Kurtz Powder Technique Basic Paper
Kurtz Powder Technique Basic Paper
Kurtz Powder Technique Basic Paper
An experimental technique using powders is described which permits the rapid classification of materials
according to
(a) magnitude of nonlinear optical coefficients relative to a crystalline quartz standard and
(b) existence or absence of phase matching direction(s) for second-harmonic generation.
Results are presented for a large number of inorganic and organic substances including single-crystal
data on phase-matched second-harmonic generation in HIO" KNbO" PbTiO" LiClO.· 3H20, and CO(NH2)..
Iodic acid (HIO,) has a nonlinear coefficient d14'''-'1.5 Xd'l LiNbO,. Since it is readily grown from water solu-
tion and does not exhibit optical damage effects, this material should be useful for nonlinear device applica-
tions.
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EVALUATION OF NONLINEAR OPTICAL MATERIALS 3799
harmonic is detected by a photomultiplier and displayed packing. Particle sizes were checked by standard
on an oscilloscope. A reference beam is obtained by use optical microscopy techniques. In the case of materials
of beam splitter placed ahead of the sample. This such as quartz, zinc oxide, and other materials having
enables the intensity of the fundamental or second known nonlinear coefficients and available in single-
harmonic generated in a reference sample8 (including crystal form, powders were made from single crystals
variations in the intellsities and mode structure of using a Spex vibrating ball mill, and then graded using
successive pulses) to be monitored by displaying both standard sieves. A similar procedure was followed in the
signals simultaneously on a dual-beam scope. The case of minerals of polycrystalline or single-crystalline
system also permitted the insertion of narrow-pass character.
filters at the second-harmonic wavelength between the
Shott filters and the photomultiplier to eliminate C. Single-Crystal SHG Apparatus
spurious signals (such as those due to multiple-photon A block diagram of the apparatus being used for
absorption) . In order to improve the efficiency of second- single-crystal SHG measurements is shown in Fig. 2.
harmonic collection at the detector, a parabolic reflector This system is an improved version of one developed by
was placed directly in front of the sample (i.e., between Smith9 for second-harmonic generation studies. This
the laser and the sample) with a small access hole for system differs from previous systems in that continuous
the laser beam. In certain cases described in the next recording of second-harmonic intensity as a function of
section this reflector was replaced with an integrating angle, temperature, or other variables is possible even
sphere, the sample being mounted at the center of the in crystals having nonlinear coefficients two orders of
sphere. For most of the work a Nd:CaW04 laser was magnitude or more smaller than quartz. It thus permits
used having peak pulse powers in the vicinity of 100 W a rapid and accurate measurement of the nonlinear
and pulsewidths of 200 fJ.sec (i.e., 0.02 J). The repetition coefficients and phase-matching properties of new
rate for this laser was limited to a maximum of one shot materials, thereby providing both a verification of the
every 30 sec. The beam diameter at the sample was powder SHG results and a more detailed description of
,...,,5 mm. A Nd:YAG laser was also used. It was Q- the nonlinear optical properties once single crystals of a
switched by a rotating mirror at a rate of 400 Hz. new material become available.
Peak powers in this case were,...,,1 kW with pulsewidths
of ,...,,200 nsec. III. EXPERIMENTAL RESULTS
B. Sample Preparation and Mounting In order to determine the essential features of second-
harmonic generation in thin powder layers, a series of
Samples were prepared and mounted using several
experiments were performed to measure the dependence
techniques. For qualitative results and for most initial
of second-harmonic intensity on the following param-
survey work a thin layer (,...,,0.2 mm) of ungraded
eters (see Fig. 3): (1) angle between detector and
powder was placed on a microscope slide and held in
direction of incident light beam (=0), (2) powder-layer
place with transparent tape. For quantitative work,
thickness (= L), (3) average particle size (=r), and
powders were graded by use of standard sieves to the
(4) laser-beam diameter (=D).
desired range of particle sizes (usually 75-150 fJ.) and
Within certain ranges of layer thickness and particle
loaded into a quartz cell of known thickness (0.2 mm,
size it was found that a fairly simple and reproducible
1 mm, etc.) with the aid of a vibrator to assure uniform
dependence on each of the above parameters could be
obtained. The following discussion is limited to this
domain. Basically the region is one in which r«L«D.
This ordering insures that the fundamental beam
strikes a large number particles of random orientation,
thereby performing a significant statistical average. In
addition a planar geometry is retained. Taken together
these factors result in a considerable simplification of
TEKTRONIX the theoretical analysis which is given in the next
section.
®CH I
CH 2
2w)
Iw)
A. Angular Distribution
When the average particle size, layer thickness, and
beam diameter satisfy the inequality given above, it is
found that the angular distribution of second-harmonic
intensity for the powder in air is nearly cosinusoidal in 0
FIG. 1. Apparatus used for study of second-harmonic generation
in powders. for both the forward and backward scattering directions.
8 J. Ducuing and N. Bloembergen, Phys. Rev. 133, 1493 (1964); 9 R. G. Smith and M. F. Galvin, J. Quantum Electron. QE-3,
see also Ref. 1S, Chap. V, p. 131. 406 (1967).
