Diffraction of optical communication Gaussian beams by volume gratings: comparison of simulations and experimental results

Transcription

1 Diffraction of optical communication Gaussian beams by volume gratings: comparison of simulations and experimental results Pierpaolo Boffi, Johann Osmond, Davide Piccinin, Maria Chiara Ubaldi, and Mario Martinelli The diffraction effects induced by a thick holographic grating on the propagation of a finite Gaussian beam are theoretically analyzed by means of the coupled-wave theory and the beam propagation method. Distortion of the transmitted and diffracted beams is simulated as a function of the grating parameters. Theoretical results are verified by experimentation realized by use of LiNbO 3 volume gratings read out by a 1550-nm Gaussian beam, typical of optical fiber communications. This analysis can be implemented as a useful tool to aid with the design of volume grating-based devices employed in optical communications Optical Society of America OCIS codes: , , , , Introduction Holography is a well-known technology employed for optical processing. Nowadays thick holographic gratings find applications in the field of optical fiber communications in the implementation of devices 1 3 such as optical filters, wavelength demultiplexers, optical interconnects, and storage media. For all these applications, the input output coupling to transmission fibers appears to be a critical constraint and conditions their design and performance assessment. Classical study of holography based on the use of plane waves cannot be utilized for fiber communication devices that involve Gaussian waves that come from the fiber. Some previous theoretical studies 4 9 have predicted optical signal distortion at the output of volume gratings, causing a loss in fiber coupling. A theoretical and experimental study of the different output beam profiles and their distortion appears to be necessary to determine the influence of a thick holographic grating on Gaussian beam propagation The authors are with the CoreCom Consortium for Research in Optical Processing and Switching, Via G. Colombo 81, Milan 20133, Italy. M. Martinelli is also with the Dipartimento di Elettronica e Informazione, Politecnico di Milano, Piazza Leonardo da Vinci 32, Milan 20133, Italy. P. Boffi s address is boffi@ corecom.it. Received 17 October 2003; revised manuscript received 31 March 2004; accepted 6 April $ Optical Society of America and more generally to optimize the insertion loss of grating-based devices. Here, a theoretical and experimental analysis of the Bragg diffraction of finite Gaussian beams by volume gratings is reported. In Section 2 we introduce the different theories used in our study. In Section 3 we provide through simulations the theoretical evolution of distortions that are due to the thick grating diffraction as a function of grating parameters. Simulation results obtained with both theories taken into account are then presented. Moreover, in Section 4 we show the experimental results and their comparison with simulations. In our experimentation, a thick holographic grating is written in a standard photorefractive crystal LiNbO 3 by means of Ar laser plane waves at 488 nm a wavelength that corresponds to the maximal photosensitivity of our material. In contrast, the reading wave is a Gaussian beam at a different wavelength. In our case, we used a reading 1550-nm beam to take into account the real application of holographic gratings in the field of optical communications. We obtained good agreement between the experimental results and the simulations by demonstrating the capability to simulate and foresee the Gaussian beam propagation in a volume grating. 2. Propagation of Gaussian Beams in Thick Gratings: Theory In the following we theoretically analyze the diffraction of a Gaussian beam that is due to a thick grating. The diffraction conditions are shown in Fig. 1. We used the classic layout configuration APPLIED OPTICS Vol. 43, No July 2004

2 where the amplitude of refractive-index modulation n 1 is small compared with n 0. We assume a single unslanted grating and a reading TE-polarized Gaussian incident wave, whose beam waist is large in comparison with the free-space wavelength of reading beam 1.55 m. Only the transmitted and diffracted waves are considered 4,5 here because the grating thickness values used in this study are quite high; thus other orders of diffraction can be neglected. We also do not take absorption into account. Two different approaches are used in our theoretical analysis. In Subsection 2.A we introduce the Kogelnik coupled-wave analysis 4 CWA for the theoretical study of thick grating diffraction and distortion of the output beam profile as a function of the grating and the Gaussian input beam parameters. A detailed description of the CWA is justified to understand diffraction behavior as a function of different parameters. In Subsection 2.B the so-called BPM beam propagation method 10 algorithm is also considered to confirm the accuracy of the CWA by comparison of the simulation results obtained with the two methods. These simulations will also give an outline of the different distortion types observed for both transmitted and diffracted beams that will be experimentally analyzed further Section 3. Fig. 1. Model of a thick grating with unslanted fringes in the Bragg diffraction regime. B is the Bragg angle of incidence of the reading beam in the medium defined by 2 sin B. Grating is assumed to extend infinitely in the x y plane and to be thick: the Q parameter, Q 2 d n 0 2, provides an evaluation of the grating thickness with respect to the condition 4 Q 1, where is the grating period, is the free-space wavelength of the reading beam, d is the grating thickness, and n 0 is the average refractive index of the medium outside the grating. The refractive index is sinusoidally modulated in the x direction and in the region 0 z d by n x n 0 n 1 cos 2 x n 0 n 1 f x, (1) A. Kogelnik s Coupled-Wave Theory The phase curvature of the Gaussian beam profile is assumed to be negligibly small in the grating region, a condition satisfied if the number of grating periods across the Gaussian spot size is sufficiently large. 