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ELECTRO-OPTIC PROPERTIES of KH2PO4 and ISOMORPHS, KDP, KD*P / DKDP, ADP, AD*P
(Table of Material Properties appears below)

THE POCKELS ELECTRO-OPTIC EFFECT

Crystals which belong to twenty symmetry classes which lack a center of symmetry can show a linear electro-optic effect, that is, a change in refractive indices directly proportional to an applied voltage. The symmetry conditions for the occurrence of this effect are exactly the same as for the occurrence of the piezoelectric effect. Thus, there is an exact symmetry analogy between the linear electro-optic effect (refractive index a linear function of electric field) and the converse piezoelectric effect (geometric deformation a linear function of electric field). The linear electro-optic effect has the same relation to the Kerr effect (refractive index a quadratic function of electric field) as converse piezoelectricity has to electrostriction (geometric deformation a quadratic function of electric field). The linear change in refractive index obtained at room temperature with practical electric fields (up to 20kV/cm) is only of the order of 10-4. Although this is too little to change refraction angles for most practical purposes78, it is sufficient to produce retardations of the order of one wavelength and hence lead to interference phenomena. These interference phenomena are used to modulate light phase or intensity. A one-half wavelength relative retardation can change the transmission of polarized light from 0 to 100 percent. An ac voltage producing a peak retardation of one fourth wavelength can give 100 percent modulation of the carrier.

The linear electro-optic effect may be regarded as a special case of second order (non-linear!) electric interaction in the crystal: The action of a low frequency applied field and the electric field of the optic-frequency electro-magnetic wave combine to cause electric polarization at the optic frequency.

Linear electro-optic phenomena were discovered by Roentgen in quartz and thoroughly investigated in several crystals before the turn of the century by Pockels1,2, in whose honor the effect is now generally called the Pockels effect. The broader study of higher order interaction in crystals began with Franken's discovery of frequency doubling of laser beams in quartz and KH2PO43. In many linear electro-optic devices the longitudinal effect is used, that is, the light beam and electric field are parallel. Longitudinal effect devices are particularly useful for light beams of large cross-sectional area. Other electro-optic devices use the transverse effect with the light beam perpendicular to the applied field. Transverse effect devices avoid the use of transparent electrodes in the light path. In addition, the voltage required for a given retardation can be reduced by increasing the ratio of the light path length to the electrode spacing, whereas in longitudinal effect devices the required voltage is independent of the dimensions of the crystal. It can be shown from symmetry considerations that a longitudinal effect free of background birefringence and optical activity is obtained only with crystals of two classes: the class 3m of the cubic system and the class 2m of the tetragonal system. Class 3m is represented by cubic zinc sulfide (sphalerite) and its isomorphs including ZnSe, ZnTe, GaAs, and CuCI. For a recent collection of data on such crystals see the compilations in Landolt-Boernstein9,10. Class 2m is represented by KH2PO4 (KDP) and its isomorphs7,8,9,10. Relatively large electro-optic effects and the availability of large crystals of high perfection have given crystals of this group continuing major importance for both "longitudinal" and transverse modulators as well as frequency doubling and mixing devices. Large strain-free crystals of KDP and a number of its isomorphs are available from Cleveland Crystals, Inc. These crystals are transparent throughout the visible and ultra-violet; one of the isomorphs (KH2P04) is transparent to below 0.18µm80. The infrared cutoff is near 1.5µm for the dihydrogen phosphates and near 2.1µm for KD2 PO4.

The electro-optic effects are traditionally expressd by the Pockels coefficients rij, where the first subscript refers to a tensor component of the index ellipsoid and the second to the electric field vector. In crystal class 2m there are only 2 coefficients, r63 and r41. The first measures the effect of a field component parallel to the optic axis. This makes the crystal optically biaxial, with vibration directions of increased and decreased refraction at ±45° to the X and Y axes. The r63 coefficient is controlling for all modulators with field and light beam parallel to the optic axis (longitudinal devices) as well as most transverse devices. Transverse devices based on r63 have the light path (length of the modulator bar) at ±45° to X. Only the ordinary ray undergoes variation of the refractive index.

