relative to the formation time of stationary resonator modes. In the case of pulses of shorter duration, a high-divergence radiation with a low degree of the spatial coherence has been demonstrated theoretically and experimentally. BOO-4018/9~/$09.50
0 1995 Elsevier Science B.V. All rights
SSD~OO30-4018(9S)OO482-3
reserved
This article is devoted to theoretical analysis of the spatial distribution and coherence of the short-pulse radiation generated in capillary gas lasers with optical feedback implemented by waveguide Fabry-Perot resonators with external mirrors. A waveguide resonator configuration has been developed which provided near diffraction-limited transverse-coherent radiation of a short-pulse waveguide &laser.
2. Spatial distribution Let us first consider a capillary gas laser operated to produce amplified spontaneous emission with an external flat back reflector. Fig. la shows the Iongitudinal cross-section of the laser with a cylindrical hollow waveguide. The respective radiate equivalent for the laser resonator is presented in Fig. 1b. The most amphfied radiation is that which originates from the points located in the plane of output laser aperture. The rays from these points which are m times reflected from the waveguide wall have been called m-type pulsed waveguide mode [ 71. Let the radiation from the waveguideend surface be isotropic (an unpolarized Lambert’s
S. V. KukhLevsky,L. Kozma /Optics Cotntnunications 122 (1995) 35-39
36
(a)WAVEGUIDE
;
(b)
MIRROR
material of the waveguide wall with the refractive index n. Here we consider the case when the capillary wall has negligible absorption coefficient. Note that in expression (2) only one m-th type pulsed mode contributes to the radiation at a given angle. It should be also mentioned that in fact we consider the optical properties of passive waveguide resonators. The number of waveguide modes M which take part in the generation depends on the angle of escape of the radiation out of the waveguide in the free space between the mirror and waveguide end (see Fig. I b) . The value of M for the maximum in the angle interval ( 1) is given by
I----------------------
--_P
Fig. 1. Schematic diagram of a capillary gas laser operated to produce amplified spontaneous emission with an external flat back reflector: (a) longitudinal cross-section of the laser; (b) radiate equivalent of the laser resonator.
source [8] with the center located in the point P, as shown in Fig. I). Then an m-type mode occupies the angle interval given by
M=&.
(5)
Let us now consider a waveguide resonator with an external flat reflector and an internal flat output mirror. It may easily be demonstrated that the power distribution in this case is given by
(1) where A is the distance between the mirror and waveguide end, L and D are the length and diameter of the waveguide, respectively. Note that at the definition of angle interval, the resonator has been considered as a waveguide with an effective length Lerr= 2(L + A ) . The output power which originates from the waveguide end surface and which falls within an element da of the solid angle around a direction specified by the angle p is given by J’(V) = EBSdficos(p) n*= 0
2,(q) RI,
(2)
with
P(rp, N) = 5 BS da cos(p) 6( 4~) (WWv, ni=o where, for an m-th mode we have:
(7) here, N= ct/2( L + A ) is the number of round-trips of light through the resonator during the generation time t, R, and R2 are respectively the reflectivity of back and output mirrors, and the value of M is given by M=
N(L+A) --A 26.
(3)
$=arcsin
cos((cl
( 1 -
rl
’
where B and S are respectively the brightness and area of the Lambert’ source; R, is the mirror reflectivity; 2,( ~0)is the Fresnel’s reflection coefficient [ 81 for an m-th mode; II/is the angle of the refracted ray in the
(6)
.
(8)
Analysis of Eqs. (l)-(8) shows that the space between mirror and waveguide end plays the role of a mode filter. The number of waveguide mode M and the divergence angle are decreased with the increasing value of A. In the case of a two-mirror resonator the beam divergence is decreased also with the increasing number of the round trips. As an example, Fig. 2a,b shows the results of a calculation of a( cp)/P(O) for a waveguide laser with the above considered resonators. The calculations have been made for an N2 laser (h = 337.1 nm) with a quartz waveguide (it = 1S7)
S. V. Kukhlevsky, L. Kozma /Optics
,p(mrad)
i
L
5
10
15
20
,p(mrad)
Fig. 2. Calculated distribution P(q) lP( 0) versus A for the oneand two- (b ) mirrorcapillary lasers.
