olography is a well-known technology employed forptical processing. Nowadays thick holographic grat-ngs find applications in the field of optical fiber com-
unications in the implementation of devices1–3 suchs optical filters, wavelength demultiplexers, opticalnterconnects, and storage media. For all these ap-lications, the input–output coupling to transmissionbers appears to be a critical constraint and conditionsheir design and performance assessment.
Classical study of holography based on the use oflane waves cannot be utilized for fiber communica-ion devices that involve Gaussian waves that comerom the fiber. Some previous theoretical studies4–9
ave predicted optical signal distortion at the outputf volume gratings, causing a loss in fiber coupling.theoretical and experimental study of the different
utput beam profiles and their distortion appears toe necessary to determine the influence of a thickolographic grating on Gaussian beam propagation
The authors are with the CoreCom �Consortium for Research inptical Processing and Switching�, Via G. Colombo 81, Milan0133, Italy. M. Martinelli is also with the Dipartimento di Elet-ronica e Informazione, Politecnico di Milano, Piazza Leonardo dainci 32, Milan 20133, Italy. P. Boffi’s e-mail address is [email protected].
nd more generally to optimize the insertion loss ofrating-based devices.Here, a theoretical and experimental analysis of the
ragg diffraction of finite Gaussian beams by volumeratings is reported. In Section 2 we introduce theifferent theories used in our study. In Section 3 werovide through simulations the theoretical evolutionf distortions that are due to the thick grating diffrac-ion as a function of grating parameters. Simulationesults obtained with both theories taken into accountre then presented. Moreover, in Section 4 we showhe experimental results and their comparison withimulations. In our experimentation, a thick holo-raphic grating is written in a standard photorefrac-ive crystal �LiNbO3� by means of Ar laser plane wavest 488 nm �a wavelength that corresponds to the max-mal photosensitivity of our material�. In contrast,he reading wave is a Gaussian beam at a differentavelength. In our case, we used a reading 1550-nmeam to take into account the real application of holo-raphic gratings in the field of optical communications.e obtained good agreement between the experimen-
al results and the simulations by demonstrating theapability to simulate and foresee the Gaussian beamropagation in a volume grating.
. Propagation of Gaussian Beams in Thick Gratings:heory
n the following we theoretically analyze the diffrac-ion of a Gaussian beam that is due to a thick grating.he diffraction conditions are shown in Fig. 1. Wesed the classic layout configuration.
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t
w
n
ibwthttt
agctt
st
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A
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Grating is assumed to extend infinitely in the x–ylane and to be thick: the Q parameter, Q � 2�d��0�2, provides an evaluation of the grating thicknessith respect to the condition4 Q �� 1, where � is therating period, � is the free-space wavelength of theeading beam, d is the grating thickness, and n0 ishe average refractive index of the medium outsidehe grating.
The refractive index is sinusoidally modulated inhe x direction and in the region 0 � z � d by
n� x� � n0 � n1 cos�2�x� � � n0 � n1 f � x�, (1)
here the amplitude of refractive-index modulation
1 is small compared with n0.We assume a single unslanted grating and a read-
ng TE-polarized Gaussian incident wave, whoseeam waist is large in comparison with the free-spaceavelength of reading beam � � 1.55 �m. Only the
ransmitted and diffracted waves are considered4,5
ere because the grating thickness values used inhis study are quite high; thus other orders of diffrac-ion can be neglected. We also do not take absorp-ion into account.
Two different approaches are used in our theoreticalnalysis. In Subsection 2.A we introduce the Ko-elnik coupled-wave analysis4 �CWA� for the theoreti-al study of thick grating diffraction and distortion ofhe output beam profile as a function of the grating andhe Gaussian input beam parameters. A detailed de-
ig. 1. Model of a thick grating with unslanted fringes in theragg diffraction regime. B is the Bragg angle of incidence of the
eading beam in the medium defined by 2� sin B � �.
cription of the CWA is justified to understand diffrac-ion behavior as a function of different parameters.
