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Noncollinear second harmonic generation in KDP
Volume
8, number
August
OPTICS COMMUNICATIONS
4
NONCOLLINEAR
SECOND
HARMONIC
GENERATION
1973
IN KDP
H. FERY and F. HERMANN I. Physikalisches Institut der Technischen UniversitCt Berlin Berlin, Germany Received
24 May 1973
This paper reports experimental and analytical results on optical second harmonic generation when a noncollinear arrangement is employed. The dependence of second harmonic power on crystal length, absorption, beam radius and the angle between the two fundamental waves is obtained. Double refraction of the crystal is taken into account, but the divergence and depletion of the fundamental waves are neglected. The results of the calculations are confirmed by experiments carried out with a KDP crystal in the near field of two intersecting TEMoo ruby laser beams. An extension of the theory to nonlinear interactions other than second harmonic generation - e.g. four-wave interaction - is possible.
1. Introduction Noncollinear index-matched second harmonic generation (SHG) has been studied experimentally by Giordmaine [l] and Golovei et al. [2]. Theoretical and experimental work has been done on noncollinear parametric noise emission [3-S], four-wave parametric amplification [6], and up-conversion [7]. Parametric oscillation with noncollinear arrangements was considered by Akhmanov et al. [8] and employed experimentally by Falk and Murray [9]. Basu and Steier report theoretical calculations for angle tuned parametric oscillators [lo] taking into account the finite aperture of the interacting waves, small angles between their wave vectors, and double refraction. To our knowledge, however, no treatment of noncollinear SHG and other nonlinear effects exists which takes into account interacting waves inclined at large angles, finite apertures, double refraction, and absorption. We have been stimulated to this work by investigations of four-wave interaction where noncollinear index matching may be of particular interest for the determination of the relative magnitude of third-order nonlinear coefficients [ 1 I].
2. Experimental considerations The experimental setup consists of a ruby laser and two mirrors which split the laser beam into two beams impinging on a KDP crystal. Both beams inclined at an angle A are used as fundamental waves creating the second harmonic (SH) in the crystal. The SH power is detected by a RCA 7265 photomultiplier and the fundamental power is monitored using a HP 4220 PIN photodiode. Both signals are displayed on a TEK 556 oscilloscope. The ruby laser is constructed in a way to permit single longitudinal TEM,, mode operation as described by Bjorkholm and Stolen [ 121. Throughout the experiments the laser worked with powers of about 100 kW having a pulse length of 20 nsec. The far field diffraction half angle of the laser beam was 3.4 X lop4 rad corresponding to a confocal parameter of 400 cm. At the locus of the KDP crystal the unfocused laser beam had a diameter of 2 mm. Beam waists of 0.6 mm and 0.25 mm were produced by the aid of lenses with 100 cm and 50 cm focal length, respectively. Care was taken of the position of the lenses with respect to the crystal. This was done in order 291
Volume
8. number
4
August
OPTIC’S <‘OM\lUNICATIONS
1973
to ensure SHG in the near field of the beams, where they can be treated as nearly plane waves. The beam diameters were measured by photometric inspection of photographic plates put into the beam path. The error of the values obtained is estimated to be YLIO%. In order to obtain two different crystal lengths using one sample the KDP was cut in the [ 1 1I] and [ 111] planes having a respective length of 10 1n111 and 20 IIIIII normal to these planes. Perfect index matching of SItG with two noncollinear fundamental waves of frequencies w, and w2 (wl = w7 = o) is described by the usual relations %=(Jl
+W? _
=h,
k, =k, +k2.
