Electro-optic sensors for electric field measurements. I. Theoretical comparison among different modulation techniques

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1 69 J. Opt. Soc. Am. B/ Vol. 19, No. 11/ November 00 Duvillaret et al. Electro-optic sensors for electric field measurements. I. Theoretical comparison among different modulation techniques Lionel Duvillaret, Stéphane Rialland, and Jean-Louis Coutaz Laboratoire d Hyperfréquences et de Caractérisation, Université de Savoie, Le Bourget du Lac Cedex, France Received November 15, 001; revised manuscript received May, 00 We present a complete analysis of electro-optic sensors for electric field measurement based on three different modulation techniques: amplitude, phase, and polarization state modulation. We treat the most general case, considering both isotropic and anisotropic crystals and taking into account the absorption of the crystal. We derive the optimal configuration of experimental setups for the three studied modulation techniques, and we give the values of the variable physical parameters required to yield the best performance. Finally we compare the three modulation techniques and show that phase or polarization state modulations result in exactly the same performance while amplitude modulation gives slightly enhanced performance. 00 Optical Society of America OCIS codes: , INTRODUCTION Although the electro-optical (EO) effect has been known for more than a century, the use of EO sensors for electric field measurements began to increase only at the beginning of the 1980s. 1 In particular, this development has been boosted by the need for high-frequency characterization, mainly for the diagnosis of electronic circuits. EO sampling developed at that time is now a widespread technique for time-resolved electric field measurements of repetitive signals. First used for guided-wave measurements, this technique has seen recent development for free-space measurements. 3 Because of the possibility of using fully dielectric sensors, 4 EO measurements are particularly well adapted for electromagnetic compatibility diagnostics and for the determination of radiation patterns of radar and microwave antennae. Although several interesting works 5 8 have been published on the optimization of particular EO measurement systems, to our knowledge no overview study has been published until now. The aim of the present paper is to give a complete analysis of the different modulation techniques that can be used for EO measurements of electric fields. In a following paper 9 we will deal with the choice of the EO crystals and their optimal orientation. Even if the choice of an EO crystal is linked to the modulation technique used for the measurements, these two problems can be treated separately. We have solved them in the most general case, and we have determined the optimal values of the variable parameters which yield the highest measurement performance. This will allow the reader to design optimized EO systems to probe either guided or freely propagating fields. The Pockels effect is the main physical phenomena used in EO sensors. As this effect is very weak, a highly optimized design is necessary not only of the EO sensor but of the whole experimental setup, even to measure strong electric fields. This is especially true for singleshot measurements 10 for which the signal-to-noise ratio (SNR) cannot be enhanced by the use of sampling techniques, but also in the design of compact probes 11 in which the EO crystal is small and the experimental efficiency could be weak. The remainder of the paper is in five sections. In Section we present the different modulation techniques that can be used and define the figure of merit of an EO system. In Sections 3 5 we derive the optimal configuration of the experimental setup for the three studied modulation techniques. In Section 6 we compare the performances of the three modulation techniques, i.e., their figures of merit.. POTENTIALLY USABLE MODULATION TECHNIQUES In most EO probing applications, one takes advantage of the linear EO effect, i.e., the Pockels effect, rather than the quadratic EO effect (dc Kerr effect) as the latter is orders of magnitude weaker. Whichever EO crystal is used to fabricate the sensor, its principal indices of refraction vary proportionally to the magnitude of the applied electric field. The electric field-induced variation of the refractive index could affect the amplitude, the phase, the polarization state, or the frequency of an optical beam probe passing through the crystal. The EO sensor geometry is varied depending on the beam parameter for which one wants to measure the induced variation. An optical beam linearly polarized along one of the two eigenpolarizations in the EO crystal coordinate system will be phase modulated by the applied electric field be /00/1169-1$ Optical Society of America