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3800 s. K. KURTZ AND T. T. PERRY
DC LAIIP I
IIOV=D
AC
RELAY SUPPLY I
NOBATRON
OCR 150-IOA I
I
I
'7-56 3 FILTERS
CORNING SCHOTT
FILTER RG-3
Q- SmwD
YAG
1o........-..-rlo1r"Ti I.-.-r*'.-r:'; Eill
9524
PREAMP
ADYU
ROTATING PM AI02E
IIIRROR
COOLING
HOSES I
I r--
COOLING I I
I
CONTROL I L..-_---' ,-- I
I BODINE : I HV
I MttioMR ANGLE MARKER: I SUPPLY
- 900 TO
400 ... I RBC 2505 MICROSWITCH : I -1100 VOLTS
100V
ROTATI NG I _ _ _ _ _ _ _ _ _ _ _ _ _ _ ~-
L I
IIIRROR
DRIVE POWER AIIP
-~
DYNAKIT
MARKm
500n
MOTOR CONTROL CH 2
400111 GK HELLER BASE LINE
HP AUDIO CX-25 aNTRaL
OSC
200CD START CW CH I
o 0
OFF CCW
HP MODEL -1100B DUAL CHANNEL l~=:j-----'
STRIP CHART RECORDER
Thus, the sample considered as a source of second- obtain readings proportional to the total integrated
harmonic radiation behaves like an isotropic planar harmonic flux. As we shall see, however, the powder
radiator obeying Lambert's (cos) lawlO second-harmonic intensities for non-phase-matchable
materials are low. Combined with the relatively low
1(8,<1» =10 co'BIJ,
efficiency of the integrating sphere, this leads to detec-
where 8 and <I> are spherical polar angles and the photo- tion problems unless the fundamental power densities
metric intensity 10 (energy/sec/sr) is defined as are quite high. When the particle size r approaches the
layer thickness L appreciable deviations from the cos
10= f BdS, distribution are observed, particularly for phase- match-
able materials.
Also indicated in Fig. 3 is the intensity distribution
where B is the photometric brightness and dS is a surface
when the same powder is contained in a liquid of match-
element. A typical measured angular dependence is
ing refractive index. (In the case of ADP shown here,
illustrated in Fig. 3. The photometric intensity 10 of the
which is uniaxial, the index of the liquid was chosen to
front and back surfaces of the powder layer is not in
approximately match the average of the ordinary and
general equal for powders in air. As the particle size
extraordinary indices, i.e., nliq = 1.51.) The advantage
decreases, the second-harmonic lobe in backward
of the index matching of the powder is evident in that the
direction grows while that in the forward direction
second harmonic is not significantly scattered due to
diminishes (relative to the situation where they are
reflection and refraction at particle interfaces, and
equal). Thus, sampling of the harmonic intensity in the
hence all the harmonic flux is contained in a narrow cone
forward direction alone can give misleading results if
in the forward direction.
the backward lobe is not taken into account. An
integrating Ulbicht spherell can be used in this case to B. Dependence of [2.. on Layer Thickness L
10M. Born and E. Wolf, Principles of Optics (The MacMillan The second-harmonic signal was found to vary
Company, New York, 1964), Chap. IV, fl. 182. linearly with layer thickness L for fixed particle size r
11 R. Ulbricht, Das Kugelphotometer (R. Oldenburg, Munich
and Berlin, 1920); D. G. Goebel, Appl. Opt. 6,125 (1967). after integrating over the 411" solid angle. This implies
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EVALUATION OF NONLINEAR OPTICAL MATERIALS 3801
1.o,.--------------------,
EXPERIMENT
-<>---<>-0-
- - - THEORY
0.1
• IN LIQUID
c:J IN AIR
0.0 .':-.1~~--'--'--'-'~~+.1.0,---.c.....J
..~_-'--....t.L......Jol........n-',J1Q!!-:0~-"-...L....J
90· 90·
6 Vfc
.01
.
r
10 100
0.2 /
f
I
~ --'0-<
f
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3802 S. K. KURTZ AND T. T. PERRY
and 5), where the intensity is inversely proportional to r time demonstrate how the division into SHG classes is
for this range of particle sizes. This difference in be- accomplished.
havior is due to the existence of a phase-matching We make the following basic assumptions:
direction13 in ADP and is in agreement with the theory
presented in the next section. It is in fact the striking (1) The particles are immersed in a transparent
difference in particle-size dependence which enables one liquid medium of nearly matching refractive index.
to distinguish phase-matchable materials on the basis of (2) The particles are single crystallites of nearly
second-harmonic measurements in powders. identical size, random orientation, and densely packed
(e.g., form factors, polycrystallinity, and voids are
neglected) .
IV. THEORY
(3) With appropriate modifications (such as aver-
It is obvious at the outset that a number of simplifing aging over angles) the theoretical expressions for
assumptions. are necessary to reduce a theoretical second-harmonic generation in single-crystal plane-
analysis of SHG in powders to manageable proportions. parallel slabs can be used in the solution of the problem.