(a)
with the length L= 15 cm and diameter of 0.45 mm. Fig. 2a demonstrates the power distribution for the onemirror laser with different values of A. Curves AB, AC, AD, AE, AF, AG correspond to A of L/5, L/10, L/15, L/20, L/30 and 0. respectively. Fig. 2b shows the power distribution for 5-round-trip generation of the two-mirror laser. Curves AB, AC, AD, AE, AF of the figure are calculated for A of L/5, L/ 10, L/15, L/ 20 and 0, respectively. We can notice, both in the Fig. 2a and Fig. 2b, that the divergence angle is decreased with the increasing value of A. This dependence is simply related to the decrease of the angle of escape of the radiation out of the waveguide. One also can see that the two-mirror resonator provides a beam divergence lower than that of the one-mirror laser. That is attributed to the increase of the Fresnel’s losses with the increasing number of round-trips. Also note that in the case of the one-mirror laser the value of A > L/5 is necessary to approach the divergence angles which cor-
Communications
122 (I 995) 35-39
37
responds to that of the low-order stationary modes (a few mrad [l-3]). We will not consider here optical properties of the waveguide resonators with nonflat mirrors. These properties can easily be derived along the above way. Just note that the nonflat mirrors can provide effective waveguide mode discrimination at shorter than that in flat-mirror resonators. Mode-filter properties of a two-mirror waveguide resonator we have also studied experimentally in a Nz laser (output pulse duration of about 8 ns) . The experimental procedures are described in Ref. [ 61. Parameters of the waveguide have been the same as that of Fig. 2. Fig. 3 demonstrates the experimental and calculated distributions P( cp)lP( 0) of the laser with an internal flat output mirror and an external (flat or convex) back reflector. Curve A shows the experimental distribution for the laser with a flat-back mirror located at the distance A = 60 cm. The distribution B has been observed in the laser with a short-radius-curvature ( C = 10 cm) mirror located at the distance A = 30 cm. One can see that the experimental distribution A is significantly different from the theoretical distribution C calculated on the basis of Eqs. (6)-( 8) for the lround-trip generation. For the 1-round-trip radiation the resonator model predicts the value of divergence angle lower than the diffraction limit. That is a consequence of the modeling in the frames of geometrical optics. The more rigorous wave-optics treatment of
0
02
04
06
08
1
dmrad) Fig. 3. Experimental (A. B) and calculated (C. D) power distributions P( q) lP (0) of the two-mirror capillary laser: ( A) distribution with the flat back reflector; (B) distribution with the nonflat back reflector; (C) distribution calculated for the case (A) on the basis of Eqs. (6)-(g); (D) distribution calculated for the Froungofer’s diffraction.
38
XV. Kukhlevsky, L. Kozma /Optics Communications 122 (199.5) 35-39
capillary waveguides is presented in our subsequent paper [ 91. Comparison of the experimental curves A and B with the Froungofers distribution D shows that the resonator configuration with the external mirrors has provided near diffraction-limited radiation (curves D presents the calculated distribution P( rp) lP( 0) for a plane electromagnetic wave passed through a 0.45mm aperture).