2 Duvillaret et al. Vol. 19, No. 11/November 00/J. Opt. Soc. Am. B 693 cause its transit time through the crystal depends on the applied electric field. Practically, such a phase modulation (PM) can be exploited in a Mach Zehnder interferometer in one of whose two arms an EO crystal has been inserted. 1 Thus the phase modulation is transformed into a readable intensity modulation. The electric field-dependent refractive index of the crystal can also be exploited using a Fabry Perot resonant cavity to produce an amplitude modulation (AM) of the optical beam. Any modification of the refractive index of the crystal will induce a change in the power of the optical beam both transmitted and reflected by the EO sensor. 13 This effect is greatly enhanced if the system is working close to a resonance peak of the Fabry Perot cavity. In the case of a circularly polarized optical beam, the applied electric field induces a phase retardation between the two components of the optical beam polarization aligned along the two eigendirections of polarization in the crystal. This phase retardation leads to a change in the polarization state of the optical beam. Practically, such a polarization state modulation (PSM) is converted into an optical beam power modulation by use of a polarizer. This modulation technique is by far the most used at present. Note that a frequency modulation of the optical beam is also present in all the preceding cases. 14 For an applied electric field E for which the angular frequency is much smaller than the optical carrier frequency, several sidebands separated by will appear in the spectrum of the frequency-modulated optical signal. To be exploitable, these sidebands should be distinguishable from the optical carrier. Thus this modulation technique is well adapted only for high-frequency electric field measurements, becoming unusable for low frequencies typically below a few tens to a few hundreds of megahertz. Because of this major drawback we have not considered frequency modulation in this study. We now focus on the fundamental physical feature that is to be optimized. Whichever modulation technique is chosen, the modulated optical signal is always sent to a photodetector to be converted into an electrical signal that will be acquired and processed. Prior to this conversion, the optical signal must of necessity be modulated in amplitude by the electric field to be measured. In the case of AM the optical signal is already modulated in amplitude. In the case of PM the conversion into AM is realized by making the modulated probe beam and a reference beam interfere. For PSM placing a polarizer in front of the photodetector ensures the conversion into AM. Furthermore AM, PM, or PSM of the optical probe beam originate from the electric field-induced phase retardation E through the Pockels effect. In the case of AM or PM, E represents the electric field-induced phase retardation of a probe beam linearly polarized along one of the two eigendirections of polarization in the EO crystal, whereas for PSM, E represents the electric field-induced phase retardation between the two eigenpolarizations of an optical probe beam in the EO crystal referential. Thus for the three studied modulation techniques, the problem consists first in finding the expression of the normalized optical power P impinging on the photodetector versus E. Here we define P as the detected optical power normalized with respect to the incident optical power impinging on the EO sensor. Once the expression of P versus E is obtained, one needs to find the optimal configuration of the experimental setup to maximize first, its linearity of measure to obtain quantitative and exploitable measurements, and second, its figure of merit. This optimization study, realized in Sections 3 5 for the three studied modulation techniques, has been done for only one of the two optical beams delivered by the sensor. Whichever modulation technique is considered, there are always two detection paths resulting in two modulated optical signals. For AM two light beams are reflected and transmitted by the Fabry Perot cavity. For PSM two cross-polarized light beams emerge from the EO crystal. For PM two light beams are transmitted to the two detection paths. Obviously one should take advantage of using both of these two optical signals, but the simultaneous optimization of the two signals is far more difficult. Moreover as compared to a single optical-signal detection, a double detection would lead to a gain of 3 db at the very most. We will discuss this particular point in Section 6. On the other hand, we have considered the optical absorption of the EO crystal in this optimization study whether the absorption is isotropic or not. Although the absorption coefficient of most inorganic crystals such as LiTaO 3 may be neglected, such is not the case when we consider the most promising EO crystals, e.g., organic crystals such as 4-N,N-dimethylamino-4-Nmethyl-stilbazolium tosylate (DAST) which presents a minimum absorption coefficient of 1.5 cm We now consider the figure of merit of an EO sensor. The intrinsic frequency bandwidth f c of the sensor depends on the effective transit time of the light in the EO crystal, and hence, varies inversely proportional to the crystal length L. On the other hand, the phase retardation E, proportional to L/, increases as the crystal length, and so the measurement sensitivity S E P / E varies as well. Consequently only the maximization of the product of frequency bandwidth and measurement sensitivity, which is independent of the crystal thickness, would seem to be of interest. However, maximizing such a product does not ensure realization of an experimental setup with the best performance. Indeed, increasing the measurement sensitivity for a given frequency bandwidth may increase the level of noise as well. Consequently, the figure of merit of the experimental setup should be proportional to the product of frequency bandwidth f c and SNR, this latter of course being itself directly proportional to the ratio of the measurement sensitivity S E and the noise. Although a noise expression can be easily obtained for a specific experimental setup, we cannot give an expression for a generic setup. Nevertheless, only three kinds of noise are usually encountered in EO systems: (a) (b) (c) noise that is signal independent (e.g., electronic noise in the detection circuit), noise that is directly proportional to the signal (e.g., relative intensity noise of the laser), noise that is proportional to the square root of the signal (shot noise in the photodetector).

