Quadratic Electro-Optic Effects in Bacteriorhodopsin : Measurement of γ (-ω ; 0 , 0 , ω ) in Dried Gelatin Thin Films

Quadratic electro-optic effects (dc or low frequency Kerr effect) of bacteriorhodopsin dispersed in dried gelatin thin films are examined in the near resonance region at three wavelengths: 633, 647, and 676 nm. The films show relatively large quadratic electro-optic effects compared to other molecular dispersed systems. The purple membrane is fixed within the polymerized gelatin matrix, and we show that the electronic contribution to γ dominates over possible orientational contributions. At 676 nm, the quadratic electro-optic coefficient s1133(−ω;0,0,ω) is 6.7×10−20 m2/V2 and the third order nonlinear susceptibility χ1133(3)(−ω;0,0,ω) is 7.0×10−13 cm4 statCoulomb−2, with both values obtained for a protein concentration of 6.9×1018 cm−3. The orientationally averaged second molecular hyperpolarizability 〈γ(−ω;0,0,ω)〉 determined from the quadratic electro-optic coefficients at 676 nm assuming an Onsager ellipsoidal local field factor is (10.8±5.1)×10−32 cm7 statCoulomb−2 [(1.34±0.63)×10−56 F3 m4 C−2]. The 〈γ...


I. INTRODUCTION
[3][4][5][6][7][8][9][10][11] The physiological function of this protein is to produce energy for converting ADP to ATP by pumping protons from the cytoplasmic to the extracellular side of the cell wall.The resulting pH gradient across the cell wall generates a proton motive force for synthesizing ATP.The proton pumping process is mediated by a complex photocycle which is illustrated in Fig. 1.When bacteriorhodopsin first absorbs light, it converts from the dark-adapted form to a light-adapted form ͑bR͒, which contains only an all-trans protonated Schiff base chromophore.Unless specified otherwise, all of our experiments and all similar studies in the literature were carried out on the light-adapted form.
3][14][15][16][17][18][19] The first measurement of ␤ by Huang et al. gave a value of 2500ϫ10 Ϫ30 cm 5 statcoulomb Ϫ1 , approximately ten times larger than other organic chromophores of comparable conjugation length. 19Subsequent studies yielded a comparable value of 2250Ϯ240ϫ10 Ϫ30 cm 5 statcoulomb Ϫ1 based on an analysis of the two-photon double resonance spectrum. 12Analysis of the two-photon data also indicate that ␥ should be relatively large, due in part to the large change in dipole moment upon excitation ͑type III enhancement͒. 15,20n this study, we report the direct observation of ␥ of bR dispersed in dried gelatin thin films in the near resonance region.Our experimental method is based on the technique proposed by Schildkraut for measuring linear electrooptic effects by using a reflection geometry. 21In this experiment, gelatin films containing bR were formed on ITO covered glass substrates and Au electrodes were deposited on the surface of the gelatin films.A low frequency electric field is applied to the gelatin film to induce birefringence.Low intensity laser light is then directed onto the film through the ITO coated glass substrate.The laser light propagating inside the gelatin film is then reflected back by the surface of the Au electrode.The electromagnetic field associated with the laser light is modulated with a change in the phase difference between s-and p-polarized component caused via linear electro-optic effects and/or quadratic electrooptic effect ͑also referred to as dc or low frequency Kerr effects͒.a͒ Present address: Fuji Electric Corporate Research and Development, Ltd., 2-2-1 Nagasaka Yokosuka 240-0194, Japan.b͒ Author to whom correspondence should be addressed.
Gelatin has been known to have a cross-linked, collagenlike structure with hydrogen bonds.When purple membrane is immersed in the gelatin matrix, the purple membrane is considered to be spatially fixed.The interaction between the host molecule and the gelatin matrix via the hydrogen bonds is expected to play a role in preventing induction of molecular orientation under strong modulation field.For example, Ho et al. investigated linear electro-optic effects of p-nitrophenol in a gelatin matrix and observed that the gelatin matrix maintained poling stability. 22Thus, we conclude that orientational contributions to the measured electro-optic effect can be ignored.Our experimental results provide support for this conclusion.By analyzing the wavelength dependence of ␥, we gain insight into the electronic origin of the large second order hyperpolarizability of bacteriorhodopsin.