11 Neglecting the phase curvature of the input beams, we can write the total electric field in the grating as a linear superposition of the complex amplitudes of the transmitted and diffracted waves, R x, z and S x, z respectively. At first, the study is considered under a near-field condition, which corresponds to an observation distance that is smaller than the Rayleigh length. Introducing a new coordinate system r, s defined by r z sin B x cos B and s z sin B x cos B and considering that at the input plane z 0 r s, there are no diffracted waves S 0, and the transmitted R beam is only a function of r, we obtain the transmitted and reflected amplitude expressions for input beam R 0 r and for a grating of thickness d in a near-field condition 5 : 1 R r R 0 r 1 2 R 0 r d 1 u sin B 1 1 u J 1 1 u 2 du, u 1 S s i 2 R 0 s d 1 u sin B 1 J 0 1 u 2 du, (2) where d cos B is the grating strength or a phase delay factor, n 1, J 0 and J 1 are Bessel functions of the first kind. In the case of an incident Gaussian beam with an amplitude profile R 0 r E 0 exp r 2 0 2, we obtain 5 1 R r R 0 r 1 2 E 0 exp g 1 u r u J 1 1 u 2 du, u 1 S s i 1 2 E 0 exp g 1 u s 2 1 J 0 1 u du, (3) 1 July 2004 Vol. 43, No. 19 APPLIED OPTICS 3855

3 where r r 0, s s 0, E 0 is the peak value of the electric field, 0 is the Gaussian beam 1 e 2 radius, and geometry parameter g d sin B 0. To determine the electric field intensity of the transmitted beam in the far-field zone we can use the Fraunhofer approximation that leads to the calculation of the far-field electric intensity as the squared Fourier transform of the near-field electric intensity. Another formalism also issued from the Kogelnik theory allows us to calculate the far-field intensities more easily thanks to evaluation of the transfer function of the grating. The spatial output profile in the far field could in fact be expressed as the product of the transfer function and the angular spectrum of the input beam: S ff E H S, R ff E H R, (4) where E the angular spectrum of the input beam is the Fourier transform of the input spatial beam E r at z d and H R and H S are the R transmitted and the S diffracted beam transfer functions. These transfer functions are obtained from the Kogelnik expressions of the output R and S fields 4 for a single lossless unslanted grating when the input beam is a unit amplitude, uniform, and planar wave : The diffraction efficiency is defined here as SS*, where S is the output signal at z d for an incident plane wave of unit amplitude. It can also be written as 4 sin when the Bragg condition is verified, 0, and we can obtain a complete conversion of energy for * 2 m, where m is an integer. B. Beam Propagation Method The BPM algorithm is just a recursion relationship giving expressions of the electric field from the Helmholtz scalar-wave equation at infinitesimally small axial distances z one from another. 10 The underlying assumptions of its classical use are as follows: all the angles are small to maintain the paraxial condition, a small grating modulation is assumed, and the backward reflection and its effect on the forward propagation are neglected. Paraxiality represents the major limitation of the basic paraxial BPM for the study of propagation in free space or in a grating. Enhanced BPMs that extend the validity to wide angles are available. (7) R exp i cos i sin , S i exp i sin (5) The R transmitted and S diffracted beam transfer functions H R and H S are equal to the expressions presented previously in Eqs. 5, where the dephasing term is 1, the first-order approximation 8 of the Taylor series of the dephasing term d 2 cos B dimensionless for a slight deviation of the input angle B but without wavelength detuning: 1 Kd 2 d. (6) For unslanted gratings K sin K 2 4 n 0, where is the angle of incidence of the reading beam in the medium. The spatial output profile in the near field can be expressed as the inverse Fourier transform of the product of the transfer function and the angular spectrum of the input beam. 8 We can observe that, as the R and the S beam profiles depend on only two variables and g in the near field, we have the same propriety in the far field. Besides, the S-beam profile is symmetrical in the near field around the s g axis and in the far field around the B axis. They allow us to work at angles larger than deg from the optical z axis, which is typically the limit of what can be considered paraxial. The most popular approach is referred to as the multistep Padébased wide-angle technique, which allows us to relax, to varying degrees, the paraxial approximation of the classical BPM. 12 The BPM extended to wide angles provides another method that can be used for analysis of the Bragg diffraction of thick gratings. 3. Simulations of Bragg Diffraction of Gaussian Beams by a Transmission Unslanted Volume Grating The BPM has already been used as a powerful method for analyzing the volume grating diffraction problem 10 and has been compared with the rigorous CWA 13 by calculation of the diffraction efficiency of a plane-wave input. In Subsection 3.A we present a comparison of CWA and the BPM taking into account calculated intensity profiles of both transmitted and diffracted beams in the case of a Gaussian wave input to a grating structure. For this purpose we use some significant configurations of a single unslanted grating and a TE-polarized Gaussian beam without tak APPLIED OPTICS Vol. 43, No July 2004