In Subsection 2.B the so-called BPM �beam propa-ation method�10 algorithm is also considered to con-rm the accuracy of the CWA by comparison of theimulation results obtained with the two methods.hese simulations will also give an outline of theifferent distortion types observed for both transmit-ed and diffracted beams that will be experimentallynalyzed further �Section 3�.
. Kogelnik’s Coupled-Wave Theory
he phase curvature of the Gaussian beam profile isssumed to be negligibly small in the grating region,condition satisfied if the number of grating periods
cross the Gaussian spot size is sufficiently large.11
eglecting the phase curvature of the input beams,e can write the total electric field in the grating aslinear superposition of the complex amplitudes of
he transmitted and diffracted waves, R�x, z� and�x, z� respectively.At first, the study is considered under a near-field
ondition, which corresponds to an observation dis-ance that is smaller than the Rayleigh length. In-roducing a new coordinate system �r, s� defined by� z sin B � x cos B and s � z sin B � x cos B andonsidering that at the input plane z � 0 �r � �s�,here are no diffracted waves �S � 0�, and the trans-itted R beam is only a function of r, we obtain the
ransmitted and reflected amplitude expressions fornput beam R0�r� and for a grating of thickness d in aear-field condition5:
R�r� � R0�r� � 1�2 ��1
�1
R0�r � d�1 � u�sin B�
� �1 � u1 � u�
1�2
J1� �1 � u2��du,
S�s� � �i2
��1
�1
R0�s � d�1 � u�sin B�
� J0� �1 � u2��du, (2)
here � �d�cos B is the grating strength �or ahase delay factor�, � � �n1��, J0 and J1 are Besselunctions of the first kind.
In the case of an incident Gaussian beam with anmplitude profile R0�r� � E0 exp��r2��0
here r � r��0, s � s��0, E0 is the peak value of thelectric field, �0 is the Gaussian beam 1�e2 radius,nd geometry parameter g � d sin B��0.To determine the electric field intensity of the
ransmitted beam in the far-field zone we can use theraunhofer approximation that leads to the calcula-
ion of the far-field electric intensity as the squaredourier transform of the near-field electric intensity.nother formalism also issued from the Kogelnik the-ry allows us to calculate the far-field intensitiesore easily thanks to evaluation of the transfer func-
ion of the grating.The spatial output profile in the far field could in
act be expressed as the product of the transfer func-ion and the angular spectrum of the input beam:
Sff��� � E� ��� HS���,
Rff��� � E� ��� HR���, (4)
here E� ��� �the angular spectrum of the inputeam� is the Fourier transform of the input spatialeam E�r� �at z � d� and HR��� and HS��� are the�transmitted� and the S �diffracted� beam transfer
unctions. These transfer functions are obtainedrom the Kogelnik expressions of the output R and Selds4 �for a single lossless unslanted grating whenhe input beam is a unit amplitude, uniform, andlanar wave�:
Tdoubtcps
3b
TmpCpccdtsi
he R �transmitted� and S �diffracted� beam transferunctions �HR��� and HS���� are equal to the ex-ressions presented previously in Eqs. �5�, where theephasing term is �1, the first-order approximation8
f the Taylor series of the dephasing term � � d��2os B �dimensionless� for a slight deviation of thenput angle � � B � �� but without wavelengthetuning:
�1 ��Kd
2�
��d�
. (6)
or unslanted gratings �� � K sin � �K2�4�n0��,here is the angle of incidence of the reading beam
n the medium.The spatial output profile in the near field can be
xpressed as the inverse Fourier transform of theroduct of the transfer function and the angular spec-rum of the input beam.8 We can observe that, ashe R and the S beam profiles depend on only twoariables � and g� in the near field, we have the sameropriety in the far field. Besides, the S-beam pro-le is symmetrical �in the near field around the s �axis and in the far field around the � axis�.
he diffraction efficiency is defined here as � � SS*,here S is the output signal �at z � d� for an incidentlane wave of unit amplitude. It can also be writtens4
� �sin2� 2 � �2�1�2
1 ��2
2
(7)
hen the Bragg condition is verified, � � 0, and wean obtain a complete conversion of energy for * ���2� � m�, where m is an integer.