A. Sample preparation
Purple membrane was isolated from the strain S9-P of Halobacterium salinarium by using the following procedure.The bacterium was collected by spinning down the culture containing the bacterium in a Beckman JA-10 rotor at 5000 rpm ͑15 000 g͒ for 10-15 min, then resuspending in distilled water.The total volume of the suspension was approximately 500 mL.We then added 0.15 mg/mL of DNAse I type IV ͑Sigma D-5025͒ and 0.25 g of MgSO 4 to the suspension to digest the DNA contaminant.The suspension was left over night with gentle stirring at ambient temperature, and the cell debris was removed by spinning the sample down using a Beckman JA-17 rotor at 5000 rpm for 5 to 10 min.
The supernatant and purple membrane were saved and spun down in a 45 Ti rotor at 32 000 rpm ͑109 000 g͒ for 35 min at 4 °C.Pellets of purple membrane which formed at the bottom of the centrifuge tubes were washed via resuspension in de-ionized water, followed by centrifugation as described above.This procedure was repeated until the supernatant be-came clear.Brownish cell debris found in the pellet at bottom of centrifuge tubes was removed mechanically.
The gelatin thin films dispersed with the purple membrane were prepared as follows.Gelatin ͑gelatin type A from porcine skin ͓CAS# 9000-70-8͔, Sigma Chemical Co., used as received͒ was dissolved in deionized water by heating to 60-65 °C for 40 min in a round bottom flask equipped with a water condenser on a water bath.The concentration of gelatin was 10 w/w% in de-ionized water.After the gelatin was completely dissolved, the solution was filtered with a syringe filter ͑pore size 5 m Micron Separations Inc.͒ while the solution was still hot.The suspension of bacteriorhodopsin in deionized water was spun down in the Beckman 45 Ti rotor ͑32 000 rpm, 109 000 g for 30 min͒ to yield a pellet of purple membrane.The pellet was resuspended in a minimal amount of deionized water.Then the suspension was sonicated for 30 min.The sonicator tip was directly immersed in the suspension for efficient sonication.
The gelatin solution and protein suspension were filtered with a syringe filter ͑AcetatePlus Membrane, pore size 5 m, diameter 25 mm, Micron Separations Inc.͒, heated to 60 °C, and then mixed with gentle stirring for about 30 min while maintaining the temperature at 60 °C.The resulting mixture was coated onto an indium tin oxide ͑ITO͒ coated BK-7 glass substrate via the spin coating method.The thickness of the ITO coating was approximately 1000 Å yielding a resistivity of about 1 k⍀ cm.Spin coating was carried out by pipetting 200-250 L of the protein:gelatin mixture onto the preheated ITO coated substrate so that the mixture covered the surface as uniformly as was possible.The substrate was then transferred to the spin coating apparatus, and the thickness of the final film was controlled by adjusting the rotational speed of the spin coater.By using this procedure, film thicknesses ranging from 3 to 15 m were achieved.
After the films had formed on the ITO covered glass substrates, they were dried in a closed container, in which the relative humidity was maintained at 80% to 85% and temperature was maintained at 25 °C for two days.The humidity was controlled by placing a beaker containing a saturated KCl solution inside the container.
The film thickness was measured by taking transmittance spectra from the visible to the near ir region.An interference pattern was found in the transmittance spectra when the surface quality of the films was high, and the films were of uniform thickness.All further optical studies were carried out only on those films which satisfied the above two criteria.The film thickness was determined from the interference pattern by using the following equation:

͑1͒
where l is the film thickness, ⌬m is the number of peaks ͑or valleys͒ in the interference pattern, 1 is the longest wavelength in the interference pattern, 2 is the shortest wave length in the interference pattern, is the incident angle of a probe beam and n is the refractive index of the films ͑n ϭ1.53, see below͒.The transmittances were measured using a Shimadzu UV-3101 UV-Vis-NIR spectrometer with optical film thickness measurement software.After the film thickness was confirmed, four to six round Au thin film electrodes were deposited directly onto the surface of the protein:gelatin film by using a dc sputtering process.Film thickness of the Au electrode was approximately 1000 Å and its diameter was 0.525 cm.Because the sputtering process removes moisture from the gelatin film, the gelatin film directly under the Au electrodes shrunk causing distortion of the film.To remove the distortion, the films were placed in the humidity and temperature controlled container again for at least five days.Prior to carrying out our optical measurements,the films were stored for two days at ambient temperature and about 65% relative humidity.

B. Electrooptic light modulation with quadratic electrooptic effects
Figure 2 shows the experimental setup for the electrooptic response measurement using the reflection technique.Radiation from a He-Ne laser ͑05LHP151 5 mW polarized laser head, Melles-Griot, Irvine CA͒ or a Krypton ion laser ͑Innova-301, 750 mW, Coherent Corp., Palo Alto CA͒ provided the probe irradiation.Light intensity was measured with a photodiode ͑Model 2001 optical receiver, New Focus, Mountainview, CA͒ and a digital multimeter ͑Keithley 179A TRMS͒, and all intensities were corrected by the response factors of the photodiode.Data were collected at three wavelengths: 632.8 nm ͑He-Ne͒, 647.1 nm ͑Kr ion͒ and 676.4 nm ͑Kr ion͒.These wavelengths correspond to the near absorption edge region of the ground state of the bacteriorhodopsin ͑Fig.1͒.
Polarization of the incident light was adjusted to 45 degrees with respect to the plane of incidence with a glan-laser prism so that the electric field vector of the incident light yielded an equal amount of s-and p-polarized components.
The reflected beam was passed through a zero order quarter wave plate followed by a second glan-laser prism,which was set to be cross-Nicol relative to the first prism.The quarter wave plate thus gives 90 degree phase biasing to the reflected beam.The rotation angle of the quarter wave plate was adjusted so that the maximum output power was obtained.
After the reflected beam was passed through the second aperture, it was focused to fill but not overflow the active area of the photodiode.The photodiode was carefully shielded with Cu mesh to eliminate signal due to the stray emf from the high voltage power supply and associated leads.The output signal from the photodiode was fed into a digital lock-in amplifier ͑SRS850 DSP, Stanford Research Systems, Sunnyvale, CA͒.A TTL reference signal at 500 Hz was provided to the lock-in amplifier using a synthesized function generator ͑DS345 Synthesized Function Generator, Stanford Research Systems͒.The function generator also provided a synchronized signal to the ac power supply ͑Model 1100 High voltage ac power supply, Joseph Rolfe Associates, Palo Alto Stanford, CA͒.The high voltage power supply was driven well below its rms capacity to avoid nonsinusoidal behavior and no bias was added to the signal to avoid creating a net orientational ͑poling͒ applied field.