4 Fig. 2. Comparison of the a R-beam profiles and b S-beam profiles in the near field. The results were obtained by CWA and the BPM. The following parameters were used: n , m, d m, BPM distance of observation d obs 120 m, and BPM z axis computed step z 0.04 m. Hence, the geometry and grating strength parameters are g 3.0 and 4. ing into account absorption of the Gaussian beam into the grating medium. Here we focus only on the near-field configuration because the far-field results can be deduced from the near-field results. Furthermore, the near-field condition corresponds to the reasonable position in which the fiber coupling would take place. We then provide a discussion of the grating parameters and finally conclude with the simulations and an experimental discussion. A. Coupled-Wave Analysis and Beam Propagation Method Near-Field Simulations At first it is necessary to specify that like the BPM, the CWA or coupled-mode analysis is also approximate because it involves the solution of a scalar-wave equation and neglects some boundary conditions and some spatial harmonic components. Since the two different computing solutions based on CWA rely on the same method, one using fast Fourier transform which has been previously exposed and the other an integral in the spatial domain, 5 they give similar results. Afterward, we plot only the results obtained with the spatial domain integration for the near field. With regard to the BPM, it has been used with RSoft BeamPROP 4.0 software, which implements a Padé algorithm that extends the validity of the BPM to off paraxiality. This software enables us to plot the intensity beam profiles at a desired distance of observation from the grating end face usually chosen to allow a sufficient spatial separation of the Braggscattered beam from the transmitted beam. The computed step points that represent the refractiveindex modulation are a fixed rate of 24 points per designed grating period. Since the main purpose of these simulations is a comparison of different diffracted and transmitted beam shapes and not the lateral shift not considered in experiments, we plot the BPM and CWA superposed on the theoretical lateral position obtained with CWA simulations. Assumptions exposed in Subsection 2.A are verified by our simulations, for example, the Gaussian beam waist is large in comparison to the free-space wavelength 1.55 m, and the amplitude of refractive-index modulation n 1 is small compared with the average refractive index of the LiNbO 3 :Fe crystal n used afterward in our experiments. We consider a single case of grating configuration: its average refractive index, its period, and its Bragg angle values are fixed by experiments. These latter parameters also fix the grating period, m, from which we can deduce the Bragg angle B with the Bragg condition 4 : 2 sin B N n 0, where N is an integer. It is obvious that this study is also applicable for values of other parameters. Since it has been exposed in theory and has been demonstrated by Moharam et al., 5 the spatial profiles of the transmitted and diffracted beams and the diffraction efficiency obtained with CWA can be written and presented as functions of only two normalized parameters: grating strength n 1 d cos B and geometry parameter g d sin B 0. We chose to vary refractive-index modulation n 1, beam radius 0 Gaussian beam 1 e 2 radius, and grating thickness d also called grating length to obtain representative values of the g and parameters that correspond to interesting cases to be analyzed. Figures 2 4 show three interesting but different computed simulations. These cases present meaningful distortions of diffracted and transmitted Gaussian beam profiles as functions of grating strength and geometry parameter and, as a consequence, a function of d, 0 and n 1. By observing the near-field computations of diffracted S beam or Bragg-scattered beam and transmitted R beam or reference beam beam profiles, one can observe good agreement between the two techniques, and the behavior of these profiles as a function of g and values can be readily understood. Comparing the BPM and CWA simulations, one can first observe that profiles obtained by both simulations are quite equivalent even if there are occa- 1 July 2004 Vol. 43, No. 19 APPLIED OPTICS 3857

5 Fig. 3. Same as Fig. 2 except that the following parameters were used: n , 0 50 m, d m, BPM distance of observation d obs 1200 m, and BPM z axis computed step z 0.3 m. The geometry and grating strength parameters are g 4.5 and sional minor differences. Profiles from both simulations have the same general shape. As far as shape distortion behavior due to volume grating is concerned, in the simulation related to Fig. 2 b, the S-beam profile can be seen as two Gaussian lobes that overlap. The same kind of shape has been analyzed by Chu et al. 6 Bragg-scattered first-order Gaussian beam for half-space, and it agrees with Forshaw s experiments. 