If the simplifications are sound, the rigor can be A. The Single-particle Equations
regained when more extensive and detailed experi-
mental data are available. In this section we are Using the results of Kleinman14 and Bloembergen,16
primarily concerned with deriving expressions which will one arrives at the following expression for the electric
give a reasonable semiquantitative explanation of the field amplitude of the second-harmonic wave in a cubic
experimental data presented in Sec. III and at the same crystal
The notation differs somewhat from that of Bloem- where the parameters not previously defined are
bergen15 in that the symbols have the following defini-
tions: Ie = coherence length=A/4(~-nw),
r= thickness of plane-parallel slab,
w=frequency of fundamental optical wave, d2w =nonlinear optical coefficient [d i jk 2w (see Ref. 25)J,
P .LNLS = component of nonlinear optical polarization in A= wavelength of fundamental optical signal,
direction ...L to plane of transmission, a= [(nw+ 1) /(n2w+ 1) J exp[ - (a/2)rJ,
n", = index of refraction of nonlinear medium at a = absorption coefficient at second-harmonic fre-
angular frequency w, quency.
(I. = angle of transmission for polarization wave (k.)
measured from z axis, In deriving Eq. (4), absorption at the fundamental was
(IT = angle of transmission for homogeneous wave kT assumed to be zero, and the extinction coefficient k
measured from z axis, (i.e., a=47rkIA) at the second harmonic was assumed to
z=distance along surface normal of medium be small compared to n2w. Miller16 has derived an
(z=O at surface). expression similar to Eq. (4). The primary difference
between Miller's result and present form is that "a"
Equation (3) is completely general in that nw and n2w contains exp[ -(a/2)r] instead of e-ar (where a is
can be complex [e.g., one can include the effects of defined here in the customary fashion J2w(r) = Io2wr-ar) .
optical absorption in Eq. (3)]. For normal incidence The expression for "a" given here is the correct form.17
(COsOT= cos0 =1), one can manipulate Eq. (3) to
8 The terms [4n2w2/(nw+n2",)2J and [(nw+l)/n2w+lJ can
obtain an expression for the transmitted second-har- usually be set equal to unity since n2w-nw«1 for most
monic intensity I ext2w in terms of the incident intensity materials.
I ext'" (Iext being the intensity external to the slab) Equation (4) can be generalized without too much
difficulty to nonnormal incidence. The resulting ex-
X [647rl ed2w Iextw /A(nw+ 1) 2(n2w+ 1) J2 14D. A. Kleinman, Phys. Rev. 128, 1761 (1962).
16 N. Bloembergen, Nonlinear Optics (W. A. Benjamin, Inc.,
New York, 1965), Chap. 4.
X {(l-a)2+4a sin 2[ ( 7J'/2) (rile) J}' (4) 16 R. C. Miller, Proceedings of the International Conference on
Ferroelectrics (lnst. of Physics Czech. Acad. Sci., Prague, 1966),
Vol. I, p. 407. A slightly different form is given in R. C. Miller and
13 R. W. Terhune, P. D. Maker, and C. M. Savage, Appl. Phys. A. Savage, Appl. Phys. Letters 4, 169 (1966).
Letters 2, 54 (1963). 17 R. C. Miller (private communication).
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EVALUATION OF NONLINEAR OPTICAL MATERIALS 3803
TABLE I." Comparison of calculated and measured second-harmonic powder response relative to quartz standard.
Calc. Meas.
ZnO
(noo =1.95, n..,=2.05) daa,=43, 1'3=5.3 J.I. (-) (-) 4.6 6
d'31=13, laJ=4.81 J.I. (+) (-) 4.8
dm =14, 115=6.31 J.I. (- ) (+) 5.5
(+) (+) 5.7
CdS
(noo =2.34, n2oo=2.65) d33 ,=186, 133=1. 71 J.I. ( -) (-) 6.2 10
d311 =96,131 = 1.66 J.I. (-) (+) 8.9
dm = 105, h5 = 1. 89 J.I. (+) (-) 10.6
(+) (+) 13.2
ZnS
(n=2.34) d333 = 180, 133=5.3 J.I. (-) (- ) 31 100
d311 =90 (-) (+) 47
dU3 = (90) (+) (-) 64
(+) (+) 84
GaP
(noo =3.1, n.oo=3.51) dm = 350, 1123 = 1. 29 J.I. 11.9 15
d123 = 525, 1128 = 1. 29 J.I. 14.0
ADP
(noo= 1.48, n2w = 1.53) dl,a = 2.8, 1t'3» 100 p. 14 15
LiNbO,
(noo =2.233, n..,=2.231) d333 = 320, 133 = 5.82 p. 860 600
d311 =36,131 »100 J.I.
d222 =19, 122=5.82 J.I.