C. Signal analysis
In the weak poling limit, the refractive index change associated with an external applied electric field can be written for ordinary and extraordinary rays as follows: where r 113 and r 333 are the linear electrooptic coefficients measured along in-plane and film normal directions, respectively, and s 1133 and s 3333 are the quadratic electro-optic coefficients measured along the in-plane and film normal directions, respectively.The Schildkraut derived expression for the degree of the phase modulation in the reflection geometry is When the modulation field can be represented as a simple ac field ͓E m ϭ(V m /l)cos ⍀t͔, the relationship between the phase shift, ⌫, and the modulated laser light intensity ratio, I ac /I dc , can be calculated as follows: 23,24 FIG. 2. The apparatus used to measure the nonlinear electrooptical properties of the protein thin films.The x 1 , x 2 , and x 3 axes were taken as a coordinate system fixed inside the film, where the x 3 axis is perpendicular to the film surface and x 1 and x 2 are parallel to the film surface.Also, denotes the incident angle of the laser and ␣ denotes an effective propagation angle of the laser inside the film.The relative thickness of the glass substrate is unrealistically small for graphical convenience.
where l is film thickness, V m is applied voltage, I ac is degree of modulation and I dc is defined as the dc signal obtained by subtracting the dc level with the quarter wave plate with applying no modulation voltage from the dc level without the quarter wave plate with applying no modulation voltage.The dc signal thus corresponds to 50% transmittance, and the /2 phase biasing has been added into Eq.͑5͒.The phase biasing is achieved by placing the quarter wave (/4) plate into the optical path as shown in Fig. 2. The term ⌬n abs arises from the absorption change of the medium and does not depend on the modulation field.The system is considered to be isotropic, and thus the term involving r 113 is either not detectable or negligible compared to the term involving s 1133 .Only the term depending on cos 2⍀t is detected by the lock-in amplifier at the second harmonic frequency of the modulation field.The degree of modulation at the second harmonic frequency is defined as

͑6͒
where ⌬⌫ 2⍀ is the amplitude of the phase shift modulation.
The relation between the effective propagation angle, ␣, and the incident angle, , is given by Snell's law sin ϭn sin ␣. ͑7͒ Considering Eq. ͑7͒ and assuming that both the modulation field and the modulated light amplitude are measured as rms amplitudes, Eq. ͑6͒ reduces to

A. Quadratic electrooptic effects observed in the gelatin films containing bR
Figure 3 shows a typical electrooptic response from bacteriorhodopsin dispersed in gelatin film obtained using 632.8 nm laser radiation.Signal amplitudes at the second harmonic frequency ͑1 kHz͒ were typically 2 to 3 orders of magnitude larger than those at the fundamental frequency ͑500 Hz͒.Signals at a fundamental frequency based on linear electrooptic effects were not detectable in most cases, because the protein samples are highly homogeneous and isotropic.However, the quadratic modulation voltage dependence of the signals at a second harmonic frequency were distinct and the raw signal intensity in rms voltage was on the order of 0.1-10 V, levels readily observable by the digital lock-in amplifier.The modulated signal amplitudes normalized with the dc components were on the order of 10 Ϫ4 -10 Ϫ2 .
Three consecutive experiments were carried out for each sample at each wavelength ͑Fig.3͒.Spectrophotometric studies were carried out to verify that the protein remained light-adapted during the experiment.Contributions to the quadratic electro-optic effects from gelatin itself were negligible.The normalized modulated light intensities were replotted against the quantity ͑rms amplitude of modulation voltage͒ 2 wavelengthϫfilm thickness .
From the slopes of the curves obtained with this analysis, s 1133 was calculated via Eq.͑8͒. at 632.8 nm for a range in bacteriorhodopsin concentration from 6.9ϫ10 17 cm Ϫ3 to 6.9ϫ10 18 cm Ϫ3 .The curves shown in Fig. 4 were successfully fitted with straight lines with good correlation between the above two quantities.The same curve fitting procedures were used to analyze the data at 647.1 and 676.4 nm.
Figure 5 shows the bacteriorhodopsin concentration dependence of the quadratic electrooptic effects measured at several wavelengths.These data indicate that the quadratic electro-optic effects are approximately linear with respect to concentration of bacteriorhodopsin within the concentration range investigated here.This observation implies that the second molecular hyperpolarizability can be determined from the slopes of these curves.We can thus conclude with confidence that the protein, and not the gelatin matrix or other components along the light path, is responsible for the signal.It is thus possible to equate the measured value of (3) to the molecular ␥ via the relation (3) ϭNL͗␥͘, where N is the concentration of the protein and L is a local field correction.
Neither the linear nor the quadratic electrooptic effect is expected to be dependent on the incident light intensity, but dependent only on the amplitude of the modulated electric field.However, the modulated light intensities with the quadratic electrooptic effects detected at 633 nm are proportional to the logarithm of the photon density of the incident light while the signal levels measured at 647 and 676 nm are invariant to photon density as shown in Fig. 6.
One possible explanation for the anomalous signal enhancement at 633 nm is an increased population of the O state of bacteriorhodopsin, which has an absorption maximum at ϳ645 nm ͑see Fig. 1͒.Bacteriorhodopsin has a sufficient absorptivity at 633 nm to yield a small population of photocycling molecules within the irradiated volume element, and the O state has a relatively long lifetime ͑ϳ 4 ms͒.We conclude that the quadratic electro-optic effect is likely enhanced anomalously by O state interference.Regardless of the origin of the effect, we removed the interference by ex-trapolating our data back to low light levels where such interference was minimal.