3 694 J. Opt. Soc. Am. B/ Vol. 19, No. 11/ November 00 Duvillaret et al. Thus depending on the dominant type of noise, the SNR will be respectively proportional to one of the following: (a) S E (signal-independent noise), (b) S E /P ( E 0) (signal-proportional noise), or (c) S E /P ( E 0) 1/ (noise proportional to the square root of signal). Consequently we will get three different definitions of the figure of merit, depending on the type of noise considered. These three figures of merit are respectively defined as the maximum value of f c S E nl/c, f c S E /P ( E 0) nl/c, orf c S E /P ( E 0) 1/ nl/c. The factor nl/c has been included in each case to obtain a dimensionless quantity. Obviously we will be able to compare the figures of merit of the three studied modulation techniques for the same kind of noise, but the comparison among figures of merit linked to different kinds of noise will make no sense. Having clearly established the figure of merit that is to be calculated, we can now deal with the study of the first modulation technique. 3. POLARIZATION STATE MODULATION A. Detected Power We consider an optical probe beam propagating through an EO crystal. The phase retardation between the two eigencomponents of the optical probe beam polarization in the EO crystal coordinate system will be denoted cry. Let us write also n and n as the principal indices of refraction relative to the two eigenpolarizations. In the presence of an electric field E (at the low frequency much smaller than the optical frequency), the phase retardation is given by where 0 and E are the phase retardations due respectively to the natural birefringence and the electric fieldinduced birefringence of the crystal. In order to derive the measurement sensitivity S E of a generic experimental setup, let us consider the most general case, i.e., any polarization state of the incident probe beam and the most general detection system. As we do not know what is the optimal polarization state of the light beam impinging on the polarizing beam splitter, the polarization state (1) Fig.. Representation of any polarization state of the probe beam in terms of ellipticity b/a, optical power density I, and orientation, and in terms of magnitudes A 1 and A and phase retardation. of the probe beam emerging from the crystal must be converted first into any other polarization state. For that purpose, three birefringent plates are needed as shown in Fig. 1. A first /4 plate transforms the elliptic polarization of the beam into a linear one. A second /4 plate converts this linearly polarized light into elliptically polarized light with the desired eccentricity but whose direction is not adjustable at the same time. Hence a / plate rotates the axis of this elliptical polarization state to the selected direction. To state this problem in equation form, we use the Jones matrix formalism. 16 All the calculations given hereafter are done in the eigendielectric coordinate system of the EO crystal. The Jones vector associated with the incident probe beam having any polarization state is A 1 () A expj. The optical power density I of the optical beam, its ellipticity defined as b/a, and the orientation of its polarization state (see Fig. ) in the crystal coordinate system are given by 17 I A 1 A, , A 1A sin A 1 A, tan A 1A cos A 1 A. (3) The inverse transformation leads to the magnitudes A 1 and A and the phase retardation as A 1 cos sin 1 I, A sin cos 1 I, Fig. 1. Generic detection system for the use of PSM: EO, electro-optic, crystal; PBS, polarizing beam splitter; PD, photodiode. sin sin cos sin 4 cos 4. (4)

4 Duvillaret et al. Vol. 19, No. 11/November 00/J. Opt. Soc. Am. B 695 If we consider an anisotropic absorption of the EO crystal, then its Jones matrix is given by 1/ 0 0 1/ (5) expj 0 E, where and represent the optical power attenuation of the two eigenpolarizations of the probe beam transmitted by the crystal. We pose the nonrestrictive hypothesis that the second axis of the crystal coordinate system represents the most important losses, i.e., 1. The Fresnel reflection coefficients at both sides of the crystal can be integrated into the two attenuation factors and. The Jones vector associated with the probe beam transmitted by the crystal corresponds to the product of expressions (5) and (); it is given by 1/ A 1 1/ (6) A expj, where 0 E. The Jones matrix of the system composed of the three wave plates used to adjust the polarization of the beam emerging from the crystal (see Fig. 1) is given by R M 1/ R M 1/4 R M 1/4 R, (7) where R represents a rotation matrix of angle and M 1/m is the Jones matrix of a /m plate. 