14 Other diffracted profiles present some different distortions that are the same for both simulations and that will be further analyzed. R-beam profile distortions are also the same, except for the first case seen in Fig. 2 a where the BPM simulation does not represent a small sidelobe separated by a zero from the central peak and visible on the CWA simulation. If we define the beam profile full width as the profile width at 95% amplitude, a comparison of the different widths shows that the S- and R-beams full width values obtained with the CWA and BPM simulations are quite similar. A dispersion of 3% for S beams and 6% for R beams was found when we excluded cases in which a sidelobe was visible on CWA simulation but not on the BPM. Widths are slightly larger with BPM simulations than with CWA simulations. Differences in intensity of the profiles are present because BPM simulations are more attenuated than CWA simulations, especially for R beams. We determined that with BPM simulations a small portion of the beam power is lost, and we neglected the contributions that are due to backward reflections, which could explain the difference between beam intensities. A propriety of the CWA theory mentioned above is not valid for the BPM theory. The profiles of the transmitted and diffracted beams are not the same for the same pair of parameters g, but are composed of different values of the refractive-index modulation, beam waist, and grating thickness. A source of error and difference between simulations Fig. 4. Same as Fig. 2 except that the following parameters were used: n , m, d 300 m, BPM distance of observation d obs 550 m, and BPM z axis computed step z 0.3 m. The geometry and grating strength parameters are g 3 and 2, APPLIED OPTICS Vol. 43, No July 2004

6 can be wide angles: extension of the BPM to wide angles owing to the Padé algorithm introduces errors that could vary with simulation parameters. Moreover BPM simulations require a high computed precision and the available amount of allocated memory sometimes limits the precision of the simulated profiles. More generally, we can conclude with the help of these plotted simulations that shapes, widths, and amplitude of the beam profiles are generally close between the BPM and the CWA simulations. However, some singular differences can be found, for example, singularity of the profiles sidelobe, zero... or peak intensity value dispersion. Since there is good agreement between the BPM technique and the CWA method when some approximations are taken into account, these simulation tools can equivalently predict the diffraction behavior of a Gaussian beam that propagates into a volume grating. B. Theoretical Discussion Taking into consideration a fixed configuration of a grating,, B, we found it interesting to study the evolution of diffracted and transmitted profiles and the diffraction efficiency as a function of other parameters such as n 1, d, and 0 and then as a function of g and. It could, for example, provide a solution to obtain a Gaussian profile with the highest diffraction efficiency or otherwise predict what kind of distortion could be observed for a given configuration. Some trends of the S- and R-beam characteristics as a function of g and have already been analyzed in the literature. 5 8 The major conclusion is that the profiles remain Gaussian for g 1: it means a small value of the grating thickness and or a large value of the beam radius. For example, the smaller the value of grating thickness d therefore for small values of g, the less significant the interactions and the less distorted the diffracted and transmitted beams. In this case, diffraction is comparable to the case of the plane wave, and the peak amplitude therefore follows the sin 2 function. 4 The same conclusion could be explained in the transfer function formalism: the impulse response of the grating appears to be close to an impulse and allows us to transmit the input profile undistorted. Inversely, at a higher value of the grating thickness, the impulse response widens. At a fixed value of g and g 1, the profiles are again Gaussian-like for very large values of verifying g 8 according to Moharam et al. 5. Physically, this means that, for a small Gaussian profile high value of g, the distortion is compensated by a significant amount of grating strength and therefore by a high refractive-index modulation and or by a substantial grating thickness. As the value of grating strength increases, the concentration of energy in the S beam is closer to s 0 g, and its profile is more and more Gaussian. A trade-off exists between conserving a Gaussian profile and obtaining the highest diffraction efficiency possible. In our analysis we performed a complete study related to the shape distortions of diffracted and Fig. 5. Three-dimensional plot of the near-field diffracted S-beam profile for a Gaussian wave input as a function of gamma 0;5 and s 0 2;8 with a fixed g value of 3. The normalized intensity is plotted on the vertical axis. transmitted beams by considering their behavior as a function of different parameters, taking into account grating strength and geometry parameter g in the near field. 1. S-Beam First-Order Diffracted Beam A diffracted beam can present different kinds of distortion that changes it from Gaussian-like because of different values of the grating strength and the geometry parameter. Figures 5 and 6 show threedimensional plots of the diffracted beam profile that varies with one of the two parameters grating strength and geometry parameter, respectively when the other parameter has a fixed value. We can first generally observe that these profiles have a dramatic evolution as a function of these parameters and that distortion increases with an increase in the geometry parameter value. Fig. 6. Three-dimensional plot of the near-field diffracted S-beam profile for a Gaussian wave input as a function of g 0;6 and s 0 2;12 with a fixed value of 9 4. The normalized intensity is plotted on the vertical axis. 1 July 2004 Vol. 43, No. 19 APPLIED OPTICS 3859