and the other symbols have the meanings accepted in the previous section. Now we consider a capillary laser with an external (flat or nonflat) back reflector and internal output flat mirror. It is easily demonstrated along the above way that a degree of spatial coherence of the N-round-trip radiation in the resonator configuration with a flat back reflector is given by I%*=
(2;g-yA))‘2J*(2;~A)), (11)
3. Spatial coherence with Let us consider spatial coherence of a capillary laser with optical feedback implemented by a waveguide resonator with the external mirrors. We first consider the one-mirror laser with a flat back reflector. Radiation from the waveguide entrance can be presented [ 7, lo] as the radiation from the effective circular source of radius r,, (see Fig. 1 and Fig. 4). The intensity and radius of the source are determined by Eqs. (2)) (6). We will consider the case of quasi-monochromatic radiation with the wavelength h. If the intensity distribution of the effective source is uniform, then it follows from the van Cittert-Zernike theorem [ 81, that adegree of spatial coherence of the radiation for two points separated by the diameter D of the waveguide in the plane of the output laser aperture is given by
r,, = 2Nt L +
A Mw,,,,,
(12)
and the spatial coherence of an 1-round-trip generation in the resonator configuration with a nonflat back mirror of C radius-curvature (in paraxial approximation) is given by
A
(9)
0.0 0.0
1
02
1
04
1
06
1
06
1
10
J
with r,,=2(L+A)tancp,,,
(10)
where J, is the first order Bessel’s function, pm, is the angle at which P( cp,,,) is sufficiently small in comparison with P( 0) (for instance P( cp,,,) = 0.1 P( 0) )
1 WAVEGUIDE
j
Fig. 4. Radiate equivalent of a capillary gas laser operated to produce amplified spontaneous emission with an external flat back reflector.
Fig. 5. Spatial coherence -y,* versus A calculated for the two-mirror laser: (a) resonator with the flat back reflector; (b) resonator with the convex back reflector.
39
S. V. Ktrkhlevsky, L. Kozma / Optics Communicarions 122 ( 1995) 35-39
y,?= (g&J
(13)
2g$&).
As an example, Fig. 5a,b shows calculated y,2 as a function of A for a waveguide N, laser with an internal flat mirror and an external (flat or nonflat) back reflector. Parameters of the waveguide are described in the previous section. Dependencies A, B, C, D, E, F of Fig. 5a correspond to the 1, 2, 3, 4, 5, 10 round-trips radiation in the resonator with flat back reflector (C= m). Fig. 5b demonstrates yr2 of the l-round-trip radiation of the laser with nonflat back mirror versus A. Curves A, B, C, D of Fig. 5b are calculated for C of 10 m, I m, 0.1 m and 0.01 m, respectively. One can see that the degree of spatial coherence increases with the increasing values of A, Nand 1IC. The dependence is a result of the decrease of the radius of effective circular source with the increasing values of the parameters. Also note that at the suitable values of A and C the theory predicts the coherent radiation even in the case of the 1 round-trip. Spatial coherence of the above mentioned waveguide N, laser we have studied experimentally. As an example Fig. 6 shows the measured yrZ as a function of A for the laser with flat C = x (data A) and convex C = 0. I m (data B) back reflectors. Theoretical curves
C and D calculated on the basis of (9)-( 13) for the lround-trip generation are depicted in the figure for comparison. The curves C and D correspond the generation with the flat and convex back reflectors, respectively. The experimental results show that in accordance with the theoretical predictions the degree of the spatial coherence increases with the increasing values of A and 1/C. At the suitable values of A and 1 /C the resonator configuration has provided practically coherent laser radiation.
4. Conclusions Spatial distribution and coherence of pulsed radiation generated in capillary gas lasers with optical feedback implemented by waveguide resonators with external mirrors have been analyzed. A waveguide resonator configuration has been developed which provided near diffraction-limited transverse-coherent radiation in a short-pulse waveguide NT-laser. The above mentioned analysis has been based on the geometrical optics approach. The more rigorous waveoptics treatment of capillary waveguides is presented in our subsequent paper [ 91. The results of the present work can be used for the construction of pulsed capillary gas lasers with high beam quality.
References [ II
J.J. Degnan. Appl. Phys.
II
(1976)
1.
[2] L.R. Abrams. Laser Handbook, Vol. 3. ed. M.L. Stich (NorthHolland. Amsterdam, 1979). 13 I P.W. Smith, O.R. Wood II, P.J. Maloney and C.R. Adams, N i_
-
06
IEEE J. Quantum Electron. I7 ( 198
j
..-
A
I ) I 166.