16 The product of the matrix resulting from expression (7) and the vector given in relation (6) leads, using relation (4), to the normalized optical power P 1 or P sent on the two detection paths given by where P 1 P P, P P P, 1 1 P, 4 4 P 1 1/ a cos b sin 1 1 c, 4 4 (8a) 1 cos, (8b) 1 a cos cos4 sin sin4 cos, b sin cos4, c cos cos4 cos sin4 sin. (9) The three secondary parameters (a, b, c) given in relation (9) depend on the orientations (,,) of the birefringent plates and satisfy the relation a b c 1. Consequently these three parameters are interdependent. Hence the parameter a can be suppressed in the following. It is easy to show that imposing the value of the angle will not restrain the possible values of the parameters b and c. Consequently we set down the condition 0. This signifies that the first /4 plate after the crystal creates a systematic phase retardation of / between the two eigenpolarizations in the crystal coordinate system. Such systematic phase retardation can also be created before the crystal and therefore this first /4 plate can be suppressed. For this simplified configuration, relations (8) remain unchanged but relations (9) take the simpler form a cos4 sin, b sin4, c cos4 cos. (10) B. Figure of Merit As the unique dependence of the normalized optical power upon the electric field E appears in relations (8) through the term a cos b sin, a linear response versus E will be obtained with the highest measurement sensitivity S E together with the lowest nonlinearity, providing that cos b1 c 1/. (11) This condition allows us not only to maximize S E but also to nullify P i / E (i 1 or ). Under this condition and in the limit of weak electric-field-induced phase retardation E, P can be rewritten as P E 1 1 c 1/ c 1 1. (1) 4 From now on we have to treat separately three subproblems depending on the dominant type of noise. We must also maximize the product of the frequency bandwidth f c and a quantity proportional to the SNR. As the impulse response of the sensor is an nl/c-time-duration crenel, the associated frequency bandwidth is then equal to 0.443c/nL. Because of the symmetry that exists between the two detection paths (the expression for P may be deduced from the expression for P 1 by the two operations 1/ 1/ and c c), we will treat here only detection path 1 (the upper path in Fig. 1). In the case of signal-independent noise, the figure of merit is obtained for c 0 and is equal to max( f c S E nl/c) 0.1 1/. In the case of signalproportional noise, the figure of merit, defined as max f c S E /P 1( E 0) nl/c, is equal to and is obtained for c (1 ) (1 )/(1 ) (1 ) independently of the value of. However, if we also want to have a reliable setup for which the figure of merit is mostly independent of adjustment errors of parameters c and, we can impose the requirement that the absolute values of the second partial derivatives of S E /P 1( E 0) with respect to c and be minimum. The product of these two partial derivatives takes its minimum absolute value when c (1 1/ )/

5 1 /1 cos 1 /1. (14) Using relation (10), we deduce a second relation cos4 cos 1 /1. (15) The last equation we need is derived from Eqs. (4), (10), (11), and (13) as sin4 arcsin 1/ / 1 cos 1 tan tan tan (16) 696 J. Opt. Soc. Am. B/ Vol. 19, No. 11/ November 00 Duvillaret et al. (1 1/ ). It is interesting to note that in this case the figure of merit is independent of the absorption factors 1 and. In the case of noise proportional to the square arccos 1 1 cos 0 1/, (17c) root of the signal, the figure of merit, defined as max( f c S E /P 1( E 0) 1/ nl/c), is equal to () 1/ /(1 1/3 ) 3/ and is obtained for c (1 4 arccos 1 1 1/3 )/(1 1/3 ). sin 0 We introduce a new parameter equal to 0, 1/, and 1/3, respectively, for signal-independent noise, signalproportional noise, and noise proportional to the square Relations (17) define entirely the EO setup parameters: 1 1 cos 0 1/. (17d) root of the signal. Thus we get unique expressions for The polarization state of the incident beam is given by its the optimal values of c and as polarization direction and its ellipticity, while the orientations of the /4 and / plates are, respectively,,, c 1 /1. (13) and (recall that we have already shown that 0). Using expression (13), relation (8) reduces to It is important to note that other quadruplet solutions of the system exist and also lead to an optimal configuration. However the solution expressed in relations (17) has the advantage of eliminating one of the four parameters, as we have 0 in all cases. Therefore one of the optimal polarization states of the laser beam impinging on the crystal is linear. We have now demonstrated that the optimal system of PSM detection for any kind of EO crystal isotropic or not, presenting an anisotropic absorption or not is always composed, in its most general implementation scheme, of two / plates, one /4 plate, a polarizer, and a polarizing beam splitter, as shown in Fig. 3. Obviously some of these optical elements can be suppressed, de- A solution of relation (14) exists only if the condition tan 1 is satisfied. When this condition is fulfilled, it is possible to show that relation (16) always has a solution. One of the solutions of the system formed by relations (14) (16) is then obtained by imposing the condition tan 1, which gives 0, arctan /, (17a) (17b) Fig. 3. Optimal experimental setup for the use of PSM: P, polarizer; EO, electro-optic crystal; PBS, polarizing beam splitter; PD, photodiode. Table 1. Figure of Merit, Expression of Normalized Optical Power Detected on Path 1, and Associated Optimal Values of Variable Parameters in Case of Polarization State Modulation Noise Type Independent ( 0) ( 1/) to Square Root of Signal ( 1/3) Figure of merit /3 3/ P 1( E 1) 4 1 E 1 E 1 /3 5/6 E 1 1/3 (rad) arctan( / ) (rad) (rad) 1 arccos arccos 1 1 sin 0 1 cos