(I,21
In our case. however, we have to take special care of the vector character of eq. (2). We consider type I index matching with negative uniaxial crystals (e.g. KDP, both fundamental waves ordinary, the SH extraordinary). where both fundamental waves having wave vectorsk, and kz and the SH wave vectork3 are inclined at an angle $a (see fig. 1). An increase of the angle LI must be compensated by reducing the SH refractive index fz3(0). This is achieved by turning the crystal in a sense which enlarges the angle 0 between the optic axis and k,. The dependence of 0 on a (and 62C on LI, which can be measured outside the crystal) is shown in fig. 2. The curve was computed using Irefractive index data given by Zernike in ref. [ 131. The points indicate optimum SHG at an angle 6,, for a given angle a,. All experimental values for 62c arc nearly 0.3” larger than the computed ones. This may be due to a slight deviation of the crystal’s surface from the 111 l] plane. The experimental results on the power of the generated SH and theoretical curves are given in fig. 3. A function G which equals the SH power reduced to constant fundamental input power P,, = P,,/P,; P,,except for a constant factor [see eq. (lo)] is shown with its computed dependence on n and A,. No absolute measurements have been made. The relative position of the experimental values to the theoretical ones was achieved by fitting just one experimental point to the corresponding theoretical value (a, = O”, I = 1 cm, \V = 0.3 mm). Further discussion of these results will be postponed until after the following theoretical considerations.
\FUNDAMENTAL
Fig. 1. Illustrating
noncollinear
SHG.
Volume
8, number
4
August
OPTICS COMMUNICATIONS
20”
+ 3,68” 0”
30”
a,-
4(
t
jO,55"
55”
PARAMETER
I/w
a I= 20mm,w=
0,125mm
- I= lOmm,w=
0,125mm
O I= + I= 0 I= xl=
6 2e SO”
-20”
1973
20mm,w = 0,3 mm lOmm,w= 0,3 mm 20mm,w= 1 mm lOmm,w= 1 mm
65”
70” x
I
x-
x
75”
-60”
0
‘1
13” IO”
13”
15” 16”
17” 17,25” a-
Fig. 2. Sze and 0 as function collinear SHC.
of A, and A for optimum
non-
Y55”
8”
IO”
13”
15” 16” A-
17”
Fig. 3. The function G defined in eq. (10) as a function of Ae and A for several values of w/l compared with measured values of SH power.
3. Theoretical considerations Our theoretical description of noncollinear SHG is based on an extension of the “heuristic treatment” of co19 linear SHG with double refraction given by BADK in [ 141 to noncollinear configurations. Starting from t.he wave equation it follows for unbounded plane waves that the SH amplitude p3 generated with two noncollinear fundamental waves satisfies the differential equation (3) where K = (2rr/c2) 2LI,, sin(8 t CY)sin 2@
(4)
in the case of type I SHG in negative uniaxial crystals of point group 42m and sin 2@ = 1 for the experimentally chosen crystal orientation. Since we neglect depletion of the fundamental waves, their amplitudes p1 and p2 are treated as constants. (Yis the double refraction angle of the extraordinary SH (see fig. 1). The second factor 2 in eq. (4) appears due to the presence of two fundamental waves and must be replaced by 1 when collinear SHG is concerned (p 1p2 + p2). d,, is the relevant component of the SHG tensor d defined by Boyd and Kleinman in ref. [15]. We consider now the influence of the finite extent of the fundamental waves on the generated SH power. The spatial distribution of the fundamental amplitude is assumed to be that of the near field of a TEM,, mode which approximately is given by p(r) = p exp(-.-r2/w2),
(5)
where w is the beam radius. 293
Volume
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OPTICS COMMUNICATIONS
4
August
Eq. (3) is valid for unbounded plane waves and should represent a good approximation the near field of two TEM,, modes as well. This leads to
to noncollinear
1973
SHG in
(6) with ____
When absorption
Q=
CL, I,2 wf
N=
Csin&iCos6i?SZS_L, 1.2 cos2&
w;