"Values for dijk 200 are taken from Ref. 25 and are given in units of 10-' cm/statvolt.
pressions are very complicated and will not be given we need to specify whether the second-harmonic fields
here. The two primary differences are an angular from different particles are phase-correlated or un-
dependence of the coherence length te and an additional correlated. For particle sizes much larger than an average
angular dependence due to reflection losses. coherence length (which we shall define in a moment)
we assume that the second-harmonic fields generated by
B. Summing over Particles. Angular Averages for different particles are un correlated in which case the
Materials which are not Phase-Matchable total second-harmonic intensity is just the sum of the
Let us assume for the moment that the fundamental contributions from each individual particle. Since each
beam strikes each particle at normal incidence. After individual particle is assumed to have arbitrary orienta-
traversing each particle the beam proceeds to the next tion and the fundamental traverses a large number of
in line. In traveling through the powder the beam particles we can calculate the contribution per particle
encounters approximately Llr particles, where r is the by performing a suitable average of pw [Eq. (4) ] over
average particle thickness. In order to proceed further all angles.
and calculate the resultant second-harmonic intensity The major angular dependent term in Eq. (4) is d2 w
since this is the component of a third-rank tensor whose
principal axis system coincides with the crystallographic
axes. For noncubic materials, the coherence length
I PHASE-MATCHABLE (eg.ADP)
I lc[ =A/4(n2w-nw) ] will also depend on angular orienta-
I
I tion through n2w and nw' Since we are excluding phase-
I
l-... matchable materials in this section (i.e., materials for
I which n2w-nw-+O for certain directions of propagation)
I
I we can define an average coherence length
I NON PHASE -MATCHABLE (eg.QUARTZ)
(5)
23456789101112
and proceed to deal separately with the angular aver-
aging of dijk2w. For phase-matchable materials one
cannot indulge in this simplification but must include
FIG. 7. Schematic representation of different particle-size
dependences for phase-matchable and non-phase-matchable the angular dependence of te explicitly before taking the
materials. angular averages. There is of course an intermediate
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3804 S. K. KURTZ AND T. T. PERRY
10,000"..-------
1,000 OKlO,
lOa
N
~
FIG. 8. Plot of second-harmonic intensities
~ (relative to quartz standard) as a function of
..~
3
~~
o
10
index of refraction, showing division into
SHG classes.
o.,_L~,:::--::,:----;~~~
1.4 1.5 1.6 1.7 1.8 1.9 20 2.1 2.2 23 24 25 26 27
REFRACTIVE INDEX
area of materials which are "almost" phase-matchable. It is of course very difficult to obtain powders having
The present treatment applies where the maximum an exact particle size r. In practice there will always be
birefringence is substantially less than the dispersion a distribution about some mean value r. If the rms
(e.g., in a negative uniaxial material with birefringence deviation of r from l' exceeds, le, then the oscillatory
equal to t of the dispersion, the maximum variation in behavior due to sin2[(11"/2) (r/le)J will be smeared out
coherence length is 1'0.130%). and we can replace this term by t. For particle-size
The problem of angular averages for second-order distributions having a ±1O% spread one would expect
polarizability tensors (such as dijk'2oJ) has been treated EO see some !?oderate oscillatory behavior in the region
by eyvin, Rauch, and Decius.18 By introducing the Ic~r~3 to Sic. This has indeed been observed in studies
direction cosines <PFi relating a laboratory-fixed co- of quartz powders (see Fig. 5) where noticeable minima
ordinate system F, G, H to a molecule (particle)-fixed have been observed at 21e and 41•. The dominant
system and utilizing the tensor transformation prop- particle-size dependence however comes from the
erties Ljr term in Eq. (7) which leads to an inverse rela-
tion between mean particle size and second-harmonic
intensity. F<?r r«le, one is tempted !o replace
the above authors arrive at expressions for ({3FGH 2) of the sin2[(11"/2) (1'11.) J in Eq. (7) by [(11"/2) (1'11.) J2, which
form leads to [2'" cr. rL which agrees with the observed
behavior (see Fig. 4). When the particle size is much less
({3FFF2) = 1/7 L: (3iii2 + (6/35) L: {3iii{3ijj+ (9/35) than the average coherence length the assumption of
i """j lack of correlation between harmonic generated in
x L: {3ii!+6/35 L: {3iij{3jkk+(12/35){3ijk2 (6) different particles is much more difficult to justify and
i~j ijk cyclic
may in fact be incorrect. An alternative approach which
and two similar expressions for ({3FGG2) and ({3FGH 2). leads to the same result (i.e., [2'" cr.rL) assumes that the
Using their results one can obtain expressions for second-harmonic fields from different particles are
« d2w) 2) in the laboratory reference frame for the various correlated provided the particles are separated by a
crystal-symmetry classes exhibiting second generation. distance which is less than a coherence lengtl];. Breaking
Some of these results for « d2w) 2) are given in the the cell thickness up)nto N regions (N=L/l.), each of
Appendix. Replacing (d2oJ)2 in (4) by its appropriate which contains N' = lel? particles, we treat the nonlinear
average and setting a=O we obtain the total second- polarization pNLS within each of the N regions by a
harmonic intensity from the powder for r»le one-dimensional random-walk model where the step
length is proportional to rJ2w. The net contribution of
3211" [6411"Iext"']2 each region to second-harmonic intensity is then pro-
Itota12",:::, -C- «rJ2w)2) }..(n",+1)2(n2",+1) portional to «d2",)2)IN' giving a total intensity from all
N regions
(7)
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EVALUATION OF NONLINEAR OPTICAL MATERIALS 3805
Refractive
Material Pt. group Part. size Color 12w/IQu8rtz'k.> SHG class index Comments
LiNbOa 3m 75-150 I-' white 600 A No=2.286 Known: for comparison only
Ne=2.200
HIOa 222 75-1501-'8 white b 300 A Nx=2.0103 101 cleavage (J. P. Remeika)
KIOa·2HIOa ungraded white 40 A(or B?) Ny=1.9829
Nz= 1.8547
KIOa 75-150 I-' white 1200-2400 A Nx= 1. 700
Ny=1.828
Nz=1.832
LiIOa 6 75-150 I-' white 300 A Difficult to grow 0. G. Berg-
man)
Ag2HgI4 42m 5-10 I-' yellow 1500 A 0. G. Bergman)
CU2HgI4 42m 5-10 Org. red 200 A
KNbO a mm2 75-1501-'" white b 2000 A (N~2.4) Mostly single domain (A.