. Beam Propagation Method
he BPM algorithm is just a recursion relationshipiving expressions of the electric field from the Helm-oltz scalar-wave equation at infinitesimally smallxial distances �z one from another.10 The under-ying assumptions of its classical use are as follows:
all the angles are small to maintain the paraxialondition,
a small grating modulation is assumed, andthe backward reflection and its effect on the for-ard propagation are neglected.
Paraxiality represents the major limitation of theasic paraxial BPM for the study of propagation inree space or in a grating. Enhanced BPMs thatxtend the validity to wide angles are available.
hey allow us to work at angles larger than �10–15eg from the optical z axis, which is typically the limitf what can be considered paraxial. The most pop-lar approach is referred to as the multistep Pade-ased wide-angle technique, which allows us to relax,o varying degrees, the paraxial approximation of thelassical BPM.12 The BPM extended to wide anglesrovides another method that can be used for analy-is of the Bragg diffraction of thick gratings.
. Simulations of Bragg Diffraction of Gaussian Beamsy a Transmission Unslanted Volume Grating
he BPM has already been used as a powerfulethod for analyzing the volume grating diffraction
roblem10 and has been compared with the rigorousWA13 by calculation of the diffraction efficiency of alane-wave input. In Subsection 3.A we present aomparison of CWA and the BPM taking into accountalculated intensity profiles of both transmitted andiffracted beams in the case of a Gaussian wave inputo a grating structure. For this purpose we use someignificant configurations of a single unslanted grat-ng and a TE-polarized Gaussian beam without tak-
ng into account absorption of the Gaussian beamnto the grating medium. Here we focus only on theear-field configuration because the far-field resultsan be deduced from the near-field results. Further-ore, the near-field condition corresponds to the rea-
onable position in which the fiber coupling wouldake place. We then provide a discussion of the grat-ng parameters and finally conclude with the simu-ations and an experimental discussion.
. Coupled-Wave Analysis and Beam Propagationethod Near-Field Simulations
t first it is necessary to specify that like the BPM,he CWA �or coupled-mode analysis� is also approxi-ate because it involves the solution of a scalar-wave
quation and neglects some boundary conditions andome spatial harmonic components. Since the twoifferent computing solutions based on CWA rely onhe same method, one using fast Fourier transformwhich has been previously exposed� and the other anntegral in the spatial domain,5 they give similar re-ults. Afterward, we plot only the results obtainedith the spatial domain integration for the near field.With regard to the BPM, it has been used with
Soft BeamPROP 4.0 software, which implements aade algorithm that extends the validity of the BPMo off paraxiality. This software enables us to plothe intensity beam profiles at a desired distance ofbservation from the grating end face �usually choseno allow a sufficient spatial separation of the Bragg-cattered beam from the transmitted beam�. Theomputed step points that represent the refractive-ndex modulation are a fixed rate of 24 points peresigned grating period.Since the main purpose of these simulations is a
omparison of different diffracted and transmittedeam shapes and not the lateral shift �not consideredn experiments�, we plot the BPM and CWA super-osed on the theoretical lateral position obtainedith CWA simulations. Assumptions exposed inubsection 2.A are verified by our simulations, for
ig. 2. Comparison of the �a� R-beam profiles and �b� S-beam profihe following parameters were used: n1 � 5 � 10�3, �0 � 9.21 �maxis computed step �z � 0.04 �m. Hence, the geometry and g
xample, the Gaussian beam waist is large in com-arison to the free-space wavelength �� � 1.55 �m�,nd the amplitude of refractive-index modulation n1s small compared with the average refractive indexf the LiNbO3:Fe crystal n0 � 2.125 used afterwardn our experiments. We consider a single case ofrating configuration: its average refractive index,ts period, and its Bragg angle values are fixed byxperiments. These latter parameters also fix therating period, � � 0.94274 �m, from which we caneduce the Bragg angle B � 22.759° �with the Braggondition4: 2� sin B � N���n0�, where N is an in-eger�. It is obvious that this study is also applicableor values of other parameters.
