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Optical second-harmonic generation in a centrosymmetric crystal involving the spatial gradient of the electric field
Volume 16, number 2
OPTICS (7OMMUNICATIONS
February 1976
OPTICAL SECOND-HARMONIC GENERATION IN A CENTROSYMMETRIC CRYSTAL INVOLVING THE SPATIAL GRADIENT OF THE ELECTRIC FIELD L. ORTMANN and tt. VOGT 11. t'hvsikalisches h~stitut der Universiti)t Ki)IH, 5 K6ln 41, UHirersitiitsstrasse 14, Fed. Rep. Germany
Received 22 October 1975
Phase-matched second-harmonic generation has been observed in centrosymmetric sodium nitrate (NaNO3) as function of direction. Experimental results can be adequately described by a nonlinear source term being proportional to the product of the fundamental electric field and its spatial gradient.
1. Introduction Optical second-harlnonic generation (StiG) in centrosynrmetric crystals has been studied rather rarely. Terhune et al. [11 and later oll Bjorkhohn and Siegman [2] observed phase-matched SttG in centrosymmetric calcite (CaCO3). They focused their attention on the increase of the second-ha> monic intensity, when disturbing the inversion symmetry by an externally applied, static electric field. In the course of studies on structural phase transitions StlG was sometimes found not only in the non-centrosymlnetric low-temperature phase, but also in the centrosymmetric one above the transition temperature [3 - 5]. Explanations of this phenomenon have been suggested in the case of NIt4CI [4] and NaNO 2 [5]. In simple centrosymmetric crystals like the alkali halides, weak SHG was also detected [6]. It has been shown, that this effect is only partly typical of the crystal bulk, but mainly arises from discontinuities at the crystal surfaces. Recently Kielich and Zawodny 17] have summarized existing theoretical and experinrental results on SHG in the case of inversion symmetry. The present paper deals with phase-matched SHG in single crystals of sodium nitrate (NaNO3), which has the same centrosymmetric crystal structure of point group 3m as calcite. Our experimental results allow the conclusion, that the observed SHG is an effect of spatial dispersion involving the gradi234
enl of the fundamental electric field.
2. Crystal structure of NaNO 3 and origin of SHG The rhombohedral bmrolecular unit cell of NaNO 3 is drawn in fig. 1. The structural feature of most concern to SHG is the alternating orientation of the NO 3-ions. Within the array of NO,--ions along the three-fold c-axis each NO,--ion is rotated by 60 ° or equivalently 180 ° with respect to its neighbours. Due to this antiparallel ordering the intermediate Na+-sites become centres of inversion, although the individual NO3-ions have the symmetry of equilateral triangles and hence are not centrosymmetric. As the electronic polarizabilities of NO 3 are much larger than those of Na + the second-harmonic light may be assumed to arise from nonlinear multipole moments induced in a NO3-ion. The lowest order term to be considered is a dipole m o m e n t of the form pNL = 13 : F(1)/7(1) ,
(1)
where the third-rank tensor 13 represents the nonlinear polarizability and F (1) is the h)cal electric field associated with the incident laser radiation. Because of the antiparallel ordering, the nonlinear polarizabilities 13 and hence the dipole moments pNL of neighbouring NO,--ions have opposite signs
Volume 16, number 2
OPTICS COMMUNICATIONS
February 1976
On the other hand the opposite dipole moments
_+~ : F~I)(0) F(I)(0) of neighbouring NO~--ions represent a quadrupole moment, leading to a macroscopic nonlinear quadrupole moment density: QNL = xQ : E l l ) E ( 1 ) .
(4)
Both source terms (3) and (4) determine the nonlinear part D NL of the dielectric displacement [8] : D N L = 4 r r ( P NL
v.QNL)=4rcXiE
Ill ~ E (1).