Linz)
PbTiOa 4mm 75-1501-" off-white c 2400 A Gerson (J. Barshataky)
[C SH 4 (N02).] mm2 25 I-' light yellow 500 A Nx= 1.432 M-dinitrobenzene
Nz=1.839 (010) optic plane
42m 75-1501-" white b 400 A N=1.484 urea
N=1.602 (K. Nassau)
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3806 S. K. KURTZ AND T. T. PERRY
Refractive
Material Pt. group Part. size Color R/quartz SHG class index Comments
Equation (13) is independent of particle size r, When the nonlinear coefficients dii"'k and coherence
consistent with the experimental results of Sec. III (see lengths t. are known, it is possible to evaluate the ratio
Fig. 6) for larger. Forr~rpM/sinOm, Eqs. (9)-(13) are of second-harmonic intensities for powders of any two
no longer valid approximations. While a detailed solu- materials using the results given in Eqs. (13) and (14).
tion for this region has not been obtained, it can be Assuming identical particle size r(»l.) and funda-
shown by arguments similar to those used in deriving mental intensity I ext"', this intensity ratio has been
Eq. (8) that Iext 2'" decreases with decreasing l' in this calculated (relative to quartz) for six materials. The
region. The other nonlinear coefficients di ;k2", (for which results are shown in Table I and are seen to be in reason-
phase matching is not possible) will also contribute to able agreement with experiment.
the second-harmonic intensity but in the form derived
for non-phase-matchable materials, namely, D. Division into SHG Classes
Iext 2'" = {(321l'/ c) [64Iext'" /A(n",+ 1)2 (n2'" +1) ]2) Based on the foregoing analysis of SHG in powders
one can develop a useful division of nonlinear optical
X (d ijk )[L(1.F/2r].
2
(14)
materials into five distinct classes. Crude definitions of
For sufficiently large particle sizes (1'»1.) this contribu- these five classes were given in Sec. I. In order to be
tion will be small compared to that in Eq. (13) and can more specific let us write down the expressions for the
be neglected. intensities in both the phase-matched and unphase-
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EVALUATION OF NONLINEAR OPTICAL MATERIALS 3807
Refractive
Material Pt. group Part. size Color R/quartz SHG class index Comments
matched cases. We shall omit all parameters which are right of the vertical dashed line. In this region the
identical by virtue of using the same experimental phase-matchable materials have attained their maxi-
apparatus, powder size, etc. mum second-harmonic intensity independent of particle
size, and un-phase-matchable materials, while still
not phase-matchable
varying inversely with particle size, are well away from
7»1 (15) the region near 1. where substantial structure in the
second-harmonic intensity occurs (see Fig. 5).
Iext2wcr. (dPM2w)2[(~/4)rpMJ phase-matchable The second fact which is needed for the division into
7»rpM/sinOm • (16) SHG classes is a relation between J2w and refractive
index. Miller22 has shown from experimental data (and
Two additional facts are essential to obtain the the Garrett23-Bloembergen24.....Robinson25 anharmonic
desired division into SHG classes. First we note that a oscillator model for SHG have confirmed) that
particle size r of 100 J.I. generally meets the requirement
that r be much greater than the average coherence d2w cr. (n2.,2-1) (nwL1)2'"'-'(n2-1) 3. (17)
length. In the case of phase-matchable materials it also
satisfies the condition 1'»rpM/sinOm [see Eq. (13)J Thus in Eqs. (15) and (16) we can replace «(J2w)2) and
since typical values of rpM are in the range 1-10 J.I.. It is (dPM 2w )2 by (n2_1)6. After making this substitution
convenient in this respect to treat the quantity rPM/
sinOm as a fictitious average coherence length 1. for the 22 R. C. Miller, Appl. Phys. Letters 5, 17 (1964).
phase-matchable case. Thus, for an average particle 2' C. G. B. Garrett, IEEE J. Quantum Electron. QE-4, 70
(1968) .
size of 100 J.I. the inequality 1'»1. is satisfied for both 24 Ref. 15, Chap. I.
classes of materials. This region is shown in Fig. 7 to the 26 F. N. H. Robinson, Bel! System Tech. J. 46, 913 (1967).
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3808 s. K. KURTZ AND T. T. PERRY
V. MATERIALS SURVEY
A. Experimental Results
:I.
o
II) The survey results for about 100 inorganic and
:1.::1..:1.'1
.t-1.I')Otl')
<"')N....-I,t-...
organic compounds as well as a number of naturally
V occurring minerals are shown in Tables II-VI.