The r41 effect is, in first order, a rotation of the index ellipsoid around an electric field in the X direction and leads to maximum index change for light travelling at 45° to the optic axis. In spite of the high r41 coefficients of KDP type crystals such devices have found only very limited application. The Pockels coefficients rij are related to observable refractive index changes by:

 
Δn = f rij n3 E          (1)
where f is a numerical factor depending on crystal symmetry and device orientation, n is the appropriate refractive index, and E is the applied field strength. For KDP type crystals in longitudinal setup f = 1 and: 
Δn = f r63 no3 E          (2)
Here no is the ordinary index of refraction and n the difference in index for components polarized at + and - 45° to the X axis. The path difference in fractions of a wavelength in air then becomes: 
Δn x/λ = r63 no3 E x/λ = r63 no3 V/λ          (3)
where X is the crystal thickness and V the applied voltage. The voltage required to obtain one-half wavelength relative retardation then follows: 
Vλ/2 = λ/[2 r63 no3]         (4)
This magnitude has been widely used to characterize crystals for modulator use and is listed in our tables; note, however, that it is inverse to the strength of the electro-optic effect and that it depends explicitly on wavelength. The quantity r63no3 varies only a few percent over the range of transparency of these crystals.

Because KDP type crystals are uniaxial with fairly high birefringence, longitudinal effect devices made of them have a limited angular aperture. The intensity of a light beam transmitted through basal section (Z-cut) placed between crossed polarizers is: 

I = Io sin2{[x(n02-ne2)sin(2Θ)]/[2λnone2]}          (5)
where Io is the intensity of the incident beam polarized at 45° to its plane of incidence, x is the plate thickness, ne is the extraordinary index of refraction, and Θ is the angle of incidence, outside the crystal. For a KDP crystal plate 2mm thick and λ = 0.589µm, equation (5) shows a first maximum for the transmitted intensity at Θ=5.1°. For linear polarized light from a uniformly radiating point source limited by an iris to a divergence of half-angle Θmax, integration of equation (5) leads to an average transmitted intensity of: 
Iave = (Io/6) {[x(no2-ne2)max2]/[2none2]}2  for max << 1           (6)
If Iave/Io for a 2mm Z-cut KDP crystal in diverging 0.589µm light is to be less than 0.005, equation (6) shows that max must be less than 1.6°. For a prescribed transmission ratio both  and max are proportional to the inverse square root of the crystal thickness. The restricted angular aperture of Pockels effect modulators can be an important limitation in certain applications, such as photographic shutters. In laser devices the angular aperture of longitudinal KDP type modulators is generally ample.

For transverse KDP type devices the factor f in eq. (2) becomes 1/2 and the induced path difference is: 

nx/ = r63no3xE/(2) = r63no3xV/(2t)          (7)
where V is the applied voltage and t the crystal thickness between the electrodes. In a transverse modulator rod a path difference(no - ne)x amounting to many wavelengths is present before the application of an electric field; this may be compensated by using two exactly equal rods in tandem and rotated at 90° to each other, with appropriate phasing of the applied voltage. The allowable angular aperture is less than for a longitudinal device but this is not a serious problem in highly collimated laser devices.

The symmetry analogies of the linear electro-optic and the converse piezoelectric effects have already been mentioned. More than that, a part of the linear electro-optic effect may be regarded as secondary: The electric field causes a deformation of the crystal by the converse piezoelectric effect and this deformation in turn produces a change in the refractive indices through the elasto-optic effect. This secondary effect may be suppressed by clamping the crystal. The total (free) electro-optic coefficients, r63T and r41T are related to the clamped coefficients r63S and r41S by: 

r63T = r63S + p66 d36 ,  and  r41T = r41S + p44 d14          (8)
where p44 and p66 are elasto-optic (strain-optic) constants, and d14 and d36 the piezoelectric constants. In ADP, the the clamped effect r63S amounts to only about 60% of the free effect, while in KDP the clamped r63S is near 90% of the free r63T. For ADP and KDP X-cuts the coefficients r41S and r41T are practically identical because the piezo-electric coefficients d14 are very small; but this is not true for the arsenates.

The clamped condition may be realized by mechanical constraints or, more effectively. by operation at frequencies well above the elastic resonances of the device, typically about 1 MHz. Near the elastic resonance frequencies, on the other hand, elastic strain may be a multiple of d36E. and an enhanced elasto-optic contribution to the observed optic effect results. ADP and KDP X-cut devices have been used where a flat frequency response is essential. The clamped electro-optic constant r63S as well as the clamped dielectric constant of KDP have been shown to be independent of frequency up to at least 10 GHz5.