͑9͒
A relation between the macroscopic third order nonlinear susceptibility of the films and the second molecular hyperpolarizability of the protein can be derived by taking local field factors into account.However, the choice of a local field factor is not obvious, because we are dealing with a chromophore imbedded inside a complex binding site ͑see Fig. 7͒. 9,25For this reason, we will investigate here a variety of local field factors.The local field factor introduced by Onsager has been used extensively for studies of nonlinear optical properties of organic molecules ͑Appendix A͒.The Onsager's local field factor was derived assuming that the molecule is a dipole placed in a virtual spherical cavity in a continuous dielectric matrix and the local field is a sum of the field inside the cavity and the reaction field resulting from an interaction between the dipole moment of a molecule in the cavity and a polarization induced in surrounding matter by an external field.However, the retinal chromophore considered here is a rod shaped polyene.Hence, an ellipsoidal cavity is considered to be more suitable in this case ͑Fig.7͒. 26The expressions of the local field factors for the ellipsoidal cavity at a static field f ellipse (0) and at an optical frequency f ellipse () are given in ͑A6͒ and ͑A7͒ in Appendix A, respectively.Also, the expressions of the local field factors for the spherical cavity at a static field f sphere (0) and at an optical frequency f sphere () are given in ͑A8͒ and ͑A9͒ in Appendix A, respectively.For example, f ellipse (0) is given as follows: FIG. 6. Photon density dependence of the modulated light intensity detected at the second harmonic of the modulation frequency.The modulation voltage was fixed at 200 V rms, the film thickness was 15.3 m, protein concentration was 6.9ϫ10 18 cm Ϫ3 and the modulation frequency was 500 Hz.
An alternative, but widely accepted, approximation to the local field correction was developed by Lorentz.In the Lorentz's approximation, the reaction field is not taken into account.For comparison, the Lorentz-type local field factors for ellipsoidal and spherical cavities were also considered as the local field correction.The local field factors derived using the Lorentz approximation are given in ͑A14͒ and ͑A15͒ in Appendix A. For example, f sphere () is given as follows: The permittivity of gelatin measured at 1 kHz (⑀ 2 (0) ϭ2.07) was used as the static field permittivity of the polymer matrix.The refractive indices of the protein and the gelatin matrix at optical frequencies were taken from Song et al. 27 and were used for the calculation of the permittivities assuming ⑀ 1 ()Хn 2 ϭ(1.53) 2 for bacteriorhodopsin and ⑀ 2 ()Хn 2 ϭ(1.54) 2 for gelatin.The refractive index of op-sin ͑the protein without the chromophore͒ was assumed to be equal to that of the gelatin matrix based on the experimental result for the effective dielectric constant of the protein in the M state (⑀ 2 (0)ϭ2.2) obtained by Dioumaev et al. 28 The resulting local field factors are tabulated in Table I.
The molecular hyperpolarizability, ␥, can now be related to the third order nonlinear susceptibility, (3) , by using the following relationship:

͑10͒
where ͗␥(Ϫ;0,0,)͘ is an orientational average of ␥ over the molecular coordinates and N is concentration of the guest molecule.The subscripts i, j, k, and l denote film coordinates and i jkl (3) (Ϫ;0,0,) is a tensor component of (3) .
The calculation of ͗␥(Ϫ;0,0,)͘ is discussed in Appendix B. If the films are isotropic, ͗␥(Ϫ;0,0,)͘ 3333 is also an average value over the film coordinates, and the average value of ␥ over the film coordinate is given by ͗␥͘ϵ͗␥͘ 3333 ϭ3͗␥͘ 1133 .Thus, ͗␥(Ϫ;0,0,)͘ can be calculated from 1133 (3) ͑Ϫ;0,0,͒ by using Eqs.͑10͒ and ͑11͒.The results are presented in Table IV.We consider the values for the Onsager elliptical local field factors to be the most experimentally relevant.The difference between the Onsager and Lorentz elliptical values, are, however much smaller than the experimental error.It would seem clear from an examination of Fig. 7 that an elliptical cavity is much more appropriate than a spherical cavity approximation.In these formulas, quantities with * denote local field cor- rected quantities and A 2 and A 4 are orientational order parameters.The factors A 2 and A 4 are zero for an isotropic system, k is a microscopic elastic modulus constant in dyn cm, and k is defined as a force constant of the restoring force F acting on the chromophore upon applying an electric field