The measurements given in Tables II-VI were made
for powders in air. While the theory developed in the
preceding section applied to the index-matched liquid
case, the experimental results for ZnO (see Sec. III and
N .....
Figs. 4 and 5) and other materials indicate that
measurements in air are adequate (i.e., the correct ratio
to the quartz standard is retained) for assigning
materials to the various SHG classes provided one
0 takes care to collect all the second-harmonic flux.
II:
c.:. In the case of transparent materials the assignment
~
.;:: S-
0
to SHG classes is relatively straightforward using the
....'"oS II: limits set for the various classes in Sec. IV. For absorbing
::s 6 0Z sII: II: 0"
J ~
:l
q~ q
materials no adequate theory has been formulated. The
@ 0 ~
~ ~ results with absorbing materials having known non-
Z
OoS~
.~
0
u
~
:l U
ti ; ~~~~~ '" 0..0
linear properties (e.g., Se, Te) suggest that only
VSC!)f:Q '"
f:Q U'" .:t3 U u C!)~~c3 materials in classes A-C will exhibit detectable second
harmonic in the powder measurement. Very thin
layers7 were also found to be more desirable in this case
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TABLE VI. Summary of powder survey results: SHG class E.
Material Pt. group Part. size Color R/quartz SHG class Refractive index Comments
wulfenite
<:
PbMoO, 4/m ungraded off-white 0 E No=2.4053 :.-
Ne=2.2826 t-<
Gd2Ti20 7 ungraded white 0 E (L. Mercer) c:::
Pb.CI (VO,) 3 6/m 25 J1. yellow-orange 0 E No=2.628 vanadinite :.-
Ne=2.505 (K. Nassau) i-l
.....
ZnWO. 2/m ungraded white 0 E (L. G. Van Uitert) 0
Ba(N03). m3 ungraded white 0 E N=1.57 Z
Pb3MgNbaOp ungraded yellow 0 E (L. G. Van Uitert) 0
"%j
Hg(CHaCOO)2 ungraded light yellow 0 E mercuricacetate
BiPO, 622 ungraded white 0 E Z
Ba(Br03h 2/m 75-125 J1. white 0 E 0
ungraded white 0 E
Z
Pb(SCNh t-<
CJI,CI.C02H I ungraded white 0 E Nz-Nx=0.280 P-chlorobenzoic acid .....
CsPbCla ungraded white 0 E (H. J. Guggenheim) Z
i:%j
MgTi03 3 ungraded white 0 E No=2.31 (J. G. Bergman) :.-
Ne=1.95 :;d
MgC03 3m 25-125 J1. white 0 E No=1. 700 magnesite
Ne=1.509 (K. Nassau) 0
'"d
MnC03 3m 25-125 J1. pink 0 E No=1.817 rhodochrosite
Ne=1.597 (K. Nassau) i-l
.....
Pb3(PO.) 2 6/m 25 J1. white 0 E No=1.970 (K. Nassau) l.l
Ne=1.926 :.-
ungraded pink 0 E No=1.664 friedelite t-<
9(Mn, Fe)O·8SiO.·MnCh·7H2O
Ne=1.629 (K. Nassau)
yellow 0 E No=2.263 mimetite
a::
(F, CI) (PbAsO')3Pb2 6/m 75-150 J1.
Ne=2.239 (K. Nassau)
:.-
i-l
AbSiO. I 150 J1. white 0 E Nx=1.713 kyanite trl
Ny=1. 722 (K. Nassau) :;d
Nz=1.729 .....
PbO mmm ungraded yellow 0 E Nx=2.51 masicot :.-
Ny=2.61 t-<
Nz=2.71
en
PbO 4/mmm ungraded red 0 E No=2.665 litharge
Ne=2.535
Pb(IOa). mmm ungraded white 0 E Nx=2.15
Ny=2.15
Nz=2.18
WOa 2/m ungraded white 0 E V>
00
0
'0
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3810 S. K. KURTZ AND T. T. PERRY
ROTATION AXIS
W
~~,E'" em
, I I I I I I 1 I 1 I
(~100 f.L). At present only qualitative information can The converse situation of crystals belonging to well-
be obtained from the results on highly absorbing mate- established acentric point groups is of interest since if
rials. Rather than develop a theoretical model for this the powder technique is capable of detecting second
case an increase in the wavelength of the fundamental harmonic in any acentric structures, then it can be used
to a region where the material is transparent at both as a sensitive and reliable method for establishing the
fundamental and harmonic would appear to be the best presence or absence of a center of symmetry. The test
solution. A system which will operate in the 1.5 f.L (2w) most commonly used at present for this determination
to 3 f.L (w) region is currently being constructed for this is the Giebe-Scheibe28 piezoelectric test. While further
purpose. work is needed to establish the relative merits of the
Materials in SHG classes A and B are potentially SHG powder technique several observations are possible.