I41 L.A. Newman, Appl. Phys. Len. 33 ( 1978) 501.
IS I C.P.
04j
Christensen, C. Gordon
111, C. Mountoulas
Feldman, Optics Lett. 12 ( 1987)
161 S.V. Kukhlevsky and L. Kozma. Opt. Eng. 32
[ 7 I S.V. 009,1"'1"'1" 00 01
and B.J.
169.
( 1993) 2464. 1IS ( 1995)
Kukhlevsky and L. Kozma. Optics Comm.
425. 02
03
04
05
06
1 cm)
18 I M. Born and E. Wolf, Principles of Optics ( Mac-Millan.
New
York. 1964). 191 S.V.
Kukhlevsky
and L. Kozma.
Pulsed-waveguide-mode
treatment of straight and tapered capillary waveguides. J. Opt. Fig. 6. Experimental
(A, B ) and calculated (C, D) dependencies of
the spatial coherence ylZ as a function of for the capillary laser with flat (A, C) and convex (B, D) back reflectors.
Sot. Am. B, to be published. [ IO] S.T. Kornilov and E.D. Procenko, Volnovodnie gazovie lazery ( rezonatoty
)
( MIFI,
Moscow. 1987).
OPTICS COMMUNICATIONS Optics Communications 122 ( 1995) 35-39
ECSEVIER
Diffraction-listed transverse-coherent radiation of pulsed capillary gas lasers with waveguide resonators S.V. Kukhlevsky, L. Kozma Janus Pannonius University, Department
of'Physics.Ifjusag u. 6. Pets H-7624. Hungary
Received 3 October 1994; revised version received 19 January 1995
Abstract The spatial d~st~bution and coherence of pulsed radiation generated in capillary gas lasers with optical feedback implemented by waveguide resonators with external mirrors have been analyzed. A waveguide resonator configuration has been developed which provided near diffraction-limited transverse-coherent radiation of a short-pulse waveguide N,-laser.
1. Introduction
Optical feedback in CW and pulsed capilary gas lasers with values of the Fresnel’s numbers of the discharge channel less than about 1 is usuaily implemented by waveguide Fabry-Perot resonators [ l--6]. Waveguide resonators are modeled by considering each end of the waveguide as a radiating source whose electric field distribution is a stationary solution of Maxwell’s equations for the waveguide boundary conditions, i.e., a linear combination of stationary waveguide modes. In the work [7] we have developed a method for the calculation of the spatial distribution and coherency of pulsed waveguide modes generated in capillary gas lasers with waveguide resonators. The results of the work show that the employment of waveguide FabryPerot resonators with internal mirrors for the formation of the spatially-coherent low-divergence radiation is reasonable only if a pulse duration is large relative to the formation time of stationary resonator modes. In the case of pulses of shorter duration, a high-divergence radiation with a low degree of the spatial coherence has been demonstrated theoretically and experimentally. BOO-4018/9~/$09.50
0 1995 Elsevier Science B.V. All rights
SSD~OO30-4018(9S)OO482-3
reserved
This article is devoted to theoretical analysis of the spatial distribution and coherence of the short-pulse radiation generated in capillary gas lasers with optical feedback implemented by waveguide Fabry-Perot resonators with external mirrors. A waveguide resonator configuration has been developed which provided near diffraction-limited transverse-coherent radiation of a short-pulse waveguide &laser.