is taken into account eq. (6) becomes
+G,P~ dp3 (x, y, z)/dz = -~~~
__ exp[-y3(zpz,)/2 k, cos(F3fcY) COSC?
cos(sj
+a)] exp[-F(z-z,)/2]
exp[-x2L-z2M-y2Q+2xzN] (7)
with F = y/cos6 t + y/cosfi7 ~~y3/cos(63 + a). y and y3 are the absorption coefficients of the fundamental and SH waves, respectively. The expressions COS~~,/CO~~~ take into account the deformation of the beam cross sections when they are refracted at the crystal’s surface. To integrate eq. (6) we follow the arguments of BADK [ 141: the SH field amplitude at an observer point x2, ,v2, z2 at the exit surface z2 = z, will be obtained by integrating eq. (6) along a path x=z
tan(63+cu)+xZ
(8)
which is parallel to the SH poynting vector S,. The power of the SH at the exit surface but still inside the crystal is obtained by integrating the time average of the z component of the poynting vector. &__ = (cn,@)/%r)
p; cos(k 3 + cl) cos CY
(9)
over x2 and ~2. Finally we find for the reduced SH power outside the crystal
P3r=P3e/P1,P2c =32cw;K2n-'G, with G = [gfg;g;
x JJ _m
cos63e]/[cOSq63 t(u) co&] wc2wy2‘13-2 (0) exptrz,)
exp[-y3Z/cos(63
+a)] 2
dx, dv2
7 exp(-Fz/2) I_-ii
exp[-{z
tan(63+o)+x2}2L-z2nii~_~~Q+2z(z
tan(63+ol)
.
+x2)N]dz i
The field amplitudes p1 and p2 inside the crystal have been transformed to the fundamental powers P,,and P,, before entering the crystal. gt, g2 and g3 are the transmission coefficients of the three interacting waves, which strongly depend on the orientation of the crystal. It should be pointed out that eq. (10) with some modifications can be applied to any case of three waves interaction when small conversion rates are employed and the divergence of the waves can be neglected. Furthermore eq. (10) can be extended to account for four wave interactions. This will be reported elsewhere.
294
Volume
8, number
4
August
OPTICS COMMUNICATIONS
1973
4. Experimental results The dependence of the function G [eq. (lo)] on the angle A for various parameters is shown in fig. 4 where we have assumed w l “=w2 = w and -z = z, = :Z (the beam axes cross in the middle of the crystal). Curve (a) is obtained for a nonlkear material embedded in an isotropic linear medium of the same refractive index (81 = g2 =g3 = 1, hi = 6 ie). The SH is assumed to propagate in the direction of the z axis (h3 = O”, 6 I = -6 2) and double refraction is neglected (o = 0). In this case the reduction of the interaction volume with increased A leads to a decrease of SH power as shown in fig. 4, curve (a). Curve (b) shows the dependence of G on A in the special case of a KDP crystal cut in the [ 11 l] or [ 11 i] plane. Double refraction,, reflexion losses, beam deformation and turning of the crystal is included in the computations. The decrease of the SH power with increasing A is not as strong as in curve (a) because the interaction volume of the waves is lengthenend when 0 must be enlarged by turning the crystal to match SHG with larger angles A. The collinear value gets smaller mainly due to reflexion losses. Curve (c) shows what should be observed experimentally: an additional factor sin28/sin28 eoll is included which originates from the coefficient K [eq. (4) with cy< 01. The factor is normalized to the collinear value. Curves (d) and (e) are similar to curve (c), but other parameters l/w are used. Curves (f), (g) and (h), (i) show the influence of two different absorption coefficients on curve (d) and (e), respectively. If absorption is included a parameter Z/w cannot be used, the values for 1 and w must be given explicitly. The curves in fig. 3 have been computed with the experimental values for I and w taking y = y3 = 0. For Z/w 2 20 the SH power is independent of 2 and w if A exceeds a proper value. This may be explained as follows: both fundamental1 intensities J, and J, vary as l/w 2. One may define an interaction volume V which is formed by the two crossing fundamental beams. This volume I/ is proportional to w2Z1,, where 1, is the extension of V in the direction ofK . I for his part varies as w. If I’ is fully contained within the crystal the SH power is roughly proportional to I$ dlJlJ2 - lyVJIJ2 - w4J1J2 and there will be no change in SH power when w is enlarged or 1 is reduced. The value for A mentioned above is reached when V is within the crystal. Furthermore should be noted the change of a quadratic to a linear dependence of collinear (A = 0”) SHG on Z/w at l/w x 61. This is the consequence of the “wdk-off effect” which has been discussed by many authors (e.g. in ref. [ 141). Comparing the experimental results with the computed curves one finds small discrepancies. The experimental SH power for the unfocused beams (Z/w = 10 and Z/w = 20) are smaller than theoretically predicted. In this case, however, the KDI’ crystal is not placed at the beam waists of the fundamental waves and divergence may cause a reduction of the SH power (see e.g. [ 15,161). Furthermore the experimental collinear values for the 2 cm length
O0
5O
d
I-lcm,w=0,05cm,~=~3=0
f g
I= Icm,w=0,05cm.~
I=1cm,w=0,05cm,~=~~0,01cm-' =b,=O.O5cm-'
IO"
15"
20"
AFig. 4. The function
G as a function
of A for
several values of 1,W and 7. 295
Volume
8. number
4
OPTICS COMMUNICATIONS
August
1973
crystal compared to those of 1 cm length are too small. This may be attributed to a small but not negligible absorption (see fig. 4) which has not been accounted for in the computed curves of fig. 3. In spite of the deviations between theory and experiment, the theory depicted seems to give a good description of noncollinear SHG.