The coefficients r63, and to a lesser extent r41, are strongly temperature dependent. Increases by a factor 1000 have been measured as the temperature is lowered close to the Curie point. This dependence for r63 may be expressed by a Curie-Weiss law: 

r63 = 63(33-0) = 630[C/(T-Tc) -1]          (9)
Here 33 and 0 are the permittivities of the crystal and free space, C is the dielectric Curie constant, Tc is the Curie temperature, and 63 is an electro-optic constant for dielectric polarization, an independent variable. The 63 coefficients are not only nearly independent of temperature but also show a much reduced variation between members of the KDP group, and between free and clamped condition. 63T is 0.059 m2/C for KH2P04,0.061 for KD2P04, 0.062 for KH2As04, and 0.063 for NH4H2P049.

As 63 coefficients measure the charge density, and hence the current required to attain a prescribed level of light modulation, the current requirement is practically independent of temperature or of the choice among KDP type crystals. The differences in room temperature electro-optic properties of KH2PO4 and KD2PO4 can be fully accounted for by the 100° difference in Curie points. The principal reason for specifying the more costly KD2PO4 has been the resulting higher dielectric constant and hence lower voltage requirement. In applications involving light in the 0.9 to 1.81µm region the lower absorption of KD2PO4 is likely to be decisive.

The most important publications on the electro-optic properties and applications of KDP type crystals have recently been reprinted.5 

NON-LINEAR RESPONSE TO COHERENT LIGHT

In its most general terms, second-order non-linear dielectric interaction is the generation of electric polarization proportional to the product of two electric field components. The non-linear polarization contains the sum and difference frequencies of the input signals. In the Pockels effect, one component is at optic frequency (the incident beam) while the other is at much lower frequency, well below the infrared absorption region, or even d.c. The other case of major interest is the interaction of two coherent optic-frequency signals of the same frequency, and especially the interaction of a single input component with itself. This leads to generation of an output component at doubled frequency and a d.c. electric polarization. All these effects can be expressed by non-linear optic coefficients dijk, where any two subscripts relate to the input and the third to the output. The value of the dijk depends on each of the three frequencies involved. For KDP type crystals (class 2m) the only dijk coefficients different from zero are those with all subscripts different. For frequency doubling these are d312 (21, 1, 1) and d132 = d231(21, 1, 1), or in contracted form, d36 (2,) and d14 = d25 (2,). If the generating and resulting frequencies are all well within the optic transmission band, the order of the subscripts affects the value of dijk to an extent that is insignificant compared to other factors determining second harmonic or sum/difference frequency intensity. Then KDP type crystals may be characterized by a single non-linear coefficient d123 = d14  d36 (Kleinmann relation).

The relative values of dijk for different orientation and temperature and for different substances can be obtained with much better accuracy than absolute values. Therefore the literature contains many values relative to d36 of KDP. Absolute values were calculated from relative values by Zernike and Midwinter6 and also more recently by Kurtz, Jerphagnon, and Choy4.

For one input or output near zero frequency (Pockels effect or rectified output) we have non-linear coefficients related to the Pockels coefficients: 

4d36(O,) = -no4 r63T  and  4d14(O,) = -no2 ne2 r41T          (10)
(ref. 5, eq. 17,.p9 59). The numerical factor with the dij is in controversy79. 
PHASE MATCHING (ref. 6, chapter 3)

For substantial generation of sum and difference frequencies, and especially for frequency doubling, it is required that the incoming and generated light are in a defined phase relation along the path of interaction. Due to normal dispersion the double-frequency light generated sees a refractive index a few hundredths higher than the fundamental frequency. This would bring the two waves out of phase over a path of about /[n(2) - n()], in the order of 10 to 50 microns. Fortunately the birefringence of many non-linear crystals, and the KDP group in particular, allows one to overcome this limitation by phase matching. The common way of achieving this (type I phase-matching) with KDP type crystals (class 2m, no>ne) uses an ordinary wave exciting an electric vector in the basal plane at 45° to the X and Y axes. The non-linear response is electric polarization in the Z direction which builds up to an extraordinary wave. Cumulative energy transfer occurs along a common wave normal if the two waves see the same refractive index. For KDP and ADP at room temperature ne(2) = no() for input wavelengths near 500 nm. Type I phase matching is obtained for the indicated wavelength with wave propagation perpendicular to the optic axis, i.e. 90° phase matching. The condition ne(2)= no() occurs for longer wavelengths with KDP type rubidium and cesium salts, which have lower birefringence. This is their principal merit for optic devices. Some tuning is possible by temperature variation; no-ne decreases with increasing temperature.

