Diffraction from tunable periodic structures. II ... - OSA Publishing

Diffraction from tunable periodic structures. II. Experimental observation of electric field–induced diffraction peaks Xi Yang, Markus Aspelmeyer, Lowell T. Wood, and John H. Miller, Jr.

As previously predicted 关Appl. Opt. 40, 5583 共2001兲兴, we have now observed electric field–induced diffraction peaks in transmission and reflection experiments by use of a LiNbO3 sample with interdigital planar electrodes that serve as a diffraction grating. The magnitudes of the new peaks in the reflection experiments are ten times larger than those in the transmission experiments. We interpret these effects in terms of a field-induced refractive-index change produced by the linear electro-optic effect. The positive and negative changes in the refractive index produce two diffraction gratings that are period doubled with respect to the original grating and that have a phase difference between them. The superposition of the diffracted light from these gratings is shown to account for the new peaks. From the relative magnitude of the new peak to that of the central peak, we estimate the refractive-index change to be 0.004. © 2002 Optical Society of America OCIS codes: 050.1950, 050.1960, 160.2100, 230.2090.

1. Introduction

The importance of electro-optic thin-film materials as phase and amplitude modulators in photonic devices has led to the development of several methods for characterizing such materials. See Refs. 1–5 and references therein for details concerning such techniques. Recently, Trivedi et al.6 and Tayebati et al.7 used a diffraction-based technique to measure the electro-optic coefficients for Sr0.6Ba0.4Nb2O6 共SBN: 60兲 and Sr0.75Ba0.25Nb2O6 共SBN:75兲. Their experimental technique relied on measuring changes in the total optical power in the first-order, diffracted spot from a grating assembled with interdigital electrodes 共IDEs兲 and when a potential difference was applied to them. Yang et al.8 solved this problem by appropriately grouping periodic structures according to their phase and space correlations. Numerical calculations based on this solution clearly show that new When this research was performed the authors were with the Department of Physics, University of Houston, 617 Science and Research Building One, Houston, Texas 77204-5005. M. Aspelmeyer is now with Sektion Physik and Center for NanoScience 共CeNS兲, Ludwig-Maximilians-Universita¨t Mu¨nchen, GeschwisterScholl-Platz 1, 80539 Mu¨nchen, Germany. Received 14 March 2002; revised manuscript received 18 June 2002. 0003-6935兾02兾285845-06$15.00兾0 © 2002 Optical Society of America

peaks in the diffraction pattern arise as the electric field–induced refractive-index change increases. These new peaks are due to a period doubling of the phase grating induced by the electro-optic behavior of the material underlying the interdigital planar electrodes 共IDEs兲.1 The oppositely directed electric fields in any two of the neighboring interdigital fingers and spaces 共Fig. 1兲 cause alternating signs of the field-induced refractive-index changes, because of the linear electro-optic effect. Therefore, the period of the region with the same refractive-index distribution doubles once the electric field is applied. As a result of this period doubling, new peaks appear between neighboring diffraction peaks, and their relative strengths can be used to study the linear electrooptic behavior of the sample. 2. New-Peak Development Theory

Taking into account the alternating signs of the fieldinduced refractive-index changes 关Fig. 1共a兲兴 in a linear electro-optic sample with IDE, we consider the grating to consist of two subgratings with different refractive indices and with a doubled period 关2共a⫹b兲兴 with respect to the original grating. Here, we define the width of the electrodes as a and the spacing between them as b to include the case for which a and b are unequal. The far-field diffraction pattern is the coherent superposition of the scattered waves from the two gratings. The spatial relation between 1 October 2002 兾 Vol. 41, No. 28 兾 APPLIED OPTICS

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where ␦⬘ ⫽ k关2共a ⫹ b兲兴 sin共␪兲 and ⌽ 0 ⫽ k关2共a ⫹ b兲兴共2兲兵cos关共␪兾2兲 ⫹ ␣兴 ⫻ sin共␪兾2兲其 ⫹ k2⌬n共E兲2t.

Fig. 1. 共a兲 Electric field lines. The horizontal arrows show the in-plane electric field components; the vertical arrows show the normal field components, 共b兲 Interdigital planar electrodes. Grating groups 1 and 2 are indicated.

these two gratings is similar to that of the double slits in Young’s interference experiment. Before the electric field is applied, grating one is equivalent to grating two except for a transverse separation 共a⫹b兲 between them. The influence of the field-induced refractive-index changes inside the sample is equivalent to that of putting a glass plate in front of a slit in Young’s interference experiment, thereby introducing an additional phase shift to one of the slits. In our case, the interference pattern shifts transversely by a distance that is determined by the phase difference between the two subgratings 关Fig. 2共a兲兴. The intensity of the far-field diffraction pattern is directly proportional to the product of the diffraction term from either of the two grating groups, and the corresponding interference term between them and is given by8