6 Duvillaret et al. Vol. 19, No. 11/November 00/J. Opt. Soc. Am. B 697 pending on the laser used and on the orientation of both the crystal and the polarizing beam splitter. For example, in case a linearly polarized laser is employed, the polarizer would be suppressed. The main results are summarized in Table 1. In the case of isotropic absorption of the crystal, i.e., 1 (this actually constitutes the most common practical case), the setup configuration is always optimized for equally balanced detection paths, whichever type of noise we consider. A careful analysis of relations (17) elucidates the role of the three birefringent plates. Relation (17b) is concerned only with the anisotropy of absorption in the crystal and leads to 45 in the case of isotropic absorption. The first half-wave plate is used to give the proper orientation of the polarization of the laser beam where it enters the crystal and, if necessary, to compensate for anisotropy of absorption in the crystal. Relation (17c) shows that the axes of the quarter-wave plate must be aligned with the directions of the crystal referential when the crystal exhibits no natural birefringence. The role of the quarter-wave plate is to compensate, if necessary, for the crystal s natural birefringence. The role of the second half-wave plate, at the least, is to ensure an optimal balance of the optical power between the two detection paths. It is possible to show that the absence of compensation for anisotropic absorption ensured by the first half-wave plate would slightly reduce the figure of merit only if the shot noise is the dominating noise source. For example, the figure of merit is reduced by only 8.4% for a very high absorption anisotropy factor of In contrast, the compensation for natural birefringence ensured by the quarter-wave plate is essential if 0 is not close to a multiple of. In the absence of natural birefringence compensation the figure of merit would be reduced by a factor bounded by 1/cos 0 and (1 sin 0 )/cos 0. With respect to PSM we can say in summary that for most of the common EO crystals, the optimized detection system consists of the following: (a) a laser beam with linear polarization oriented at 45 with respect to the directions of the EO crystal coordinate system; (b) a quarter-wave plate situated after the crystal (to compensate for its natural birefringence); (c) a half-wave plate (to ensure an optimal balance of the optical power between the two detection paths). 4. AMPLITUDE MODULATION A Fabry Perot resonant cavity sensor consists of an EO crystal located between two mirrors. The electric fieldinduced variation of the crystal refractive index modifies the transmission coefficient of the cavity, resulting in amplitude modulation of the probe beam. In terms of simplicity and performance, the best arrangement consists in directly coating both ends of the crystal with reflective layers (see Fig. 4). This is the configuration we will consider here. Moreover, as we are dealing with AM, we assume that the incident optical beam is linearly polarized along one of two eigendirections of polarization in the EO crystal. Fig. 4. Fabry Pérot interferometer used for AM: EO, electrooptic crystal; M, mirror. Fig. 5. Power transmission coefficient versus optical frequency of a 10-finesse Fabry Pérot interferometer with the optimal operation point for AM. First we recall the general expression of the power transmission coefficient of a Fabry Perot etalon made of a medium with a refractive index n and an absorption coefficient, and two different mirrors without loss (see Fig. 4), 18 namely R T FP T F 1 1 F sin 0 E r, (18) where R (R 1 R ) 1/, T (T 1 T ) 1/, and r ( r1 r )/ are the mean values, respectively, of the power reflection and transmission coefficients and of the phase retardation due to reflection from the two mirrors. 0 is the electric field free-phase retardation of the light beam induced by the propagation through the crystal while E represents the relative phase retardation induced by the electric field, which is given by E L ne n0, (19) where is the optical wavelength and L is the cavity length. We further consider F as the finesse of the Fabry Perot cavity, defined by F FSR R expl f 1 R expl, (0) where f (see Fig. 5) is the FWHM of a resonance peak and FSR is the free spectral range given by FSR c nl. (1) The power reflection coefficient of the Fabry Perot cavity R FP can be obtained by writing the energy conservation law R FP T FP A FP 1. ()

7 698 J. Opt. Soc. Am. B/ Vol. 19, No. 11/ November 00 Duvillaret et al. The power absorption coefficient A FP is derived by taking into account the attenuation of the different beams resulting from amplitude division of the beam incident on the two mirrors of the cavity and writing it as A FP int 1 expl p0 T 1 R p/ int 1 R p1/ expl T 11 expl1 R expl 1 R. (3) expl In relation (3) the first term on the right-hand side 1 exp(l) represents the relative loss of power for a beam propagating through the crystal, while the summation represents the normalized powers of all the beams reflected into and from the crystal. We must now define the operation point of the sensor, both in reflection and in transmission, to ensure its highest measurement sensitivity. For measurements in transmission (respectively, in reflection), the highest sensitivity together with the lowest nonlinear response of the sensor is reached when the first derivative of T FP (respectively, of R FP ) with respect to E approaches maximum while its second derivative approaches null. Both in transmission and in reflection, we obtain the same optimal operation point (see Fig. 5): It is defined