Acknowledgements The authors wish to express their appreciation to Professor H. Eichler for the many discussions they have had and for his critical reading of the manuscript. This work was supported by the “Deutsche Forschungsgemeinschaft”.
References [ 11 [2] [ 31 [4] [5] 161 [7] [S] [9]
[lo] [ll] [ 121 [ 131 [ 141 [ 151 1161
296
J.A. Giordmainr, Phys. Rev. Lett. 8 (1962) 19. M.P. Golovei and G.I. Kosourov. JETP Lctt. 9 (1969) 132. R.G. Smith, J.G. Skinner, J.E. C&sic and W.G. Nilsen. Appl. Phys. Lett. 12 (I 968) 97. J.P. Budin, B. Godard. J. Ducuing, IEEE J. of Q.E. 4 (1968) 831. T.G. Giallorenzi and C.L. Tang, Phys. Rev. 166 (1968) 225. A.P. Vcduty and B.P. Kirsanov, JETP 29 (1969) 632. J. Warner, Opto Electronics 1 (1969) 25. S.A. Akhmanov and R.V. Khokhlow, JETP 16 (1962) 252. J. Falk and J.E. Murray, Appl. Phys. Lctt. 14 (1969) 245. R. Basu and W.11. Stcicr, IEEE J. of Q.E. 8 (1972) 693. H. Eichlcr, II. Fery and I‘. Ilcrmann, Opt. Comm. 6 (1972) 152. J.E. Bjorkholm and R.11. Stolen, J. of Appl. Phys. 39 (1968) 4043. F. Zcrnike, J. of the Opt. Sot. of Am. 54 (1964) 1215. G.D. Boyd, A. Ashkin, J.M. Dziedzic and D.A. Kleinman, Phys. Rev. 137 (1965) A 1305. G.D. Boyd and D.A. Klcinman, J. of Appl. Phy% 39 (1968) 3597. D.A. Kleinman, Phys. Rev. 128 (1962) 1761.
8, number
August
OPTICS COMMUNICATIONS
4
NONCOLLINEAR
SECOND
HARMONIC
GENERATION
1973
IN KDP
H. FERY and F. HERMANN I. Physikalisches Institut der Technischen UniversitCt Berlin Berlin, Germany Received
24 May 1973
This paper reports experimental and analytical results on optical second harmonic generation when a noncollinear arrangement is employed. The dependence of second harmonic power on crystal length, absorption, beam radius and the angle between the two fundamental waves is obtained. Double refraction of the crystal is taken into account, but the divergence and depletion of the fundamental waves are neglected. The results of the calculations are confirmed by experiments carried out with a KDP crystal in the near field of two intersecting TEMoo ruby laser beams. An extension of the theory to nonlinear interactions other than second harmonic generation - e.g. four-wave interaction - is possible.