2. Spatial distribution Let us first consider a capillary gas laser operated to produce amplified spontaneous emission with an external flat back reflector. Fig. la shows the Iongitudinal cross-section of the laser with a cylindrical hollow waveguide. The respective radiate equivalent for the laser resonator is presented in Fig. 1b. The most amphfied radiation is that which originates from the points located in the plane of output laser aperture. The rays from these points which are m times reflected from the waveguide wall have been called m-type pulsed waveguide mode [ 71. Let the radiation from the waveguideend surface be isotropic (an unpolarized Lambert’s
S. V. KukhLevsky,L. Kozma /Optics Cotntnunications 122 (1995) 35-39
36
(a)WAVEGUIDE
;
(b)
MIRROR
material of the waveguide wall with the refractive index n. Here we consider the case when the capillary wall has negligible absorption coefficient. Note that in expression (2) only one m-th type pulsed mode contributes to the radiation at a given angle. It should be also mentioned that in fact we consider the optical properties of passive waveguide resonators. The number of waveguide modes M which take part in the generation depends on the angle of escape of the radiation out of the waveguide in the free space between the mirror and waveguide end (see Fig. I b) . The value of M for the maximum in the angle interval ( 1) is given by
I----------------------
--_P
Fig. 1. Schematic diagram of a capillary gas laser operated to produce amplified spontaneous emission with an external flat back reflector: (a) longitudinal cross-section of the laser; (b) radiate equivalent of the laser resonator.
source [8] with the center located in the point P, as shown in Fig. I). Then an m-type mode occupies the angle interval given by
M=&.
(5)
Let us now consider a waveguide resonator with an external flat reflector and an internal flat output mirror. It may easily be demonstrated that the power distribution in this case is given by
(1) where A is the distance between the mirror and waveguide end, L and D are the length and diameter of the waveguide, respectively. Note that at the definition of angle interval, the resonator has been considered as a waveguide with an effective length Lerr= 2(L + A ) . The output power which originates from the waveguide end surface and which falls within an element da of the solid angle around a direction specified by the angle p is given by J’(V) = EBSdficos(p) n*= 0
2,(q) RI,
(2)
with
P(rp, N) = 5 BS da cos(p) 6( 4~) (WWv, ni=o where, for an m-th mode we have:
(7) here, N= ct/2( L + A ) is the number of round-trips of light through the resonator during the generation time t, R, and R2 are respectively the reflectivity of back and output mirrors, and the value of M is given by M=
N(L+A) --A 26.
(3)
$=arcsin
cos((cl
( 1 -
rl
’
where B and S are respectively the brightness and area of the Lambert’ source; R, is the mirror reflectivity; 2,( ~0)is the Fresnel’s reflection coefficient [ 81 for an m-th mode; II/is the angle of the refracted ray in the
(6)
.
(8)
Analysis of Eqs. (l)-(8) shows that the space between mirror and waveguide end plays the role of a mode filter. The number of waveguide mode M and the divergence angle are decreased with the increasing value of A. In the case of a two-mirror resonator the beam divergence is decreased also with the increasing number of the round trips. As an example, Fig. 2a,b shows the results of a calculation of a( cp)/P(O) for a waveguide laser with the above considered resonators. The calculations have been made for an N2 laser (h = 337.1 nm) with a quartz waveguide (it = 1S7)
S. V. Kukhlevsky, L. Kozma /Optics
,p(mrad)
i
L
5
10
15
20
,p(mrad)
Fig. 2. Calculated distribution P(q) lP( 0) versus A for the oneand two- (b ) mirrorcapillary lasers.