7 For some values of g and for example, g 3 and 2.1, some sidelobes appear around the principal peak see Fig. 4 b. The position of the sidelobes varies as a function of the considered parameters as shown in Fig. 5 for g 3. Another type of distortion has already been observed in the past by Forshaw 14 and predicted by Chu et al. 6 As far as we are concerned, this distortion, called hole burnt into the profiles, 6 resembles two Gaussian shapes that overlap corresponding to different diffracted contributions. Figure 2 b shows this effect for 4. The burnt hole is more visible as g increases. For very small values of 1 the diffraction efficiency is small because of a small refractive-index modulation and therefore has a uniform diffraction over a large length. The profiles have a rectangular shape Fig. 3 b and this effect is more visible as the value of g increases for a fixed value of. This case corresponds to a large value of the profile width. More generally, as the g value increases, the S-beam profile width increases; at a fixed value of g, as the value increases the width decreases. In any case, the S beam as well as the R beam for the same reason is confined to a defined region. The 1 e amplitude width of the S beam or of the R beam is bounded by a boundary width value of 9 g d tan B (8) cos B However, for a given value of g, a large value exists for which the S beam full width is much smaller than the boundary width value. As has been mentioned in Subsection 2.A, even if the S-beam profiles are distorted they are constantly symmetrical around the s 0 g axis. This position is the center of the S-beam pattern for small values of g. At a high fixed value of g, the central peak tends to reach this position when the value increases. Considering that the medium ends at z d at the same position as the grating, the beam position outside the medium z d can be written as 9 x d tan B z d tan for the R beam, (9) x d tan B z d tan for the S beam, (10) where is the Bragg angle of the reading beam outside the medium obtained from the Bragg angle in the medium by Snell s law. Even if optimization of the signal coupling at the end face of the volume grating is not important for far-field behavior, it would be interesting to consider it. Usually from near-field considerations we can deduce some properties for the far-field profiles and inversely as the intensity in the far field is the squared amplitude of the Fourier transform of the amplitude in the near field. In general, for the farfield distance we consider the S-beam profiles as a central peak with sidelobe ripples. 14 These ripples occur in all the far-field profiles and are symmetrical Fig. 7. Three-dimensional plot of the transmitted R-beam profile for a Gaussian wave input as a function of 0; 5 and r 0 2; 8 with a fixed g value of 3. The normalized intensity is plotted on the vertical axis. around the Bragg angle position B. The smaller the values of grating thickness d therefore, for small values of g, the less distorted the diffracted beam, the smaller the sidelobe ripples, and the less broadened is the S-beam profile width. In fact, most of the energy is associated with a plane wave, which respects the Bragg condition. At a high value of the grating thickness, the impulse response widens, the number of components of the incident beam angular spectrum with respect to the Bragg condition decreases, and the sidelobe ripples are more important the diffracted beam is more distorted. 2. R-Beam Transmitted Beam The major distortion of the R beam consists of the presence of sidelobes but generally it is not too distorted as could be the case for the S beam and a Gaussian profile is recognizable see Figs. 2 a and 3 a. In fact, the increased value required to obtain a Gaussian-like R beam starting with a distorted configuration is lower than the increased value required for the S beam. Inversely, we also have a Gaussian-like R beam for high values of g and for 1, which corresponds to a rectangular shape of the S beam see Fig. 3 a. In some cases, the R beam can be split into two beams, an ordinary transmitted and a forward diffracted, an effect that can be observed, for example, in Fig. 8. The more the g value increases, the more distorted is the R-beam profile and the less it resembles a Gaussian profile; this evolution can be seen in Fig. 8. The distortion is due to the coupling of energy between the R and the S beams during the Bragg scattering process. 7 The R-beam profile width is greater than the incident beam width because, even if the transfer function of the grating is impulselike, it always has a broadening. As values of increase see Fig. 7 or values of g decrease, the profile width values increase. As well as what has 3860 APPLIED OPTICS Vol. 43, No July 2004

8 Fig. 8. Three-dimensional plot of the near-field transmitted R-beam profile for a Gaussian wave input as a function of g 0,6 and r 0 2,12 with a fixed value of 9 4. The normalized intensity is plotted on the vertical axis. been explained for the S beam, the full width of the R beam is bounded. The peak of the beam profile shifts spatially as the grating strength increases. At a fixed value of g, as the value of increases, the energy concentrates around the r 0 g axis and the shift increases Fig. 7. However for g 1, the R beam is symmetrical around the r 0 axis whatever the value of. The R-beam far-field pattern is confined to a small angular range and centered at the negative Bragg angle of B. The major distortion of the R beam consists in a dip or even a deep null in the middle of the R-beam profile. 