(5)
Here X is a tensor of fourth rank, existing in centrosymmetric as well as non-centrosymmetric crystal phases. For later use it is convenient to define the four subscripts of X according to DNL=47r ~ X//m,,E,(, I)3/f~'l) l, tn .tt 3Xl
Fig. 1. Unit cell of NaNO3. Note: The vertical mirror planes of the NO]-ions are rotated by 30° with respect to the glide planes built up by the e-axis and the rhombohedral basis vectors. if they are referred to a crystal coordinate system. Thus they cancel each other and no corresponding macroscopic polarization can result being directly proportional to the square of the electric laser field. Due to the finite wavelength of light, however, F (1) slightly varies along the distance z 0 = 2c = 8.41 A between neighbouring ions. This variation may be described by an expansion o f F (1) in terms o f z 0 as:
1
F(1)(z0) = F ( I ) ( 0 ) + \ ~ z ~ ] 0 z 0 + . . . .
(2)
Insertion of (2) into (1) reveals, that the sum o f the dipole moments of neighbouring NO~--ions does not vanish any longer, but becomes proportional to the product of F(1)(0) and its spatial gradient (VF(1))0" The corresponding macroscopic polarization may be written as: pNk = XN • E ( 1 ) V E t 1 ) , where E ft) is the macroscopic electric field, i.e. an appropriate average over F (1).
(3)
(6)
where the order of 1 and m deviates from the usual dot product convention [9]. II-EII I~ exp (ik I1 )r), one can write D/NL=47r ~
efr A l l /.,{l) X/mn L m
l~l, II
(7) elf ~ with X/ran = i
. z.tl ) X/lmn"l •
The effective nonlinear susceptibility Xelf depends on the wavevector k (1) of the incident laser light and may therefore be interpreted as a spatial dispersion term. It should be noted that the above suggestions on the molecular origin of X only represent the simplest approximation. Additional multipole moments of NO~ being of higher order than (1) have to be taken into account. The macroscopic description by ( 5 ) - ( 7 ) , however, remains valid in any case.
3. Experimental procedure and results As source of fundamental radiation we used a Qswitched N d 3 + - Y A G laser with an output power of about 20 kW and a repetition rate of 13 kHz. Because of the small SHG efficiency our investigations had to be restricted to collinear phase-matching (PM) between fundamental and second-harmonic wave. 235
Volume 16, number 2
OPTICS COMMUNI(TATIONS c- oxis
type II y (1)÷I~(11_ ~{21 c e- e
, "\
......\
~I
eI
¢:30 ° S
_1o
@o
+1°
~-®I
fig. 2. Upper part: cones of co]linear phase-matching in NaNO3. Lower part: phase-matching maximum at a fixed value of @. As shown by fig. 2 the PM directions form two concentric cones around the c-axis with apertures 01 = 17. I ° and 011 = 23.5 °, respectively. The corresponding wavevector conditions are 2k(o1) =k{e 1)
(inner cone, PM of type 1)
(8a)
k(l)+k~l)o =k~ 2) (outer cone, P M o f t v p e I I ) .
(8b)
and
ltere the ordinary and the extraordinary waves are indicated by the subscripts o and e, while the fundamental and the harmonic frequencies are distinguished by the superscripts (1) and (2). The apertures 01 and 011 of the PM cones derived from (8a), (8b), and the refractive indices of NaNO 3 are in agreement with the experimental values already quoted above. The refractive indices are:
February 1976
the SHG efficiency as a function of the azimuth angle 0 for both types of PM. Planar crystal slabs with a thickness of about 4 mm were used. Their front faces were perpendicular to the c-axis. A goniometer head allowed to adjust the direction of the laser beam inside tile crystal. In particular the direction angles 0 and g5 (fig. 2) could be varied independently by rotating the sample either around an axis perpendicular to c, or around c itself. The starting direction 0 = 0 was chosen parallel to one of the three two-fold axes perpendicular to c. The lower part of fig. 2 shows the second-harmonic intensity as a function of 0 at a fixed value of 4~. Tire half-width of the maximum is de termmed by the divergency of the laser beam slightly focused onto the sample by a lens of./`.= 14 cm. The area under the maximum proved to be a reproducible measure of the SttG efficiency in the PM direction specified by 01,i1 and ~). The dots in fig. 3 represent the nreasured SHG efficiency as a function o f ~ for both types of PM. As refers to a three-fold rotation axis its range has been confined to 120 °. The maxima of both curves have been set equal to 1. Their true ratio is given by •(2)
/i(2) =9+ ~ I1 max - - •
I max /
(10)
In deriving this value the walk-off and focusing effects [11] have already been taken into account. Due to the multi-mode structure and the oblique incidence of the laser radiation the maximum secondharmonic powe~ has been only about one photon per laser pulse. This order of magnitude is in line with an estimation based on section 2, according to which the nonlinear dielectric displacement in a centrosymmetric crystal is smaller than that in a non-centrosymmetric one by a factor of k (1) z 0 10 3 10 2 [2].