(a)
with the length L= 15 cm and diameter of 0.45 mm. Fig. 2a demonstrates the power distribution for the onemirror laser with different values of A. Curves AB, AC, AD, AE, AF, AG correspond to A of L/5, L/10, L/15, L/20, L/30 and 0. respectively. Fig. 2b shows the power distribution for 5-round-trip generation of the two-mirror laser. Curves AB, AC, AD, AE, AF of the figure are calculated for A of L/5, L/ 10, L/15, L/ 20 and 0, respectively. We can notice, both in the Fig. 2a and Fig. 2b, that the divergence angle is decreased with the increasing value of A. This dependence is simply related to the decrease of the angle of escape of the radiation out of the waveguide. One also can see that the two-mirror resonator provides a beam divergence lower than that of the one-mirror laser. That is attributed to the increase of the Fresnel’s losses with the increasing number of round-trips. Also note that in the case of the one-mirror laser the value of A > L/5 is necessary to approach the divergence angles which cor-
Communications
122 (I 995) 35-39
37
responds to that of the low-order stationary modes (a few mrad [l-3]). We will not consider here optical properties of the waveguide resonators with nonflat mirrors. These properties can easily be derived along the above way. Just note that the nonflat mirrors can provide effective waveguide mode discrimination at shorter than that in flat-mirror resonators. Mode-filter properties of a two-mirror waveguide resonator we have also studied experimentally in a Nz laser (output pulse duration of about 8 ns) . The experimental procedures are described in Ref. [ 61. Parameters of the waveguide have been the same as that of Fig. 2. Fig. 3 demonstrates the experimental and calculated distributions P( cp)lP( 0) of the laser with an internal flat output mirror and an external (flat or convex) back reflector. Curve A shows the experimental distribution for the laser with a flat-back mirror located at the distance A = 60 cm. The distribution B has been observed in the laser with a short-radius-curvature ( C = 10 cm) mirror located at the distance A = 30 cm. One can see that the experimental distribution A is significantly different from the theoretical distribution C calculated on the basis of Eqs. (6)-( 8) for the lround-trip generation. For the 1-round-trip radiation the resonator model predicts the value of divergence angle lower than the diffraction limit. That is a consequence of the modeling in the frames of geometrical optics. The more rigorous wave-optics treatment of
0
02
04
06
08
1
dmrad) Fig. 3. Experimental (A. B) and calculated (C. D) power distributions P( q) lP (0) of the two-mirror capillary laser: ( A) distribution with the flat back reflector; (B) distribution with the nonflat back reflector; (C) distribution calculated for the case (A) on the basis of Eqs. (6)-(g); (D) distribution calculated for the Froungofer’s diffraction.
38
XV. Kukhlevsky, L. Kozma /Optics Communications 122 (199.5) 35-39
capillary waveguides is presented in our subsequent paper [ 91. Comparison of the experimental curves A and B with the Froungofers distribution D shows that the resonator configuration with the external mirrors has provided near diffraction-limited radiation (curves D presents the calculated distribution P( rp) lP( 0) for a plane electromagnetic wave passed through a 0.45mm aperture).
and the other symbols have the meanings accepted in the previous section. Now we consider a capillary laser with an external (flat or nonflat) back reflector and internal output flat mirror. It is easily demonstrated along the above way that a degree of spatial coherence of the N-round-trip radiation in the resonator configuration with a flat back reflector is given by I%*=
(2;g-yA))‘2J*(2;~A)), (11)
3. Spatial coherence with Let us consider spatial coherence of a capillary laser with optical feedback implemented by a waveguide resonator with the external mirrors. We first consider the one-mirror laser with a flat back reflector. Radiation from the waveguide entrance can be presented [ 7, lo] as the radiation from the effective circular source of radius r,, (see Fig. 1 and Fig. 4). The intensity and radius of the source are determined by Eqs. (2)) (6). We will consider the case of quasi-monochromatic radiation with the wavelength h. If the intensity distribution of the effective source is uniform, then it follows from the van Cittert-Zernike theorem [ 81, that adegree of spatial coherence of the radiation for two points separated by the diameter D of the waveguide in the plane of the output laser aperture is given by
r,, = 2Nt L +
A Mw,,,,,
(12)
and the spatial coherence of an 1-round-trip generation in the resonator configuration with a nonflat back mirror of C radius-curvature (in paraxial approximation) is given by
A
(9)
0.0 0.0
1
02
1
04
1
06
1
06
1
10
J
with r,,=2(L+A)tancp,,,
(10)
where J, is the first order Bessel’s function, pm, is the angle at which P( cp,,,) is sufficiently small in comparison with P( 0) (for instance P( cp,,,) = 0.1 P( 0) )
1 WAVEGUIDE
j
Fig. 4. Radiate equivalent of a capillary gas laser operated to produce amplified spontaneous emission with an external flat back reflector.