C. Analysis with the two-state model
In ͑13͒, Ϫ 0 is an angle variation between the direction of the applied electric field and the static dipole moment of the chromophore before and after applying the electric field.Alternatively, k can also be estimated by multiplying the mo-lecular volume of all-trans retinal, V, by the bulk modulus of the gelatin polymer, K, assuming that the purple membrane patches are strongly interacting with the gelatin matrix.For the present study, we use an ellipsoidal cavity volume instead of the molecular volume ͑see discussion below͒.Equations ͑12͒ and ͑13͒ were evaluated assuming A 2 ϭA 4 ϭ0 ͑isotropic case͒ based on a bacteriorhodopsin concentration of 6.9ϫ10 18 cm Ϫ3 .The results are tabulated in Table II.All quantities not evaluated by us are available from the literature [30][31][32][33] and all relevant parameters are tabulated in Table III.Subtraction of s 1133 sor and s 1133 or from the experimentally measured value of s 1133 should give a pure electronic contribution to the quadratic electrooptic effect.The total orientational correction for s 1133 is consequently 8.5 ϫ10 Ϫ26 m 2 /V 2 , which is negligible compared to the experimental result of s 1133 obtained for this sample, 6.7 ϫ10 Ϫ20 m 2 /V 2 .Therefore, the orientational contribution is negligible, and will be ignored in subsequent analyses.
Our experimental values of ␥ as a function of energy can be fit with good precision by using a very simple two variable nonlinear equation: A/( B Ϫ) 3 , where is the photon frequency and A and B are the two variables.For reasons that will be clear from the subsequent discussion, this behavior is an indication that a single excited electronic state dominates the resonant enhancement of ␥.This permits the use of a simple two level model of the process where 0 ϭ ng corresponds to a transition frequency from a ground state to an excited state of interest, I denotes the average of all terms generated by permuting , 1 , 2 , and 3 and K is a numerical factor that depends on the presence of zero frequencies and repeated frequencies in the set , 1 , 2 , and 3 .In addition, i Ј is a ground state dipole moment of the chromophore along iЈ direction in the molecular coordinate, and ͉n͘ and ͉g͘ denote state vectors  for a nth state and a ground state of the chromophore, respectively.Equation ͑14͒ is a reduced expression of Eq. ͑B6͒ given in Appendix B.
Because the barred matrix element in Eq. ͑14͒ is defined as this term represents the dipole moment difference ⌬ gn between the ground state and the excited state.Also,

͑16͒
where the subscript ''1'' is used to represent the excited state, ''0'' represents the ground state and a damping term, D , is added to allow the use of this equation in fitting the experimental data ͑see discussion in Ref. 12͒.The cross terms that appear in Eq. ͑16͒ approximately cancel out and can be safely ignored. 29The most appropriate value for D is 250 cm Ϫ1 , an upper limit to the homogeneous linewidth in bR 34 ͓see also discussion following Eq.͑36͒ in Ref. 12͔.Fortunately, our subsequent calculations are not overly sensitive to assignment of the damping term provided this term is below 1000 cm Ϫ1 .It was our hope that we could use Eq.͑16͒ to examine the molecular electronic properties responsible for the large second order hyperpolarizability of bacteriorhodopsin.We therefore used weighted nonlinear leastsquares procedures to fit Eq. ͑16͒ to the Onsager elliptical experimental data.The weighting factors of the data points were assigned to be linearly proportional to the protein concentration.However, as might be anticipated by an examination of Eq. ͑16͒, the values of 10 and ⌬ 10 are highly correlated.That is, a unique simultaneous fit of these two variables is not possible.We explore their inter-relationship in Fig. 8, which presents the error contours as a function of the oscillator strength ͑y axis͒ and the change in dipole moment ͑x axis͒.The minimum error contour is shown with a gray dotted line, and is given by the equation:  12 These results suggest that in the near-resonant region, the 1 B u * ϩ -like state is the dominant contributor to ␥(Ϫ;0,0,).This conclusion follows from considerations of energy, oscillator strength, and dipole moment change, all three of which favor participation of the 1 B u * ϩ -like state in enhancing ␥(Ϫ;0,0,).Simulations based on a three state model and the above parameters indicate that the contribution of the 1 B u * ϩ -like state relative to the 1 A g * Ϫ -like state increases from ϳ100 at 676 nm to ϳ325 at 633 nm.This is further evidence that the two-state model is adequate for analyzing the results in this region of the spectrum.The two state model predicts ⌬ 10 ϭ12.8 D and 10 ϭ2.117 eV ͑Table IV͒ assuming f 10 ϭ0.8( 10 ϭ9.98 D). 12 If we include the error range in f 10 and experimental error in our experimental measurements, the error contour analysis presented in Fig. 8 predicts that the dipole moment difference between the 1 B u * ϩ -like ,* state and the ground state is 12.8Ϯ1.2D. Our assignment is in excellent agreement with the value determined from Stark measurements ͑12.4 D͒ 35 and two-photon spectroscopy ͑13.5 Ϯ0.8 D͒. 12 We anticipate that the close agreement between the present measurement of ⌬ 10 and Ponder's Stark measurement 35 is largely fortuitous given the error bars.Because the two-photon method can explicitly measure ⌬ 10 for the lowest-lying 1 B u * ϩ -like state, removing all contribu- tions from other participating states, we anticipate that the two-photon value (⌬ 10 ϭ13.5Ϯ0.8D) is the most accurate.
There is one aspect of Eq. ͑16͒ which may not be obvious upon first inspection, but which can have an interesting effect on both the magnitude as well as the frequency depen- dence of ␥(Ϫ;0,0,).The denominator of the first term decreases in magnitude faster than the denominator of the second term as approaches 10 .If the molecule has a low-lying excited singlet state that is both strongly allowed and undergoes a large change in dipole moment upon excitation, the first term will dominate and we will experience type III enhancement ͑Fig.9͒. 36Symmetric, non-polar molecules with strongly allowed low-lying excited singlet states will yield ␥(Ϫ;0,0,) values that are enhanced primarily via the second term.This type of enhancement is called ''type I'' enhancement ͑Fig.9͒. 36Clearly, the second hyperpolarizability of bacteriorhodopsin is dominated by type III enhancement, because ⌬ 10 Ͼ 10 . 15,20If the two-state approximation were rigorously accurate, type III enhancement could be verified experimentally by observing a sign change in ␥(Ϫ;0,0,) at the frequency However, in the present case, this frequency is in the ultraviolet ͑ϳ5.6 eV, ϳ220 nm͒, and other electronic states would dominate ␥(Ϫ;0,0,).