1. Introduction Noncollinear index-matched second harmonic generation (SHG) has been studied experimentally by Giordmaine [l] and Golovei et al. [2]. Theoretical and experimental work has been done on noncollinear parametric noise emission [3-S], four-wave parametric amplification [6], and up-conversion [7]. Parametric oscillation with noncollinear arrangements was considered by Akhmanov et al. [8] and employed experimentally by Falk and Murray [9]. Basu and Steier report theoretical calculations for angle tuned parametric oscillators [lo] taking into account the finite aperture of the interacting waves, small angles between their wave vectors, and double refraction. To our knowledge, however, no treatment of noncollinear SHG and other nonlinear effects exists which takes into account interacting waves inclined at large angles, finite apertures, double refraction, and absorption. We have been stimulated to this work by investigations of four-wave interaction where noncollinear index matching may be of particular interest for the determination of the relative magnitude of third-order nonlinear coefficients [ 1 I].
2. Experimental considerations The experimental setup consists of a ruby laser and two mirrors which split the laser beam into two beams impinging on a KDP crystal. Both beams inclined at an angle A are used as fundamental waves creating the second harmonic (SH) in the crystal. The SH power is detected by a RCA 7265 photomultiplier and the fundamental power is monitored using a HP 4220 PIN photodiode. Both signals are displayed on a TEK 556 oscilloscope. The ruby laser is constructed in a way to permit single longitudinal TEM,, mode operation as described by Bjorkholm and Stolen [ 121. Throughout the experiments the laser worked with powers of about 100 kW having a pulse length of 20 nsec. The far field diffraction half angle of the laser beam was 3.4 X lop4 rad corresponding to a confocal parameter of 400 cm. At the locus of the KDP crystal the unfocused laser beam had a diameter of 2 mm. Beam waists of 0.6 mm and 0.25 mm were produced by the aid of lenses with 100 cm and 50 cm focal length, respectively. Care was taken of the position of the lenses with respect to the crystal. This was done in order 291
Volume
8. number
4
August
OPTIC’S <‘OM\lUNICATIONS
1973
to ensure SHG in the near field of the beams, where they can be treated as nearly plane waves. The beam diameters were measured by photometric inspection of photographic plates put into the beam path. The error of the values obtained is estimated to be YLIO%. In order to obtain two different crystal lengths using one sample the KDP was cut in the [ 1 1I] and [ 111] planes having a respective length of 10 1n111 and 20 IIIIII normal to these planes. Perfect index matching of SItG with two noncollinear fundamental waves of frequencies w, and w2 (wl = w7 = o) is described by the usual relations %=(Jl
+W? _
=h,
k, =k, +k2.
(I,21
In our case. however, we have to take special care of the vector character of eq. (2). We consider type I index matching with negative uniaxial crystals (e.g. KDP, both fundamental waves ordinary, the SH extraordinary). where both fundamental waves having wave vectorsk, and kz and the SH wave vectork3 are inclined at an angle $a (see fig. 1). An increase of the angle LI must be compensated by reducing the SH refractive index fz3(0). This is achieved by turning the crystal in a sense which enlarges the angle 0 between the optic axis and k,. The dependence of 0 on a (and 62C on LI, which can be measured outside the crystal) is shown in fig. 2. The curve was computed using Irefractive index data given by Zernike in ref. [ 131. The points indicate optimum SHG at an angle 6,, for a given angle a,. All experimental values for 62c arc nearly 0.3” larger than the computed ones. This may be due to a slight deviation of the crystal’s surface from the 111 l] plane. The experimental results on the power of the generated SH and theoretical curves are given in fig. 3. A function G which equals the SH power reduced to constant fundamental input power P,, = P,,/P,; P,,except for a constant factor [see eq. (lo)] is shown with its computed dependence on n and A,. No absolute measurements have been made. The relative position of the experimental values to the theoretical ones was achieved by fitting just one experimental point to the corresponding theoretical value (a, = O”, I = 1 cm, \V = 0.3 mm). Further discussion of these results will be postponed until after the following theoretical considerations.
\FUNDAMENTAL
Fig. 1. Illustrating
noncollinear
SHG.