If KDP or ADP is to be used to double red or near infrared light, angle tuning is resorted to: The wave normal is inclined to the optic axis so that the generated extraordinary ray sees a refractive index intermediate between no and ne according to: 

1/n2(,2) = sin2()/ne2(2) + cos2()/no2(2)= 1/no2()          (11)
Phase matching by angle tuning has three disadvantages compared to temperature tuning:

1) Ray directions are not parallel to the wave normal; this limits the practical interaction length.
2) The effective non-linear coefficient is reduced.
3) The angular tolerance (beam angular aperture) is reduced.

Problems 2) and 3) may be ameliorated to some extent by resorting to type II phase-matching. For KDP type crystals this employs an incoming beam in the XZ plane polarized at 45° to this plane. The interaction of the resulting ordinary and extraordinary components of the fundamental creates an extraordinary ray at doubled frequency. The effect of birefringence is reduced by one-half, and the maximum effective d coefficient now occurs for a beam at 45° to the optic axis. Phase matching of ADP or KDP for doubling of 1060nm light occurs not far from this angle.12,13

TEMPERATURE LIMITATIONS

A lower temperature limit for the optic use of KDP type crystals is set by their Curie points (see table), below which optic properties are affected by domain structure which may also lead to cracking of the crystal. Operation just above the Curie point is attractive by the extremely high response to applied voltage, but this is penalized by high temperature coefficients. Most KDP type crystals convert to an alternate, inactive crystal structure at high temperatures. For all compounds except RDP, conversion does not take place below 100°C*. Chemical decomposition for the alkali metal salts does not occur below 200°C, but is noticeable for the ammonium salts at about 120°C. KDP type crystals show a small protonic conductivity with an activation energy of about 0.6 eV resulting in a positive temperature coefficient of conductivity of about 8%/°C near room temperature.14,15,16 Resistivity of the best available ADP and KDP crystals is from 10 to 20 x 108 Ohm-meter at 25°C. Ionic impurities, such as HSO4- substituting for H2PO4-, can increase the conductivity many fold. Resistivity so far achieved with the arsenates is 1 to 3 powers of 10 lower than in ADP and KDP8. The negative temperature coefficient of resistance can lead to a runaway breakdown if high electric field (kV/cm) is applied continuously.

* Fully deuterated KD2PO4 is actually only metastable near 100°C, but crystal elements have not been observed to transform. 

Propeties of KDP, KD*P, ADP, and AD*P 
 
 KH2PO4  
(KDP)		 KD2PO4  
(KD*P)		 NH4H2PO4 
(ADP)		 ND4D2PO4 
(AD*P)
Crystal Data
 
 
 99 atom% D  
 
 99 atom% D 
Crystal Symmetry and Class Tetragonal,2m Tetragonal,2m Tetragonal,2m Tetragonal,2m
Space Group I2d I2d I2d I2d
Lattice Constants47(angstroms) a=7.4529 
c=6.9751		 a=7.4697 
c=6.9766		 a=7.4991 
c=7.5493		 a=7.5193 
c=7.5400
Density, g/cc 2.3325 2.3555 1.799 1.885
Cleavage none none none none
Optical Properties 
 
 
 
 
 
 
 
Micron region of >50% transmission 
(2mm thickness)		 0.176(80) to 1.55(25) <0.2 to 2.15(25) 0.184(80) to 1.5(41) <0.2 to >2.0 *
Indicies of refraction (In 22°C air) 
 