4. Interpretation
,,(o1)=1.567,
,,(e1)= 1.331
(X(I)= 1 , 0 6 , m ) ,
n}2)=1.595,
n(e2)=1.337
0t(2)=0.53/~m).
(9)
They have been obtained from a Sellmeier dispersion formula fitted to existing experimental data [ 10]. In order to demonstrate that the observed SHG can be adequately described by (5) we have measured 236
Inserting (5) into Maxwell's equations we have obtained formulas describing the efficiency of phasematched SHG in NaNO 3 due to the fourth-rank tensor X. The principal terms may be written as: PM of type 1: l}2)~(A +Bsin3•) 2,
(lla)
Volume 16, number 2
1
0,5
~ •
.
0o
.
.
OPTICS COMMUNICATIONS
°
.
a s e - matching
.
20 °
type 1
. . . .
LO °
60 °
80 °
100 °
120 °
February 1976
Here oz(1) and &(2) denote the angles between electric field and dielectric displacement of the extraordinary fundamental and harmonic wave, respectively. The components of X are referred to a crystal coordinate system, the axes of which are parallel to one of the three C~-axes of the NO~-ions (x 1), the projection of one of the three rhombohedral basis vectors onto the NO~-plane (x2), and the c-axis (x3). In a good approximation X/htm may be assumed to be symmetric in its last two indices. It is divided into parts X(S) and X(A) being symmetric and antisymmetric in their first two indices: x(S) 1 i:m. = ~ (Xjlm. + X:/m.) an d
(12) =
0
i 20 °
0o
• ;0 o
60 °
80 °
100 ° ¢
120° )
Fig. 3. Efficiency of phase-matched SHG in NaNO3 as function of the azimuth angle ¢. For 4) = 0 ° the wavevector of the laser radiation lies in one of the vertical mirror planes of the NO~-ions. The glide planes of the crystal structure correspond to 0 = 30° and 90 °. Dots: experimental values; full lines: theory.
with A = I(x~S~I 1 + v(S) asin ot(2) ~-1122 j + t - (S) _v(S) ~sin(20 I + a ( 2 ) ) 2(X3311 /H122 j and B = v (A) cos ,~(2) + v(S) cos(201 + o~(2)) A2311 ~ ~-2311 PM of type II: ( c cos 3 , ) 2 ,
(lib)
with C = v(A) cos ot(2) cos (0ii + ot(1)) ~-2311 + v(s) ~-2311 c°s(201I + °~(2)) c ° s ( 0 I I + a(1)) (n(2)) 2 + .,(S) *-1123
(n(2)) 2 (n(2)~ 2 O
--'-
e
l
sin (Oil + c / t ) )
-
X,:m.).
Apart from a different sign and from omitting X(A) formula (1 la) has already been quoted by Pershan [8]. This author also suggested to measure the angular dependence o f l (2) in order to prove that the observed SHG is in fact due to the nonlinear source term (5). The full lines in fig. 3 show i(2)(¢) as follows from (I la) and (1 lb). The agreement between experimental points and theoretical curves indicates that the above interpretation of SHG in NaNO 3 is correct. There might be one objection against this conclusion. Internal strains and defects may destroy the centrosymmetry of the sample under study. The observed SHG would then arise from the usual third-rank tensor relating the nonlinear polarization to the square of the electric laser field. The dependence of I (2) on * would be the same as in fig. 3, if strains and defects change the point group symmetry of the real crystal from 3m to 3m or 3 [12]. In that case, however, the maxima '/~I2) minx and /}2)max would be nearly equal and would not differ by an order of magnitude as has been found experimentally according to (10). Thus the observed SHG cannot be due to deviations from the 3m symmetry, but results from the nonlinear source term (5) involving the spatial gradient of the electric field.