by a relative transmission coefficient of 75% in the asymptotic limit of high finesses. Note that this operation point depends on the laser wavelength. For this operation point, the measurement sensitivity of the Fabry Perot cavity is the same in reflection and transmission and is given by S E T FP E R FP E T F R F. (4) Although this relation is exact only in the limit of high finesses, it still gives a very precise value of the sensitivity with an error of less than one percent as far as F 10. For a weak nonlinearity, i.e., E 1, the normalized transmitted and reflected optical powers (respectively, P t and P r) at the optimal operation point are determined from relations (18), (), (3), and (4) as P t 3 4 T F 3F 1 R R E (5a) P r 1 T 11 expl1 R expl 1 R expl 3 4 T F 3F 1 E. (5b) What should the reflection coefficients of the mirrors be to maximize the performance of the setup? To answer to this question, we will first calculate the frequency bandwidth of the Fabry Perot cavity. For that purpose we consider the impulse response of the resonator to a Heaviside step function, i.e., a cw optical excitation of the device for t 0 that is switched off for positive times. Let us Fig. 6. Case of signal-independent noise: optimal reflection coefficient R (solid curve) of the cavity mirrors and associated finesse F (dashed curve) of the interferometer versus absorption factor L of the EO crystal. Fig. 7. Case of signal-independent noise: figure of merit of AM versus absorption factor L of the EO crystal. call I 0 the optical intensity stored in the cavity at t 0. The optical intensity transmitted for positive times is given by I t t 0 I 0 1 R R inttl/l 1 R intt/l expt, (6) with c/n. This function exhibits several temporal steps. In the case of high finesse cavities, the function is smoothed and simplifies itself into I t t 0 I 0 1 R R expt/ nl/c L ln R. (7) This impulse response is similar in principle to the electric discharge of a capacitor. Its frequency bandwidth is given simply by f c 1 FSRL ln R. (8) Now we deal with the calculation of the figure of merit, considering the three different kinds of noise. Whichever type of noise is being considered, the optimization problem has no analytical solution except for 0. Therefore we will give numerical results in what follows. In the case of signal-independent noise, the figure of merit presents the same expression for measurements

8 Duvillaret et al. Vol. 19, No. 11/November 00/J. Opt. Soc. Am. B 699 performed either in reflection or in transmission, and the optimal configuration of the setup is obtained for identical cavity mirrors. Figures 6 and 7 show, respectively, the optimal reflection coefficient of the mirrors and the figure of merit versus the absorption factor L. As seen in Fig. 7, the figure of merit presents a very slight variation with L, and so with the Fabry Perot finesse. Indeed, the increase of SNR with finesse is nearly completely counterbalanced by the concomitant decrease in frequency bandwidth. In the case of signal-proportional noise, the optimal configuration of the setup is obtained for identical cavity mirrors when measurements are performed in transmission. For measurements in reflection, however, the two cavity mirrors must present different reflection coefficients, namely R R 1 1 R 1 R L. (9) Fig. 10. Case of noise proportional to the square root of the signal: optimal mean reflection coefficient R (solid curve) of the cavity mirrors and associated finesse F (dashed curve) of the interferometer versus absorption factor L of the EO crystal. Heavy curves are relative to measurements performed in reflection while light curves are relative to measurements performed in transmission. Fig. 8. Case of signal-proportional noise: optimal mean reflection coefficient R (solid curve) of the cavity mirrors and associated finesse F (dashed curve) of the interferometer versus absorption factor L of the EO crystal. Heavy curves are relative to measurements performed in reflection while light curves are relative to measurements performed in transmission. Fig. 9. Case of signal-proportional noise: figure of merit of AM versus the absorption factor L of the EO crystal for measurements both in reflection (heavy curve) and in transmission (light curve). Fig. 11. Case of noise proportional to the square root of the signal: figure of merit of AM versus the absorption factor L of the EO crystal for measurements both in reflection (heavy curve) and in transmission (light curve). Note that the reflective coatings of the two mirrors must actually be different only in the case for which L is not much lower than unity. Figures 8 and 9 show, respectively, the optimal mean reflection coefficient of the mirrors and the figure of merit versus the absorption factor L. Once again the figure of merit is almost independent of the cavity finesse. Otherwise note that the figure of merit can be enhanced up to three times when performing measurements in reflection instead of transmission. Moreover the requirements for the optical quality of the cavity mirrors are much less strict for measurements in reflection. For example, with an absorption factor L of 10 4, the required reflection coefficient of the cavity mirrors is 90% for measurements performed in transmission while it is only 76% for measurements performed in reflection. Moreover, these latter measurements would lead to a figure of merit approximately three times greater. In the case of noise proportional to the square root of the signal, the optimal configuration of the setup is again obtained for identical cavity mirrors when measurements are performed in transmission, while the mirrors should be different for reflection measurements, as seen in