 99 atom% D 
 
 95 atom% D
0.266µm no=1.5599 
ne=1.5105		 no=1.5539 
ne=1.5071		 no=1.5797 
ne=1.5261		 no=1.5701 
ne=1.5191
0.3547µm no=1.5318 
ne=1.4864		 no=1.5250 
ne=1.4834		 no=1.5485 
ne=1.49895		 no=1.5403 
ne=1.4942
0.532µm no=1.5129 
ne=1.4709		 no=1.5071 
ne=1.4683		 no=1.52775 
ne=1.4815		 no=1.5211 
ne=1.4776
0.5893µm no=1.5098 
ne=1.4687		 no=1.5045 
ne=1.4661		 no=1.52435 
ne=1.4789		 no=1.51815 
ne=1.4751
0.6328µm no=1.5079 
ne=1.4673		 no=1.50285 
ne=1.4648		 no=1.5222 
ne=1.4773		 no=1.5163 
ne=1.47365
0.6943µm no=1.5055 
ne=1.4658		 no=1.50095 
ne=1.4633		 no=1.5195 
ne=1.4754		 no=1.5142 
ne=1.4719
1.064µm no=1.4944 
ne=1.46035		 no=1.4934 
ne=1.4583		 no=1.5068 
ne=1.4681		 no=1.5052 
ne=1.4658
dn/dT, 10-6/°C 
(20 to 40°C)		 
 
 99 atom% D 
 
 95 atom% D
0.266µm dno/dT= -38 
dne/dT= -29		 dno/dT= -37 
dne/dT= -24		 dno/dT= -52 
dne/dT= -2		 dno/dT= -24 
dne/dT= +2
0.532µm dno/dT= -40 
dne/dT= -29		 dno/dT= -37 
dne/dT= -24		 dno/dT= -56 
dne/dT= -2		 dno/dT= -28 
dne/dT= +2
1.064µm dno/dT= -48 
dne/dT= -31		 dno/dT= -44 
dne/dT= -24		 dno/dT= -81 
dne/dT= -3		 dno/dT= -24 
dne/dT= +5
Fresnel Refection Loss per surface 
 
 99 atom% D 
 
 95 atom% D
0.266µm e-ray: 4.8% 
o-ray: 4.1%		 e-ray: 4.7% 
o-ray: 4.1%		 e-ray: 5.05% 
o-ray: 4.3%		 e-ray: 4.9% 
o-ray: 4.25%
0.532µm e-ray: 4.2% 
o-ray: 3.6%		 e-ray: 4.1% 
o-ray: 3.6%		 e-ray: 4.4% 
o-ray: 3.8%		 e-ray: 4.3% 
o-ray: 3.7%
1.06µm e-ray: 3.9% 
o-ray: 3.5%		 e-ray: 3.9% 
o-ray: 3.5%		 e-ray: 4.1% 
o-ray: 3.6%		 e-ray: 4.1% 
o-ray: 3.6%
NLO Susceptibility d36, 10-12m/V 
 
 
 
 
 
 
 
0.6943µm 0.70(4) 0.53(4) 0.85(4) 0.77(4)
1.064µm 0.63(4) 0.42(4) 0.762(4) --- 
90° Phasematching Wavelength(µm) and Temp(C) 0.518, 25° 
0.527, 120°		 --- 0.515, -10° 
0.557, 120°		 ---
Electrical Properties 
 
 
 
 
 
 
 
Resistivity (106 ohm-cm) 10-20 10 10-20 10
Relative dielectric constant at 25C and 1KHz 
 
 99 atom% D 
  
 99 atom% D
11T/o 43.2(82) 65(64) 56(82) 70(8)
11S/o 42.5(82) 62.5(64) 55.5(82) ---
33T/o 20.8(82) 50(25) 15.5(82) 26(8)
33S/o 20.0(82) 48(82) 15.0(82) ---
Piezoelectric Coefficients, (pC/N) 
 
 
 
 
 
 
 
d14 1.3(63) 3.4(74) ~98 at% D 1.8(83) 10(67) ~98 at% D
d36 21(63) 58(25) 99 at% D 48.3(83) 75(67) 99 at% D
Electromechanical Coupling Factors 
 
 99 atom% D 
 
 99 atom% D
k14 0.008(63) --- 0.006 ---
k36 0.121(63) 0.22(25) 0.33 --- 
 
 
               
 
               
 
               
 
               
Unreferenced data are unpublished data of Cleveland Crystals, Inc. 
Where not listed, deuteration levels are unknown. 
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