I共␪兲 ⫽

再

sin关ka sin共␪兲兾2兴 ka sin共␪兲兾2

⫻ 关cos共⌽兾2兲兴 2, 5846

冎再 2

sin关m共␦⬘兾2兲兴 m sin共␦⬘兾2兲

冎

2

(1)

APPLIED OPTICS 兾 Vol. 41, No. 28 兾 1 October 2002

If a ⫽ b, the result in Eq. 共1兲 is identical to that given by Yang et al.8 The first term in Eq. 共1兲 is due to diffraction from a single slit, the second term is due to interference from a multislit subgrating, and the third term is due to interference between the two subgratings. The interference term has minimum values in the alternating peak positions of the diffraction term before the electric field is applied 关Fig. 2共b兲兴. These minimum values can be explained by noting that the peak-to-peak distance of the interference term is twice that of the diffraction term. The alternating peaks in the diffraction term are suppressed by the minima of the interference term before the electric field is applied, and the far-field diffraction pattern is shown in Fig. 2共c兲. After the electric field is applied, the minima of the interference term are moved away from the alternating diffraction peaks because of the field-induced interference pattern shift. The suppressed diffraction peaks grow and are referred to as new peaks 关Fig. 2共d兲兴. The new peaks reach maxima when the interference pattern shifts one half of a period. Afterward, the new peaks decrease and reach their minima when the interference pattern shifts one period 共Fig. 3兲. Usually, ⌬n is in a range of 10⫺4, resulting in relative intensities of the new peaks of approximately 10⫺3 with respect to the zero-field, central-peak intensity in the transmission experiment. Value ⌬n is calculated from the equation ⌬n ⫽ 共1兾2兲n 03r effE,

(2)

where E ⫽ V兾b is the in-plane electric field that is the main contributer to ⌬n, and reff is an effective electrooptic coefficient that has its primary contributions from r51. V is the applied voltage, and b is the width of the grating space. The refractive indices of LiNbO3 are 2.3188 共no兲 and 2.2241 共ne兲 at a laser wavelength of 532 nm. Therefore, the maximum refractive-index change is 2.4 ⫻ 10⫺4 when the field strength is 1.25 ⫻ 106 V兾m. Use of Eq. 共1兲 yields a 0.001 ratio of the strength of the new peak to that of the original central peak. 3. Sample Preparation

We chose LiNbO3 as the sample for the following reasons: 共a兲 LiNbO3 shows a linear electro-optical effect that is due to its noncentrosymmetric crystal structure; 共b兲 the electric field–induced refractiveindex change is large enough to make the new peaks detectable; 共c兲 no other effects, e.g., piezoelectricity, dominate the linear electro-optic effect. IDE consisting of 7-␮m conducting lines separated by 8-␮m gaps were sputter deposited on one side of a two-sided polished, 共001兲-oriented LiNbO3 single crystal 共10.5 ⫻ 9.5 ⫻ 0.5 mm3兲 so that alternating fingers could have opposite potentials. 共Unequal values for

Fig. 3. Intensity of the first new peak versus the refractive-index change 共solid curve兲 and of the center diffraction peak versus the refractive-index change 共dashed curve兲.

a and b were chosen to obtain a more general result.兲 We obtained the in-plane electric field in the 关100兴direction 共x兲 by aligning the interdigital fingers in the 关010兴-direction 共y兲. Therefore, the x and the z components of the applied electric field change the refractive indices by means of the maximum electro-optic coefficients r51 and r33. Each line consists of a thin ˚ 兲, and external contacts were layer of platinum 共500 A made with soft indium 关Fig. 1共b兲兴. 4. Experiments and Results

The schematic diagram of the transmission experiment is shown in Fig. 4. We used a lock-in amplifier and a photodiode to detect the weak optical signals. A lens with a focal length of 48 mm was used to obtain a 1° divergent beam so that we could vary the number

Fig. 2. 共a兲 Interference term 共Iint兲 between two gratings. The solid curve represents the interference before the electric field is applied, and the dashed-dotted curve represents the interference after the electric field is applied. 共b兲 Diffraction term 共Idiff兲 from one of the two gratings. 共c兲 Far-field diffraction pattern 共Iint ⫻ Idiff兲 before the electric field is applied; this pattern is proportional to the production of the interference term shown in the solid curve of 共a兲 and the diffraction term shown in 共b兲. 共d兲 Far-field diffraction pattern after the electric field is applied; this pattern is proportional to the production of the dashed curve in 共a兲 and the curve in 共b兲.

Fig. 4. Experimental setup of the LiNbO3 sample in the transmission mode. 1 October 2002 兾 Vol. 41, No. 28 兾 APPLIED OPTICS

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Fig. 6. Experimental setup of the LiNbO3 sample in the reflection mode.