Volume
8, number
4
August
OPTICS COMMUNICATIONS
20”
+ 3,68” 0”
30”
a,-
4(
t
jO,55"
55”
PARAMETER
I/w
a I= 20mm,w=
0,125mm
- I= lOmm,w=
0,125mm
O I= + I= 0 I= xl=
6 2e SO”
-20”
1973
20mm,w = 0,3 mm lOmm,w= 0,3 mm 20mm,w= 1 mm lOmm,w= 1 mm
65”
70” x
I
x-
x
75”
-60”
0
‘1
13” IO”
13”
15” 16”
17” 17,25” a-
Fig. 2. Sze and 0 as function collinear SHC.
of A, and A for optimum
non-
Y55”
8”
IO”
13”
15” 16” A-
17”
Fig. 3. The function G defined in eq. (10) as a function of Ae and A for several values of w/l compared with measured values of SH power.
3. Theoretical considerations Our theoretical description of noncollinear SHG is based on an extension of the “heuristic treatment” of co19 linear SHG with double refraction given by BADK in [ 141 to noncollinear configurations. Starting from t.he wave equation it follows for unbounded plane waves that the SH amplitude p3 generated with two noncollinear fundamental waves satisfies the differential equation (3) where K = (2rr/c2) 2LI,, sin(8 t CY)sin 2@
(4)
in the case of type I SHG in negative uniaxial crystals of point group 42m and sin 2@ = 1 for the experimentally chosen crystal orientation. Since we neglect depletion of the fundamental waves, their amplitudes p1 and p2 are treated as constants. (Yis the double refraction angle of the extraordinary SH (see fig. 1). The second factor 2 in eq. (4) appears due to the presence of two fundamental waves and must be replaced by 1 when collinear SHG is concerned (p 1p2 + p2). d,, is the relevant component of the SHG tensor d defined by Boyd and Kleinman in ref. [15]. We consider now the influence of the finite extent of the fundamental waves on the generated SH power. The spatial distribution of the fundamental amplitude is assumed to be that of the near field of a TEM,, mode which approximately is given by p(r) = p exp(-.-r2/w2),
(5)
where w is the beam radius. 293
Volume
8. number
OPTICS COMMUNICATIONS
4
August
Eq. (3) is valid for unbounded plane waves and should represent a good approximation the near field of two TEM,, modes as well. This leads to
to noncollinear
1973
SHG in
(6) with ____
When absorption
Q=
CL, I,2 wf
N=
Csin&iCos6i?SZS_L, 1.2 cos2&
w;
is taken into account eq. (6) becomes
+G,P~ dp3 (x, y, z)/dz = -~~~
__ exp[-y3(zpz,)/2 k, cos(F3fcY) COSC?
cos(sj
+a)] exp[-F(z-z,)/2]
exp[-x2L-z2M-y2Q+2xzN] (7)
with F = y/cos6 t + y/cosfi7 ~~y3/cos(63 + a). y and y3 are the absorption coefficients of the fundamental and SH waves, respectively. The expressions COS~~,/CO~~~ take into account the deformation of the beam cross sections when they are refracted at the crystal’s surface. To integrate eq. (6) we follow the arguments of BADK [ 141: the SH field amplitude at an observer point x2, ,v2, z2 at the exit surface z2 = z, will be obtained by integrating eq. (6) along a path x=z
tan(63+cu)+xZ
(8)
which is parallel to the SH poynting vector S,. The power of the SH at the exit surface but still inside the crystal is obtained by integrating the time average of the z component of the poynting vector. &__ = (cn,@)/%r)
p; cos(k 3 + cl) cos CY
(9)
over x2 and ~2. Finally we find for the reduced SH power outside the crystal
P3r=P3e/P1,P2c =32cw;K2n-'G, with G = [gfg;g;
x JJ _m
cos63e]/[cOSq63 t(u) co&] wc2wy2‘13-2 (0) exptrz,)
exp[-y3Z/cos(63
+a)] 2
dx, dv2
7 exp(-Fz/2) I_-ii
exp[-{z
tan(63+o)+x2}2L-z2nii~_~~Q+2z(z
tan(63+ol)
.
+x2)N]dz i
The field amplitudes p1 and p2 inside the crystal have been transformed to the fundamental powers P,,and P,, before entering the crystal. gt, g2 and g3 are the transmission coefficients of the three interacting waves, which strongly depend on the orientation of the crystal. It should be pointed out that eq. (10) with some modifications can be applied to any case of three waves interaction when small conversion rates are employed and the divergence of the waves can be neglected. Furthermore eq. (10) can be extended to account for four wave interactions. This will be reported elsewhere.