6 A deep null appears in the center position when * 2 m, where m is an integer. The central portion of the Gaussian spectrum of the transmitted beam has completely converted its energy into a Bragg-scattered wave, which results in a depletion of energy from its beam-center position and corresponds to the split of the R beam into two beams in the near field see Fig. 7. For some different values of, we could expect only a small dip. 3. Diffraction Efficiency and Conversion of Energy For a plane wave, the Kogelnik theory 4 predicts a complete conversion of the input beam into a diffracted beam for values of grating strength * 2 m, where m is an integer. By using finite beams we can observe that R beams still contain an amount of energy for these values of the grating strength. Considering the diffraction efficiency that has been defined in Subsection 2.A as SS* where S is the output signal at z d for an incident plane wave of unit amplitude, the value of a Gaussian beam is always less than the value of a plane wave. 7 A total conversion cannot therefore occur with a Gaussian beam. It has been interesting and innovative for us to plot in three dimensions diffraction efficiency defined previously see Fig. 6 as a function of g and see Fig. 9. From Fig. 9 we can observe that, as g Fig. 9. Three-dimensional plot of the diffraction efficiency of a Gaussian wave as a function of g 0,6 and 0,5. increases, the diffraction efficiency decreases, for a fixed value of g, as increases, the diffraction increases and tends to reach the plane-wave diffraction efficiency 4 sin 2. To obtain a maximum diffraction efficiency value, the grating strength value had to be equal to a * value. Even though that is sufficient for g 1, for larger values of g a large value of * is also necessary. C. Theoretical Conclusion From the above analyses we can conclude that, for g 1or 1 and 8 g, R and S beams are Gaussian-like. For intermediate values of g and, the R and S intensity profiles are no longer Gaussian and contain some distortion. The distortion increases when the g values increase also for larger values of the grating thickness and or for smaller values of the beam radius and the diffraction efficiency decreases. There is also a loss in the efficiency of converting energy from the input beam into the S beam. We now turn our attention to experimentation for the purpose of comparing experimental results with theoretical predictions. 4. Experimentation of Bragg Diffraction of Gaussian Beams by a Transmission Unslanted Thick Grating Forshaw first reported a study of the diffraction of a narrow laser beam by a thick hologram, 14 but this kind of analysis has never been carried out experimentally. In other published papers, the principal experimental interest focused on grating selectivity and diffraction efficiency results. Here we present our experimentation of Gaussian beam diffraction with a thick holographic grating by taking into account the diffracted beam distortion with regard to fiber coupling. A. Experimental Setup The thick gratings that we checked in our experimentation were holographic, recorded by a fairly common setup, and could be used for optical storage experi- 1 July 2004 Vol. 43, No. 19 APPLIED OPTICS 3861

9 Fig. 10. Experimental setup of the grating recording. ments. The recording material we used was a 1 cm 1 cm 2 cm photorefractive crystal of LiNbO 3 :Fe 0.05 mol% of Fe doping produced by Deltronics. The crystal was a 0 -cut and the beams were horizontally polarized to achieve maximal modulation of the refractive-index recording. The experimental setup is shown in Fig. 10. The 488-nm emission of an Ar-ion laser was expanded by a lens and filtered by a spatial filter pinhole. This first part of the setup was used to obtain a plane wave. The horizontal polarized light was obtained by a 2 wave plate. The incident beam was then divided into two beams of equal intensity by a 50:50 beam splitter. Together they produced an interference pattern in the recording material. We produced the grating by exposing the LiNbO 3 :Fe crystal to a spatially varying pattern of light intensity. The position of the mirrors induced an equal path length and an equal incident angle of 15 for both recording beams. This geometric arrangement led to a fringe period of nm, if we consider an error of 0.01 on the incident recording angles. Refractiveindex modulation n 1 was controlled by the amount of exposure time. 14 The crystal was placed on a translating support with a goniometer and a rotating stage with a measurement accuracy of The lighted areas on the material were 1 cm 1 cm squares. By measuring the total area of illumination on the input face of the crystal, we deduced the geometric form of the grating. To obtain a grating with a mostly rectangular shape, we canceled the back part of the original hexagon shape of the grating by means of an incident Ar beam of incoherent light with a suitable spatial window. This technique was also used to control grating thickness. Read out is based on the so-called two- method, with a Gaussian beam as readout radiation at 1550 nm, the well-known third window of optical communications, to analyze the experimental behavior of interesting cases for their future use in the field of optical communications. The crystal was placed on a rotating stage oriented at 55.2 angle in air corresponding to the Bragg angle in the medium from the laser reading source as shown in Fig. 11. The light source was a semiconductor laser 1550 nm pigtailed to a fiber whose end is placed on an x y scanning device, which allowed us to analyze the entire surface of the recorded volume grating. Moreover its position in combination with Fig. 11. Experimental setup for the grating analysis APPLIED OPTICS Vol. 43, No July 2004