9 700 J. Opt. Soc. Am. B/ Vol. 19, No. 11/ November 00 Duvillaret et al. R R R 1 R L. (30) Figures 10 and 11 show, respectively, the optimal mean reflection coefficient of the mirrors and the figure of merit versus the absorption factor L. The figure of merit is almost independent of the cavity finesse and can be enhanced up to a factor of 3 when performing measurements in reflection instead of transmission. Here again the requirements for the quality of the cavity mirrors are less severe for measurements in reflection. The main results to this point are summarized in Tables and 3 for measurements performed, respectively, in transmission and in reflection. Whatever type of noise Table. Figure of Merit, Expression of Normalized Optical Power, and Associated Optimal Values of Variable Parameters for Measurements Performed in Transmission in Case of Amplitude Modulation Noise Type Independent to Square Root of Signal Figure of merit See Fig. 7 See Fig. 9 See Fig. 11 Figure of merit for L 0 P t( E 1) TF R 3 3F 1 E nl m Z m r arcsin/ 3F R 1 R R SeeFig.6 SeeFig.8 SeeFig is considered, measurements in reflection lead to better performance than measurements in transmission. 5. PHASE MODULATION For phase modulation studies the EO crystal is now located in a Mach Zehnder interferometer (see Fig. 1). The applied electric field modifies the interference pattern of the interferometer and thus creates a power modulation of the transmitted light. For a probe beam linearly polarized along one of the two eigendirections of polarization in the crystal, the optical phase retardation between the two arms of the interferometer, measured in the principal detection path, is given by n 1L 0, (31) where 0 is the phase retardation that exists in the absence of the EO crystal. Losses in the two beam splitters can be neglected practically, but not always the losses occurring in the EO crystal, which can be the result of either Fresnel reflections at its two faces or of its intrinsic absorption. Consider now, the power transmission coefficient of the crystal, and R 1 and R, the power reflection coefficients, respectively, of the first and second beam splitters. The normalized optical powers detected in the principal and complementary detection paths are given, respectively, by P p R 1 1 R R 1 R 1 R 1 R 1 R 1 1 R 1/ cos, (3a) P c 1 R 1 1 R R 1 R R 1 R 1 R 1 1 R 1/ cos. (3b) Table 3. Figure of Merit, Expression of Normalized Optical Power, and Associated Optimal Values of Adjustable Parameters for Measurements Performed in Reflection in Case of Amplitude Modulation Noise Type Independent to Square Root of Signal Figure of merit See Fig. 7 See Fig. 9 See Fig. 11 Figure of merit for L 0 P t( E 1) TF 3 3 R 1 3F E nl m r arcsin/ 3F m Z R 1 R R1 1 R 1 R L R1 4 1 R 5 1 R L R R R1 1 R 1 R L R1 4 1 R 5 1 R L R (R 1 R ) 1/ SeeFig.6 SeeFig.8 SeeFig.10

10 Duvillaret et al. Vol. 19, No. 11/November 00/J. Opt. Soc. Am. B 701 1/(1 1/ ) gives a figure of merit almost independent of small variations in the reflection coefficients R 1 and R ); (3) for noise proportional to the square root of the signal, the figure of merit is equal to / /(1 1/3 ) 3/ and is obtained for R 1 1/3 /(1 1/3 ) and for R 1/(1 1/3 ). Fig. 1. Mach Zehnder interferometer used for PM with power transmission and reflection coefficients of the two beam splitters: EO electro-optic crystal; M, mirror; BS, beam splitter. Table 4. Figure of Merit, Expression of Normalized Optical Power Detected on Principal Path, and Associated Optimal Values of Variable Parameters in Case of Phase Modulation Noise Type Figure of merit P p( E 1) 0 (rad) R 1 Independent ( 0) ( 1/) E 4 ne 0 1L 1 E 1 to Square Root of Signal ( 1/3) /3 3/ /3 5/6 E m 1 1 1/3 m Z What value should the adjustable parameter 0 have to ensure a linear response and the highest sensitivity of the sensor versus E? Whichever detection path is considered, these requirements are simultaneously satisfied when cos 0, i.e., for 0 ne 0 1L. (33) We have now to calculate the figure of merit, i.e., to find the optimal values of R 1 and R that maximize the product of frequency bandwidth f c and a quantity proportional to the SNR. As was true for PSM, the intrinsic frequency bandwidth of the sensor depends only on the EO crystal length L and is given by 0.443c/nL. For this particular modulation technique the solution of the problem is especially simple and leads to the following results for the principal detection path: (1) For signal-independent noise, the figure of merit 0.1 1/ and is obtained for R 1 R 1/; () for signal-proportional noise, the figure of merit is equal to and is obtained for R 1 R /(1 R R ). (A brief analysis shows that R 1 R 1 1 Table 4 recapitulates the optimal values of the adjustable parameters and gives the figure of merit and the expression for the normalized optical power in the principal detection path in the case of a fully optimized setup. 