D. Comparison with theory
Time-dependent Hartree-Fock molecular orbital theory represents one of the more accurate methods available for calculating molecular hyperpolarizabilities. 37Kurtz and Korambath have implemented versatile iterative procedures for calculating ␣, ␤, and ␥ based on time-dependent methods within the MOPAC93 package. 38,39We used these methods to calculate ␥(Ϫ;0,0,) as a function of energy using for comparative purposes the AM1, PM3, and MNDO Hamiltonians.The iterative procedures are highly computationally intensive, particularly under resonance conditions.Despite the use of semiempirical Hamiltonians, we had to limit the calculation to the all-trans protonated Schiff base chromophore to achieve convergence at energies above 1.95 eV.The results are presented in the bottom three rows of Table IV.
The MNDO calculations do a relatively good job of reproducing the observed second-order hyperpolarizability considering the level of approximation and the fact that we are limiting the calculation to the chromophore.The fact that all three semiempirical parameterizations underestimate ␥͑Ϫ;0,0,͒ may also be due to the inherent tendency of all three methods to underestimate dipole moment changes and inaccurately represent transition energies into the low-lying excited singlet states. 25,40We note that all calculations were carried out using the standard parameterization optimized for the ground state.Nevertheless, we consider the results from the MNDO calculations to be worth noting and suggest that this Hamiltonian is the best choice for calculating the hyperpolarizabilities of molecules using time-dependent Hartree-Fock methods with MOPAC.

IV. COMMENTS AND CONCLUSIONS
We have examined the quadratic electrooptic effects ͑dc or low frequency Kerr effect͒ of bacteriorhodopsin dispersed in dried gelatin thin films.Data were collected at three wavelengths: 632.8 nm ͑He-Ne͒, 647.1 nm ͑Kr ion͒ and 676.4 nm ͑Kr ion͒.These wavelengths correspond to the near absorption edge region of the ground state of the bacteriorhodopsin ͑Fig.1͒.The films show relatively large quadratic electrooptic effects compared to other molecular dispersed systems.The purple membrane is fixed within the polymerized gelatin matrix, and we show that the electronic contribution to ␥ dominates over possible orientational contributions.At 676 nm, the quadratic electrooptic coefficient s 1133 (Ϫ;0,0,) is 6.7ϫ10 Ϫ20 m 2 /V 2 and the third order nonlinear susceptibility   1) denotes the lowest-lying, strongly-allowed 1 B u * ϩ -like ,* state.The g*(i) set of states includes all the higher energy 1 A g *-like ,* states with non-negligible transition mo- ments with the lowest-lying 1 B u * ϩ -like state.Arrows between different states correspond to electronic transition moments.Arrows that return to the same state indicate electron reorganization resulting in dipole moment changes of that state relative to the ground state.The principal terms responsible for the enhancement based on the two-state ͑types I and III͒ and expanded multistate ͑type II͒ approximations are shown below the relevant diagrams. (3)(Ϫ;0,0,) is 7.0ϫ10 Ϫ13 cm 4 statCoulomb Ϫ2 , with both values obtained for a protein concentration of 6.9 ϫ10 18 cm Ϫ3 .The value of s 1133 (Ϫ;0,0,) varies linearly with protein concentration indicating that the protein, and only the protein, is responsible for the observed hyperpolarizability.The orientationally averaged second molecular hyperpolarizability ͗␥(Ϫ;0,0,)͘ determined from the qua- dratic electro-optic coefficients at 676 nm assuming an Onsager ellipsoidal local field factor is (10.8Ϯ5.1)ϫ10 Ϫ32 cm 7 statcoulomb Ϫ2 .The value of ͗␥(Ϫ;0,0,)͘ increases tenfold to (103Ϯ65)ϫ10 Ϫ32 cm 7 statcoulomb Ϫ2 , when the probe wavelength is decreased to 633 nm.
Quadratic electrooptic effects measured at 633 nm are enhanced at high incident light intensities.We suggest that this enhancement is due to the stationary state accumulation of the O intermediate of bacteriorhodopsin, which strongly absorbs the irradiation at 633 nm.We were able to compensate for this source of error by extrapolating the data back to low photon densities.
The ␥(Ϫ;0,0,) values as a function of were fit to a two-state model of the form where ⌬ 10 is the change in dipole moment, 10 is the transition moment, and 10 is the transition energy of the lowestlying allowed 1 B u * ϩ -like ,* state.Our weighted nonlin- ear least squares fit indicates that ␥(Ϫ;0,0,) is strongly enhanced via type III processes.That is, the magnitude of ␥(Ϫ;0,0,) is dominated by the first term.We calculate that the dipole moment difference between the 1 B u * ϩ -like ,* state and the ground state is 12.8Ϯ1.2D, in good agreement with the value determined from Stark measurements ͑12.4 D͒ and two-photon spectroscopy (13.5 Ϯ0.8 D).Time-dependent Hartree-Fock methods based on the MNDO Hamiltonian yield reasonable agreement with experiment, underestimating ␥(Ϫ;0,0,) by factors of only 2-4, with the error increasing as the frequency approaches resonance.The AM1 and PM3 calculations were much less reliable, and thus we recommend use of the MNDO Hamiltonian for calculating hyperpolarizabilities within the MOPAC package.