10 Fig. 12. Comparison of the CWA simulated S-beam intensity profile with the experimental S-beam intensity profile for g 6 and 2.1 in a the near field and b the far field. the free-space propagation until the input into the crystal and the choice of collimator enabled us to control the beam radius values in the crystal. We also measured the powers and diffraction efficiency with some powermeters, and we observed the beam spots with an infrared camera. Our observations were made for both the near field and the far field. It is important to stipulate that these experimental setups enabled us to change different parameters easily to achieve a large range of g and values that were useful for our analyses. B. Experimental Results Many representative cases related to particular sets of g and values have been experimentally tested. Here we present only a few examples. As in the Kogelnik theory, we consider a zeroth-order beam and one first-order beam whose profiles have been registered in near-field and far-field conditions. We applied the Kogelnik CWA and BPM simulations to each set of parameter values both near-field and far-field solutions, which we then compared with the experimental results. Here we present only the comparison with the CWA simulations because the BPM simulations do not provide additional information. For each output beam we consider the direction along which the beam has been diffracted which has been defined in the Kogelnik theory as the s axis. The experimental profile intensities are normalized over the CWA profiles. The beam radius parameter at the input face of the crystal is equal to m in all the experiments, and the far-field condition is reached for z z Rayleigh where z Rayleigh 180 mm for this configuration; the observation distance in the far field was always chosen to be 350 mm. 1. Sidelobe Example: g 6, 2.1 In addition to the incident beam radius being preset, the other experimental parameters are an index modulation of n and a grating thickness of d 5000 m. These parameters lead to a relatively high value of g when the value is not high enough to obtain Gaussian profiles, which is the case for the non-gaussian shape of the S-beam intensity profile a central peak accompanied by sidelobe ripples. Figure 12 a shows the kind of distortion that affects a diffracted beam. In the horizontal direction, the profile of the S beam is not Gaussian but is composed of a central peak and two sidelobes, in agreement with the theoretical shape both CWA or BPM. However, we could observe some differences in the profiles with theory: the repartition of intensity in the three different lobes is not the same, and the experimental profile is not symmetrical around the s 0 g 6 axis. Besides, the two profiles do not have the same dimensions and the experimental profile is not as broad as the simulated profile. This major difference could be explained by the supposed poor quality of the grating recording. To obtain a deep refractive-index modulation such as the predicted , an exposure time of several minutes is necessary. During such a long exposure, problems of stability and perturbations become critical and can affect the grating uniformity. By considering the same diffracted beam but in far-field observation, we can see in Fig. 12 b that distortion is also visible in that position an undistorted profile in the near field would also present a Gaussian shape in the far field. The experimental profile respects globally the predicted model: the shape is nearly the same and the same broadening occurs. But not all the small collateral sidelobes are observed in the experimental profile and the depth value of the hole in the middle of the central peak is more marked in the simulated profile. If we consider the transmitted beam, we can see that it is also distorted. In Fig. 13 a we observe that the R-beam profile is theoretically composed of a central peak, a major sidelobe, and a small sidelobe on the other side. The first two components can be observed in the experimental profile with less intensity in the sidelobe. This lack of intensity is counterbalanced by a minor deep hole. The global 1 July 2004 Vol. 43, No. 19 APPLIED OPTICS 3863

11 Fig. 13. Comparison of the CWA simulated R-beam intensity profile with the experimental R-beam intensity profile for g 6 and 2.1 in a the near field and b the far field. experimental R-beam profile is narrower than the theoretical profile. This R-beam profile in the horizontal direction is not symmetrical either in the simulation or in the experiment. In the far field, the shape of the central peak and its width are maintained as the experimental result, whereas all the sidelobe peaks are reduced see Fig. 13 b. We have also taken into consideration that in our experimentation the grating recording is not homogeneous along the z axis because the power of the recording beams decreases when they progress inside the crystal from absorption losses. The grating strength presents the same evolution because the refractive-index modulation decreases, and, hence, the achieved profiles are not symmetrical and not a perfect match with respect to the theoretical profiles. Finally, even with these differences, the measured diffraction efficiency equals the value predicted by theory 1.55 m Apart from this relatively small value of the diffraction efficiency, a fiber coupling would be largely compromised in this configuration because of the significant loss of power that is due to the presence of the sidelobes, and the central peak of the S-beam intensity profile is also wider than the Gaussian profile. We can define an estimated value of the coupling efficiency as the central value of the cross-correlation function between the diffracted intensity function and the incident Gaussian function. If we make the approximation of plane phase fronts, we obtain an estimate of 9.3% of the coupling coefficient. 