Fig. 5. 共a兲 Signals with modulating electric field on. 共b兲 Intensities of the corresponding positions in 共a兲 with electric field off. 共c兲 Normalized signals.

of illuminated grating lines by changing the distance between the lens and the sample. The source-tosample and sample-to-detector distances remained large enough so as not to violate the Fraunhofer criterion. We illuminated eight grating lines and applied a 10-V potential difference across the IDE, giving rise to a field strength of 1.25 ⫻ 106 V兾m, at a frequency of 193 Hz. The detector was placed 875 5848


mm away from the sample, and we took the data by moving the detector transversely in 1-mm steps, while averaging each point over 10 s. The total movement of the detector in the experiment was 120 mm, a distance that covered the central diffraction peak and most of the two first-order diffraction peaks. The results are shown in Fig. 5共a兲. Here, the detected signal arises from the change in the intensity distribution of the diffraction pattern produced by the modulating electric field. To see the new features in the electric feld–induced intensity distribution change, we compare these signals to the intensity distribution without the field applied. To obtain the intensity distribution without the field applied, we used an optical chopper to provide an ac-modulated light signal for the intensity measurements at the same spatial positions as those in Fig. 5共a兲. This optical chopper ensures that these two diferent measurements, one obtained with the electric field on and one with it off, were obtained under the same experimental conditions. The intensity measurements with the field off are shown in Fig. 5共b兲 and were used to normalize the signals in Fig. 5共a兲. The normalized signals are shown in Fig. 5共c兲, which is a pointby-point division of the points in Fig. 5共a兲 by those in Fig. 5共b兲 and can be used to obtain the relative intensity of the new peaks to that of the central peak, as will be discussed at the end of this section. The schematic diagram of the reflection experiment is shown in Fig. 6. Again, we illuminated eight grating lines and used the same experimental procedures that we used in the transmission experiments. The angle of incidence was 32°. The results were obtained in the same way as those in Figs. 5共a兲–5共c兲 and are shown in Figs. 7共a兲–共c兲.

ten times larger than those in the transmission experiment 关Fig. 5共c兲兴. To explain this observation, we should understand the formation of the diffraction pattern. The diffraction patterns in the above experiments were formed by diffracted waves from the periodic grating. The ratio of the strength of the boundary diffraction waves to the total strength of the incident beam determines the magnitude of the diffraction peaks. This argument can also be applied to the new peaks because they are treated as the suppressed diffraction peaks before the electric field is applied. In the reflection experiment, the boundary diffraction waves, which come from the edges of the grating lines, include contributions from both the grating line side 共because of reflections from the Pt grating lines兲 and the grating space side of each edge 共because of reflections from the interface兲. In addition, the secondary beams 共reflected three times inside the sample兲 also contribute to the boundary diffraction waves owing to the high reflectivity of the Pt grating lines. The intensity of the secondary beams was approximately 15% of the primary reflected beam at an incident angle of 32°. It was obvious that the contribution of the boundary diffraction waves in the reflection experiment was much higher than that in the transmission experiment. It would be quite difficult to obtain a quantitative ratio of the boundary diffraction waves in these two experiments. Because Eq. 共1兲 is normalized to the central-peak intensity, to use it to determine the change in refractive index ⌬n requires that the new-peak intensities in Fig. 5共c兲 be renormalized to the central-peak intensity in Fig. 5共b兲. To obtain the relative new-peak intensity to the central-peak intensity in Fig. 5共b兲 requires that the new-peak position be determined from Fig. 5共c兲. The new-peak intensity from Fig. 5共a兲 divided by the central-peak intensity in Fig. 5共b兲 was used to estimate ⌬n. Then ⌬n is varied in Eq. 共1兲 until the correct ratio of the new-peak intensity to the central-peak intensity is produced. Value ⌬n was found to be 0.004. 5. Conclusion

Fig. 7. 共a兲 Signals with modulating electric field on. 共b兲 Intensities of the corresponding positions in 共a兲 with electric field off. 共c兲 Normalized signals.

In both the transmission (Fig. 5) and the reflection experiment 共Fig. 7兲, two new peaks appeared between two neighboring diffraction peaks. These observations support the new-peak prediction given by Yang et al.8 The normalized magnitudes of the new peaks in the reflection experiment 关Fig. 7共c兲兴 were almost

We experimentally observed new peaks produced by electric field–induced period doubling in both transmission and reflection experiments and estimated a refractive-index change of 0.004, an order of magnitude larger than predicted by Eq. 共2兲. This larger order of magnitude means that for this method to be useful for measuring electro-optic coefficients, it needs to be refined further. Also we observed that the normalized magnitude of the new peak in the reflection experiment was almost ten times larger than that found in the transmission experiment and proposed that a qualitative difference between the boundary diffraction waves in the reflection experiment and those in the transmission experiment was responsible for this magnitude difference. Additionally, we provided an intuitive physical explanation of the new peaks discussed in Ref. 8 by using the sim1 October 2002 兾 Vol. 41, No. 28 兾 APPLIED OPTICS

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ilarity between Young’s double slits and the fieldinduced gratings. The authors thank Professor F. Hussain for the use of his Nd:YAG laser. The authors acknowledge support from the Texas Higher Education Coordinating Board Advanced Research Program and Advanced Technology Program and from the Robert A. Welch Foundation 共E-1221兲.

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