useful in nonlinear optical devices. Single crystals of a The optical method will work on very fine powders-
number of these materials have been grown or obtained whereas the piezoelectric test does not work in general
from the sources indicated in the comments column of when the particle size goes below some specific limit
Tables II-VI. The results to date have confirmed the which varies from material to material. 29 Electrical
assignments given in Tables II-VI. conductivity has no adverse effects on the optical
measurement (provided that free-carrier absorption is
B. Powder Method as a Test for Center of Symmetry
not significant). The optical method does however
In addition to those materials which were suggested depend on the transparency of the sample at the funda-
by calculations5 based on recent theoretical models4 of mental and second-harmonic wavelengths (e.g., a
the nonlinear coefficients, we purposely included in the negative SHG response is inconclusive when the material
survey several materials (e.g., PbO, MgTi03, PbMo04, is strongly absorbing at either of the aforementioned
etc.) which have a center of symmetry and hence wavelengths) whereas the piezoelectric test is independ-
belong to class SHG-E (all third-rank tensor elements ent of the optical properties of the material.
vanish26 identically due to symmetry restrictions). There are three crystallographic point groups which
Some of these latter materials were chosen in that they present problems for the center of symmetry test. These
met all the other requirements for a class SHG-A three point groups (432, 422, and 622) do not contain
material (see Sec. IV) except symmetry. Thus, any the inversion operator (i.e., materials in these classes
slight accentric distortion should cause a detectable lack a center of symmetry). In spite of this acentric
signal at the second harmonic. This check was con- character, materials in these classes are not expected to
sidered necessary for guarding against spurious effects exhibit SHG. In the case of cubic point 432, the piezo-
due to strain, bulk defects, surface phenomena, etc. electric moduli also vanish so that the Giebe-Scheibe
No case was found in which a material having a well- technique suffers from the same limitation. In the case
established crystal structure with a center of symmetry of the remaining two point groups 422 and 622, however,
also exhibited second-harmonic generation. The experi- the vanishing of the SHG moduli diik (2",) is a consequence
mental powder system is capable of operating over a of Kleinman's permutation rule,30 which does not apply
wide range, the dynamic range being currently used is to the piezoelectric case. Thus the SHG powder test
about eight orders of magnitude in second-harmonic may be definitive for absence of a center of symmetry
intensity, the lower limit of which is approximately (classes A-D) but the converse (class E) is only
10-2 of the SHG signal from the quartz standard. Any definitive for the trigonal, orthorhombic, monoclinic,
substantial increase in the sensitivity at the lower end and triclinic crystal classes. Further work is in progress
will eventually lead to the detection of second-harmonic to develop the optical powder method for use in deter-
generation due to higher-order effects (e.g., quadrupole mining the presence or absence of a center of symmetry
interactions27 ) even in centrosymmetric compounds. in naturally occurring minerals.31
Several examples of materials exhibiting pronounced
26 J. F. Nye, Physical Properties of Crystals, (Oxford University 28 E. Giebe and A. Scheibe, Z. Physik 33, 760 (1925).
Press, London, 1964). 29 J. P. Remeika (private communication).
27 R. W. Terhune, P. Maker, and C. M. Savage, Phys. Rev. 30 D. A. Kleinman, Phys. Rev. 126, 1977 (1962).
Letters 8, 21 (1962). 31 S. K. Kurtz and K. N. Nassau (unp\\bllshcd).
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EVALUATION OF NONLINEAR OPTICAL MATERIALS 3811
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3812 S. K. KURTZ AND T. T. PERRY
Some birefringence is present at the phase-matching the multidomain character of the crystals and absorp-
angles [i.e., p;;cO, see Eq. (l1)J but, using the beam tion at 2w (of unknown origin), making the results
crossing scheme of Ashkin et al.,45 one should in principle somewhat uncertain. It should be noted that the bire-
be able to get around this limitation. fringence data of Shirane et al. 5a (An""'O.Ol) at room
Iodic acid is optically active, exhibiting a rotation46 temperature are certainly inconsistent with the pos-
of 58.7°/mm at 5461 A and 74S/mm at 4360 A. For sibility of phase matching, making further study of the
propagation directions well away from the optic axes SHG properties essential before any conclusions
(tens of degrees) these rotations are greatly reduced concerning SHG class can be made.
and should not present any major difficulty for non-
linear applications. Additional work on the nonlinear VII. CONCLUSIONS
properties of iodic acid can be found elsewhereP The
The study of second-harmonic generation in powders
single-crystal results clearly establish that the class A
has been shown to yield useful information on the non-
assignment of HIOa is correct. linear optical properties of solids. An analysis has shown
C. KNbO a that the SHG powder data can be used to assign materi-
als to one of five possible classes. Two of these classes
A crystal cut from a boule of KNbO a grown by contain the phase-matchable materials. The ability of
Linz43 by pulling from a melt was found to contain some the technique to sort out new phase-matchable materials
nearly single domain regions. Phase-matched SHG was having large nonlinear coefficients makes it attractive
observed in these regions with a d2 w estimated to be for materials survey work particularly since the
"",4Xd311LiNbO a• We note that Wood49 has estimated difficult requirement of obtaining single crystals of
that the birefringence along the "c" axis is around 0.2, optical quality has been removed. The technique also
which is consistent with a phase-matching angle Om"",90°. promises to be a useful adjunct to the piezoelectric test
for a center of symmetry.