2. Hole Burnt Example: g 2 2 The diffraction of a Gaussian beam by a volume holographic grating has been extensively studied theoretically but only a few experiments have been reported. An experiment carried out by Forshaw 14 was made with a thick holographic transmission grating and the diffraction of a narrow laser beam was studied. He observed a hole burnt into the zeroth-order transmitted R far-field patterns and a central maximum twice the width of the hole in the zero-order beam for the first-order diffracted S farfield pattern. The position of the hole in the far-field Fig. 14. Comparison of the CWA simulated S-beam intensity profile with the experimental S-beam intensity profile for g 2 and 2 in a the near field and b the far field APPLIED OPTICS Vol. 43, No July 2004

12 zeroth-order beam varies as the incident angle changes. These conclusions are in qualitative agreement with the Chu et al. theory and computation. 6 The same kind of distortion of the transmitted intensity profile is experimentally obtained by us Fig. 14 a. The experimental parameters are refractive-index modulation n and grating thickness d 1700 m. Figure 14 a shows that the experimental intensity profile of the S beam in the near field is in good agreement with the theoretical CWA and BPM intensity profile. The slight differences could be due to a lack of homogeneity of the recorded crystal, which is confirmed by the fact that they are not symmetrical around the s* 0 g 2 axis as they would be theoretically. Compared with the case g 6 and 2.1, the major difference is that the value of the grating thickness is smaller here. The smaller the values of grating thickness d therefore for small values of g, the less important are the interactions and the less distortion emerges from the diffracted beam. For farfield observations we can see that the sidelobe ripples are smaller here than in the previous case Fig. 14 b. As was determined by Forshaw, 14 we also observed a distortion on the R-beam profile composed of a central peak separated from a sidelobe by a zero. The cross-correlation plot between the incident Gaussian beam and the profile of the S beam in the near field gives an estimate of 20% for coupling coefficient, with respect to a diffraction efficiency of 46% for a Gaussian wave and 100% for a plane wave. The same conclusions are valid if we consider the profiles in the vertical direction for both the R and the S beams. 5. Conclusion The effects of volume grating diffraction on a finite Gaussian beam have been investigated experimentally and by simulations. The good agreement between the simulations and the experimental measurements allowed us to conclude that the coupled-wave theory and the BPM are accurate tools to simulate the propagation of Gaussian beams in a thick holographic grating. The two methods give similar results except for some minor differences in the profile intensities. The methods can be exploited to predict some distortions of transmitted or diffracted beams. The results indicate that, for specific values of grating parameters g and, the diffracted and transmitted beams remain Gaussianlike. On the other hand, hard distortions and efficiency loss can occur. The slight differences between simulated and experimental beam profiles obtained in our analysis have been explained by several experimental nonideal conditions, such as temporal writing beam instability, optical absorption loss of the grating material, and inhomogeneity of the grating strength distribution. By means of such predictions, it is possible to design optical devices based on volume holography optimized in terms of fiber coupling, which would make them quite attractive for use in the field of optical fiber communications. References 1. G. A. Rakuljic and V. Leyva, Volume holographic narrowband optical filter, Opt. Lett. 18, S. Breer and K. Buse, Wavelength demultiplexing with volume phase holograms in photorefractive lithium niobate, Appl. Phys. B 66, P. Boffi, M. C. Ubaldi, D. Piccinin, C. Frascolla, and M. Martinelli, 1550-nm volume holography for optical communication devices, IEEE Photon. Technol. Lett. 12, H. Kogelnik, Coupled-wave theory for thick hologram gratings, Bell Syst. Tech. J. 48, M. G. Moharam, T. K. Gaylord, and R. Magnusson, Bragg diffraction of finite beams by thick gratings, J. Opt. Soc. Am. 70, R. S. Chu, J. A. Kong, and T. Tamir, Diffraction of Gaussian beams by a periodically modulated layer, J. Opt. Soc. Am. 67, R.-S. Chu and T. Tamir, Bragg diffraction of Gaussian beams by periodically modulated media, J. Opt. Soc. Am. 66, M. R. Chatterjee and D. D. Reagan, Examination of beam propagation in misaligned holographic gratings and comparison with the acousto-optic transfer function model for profiled beams, Opt. Eng. 38, M. R. Wang, Analysis and observation of finite beam Bragg diffraction by a thick planar phase grating, Appl. Opt. 35, D. Yevick and L. Thylén, Analysis of gratings by the beampropagation method, J. Opt. Soc. Am. 72, A. E. Siegman, Bragg diffraction of a Gaussian beam by a crossed-gaussian volume grating, J. Opt. Soc. Am. 67, I. Ilic, R. Scarmozzino, and R. M. Osgood, Jr., Investigation of the Pade approximant-based wide-angle beam propagation method for accurate modeling of waveguiding circuits, J. Lightwave Technol. 14, S. Ahmed and E. N. Glytsis, Comparison of beam propagation method and rigorous coupled-wave analysis for single and multiplexed volume gratings, Appl. Opt. 35, M. R. B. Forshaw, Diffraction of a narrow laser beam by a thick hologram: experimental results, Opt. Commun. 12, July 2004 Vol. 43, No. 19 APPLIED OPTICS 3865