ACKNOWLEDGMENTS
This work was supported in part by the National Institutes of Health ͑GM-34548͒.Specialized instrumentation was funded by the W. M. Keck Foundation.The authors thank Dr. Charles Martin and Igor Barani for their help with the theoretical calculations, and Bryan Vought and Constance Birge for help in preparing purified bacteriorhodopsin.We thank Jeff Stuart, Jack Tallent and Eric Tan for many helpful discussions.

APPENDIX A: THE ELLIPSOIDAL LOCAL FIELD FACTORS
The local field factor is defined as a ratio of an amplitude of the internal field E i to that of the external field E 0 .The local field factor for a dipole placed in an ellipsoidal cavity was derived based on the theory of electric polarization described by Onsager 18 and Bo ¨ttcher. 41e have two major effects to be considered to obtain an approximate field strength interacting with a dipole inside of the ellipsoidal cavity, 42 namely, ͑1͒ The cavity field: E c , ͑2͒ Reaction field: R. The internal field E i should be a sum of the cavity field and the reaction field, that is, Additional field that arises from orientation of permanent dipoles and is called the directional field E d should be considered if one works with liquid or gas phase.However, purple membrane is fixed in the gelatin matrix by the cross linked structure of the gelatin with hydrogen bonds, and the purple membrane is considered to be randomly oriented even under the external electric field.The effect from the directional field is therefore ignored here.The cavity field for the ellipsoidal cavity is given by 43 ␣͕a,b,c͖ ͑A2͒ ⑀ 2 is dielectric constant of the matrix and A ␣ is an ellipsoidal shape factor given by where a, b, and c are semimajor axes of the ellipsoidal cavity.
Meanwhile, the linear polarizability ␣ a , the reaction field factor f a and the reaction field R a in the a direction are given as Assuming the direction of the external field E 0 is along the semimajor axis of the ellipsoid, a, substitution of ͑A4͒ into ͑A1͒ yields From ͑A5͒, the local field factor at a static field for the ellipsoidal cavity is obtained as by letting ⑀ 1 be a permittivity at an optical frequency ⑀ 1 () and ⑀ 2 a permittivity at static field ⑀ 2 (0).For a field at an optical frequency, by assigning ⑀ 1 and ⑀ 2 to represent permittivities at optical frequencies ⑀ 1 () and ⑀ 2 (), respectively, we obtain For a spherical cavity, A a ϭ1/3.In such a case ͑A6͒ is reduced to which is identical to the Onsager's local field factor for a spherical cavity.Similarly for the ellipsoidal local field factor at optical frequency, a spherical local field factor at an optical frequency is obtained from ͑A7͒ In a more simple physical picture than the Onsager's model, where the reaction field is not taken into account, the local field can be written as sum of an external field and a polarization field inside an ellipsoidal cavity or a spherical cavity.The extent to which the polarization P 2 affects the internal field is given by the depolarizing tensor L E i ϭE 0 ϩ4L-P 2 .

͑A10͒
Substituting P 2 ϭ(⑀ 2 0 /4, where ⑀ 2 is a permittivity of the dielectric, into ͑A10͒ and rearrangement yields The diagonal components of L, L ␣ , are equal to A ␣ when the shapes of the molecule and the cavity are ellipsoidal.Consequently, the internal field is given by Assuming the direction of the external field E 0 is along the semimajor axis of the ellipsoid, a, the Lorentz-type local field factor for the ellipsoidal cavity is thereby defined as For a spherical cavity, A a ϭ1/3.Then ͑A14͒ is reduced to f ellipse or sphere (0) are also defined for the Lorentz-type local field factors similarly to those defined for the Onsager type local field factors APPENDIX B: RELATION BETWEEN "3… "؊;0,0,… AND ␥"؊;0,0,… A third order macroscopic polarization at an optical frequency is given by while a microscopic polarizability for each molecule at an optical frequency is given by Here, (i, j,k,l,...) denotes indices for film coordinates while (iЈ, jЈ,kЈ,lЈ,...) denotes indices for molecular coordinates for each molecule.This quantity corresponds to an induced dipole moment of a molecule.E local is a local field corrected electric field and related to external field as The third order polarization can be related to the microscopic polarizability with the local field factors Similarly, relation between the macroscopic susceptibility (3) and the microscopic susceptibility ␥ is i jkl ͑3͒ ͑Ϫ;0,0, s is a tensor that converts a molecular coordinate of sth molecule iЈ to a coordinate i fixed in the film.For simplicity, we will calculate an orientational average of the microscopic susceptibility ␥ and try to relate it to the macroscopic susceptibility (3) .The orientational average of ␥ can be obtained by examining the symmetry of permutations of the applied fields for the expression of ␥(Ϫ;0,0,) given by Orr and Ward 44 and by taking known nonzero components of ␥ for the point group of the molecule ͑C 1 in this case͒ into account.By neglecting the damping terms, their expression can be reduced in the near resonance or resonance region to