D. CO(NH 2h Using this powder technique in conjunction with
Very thin prismatic crystals of urea were grown from recently developed theories which enable one to predict
an alcohol solution by N assau.50 A broad phase-matching the magnitude of the nonlinear coefficients, one has a
peak was observed at an external angle of "",20° con- powerful combination of tools which should provide a
sistent with the large natural birefringence and short substantial increase in the number of new materials for
path length. use in nonlinear optics applications.
E. LiCIO 4' 3H 20 The results of an initial survey of approximately 100
compounds are reported. Second-harmonic generation
Prismatic crystals of this material have been grown was detected for the first time in 56 of these materials.
by Storey51 from water solution. Measurements were Twenty-seven have been assigned to the phase-match-
made on a section "",10 mmX5 mmX5 mm of moderate able categories. Single-crystal measurements on four of
optical quality. Phase matching was observed for these latter compounds have verified the existence of
Om=22.5°. The nonlinear coefficient dan has been esti- phase-matching directions for second-harmonic genera-
mated to be slightly smaller than d111 of crystalline tion. One of these materials (HIO a) shows considerable
quartz. The single-crystal measurements are thus promise for nonlinear optical device application~ since in
consistent with the class C assignment of this material. addition to being phase-matchable: (1) It is readily
F. PbTiO a grown from solution. (2) The nonlinear coefficient is
about 1.5 times larger than d31 in LiNbO a• (3) It does
Multidomain crystals of PbTiOa were grown by not exhibit optical damage effects.
Barshatky 52 'from the flux. Phase matching seems to
occur near Om""'90° but difficulty was encountered with ACKNOWLEDGMENTS
The authors wish to thank R. C. Miller and R. A.
45 A. Ashkin, G. D. Boyd and D. A. Kleinman, Appl. Phys.
Letters 6, 179 (1965). Laudise for suggesting this work and their continuing
46 International Critical Tables, (McGraw-Hill Book Co., New interest and encouragement throughout its various
York, 1929), Vol. VII, p. 353. phases.
47 S. K. Kurtz, J. G. Bergman, Jr., and T. T. Perry, Appl.
Phys. Letters 12, 186 (1968); S. K. Kurtz, J. G. Bergman, Jr., We are especially indebted to the many people who
and T. T. Perry, Bull. Am. Phys. Soc. 13, 388 (1968). supplied us with the materials which made this study
48 Grown by a modified Kyropoulos technique similar to that
used for BaTiO.; A. Linz, V. Belruss, and C. S. Naiman, Proc.
possible, particularly J. G. Bergman, J. P. Remeika, K.
Electrochem. Soc. 112, 60C (1965). Nassau, R. Storey, J. Barshatky, E. Kolb, A. A.
49 E. A. Wood, Acta Cryst. 4, 353 (1951). Ballman, H. J. Guggenheim, L. G. Van Uitert, and
60 Grown by evaporation in air at room temperature from
saturated solution of urea in methanol. A. Linz.
51 Solution growth on a rotating seed in a Holden crystallizer
(A. Holden, The Faraday Symp. on Crystal Growth, Bristol,
1949, Discussion No.5). 63 G. Shirane, R. Pepinsky, and B. C. Frazer, Acta Cryst. 9,131
II Grown from PbO flux at llOO°C and cooled at lO°C/h. (1956) .
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EVALUATION OF NONLINEAR OPTICAL MATERIALS 3813
APPENDIX: CALCULATION OF «d 2w )2) FOR (c) Point groups 6mm and 4mm have nonzero tensor
DIFFERENT CRYSTAL POINT GROUPS elements, d33l w
For unpolarized fundamental and second harmonic, d3U2w = dS222w = d2232w = d1l32w
«d2w )2) = «dFFFl"') 2)+ <(dFGG2"') 2)+ <(dFGH2w) 2) <(d2w) 2) = (19/105) (d3332w ) 2+ (26/105)d3332wd3l12w
= (19/105).L: (diii2w) 2+ (13/105) .L: dii/wdijj2w
i i,ej
+ (114/105) (dlllwP.
+(44/105) .L: (d ii/ )2+(13/105)
w (d) Point group 3m has nonzero tensor elements,
i,ej d3332,.,
The Nature of Thermally Induced Stresses in Silicon and Their Relation to Observed
X-Ray Diffraction and Birefringence Phenomena
GENE J. CARRON AND L. K. WALFORD
Research Division, McDonnell Company, St. Louis, Missouri
(Received 21 February 1968)
The purpose of this paper is to describe qualitatively the residual stress field induced in single-crystal
silicon by electron-beam melting and subsequent resolidification in microspot areas. The stress distribution
is derived by vector relationship from previously reported experimental observations by x-ray topography
and polarized ir radiation techniques. The stresses are distributed in a sixfold pseudohyperbolic form
around the microspots. This distribution correlates with the four-lobed x-ray rosettes recorded from various
x-ray reflections. The stresses are concentrated normal to the {111} planes of the parent lattice and, in
projection, all stresses in the (110) directions are tangential to the microspot. Infrared birefringence results
from stresses along the (111) directions either within the microspot or at its interface with the parent
lattice. These directions thus become uniaxial optic axes coincident with the stress direction. Recrystalliza-
tion mechanisms are discussed in relation to .the macro stacking fault observed by x-ray topography.
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