Acknowledgement The authors wish to thank the Deutsche 237
Volume 16, nunaber 2
OPTICS COMMUNICATIONS
F o r s c h u n g s g e m e i n s c h a f t for s u p p o r t o f this work.
References [ l ] R.W. Terhune, P.D. Maker and C.M. Savage, Phys. Rev. Lett. 8 (1962) 404. [2] J.E. Bjorkhohn and A.E. Siegman, Phys. Rev. 154 (1967) 851. [31 V.S. Suvorov and A.S. Sonin, Soy. Phys. JI:,TP 27 (1968) 557: G. Dolino, L. Lajzerowicz and M. Valtade, Phys. Rev. B2 (197(I) 2194. [4] 1. l:reund, Phys. Rev. Lelt. 19 (1967) 1288: 1. Freund and L. Kopf, Phys. Rev. Lett. 24 (1970) 1017.
238
February 1976
[5] 1t. Vogt, Phys. Stat. Sol. (b) 58 (1973) 705. [6] ('.(7. Wang and A.N. Duminski, Phys. Rev. Lett. 20 (1968) 668. [7] S. Kielich and R. Zawodny, Proc. of the Conf. Optical Properties of tlighly Transparent Solids (Watervilte Valley, New ttampshke, USA, 1975), to be published. [8] P.S. Pershan, Phys. Rev. 130 (1963) 919. [9] L. Rosenfeld, Theory of F,lectrons (North-Holland Publ. Comp., Amsterdam, 1951), p. 111. ll0] L.S. Ivlev and S.I. Popova, Izv. VUZ l:iz. 5 (1972) 9l; Landolt-B6rnstein Vol. I1/8 (Springer-Verlag, Berlin, 1962), p. 2 48. [111 J.E. Bjorkhohn, Phys. Rev. 142 (1966) 126. l121 J. Midwinter and J. Warner, Brit. J. Appl. Phys. 16 (1965) 1135.
OPTICS (7OMMUNICATIONS
February 1976
OPTICAL SECOND-HARMONIC GENERATION IN A CENTROSYMMETRIC CRYSTAL INVOLVING THE SPATIAL GRADIENT OF THE ELECTRIC FIELD L. ORTMANN and tt. VOGT 11. t'hvsikalisches h~stitut der Universiti)t Ki)IH, 5 K6ln 41, UHirersitiitsstrasse 14, Fed. Rep. Germany
Received 22 October 1975
Phase-matched second-harmonic generation has been observed in centrosymmetric sodium nitrate (NaNO3) as function of direction. Experimental results can be adequately described by a nonlinear source term being proportional to the product of the fundamental electric field and its spatial gradient.
1. Introduction Optical second-harlnonic generation (StiG) in centrosynrmetric crystals has been studied rather rarely. Terhune et al. [11 and later oll Bjorkhohn and Siegman [2] observed phase-matched SttG in centrosymmetric calcite (CaCO3). They focused their attention on the increase of the second-ha> monic intensity, when disturbing the inversion symmetry by an externally applied, static electric field. In the course of studies on structural phase transitions StlG was sometimes found not only in the non-centrosymlnetric low-temperature phase, but also in the centrosymmetric one above the transition temperature [3 - 5]. Explanations of this phenomenon have been suggested in the case of NIt4CI [4] and NaNO 2 [5]. In simple centrosymmetric crystals like the alkali halides, weak SHG was also detected [6]. It has been shown, that this effect is only partly typical of the crystal bulk, but mainly arises from discontinuities at the crystal surfaces. Recently Kielich and Zawodny 17] have summarized existing theoretical and experinrental results on SHG in the case of inversion symmetry. The present paper deals with phase-matched SHG in single crystals of sodium nitrate (NaNO3), which has the same centrosymmetric crystal structure of point group 3m as calcite. Our experimental results allow the conclusion, that the observed SHG is an effect of spatial dispersion involving the gradi234
enl of the fundamental electric field.