Fig. 5. Spatial coherence -y,* versus A calculated for the two-mirror laser: (a) resonator with the flat back reflector; (b) resonator with the convex back reflector.
39
S. V. Ktrkhlevsky, L. Kozma / Optics Communicarions 122 ( 1995) 35-39
y,?= (g&J
(13)
2g$&).
As an example, Fig. 5a,b shows calculated y,2 as a function of A for a waveguide N, laser with an internal flat mirror and an external (flat or nonflat) back reflector. Parameters of the waveguide are described in the previous section. Dependencies A, B, C, D, E, F of Fig. 5a correspond to the 1, 2, 3, 4, 5, 10 round-trips radiation in the resonator with flat back reflector (C= m). Fig. 5b demonstrates yr2 of the l-round-trip radiation of the laser with nonflat back mirror versus A. Curves A, B, C, D of Fig. 5b are calculated for C of 10 m, I m, 0.1 m and 0.01 m, respectively. One can see that the degree of spatial coherence increases with the increasing values of A, Nand 1IC. The dependence is a result of the decrease of the radius of effective circular source with the increasing values of the parameters. Also note that at the suitable values of A and C the theory predicts the coherent radiation even in the case of the 1 round-trip. Spatial coherence of the above mentioned waveguide N, laser we have studied experimentally. As an example Fig. 6 shows the measured yrZ as a function of A for the laser with flat C = x (data A) and convex C = 0. I m (data B) back reflectors. Theoretical curves
C and D calculated on the basis of (9)-( 13) for the lround-trip generation are depicted in the figure for comparison. The curves C and D correspond the generation with the flat and convex back reflectors, respectively. The experimental results show that in accordance with the theoretical predictions the degree of the spatial coherence increases with the increasing values of A and 1/C. At the suitable values of A and 1 /C the resonator configuration has provided practically coherent laser radiation.
4. Conclusions Spatial distribution and coherence of pulsed radiation generated in capillary gas lasers with optical feedback implemented by waveguide resonators with external mirrors have been analyzed. A waveguide resonator configuration has been developed which provided near diffraction-limited transverse-coherent radiation in a short-pulse waveguide NT-laser. The above mentioned analysis has been based on the geometrical optics approach. The more rigorous waveoptics treatment of capillary waveguides is presented in our subsequent paper [ 91. The results of the present work can be used for the construction of pulsed capillary gas lasers with high beam quality.
References [ II
J.J. Degnan. Appl. Phys.
II
(1976)
1.
[2] L.R. Abrams. Laser Handbook, Vol. 3. ed. M.L. Stich (NorthHolland. Amsterdam, 1979). 13 I P.W. Smith, O.R. Wood II, P.J. Maloney and C.R. Adams, N i_
-
06
IEEE J. Quantum Electron. I7 ( 198
j
..-
A
I ) I 166.
I41 L.A. Newman, Appl. Phys. Len. 33 ( 1978) 501.
IS I C.P.
04j
Christensen, C. Gordon
111, C. Mountoulas
Feldman, Optics Lett. 12 ( 1987)
161 S.V. Kukhlevsky and L. Kozma. Opt. Eng. 32
[ 7 I S.V. 009,1"'1"'1" 00 01
and B.J.
169.
( 1993) 2464. 1IS ( 1995)
Kukhlevsky and L. Kozma. Optics Comm.
425. 02
03
04
05
06
1 cm)
18 I M. Born and E. Wolf, Principles of Optics ( Mac-Millan.
New
York. 1964). 191 S.V.
Kukhlevsky
and L. Kozma.
Pulsed-waveguide-mode
treatment of straight and tapered capillary waveguides. J. Opt. Fig. 6. Experimental
(A, B ) and calculated (C, D) dependencies of
the spatial coherence ylZ as a function of for the capillary laser with flat (A, C) and convex (B, D) back reflectors.
Sot. Am. B, to be published. [ IO] S.T. Kornilov and E.D. Procenko, Volnovodnie gazovie lazery ( rezonatoty
)
( MIFI,
Moscow. 1987).