͑B6͒
where I denotes the average of all terms generated by permuting Ϫ , 1 , 2 , and 3 and K is a numerical factor with labels in ͑B6͒ and whose value depends on the presence of zero frequencies and repeated frequency in the set , 1 , 2 , and 3 .i Ј is a ground state dipole moment of the chromophore along iЈ direction in the molecular coordinate.͉n͘ and ͉g͘ denote state vectors for a nth state and a ground state of the chromophore, respectively.
By noting that: ͑1͒ two of the four frequencies in Eq. ͑B6͒ are equal, i.e., either 0 or , ͑2͒ Eq. ͑B6͒ allows exchange between the indices iЈ and jЈ and between kЈ and lЈ ͑3͒ all components are nonzero for C 1 symmetry, one obtains the orientationally averaged value of ␥ for a tensor component measured in the experiment with respect to the film coordinates as 45,46 ͗␥ i Ј j Ј k Ј l Ј ͑ Ϫ;0,0, ͒͘ i jkl ϭ iЈ jЈ͕x,y,z͖.͑B8͒ Using ͑B7͒, ͑B5͒ can be rewritten for isotropic medium as i jkl ͑3͒ ͑Ϫ;0,0, ͒ f ellipse ͑ ͒ ͗␥͑Ϫ;0,0,͒͘ i jkl .͑B9͒ ͗ ͘ i jkl here denotes an average value over the molecular co- ordinates x,y,z and experimentally measured value as a tensor component specified with the indices for the film coordinate i, j,k,l.Because the local field factors are constants at specific frequency, the linear relationship between (3) and ␥ found in ͑B9͒ is assured by the observation of a linear relationship between s 1133 and concentration ͑Fig.5͒.

FIG. 1 .
FIG. 1. Schematic diagram of the photocycle of light adapted bacteriorhodopsin ͑left͒ and absorption spectra of selected intermediates ͑right͒.Bold letters denote intermediates in the photocycle, and approximate absorption maxima of the intermediates are shown in nm.The abbreviation bR denotes the ground state of light adapted bacteriorhodopsin.Arrows without the label ''h'' indicate thermal decay.The three wavelengths used in our electro-optical measurements are indicated with vertical bars at right.

Figure 4
FIG. 3. Typical electrooptic response obtained from the bacteriorhodopsin dispersed in a dried gelatin thin film.The experiments were repeated three times to confirm reproducibility of the electrooptic response from the light adapted protein.Key experimental variables are as follows: modulation frequency, 500 Hz; film thickness, 15.3 m; bR concentration, 6.9 ϫ10 18 cm Ϫ3 and incident photon density, 2.4ϫ10 21 photons m Ϫ2 s Ϫ1 .

FIG. 5 .
FIG. 5. Concentration dependence of the quadratic electro-optic effects of the gelatin films containing bacteriorhodopsin at several wavelengths in near resonance region.The quadratic electrooptic signals vary linearly with respect to the concentration of bacteriorhodopsin.

͑11͒FIG. 7 .
FIG. 7. A view of the chromophore binding site of light-adapted bacteriorhodopsin based on the model proposed by Henderson and co-workers obtained from electron cryomicroscopy diffraction studies ͑Ref.9͒.The chromophore cavity used in the calculation of the Lorentz and Onsager elliptical local field calculations is superimposed.The numbers shown in parentheses give the center of mass displacements above ͑positive, out of the paper͒ and below ͑negative, into the paper͒ the chromophore polyene chain ͑in the plane of the paper͒.The putative position of calcium͑II͒ is based on two-photon studies ͑Ref.25͒.

FIG. 8 .
FIG. 8. Error contours for the weighted nonlinear least-squares fit of the Onsager ͑elliptical͒ ␥(Ϫ;0,0,) values to the two level model ͓Eq.͑16͔͒.The individual data points were weighted linearly as a function of the protein concentration.The error contours display the root-mean-square deviation in units of ␥(Ϫ;0,0,) in units of 10 Ϫ32 cm 7 statCoulomb Ϫ2 .The vertical axis represents the oscillator strength of the 1 B u * ϩ -like ,* state while the horizontal axis represents the change in dipole moment upon excitation of the same state.Note that f 10 ϭ0.8 ͓based on the two-photon analysis ͑Ref.12͔͒ dictates a value of ⌬ 10 ϭ12.8 D based on minimization of the error ͑gray dotted line͒.The error bar for ⌬ 10 (Ϯ1.2 D) is determined by folding the error range of f 10 ͑Ϯ0.07,Ref. 12͒ onto the expectation value for the experimental error ͑lowest-error contour lines͒.The probability that ⌬ 10 is within the range 11.7-14.0D is 95%.

FIG. 9 .
FIG. 9. Schematic diagrams showing the molecular electronic origins of the type I, type II, and type III second-order hyperpolarizability enhancement processes.The symbol g(0) denotes the ground state of the all-trans protonated Schiff base chromophore and u*(1) denotes the lowest-lying, strongly-allowed 1 B u * ϩ -like ,* state.The g*(i) set of states includes all the higher energy 1 A g *-like ,* states with non-negligible transition mo- ments with the lowest-lying 1 B u * ϩ -like state.Arrows between different states correspond to electronic transition moments.Arrows that return to the same state indicate electron reorganization resulting in dipole moment changes of that state relative to the ground state.The principal terms responsible for the enhancement based on the two-state ͑types I and III͒ and expanded multistate ͑type II͒ approximations are shown below the relevant diagrams.

TABLE I .
Local field factors.

TABLE II .
Induced orientation effect for the quadratic electrooptic effect found in a gelatin film containing bacteriorhodopsin.

TABLE III .
Values used to calculate s sor ,s or .
a This value was obtained for all-trans-retinal dissolved in hexane or cyclohexane based on electric field and refractive index measurements ͑Refs.30, 31͒.b This value was measured for bacteriorhodopsin dispersed in poly ͑vinyl alcohol͒ with second harmonic generation using a 1064 nm fundamental wavelength ͑Ref.19͒.c This value was measured as a Young's modulus at 20 °C and 65% relative humidity ͑Ref.32͒.A rate of loading for the tensile strength testing was 25 kg cm Ϫ2 s Ϫ1 .