2. Crystal structure of NaNO 3 and origin of SHG The rhombohedral bmrolecular unit cell of NaNO 3 is drawn in fig. 1. The structural feature of most concern to SHG is the alternating orientation of the NO 3-ions. Within the array of NO,--ions along the three-fold c-axis each NO,--ion is rotated by 60 ° or equivalently 180 ° with respect to its neighbours. Due to this antiparallel ordering the intermediate Na+-sites become centres of inversion, although the individual NO3-ions have the symmetry of equilateral triangles and hence are not centrosymmetric. As the electronic polarizabilities of NO 3 are much larger than those of Na + the second-harmonic light may be assumed to arise from nonlinear multipole moments induced in a NO3-ion. The lowest order term to be considered is a dipole m o m e n t of the form pNL = 13 : F(1)/7(1) ,
(1)
where the third-rank tensor 13 represents the nonlinear polarizability and F (1) is the h)cal electric field associated with the incident laser radiation. Because of the antiparallel ordering, the nonlinear polarizabilities 13 and hence the dipole moments pNL of neighbouring NO,--ions have opposite signs
Volume 16, number 2
OPTICS COMMUNICATIONS
February 1976
On the other hand the opposite dipole moments
_+~ : F~I)(0) F(I)(0) of neighbouring NO~--ions represent a quadrupole moment, leading to a macroscopic nonlinear quadrupole moment density: QNL = xQ : E l l ) E ( 1 ) .
(4)
Both source terms (3) and (4) determine the nonlinear part D NL of the dielectric displacement [8] : D N L = 4 r r ( P NL
v.QNL)=4rcXiE
Ill ~ E (1).
(5)
Here X is a tensor of fourth rank, existing in centrosymmetric as well as non-centrosymmetric crystal phases. For later use it is convenient to define the four subscripts of X according to DNL=47r ~ X//m,,E,(, I)3/f~'l) l, tn .tt 3Xl
Fig. 1. Unit cell of NaNO3. Note: The vertical mirror planes of the NO]-ions are rotated by 30° with respect to the glide planes built up by the e-axis and the rhombohedral basis vectors. if they are referred to a crystal coordinate system. Thus they cancel each other and no corresponding macroscopic polarization can result being directly proportional to the square of the electric laser field. Due to the finite wavelength of light, however, F (1) slightly varies along the distance z 0 = 2c = 8.41 A between neighbouring ions. This variation may be described by an expansion o f F (1) in terms o f z 0 as:
1
F(1)(z0) = F ( I ) ( 0 ) + \ ~ z ~ ] 0 z 0 + . . . .
(2)
Insertion of (2) into (1) reveals, that the sum o f the dipole moments of neighbouring NO~--ions does not vanish any longer, but becomes proportional to the product of F(1)(0) and its spatial gradient (VF(1))0" The corresponding macroscopic polarization may be written as: pNk = XN • E ( 1 ) V E t 1 ) , where E ft) is the macroscopic electric field, i.e. an appropriate average over F (1).
(3)
(6)
where the order of 1 and m deviates from the usual dot product convention [9]. II-EII I~ exp (ik I1 )r), one can write D/NL=47r ~
efr A l l /.,{l) X/mn L m
l~l, II
(7) elf ~ with X/ran = i
. z.tl ) X/lmn"l •
The effective nonlinear susceptibility Xelf depends on the wavevector k (1) of the incident laser light and may therefore be interpreted as a spatial dispersion term. It should be noted that the above suggestions on the molecular origin of X only represent the simplest approximation. Additional multipole moments of NO~ being of higher order than (1) have to be taken into account. The macroscopic description by ( 5 ) - ( 7 ) , however, remains valid in any case.