6. COMPARISON AMONG THE THREE MODULATION TECHNIQUES By comparing Table 1 with Table 4, we notice a remarkable similarity between PSM and PM; these two modulation techniques lead to exactly the same figure of merit in the limit of transparent EO crystals for any type of noise. However, in the case of absorbing crystals exhibiting no absorption anisotropy ( 1), the dependence of the figure of merit on the power absorption coefficient clearly shows that PM leads to better performance than PSM. The figure of merit for PM is 1/ 1/ and /(1 1/3 ) 3/ higher than for PSM for a signal-independent noise and for a noise proportional to the square root of the signal, respectively. For example, a value of 0.3, corresponding to a DAST crystal a few millimeters thick, would lead to figures of merit for PM 83% and 31% higher than for PSM in the case, respectively, of signalindependent noise and noise proportional to the square root of the signal. Table 5 gives a comparison between the figures of merit of the three modulation techniques versus the type of noise. In order to establish this table, we have considered an EO crystal exhibiting a negligible absorption. AM shows significantly better performance in reflection than PM and PSM for signal-dependent noise. Despite the fact that PM and PSM are slightly better than AM in the case of signal-independent noise, AM in reflection seems to be overall the best modulation technique. However, we cannot definitively conclude this as the indices of refraction involved in AM and PM on the one hand and in PSM on the other hand are not the same. AM and PM bring into play only one of the two indices of refraction (n or n ) of the crystal [see relations (19) and (31)] whereas PSM is a function of their difference n n Table 5. Figures of Merit for PSM, AM, and PM in Case of a Fully Transparent Electro-Optic Crystal Noise Type Independent to Square Root of Signal PSM AM in transmission AM in reflection PM

11 70 J. Opt. Soc. Am. B/ Vol. 19, No. 11/ November 00 Duvillaret et al. n [see relation (1)]. Consequently, depending on the EO crystal chosen [see Ref. 9], PSM could present better performance than AM in reflection. Moreover, other parameters have to be taken into account such as the straightforwardness of implementation. For example, one can use a standard laser source for PSM, but needs a high-quality polarizing beam splitter, such as a Wollaston prism, and birefringent plates. On the other hand, typical EO sensors required for PSM are of low cost and easy to produce. In the case of AM, a high-quality laser source with a very narrow line width and very precise adjustment and control of the wavelength is needed. An EO crystal with exceptional flatness and parallelism of its faces is also necessary, as well as high-quality reflective coatings. However the thickness of an AM-based sensor is considerably reduced as compared with PM- or PSMbased sensors. Hence for noninvasive applications, AM presents an outstanding advantage as highly efficient inorganic crystals such as LiTaO 3 or KNbO 3 (see Ref. 9) have very high dielectric constants, leading to a measurement-induced perturbation all the stronger with the thicker sensor. In the case of PM, a high-quality laser source with a narrow line width is required. Furthermore the fabrication of the sensor is rather difficult and costly and the adjustment to attain the optimal operation point is not straightforward. However note that the figure of merit can be doubled by inserting the same EO crystal rotated 180 in the reference arm of the interferometer. Unfortunately, in that case the measured electric field will be averaged over two different optical paths, making such a sensor unusable for diagnosis of integrated circuits. For all the reasons described above, we believe that none of the three modulation techniques presents a decisive advantage over the others. Moreover in this study we have considered only one of the two modulated optical signals, whereas the signals in the two detection paths could be exploited simultaneously. In that case, and for transparent crystals, the incident optical power is balanced between the two detection paths. Therefore both SNR and figure of merit can be enhanced by a factor of in the case of PM or PSM. In the case of AM the figure of merit is increased by only 5% for signal-proportional noise and up to a factor of for signal-independent noise when both transmitted and reflected optical signals are employed. 7. CONCLUSION We have presented a comprehensive analysis of three different modulation techniques that can be used to design any EO measurement setup. We have derived the best configurations to achieve the highest product of frequency bandwidth and SNR for the three different types of noise that are usually encountered in such experiments. For each of these configurations, we have given the values of the various parameters required to obtain the best performance. We have demonstrated that standard PM and PSM present exactly the same performance for transparent EO crystals. We have also demonstrated that an AMbased setup should result in slightly better performance than standard PM- or PSM-based setups. However, in the absolute, a PM setup with two crystals mounted head to foot in the two arms of the interferometer permits one to reach the highest figure of merit, although the applications of such an EO sensor would be reduced compared with those based on AM or PSM sensors. 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