294
Volume
8, number
4
August
OPTICS COMMUNICATIONS
1973
4. Experimental results The dependence of the function G [eq. (lo)] on the angle A for various parameters is shown in fig. 4 where we have assumed w l “=w2 = w and -z = z, = :Z (the beam axes cross in the middle of the crystal). Curve (a) is obtained for a nonlkear material embedded in an isotropic linear medium of the same refractive index (81 = g2 =g3 = 1, hi = 6 ie). The SH is assumed to propagate in the direction of the z axis (h3 = O”, 6 I = -6 2) and double refraction is neglected (o = 0). In this case the reduction of the interaction volume with increased A leads to a decrease of SH power as shown in fig. 4, curve (a). Curve (b) shows the dependence of G on A in the special case of a KDP crystal cut in the [ 11 l] or [ 11 i] plane. Double refraction,, reflexion losses, beam deformation and turning of the crystal is included in the computations. The decrease of the SH power with increasing A is not as strong as in curve (a) because the interaction volume of the waves is lengthenend when 0 must be enlarged by turning the crystal to match SHG with larger angles A. The collinear value gets smaller mainly due to reflexion losses. Curve (c) shows what should be observed experimentally: an additional factor sin28/sin28 eoll is included which originates from the coefficient K [eq. (4) with cy< 01. The factor is normalized to the collinear value. Curves (d) and (e) are similar to curve (c), but other parameters l/w are used. Curves (f), (g) and (h), (i) show the influence of two different absorption coefficients on curve (d) and (e), respectively. If absorption is included a parameter Z/w cannot be used, the values for 1 and w must be given explicitly. The curves in fig. 3 have been computed with the experimental values for I and w taking y = y3 = 0. For Z/w 2 20 the SH power is independent of 2 and w if A exceeds a proper value. This may be explained as follows: both fundamental1 intensities J, and J, vary as l/w 2. One may define an interaction volume V which is formed by the two crossing fundamental beams. This volume I/ is proportional to w2Z1,, where 1, is the extension of V in the direction ofK . I for his part varies as w. If I’ is fully contained within the crystal the SH power is roughly proportional to I$ dlJlJ2 - lyVJIJ2 - w4J1J2 and there will be no change in SH power when w is enlarged or 1 is reduced. The value for A mentioned above is reached when V is within the crystal. Furthermore should be noted the change of a quadratic to a linear dependence of collinear (A = 0”) SHG on Z/w at l/w x 61. This is the consequence of the “wdk-off effect” which has been discussed by many authors (e.g. in ref. [ 141). Comparing the experimental results with the computed curves one finds small discrepancies. The experimental SH power for the unfocused beams (Z/w = 10 and Z/w = 20) are smaller than theoretically predicted. In this case, however, the KDI’ crystal is not placed at the beam waists of the fundamental waves and divergence may cause a reduction of the SH power (see e.g. [ 15,161). Furthermore the experimental collinear values for the 2 cm length
O0
5O
d
I-lcm,w=0,05cm,~=~3=0
f g
I= Icm,w=0,05cm.~
I=1cm,w=0,05cm,~=~~0,01cm-' =b,=O.O5cm-'
IO"
15"
20"
AFig. 4. The function
G as a function
of A for
several values of 1,W and 7. 295
Volume
8. number
4
OPTICS COMMUNICATIONS
August
1973
crystal compared to those of 1 cm length are too small. This may be attributed to a small but not negligible absorption (see fig. 4) which has not been accounted for in the computed curves of fig. 3. In spite of the deviations between theory and experiment, the theory depicted seems to give a good description of noncollinear SHG.
Acknowledgements The authors wish to express their appreciation to Professor H. Eichler for the many discussions they have had and for his critical reading of the manuscript. This work was supported by the “Deutsche Forschungsgemeinschaft”.
References [ 11 [2] [ 31 [4] [5] 161 [7] [S] [9]
[lo] [ll] [ 121 [ 131 [ 141 [ 151 1161
296
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