3. Experimental procedure and results As source of fundamental radiation we used a Qswitched N d 3 + - Y A G laser with an output power of about 20 kW and a repetition rate of 13 kHz. Because of the small SHG efficiency our investigations had to be restricted to collinear phase-matching (PM) between fundamental and second-harmonic wave. 235
Volume 16, number 2
OPTICS COMMUNI(TATIONS c- oxis
type II y (1)÷I~(11_ ~{21 c e- e
, "\
......\
~I
eI
¢:30 ° S
_1o
@o
+1°
~-®I
fig. 2. Upper part: cones of co]linear phase-matching in NaNO3. Lower part: phase-matching maximum at a fixed value of @. As shown by fig. 2 the PM directions form two concentric cones around the c-axis with apertures 01 = 17. I ° and 011 = 23.5 °, respectively. The corresponding wavevector conditions are 2k(o1) =k{e 1)
(inner cone, PM of type 1)
(8a)
k(l)+k~l)o =k~ 2) (outer cone, P M o f t v p e I I ) .
(8b)
and
ltere the ordinary and the extraordinary waves are indicated by the subscripts o and e, while the fundamental and the harmonic frequencies are distinguished by the superscripts (1) and (2). The apertures 01 and 011 of the PM cones derived from (8a), (8b), and the refractive indices of NaNO 3 are in agreement with the experimental values already quoted above. The refractive indices are:
February 1976
the SHG efficiency as a function of the azimuth angle 0 for both types of PM. Planar crystal slabs with a thickness of about 4 mm were used. Their front faces were perpendicular to the c-axis. A goniometer head allowed to adjust the direction of the laser beam inside tile crystal. In particular the direction angles 0 and g5 (fig. 2) could be varied independently by rotating the sample either around an axis perpendicular to c, or around c itself. The starting direction 0 = 0 was chosen parallel to one of the three two-fold axes perpendicular to c. The lower part of fig. 2 shows the second-harmonic intensity as a function of 0 at a fixed value of 4~. Tire half-width of the maximum is de termmed by the divergency of the laser beam slightly focused onto the sample by a lens of./`.= 14 cm. The area under the maximum proved to be a reproducible measure of the SttG efficiency in the PM direction specified by 01,i1 and ~). The dots in fig. 3 represent the nreasured SHG efficiency as a function o f ~ for both types of PM. As refers to a three-fold rotation axis its range has been confined to 120 °. The maxima of both curves have been set equal to 1. Their true ratio is given by •(2)
/i(2) =9+ ~ I1 max - - •
I max /
(10)
In deriving this value the walk-off and focusing effects [11] have already been taken into account. Due to the multi-mode structure and the oblique incidence of the laser radiation the maximum secondharmonic powe~ has been only about one photon per laser pulse. This order of magnitude is in line with an estimation based on section 2, according to which the nonlinear dielectric displacement in a centrosymmetric crystal is smaller than that in a non-centrosymmetric one by a factor of k (1) z 0 10 3 10 2 [2].
4. Interpretation
,,(o1)=1.567,
,,(e1)= 1.331
(X(I)= 1 , 0 6 , m ) ,
n}2)=1.595,
n(e2)=1.337
0t(2)=0.53/~m).
(9)
They have been obtained from a Sellmeier dispersion formula fitted to existing experimental data [ 10]. In order to demonstrate that the observed SHG can be adequately described by (5) we have measured 236
Inserting (5) into Maxwell's equations we have obtained formulas describing the efficiency of phasematched SHG in NaNO 3 due to the fourth-rank tensor X. The principal terms may be written as: PM of type 1: l}2)~(A +Bsin3•) 2,
(lla)
Volume 16, number 2
1
0,5
~ •
.
0o
.
.
OPTICS COMMUNICATIONS
°
.
a s e - matching
.
20 °
type 1
. . . .
LO °
60 °
80 °
100 °
120 °
February 1976
Here oz(1) and &(2) denote the angles between electric field and dielectric displacement of the extraordinary fundamental and harmonic wave, respectively. The components of X are referred to a crystal coordinate system, the axes of which are parallel to one of the three C~-axes of the NO~-ions (x 1), the projection of one of the three rhombohedral basis vectors onto the NO~-plane (x2), and the c-axis (x3). In a good approximation X/htm may be assumed to be symmetric in its last two indices. It is divided into parts X(S) and X(A) being symmetric and antisymmetric in their first two indices: x(S) 1 i:m. = ~ (Xjlm. + X:/m.) an d
(12) =
0
i 20 °
0o
• ;0 o
60 °
80 °
100 ° ¢
120° )
Fig. 3. Efficiency of phase-matched SHG in NaNO3 as function of the azimuth angle ¢. For 4) = 0 ° the wavevector of the laser radiation lies in one of the vertical mirror planes of the NO~-ions. The glide planes of the crystal structure correspond to 0 = 30° and 90 °. Dots: experimental values; full lines: theory.
with A = I(x~S~I 1 + v(S) asin ot(2) ~-1122 j + t - (S) _v(S) ~sin(20 I + a ( 2 ) ) 2(X3311 /H122 j and B = v (A) cos ,~(2) + v(S) cos(201 + o~(2)) A2311 ~ ~-2311 PM of type II: ( c cos 3 , ) 2 ,
(lib)
with C = v(A) cos ot(2) cos (0ii + ot(1)) ~-2311 + v(s) ~-2311 c°s(201I + °~(2)) c ° s ( 0 I I + a(1)) (n(2)) 2 + .,(S) *-1123
(n(2)) 2 (n(2)~ 2 O
--'-
e
l
sin (Oil + c / t ) )
-
X,:m.).
Apart from a different sign and from omitting X(A) formula (1 la) has already been quoted by Pershan [8]. This author also suggested to measure the angular dependence o f l (2) in order to prove that the observed SHG is in fact due to the nonlinear source term (5). The full lines in fig. 3 show i(2)(¢) as follows from (I la) and (1 lb). The agreement between experimental points and theoretical curves indicates that the above interpretation of SHG in NaNO 3 is correct. There might be one objection against this conclusion. Internal strains and defects may destroy the centrosymmetry of the sample under study. The observed SHG would then arise from the usual third-rank tensor relating the nonlinear polarization to the square of the electric laser field. The dependence of I (2) on * would be the same as in fig. 3, if strains and defects change the point group symmetry of the real crystal from 3m to 3m or 3 [12]. In that case, however, the maxima '/~I2) minx and /}2)max would be nearly equal and would not differ by an order of magnitude as has been found experimentally according to (10). Thus the observed SHG cannot be due to deviations from the 3m symmetry, but results from the nonlinear source term (5) involving the spatial gradient of the electric field.
Acknowledgement The authors wish to thank the Deutsche 237
Volume 16, nunaber 2
OPTICS COMMUNICATIONS
F o r s c h u n g s g e m e i n s c h a f t for s u p p o r t o f this work.
References [ l ] R.W. Terhune, P.D. Maker and C.M. Savage, Phys. Rev. Lett. 8 (1962) 404. [2] J.E. Bjorkhohn and A.E. Siegman, Phys. Rev. 154 (1967) 851. [31 V.S. Suvorov and A.S. Sonin, Soy. Phys. JI:,TP 27 (1968) 557: G. Dolino, L. Lajzerowicz and M. Valtade, Phys. Rev. B2 (197(I) 2194. [4] 1. l:reund, Phys. Rev. Lelt. 19 (1967) 1288: 1. Freund and L. Kopf, Phys. Rev. Lett. 24 (1970) 1017.
238
February 1976
[5] 1t. Vogt, Phys. Stat. Sol. (b) 58 (1973) 705. [6] ('.(7. Wang and A.N. Duminski, Phys. Rev. Lett. 20 (1968) 668. [7] S. Kielich and R. Zawodny, Proc. of the Conf. Optical Properties of tlighly Transparent Solids (Watervilte Valley, New ttampshke, USA, 1975), to be published. [8] P.S. Pershan, Phys. Rev. 130 (1963) 919. [9] L. Rosenfeld, Theory of F,lectrons (North-Holland Publ. Comp., Amsterdam, 1951), p. 111. ll0] L.S. Ivlev and S.I. Popova, Izv. VUZ l:iz. 5 (1972) 9l; Landolt-B6rnstein Vol. I1/8 (Springer-Verlag, Berlin, 1962), p. 2 48. [111 J.E. Bjorkhohn, Phys. Rev. 142 (1966) 126. l121 J. Midwinter and J. Warner, Brit. J. Appl. Phys. 16 (1965) 1135.