Since it has been exposed in theory and has beenemonstrated by Moharam et al.,5 the spatial profilesf the transmitted and diffracted beams and the dif-raction efficiency obtained with CWA can be writtennd presented as functions of only two normalizedarameters: grating strength � � n1d�� cos Bnd geometry parameter g � d sin B��0. We choseo vary refractive-index modulation n1, beam radius0 �Gaussian beam 1�e2 radius�, and grating thick-ess d �also called grating length� to obtain represen-ative values of the g and parameters thatorrespond to interesting cases to be analyzed.
Figures 2–4 show three interesting but differentomputed simulations. These cases present mean-ngful distortions of diffracted and transmittedaussian beam profiles as functions of grating
trength and geometry parameter �and, as a conse-uence, a function of d, �0 and n1�. By observing theear-field computations of diffracted �S beam orragg-scattered beam� and transmitted �R beam or
eference beam� beam profiles, one can observe goodgreement between the two techniques, and the be-avior of these profiles as a function of g and valuesan be readily understood.
Comparing the BPM and CWA simulations, onean first observe that profiles obtained by both sim-lations are quite equivalent even if there are occa-
the near field. The results were obtained by CWA and the BPM.71.45 �m, BPM distance of observation dobs � 120 �m, and BPM
ional minor differences. Profiles from bothimulations have the same general shape. As far ashape distortion behavior due to volume grating isoncerned, in the simulation related to Fig. 2�b�, the-beam profile can be seen as two Gaussian lobes thatverlap. The same kind of shape has been analyzedy Chu et al.6 �Bragg-scattered first-order Gaussianeam for half-space�, and it agrees with Forshaw’sxperiments.14 Other diffracted profiles presentome different distortions that are the same for bothimulations and that will be further analyzed.-beam profile distortions are also the same, except
or the first case seen in Fig. 2�a� where the BPMimulation does not represent a small sidelobe sepa-ated by a zero from the central peak and visible onhe CWA simulation.
If we define the beam profile full width as the pro-le width at 95% amplitude, a comparison of theifferent widths shows that the S- and R-beams fullidth values obtained with the CWA and BPM sim-
ig. 3. Same as Fig. 2 except that the following parameters werebservation dobs � 1200 �m, and BPM z axis computed step �z �nd � 0.0858.
ig. 4. Same as Fig. 2 except that the following parameters werebservation dobs � 550 �m, and BPM z axis computed step �z � 0� 2, 1�.
lations are quite similar. A dispersion of 3% for Seams and 6% for R beams was found when we ex-luded cases in which a sidelobe was visible on CWAimulation but not on the BPM. Widths are slightlyarger with BPM simulations than with CWA simu-ations. Differences in intensity of the profiles areresent because BPM simulations are more attenu-ted than CWA simulations, especially for R beams.e determined that with BPM simulations a small
ortion of the beam power is lost, and we neglectedhe contributions that are due to backward reflec-ions, which could explain the difference betweeneam intensities.A propriety of the CWA theory mentioned above is
ot valid for the BPM theory. The profiles of theransmitted and diffracted beams are not the sameor the same pair of parameters �g, � but are com-osed of different values of the refractive-index mod-lation, beam waist, and grating thickness. Aource of error and difference between simulations
: n1 � 6.7 � 10�5, �0 � 50 �m, d � 581.5 �m, BPM distance ofm. The geometry and grating strength parameters are g � 4.5
d: n1 � 1 � 10�2, �0 � 38.7 �m, d � 300 �m, BPM distance of. The geometry and grating strength parameters are g � 3 and
used0.3 �
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an be wide angles: extension of the BPM to widengles owing to the Pade algorithm introduces errorshat could vary with simulation parameters. More-ver BPM simulations require a high computed pre-ision and the available amount of allocated memoryometimes limits the precision of the simulated pro-les. More generally, we can conclude with the helpf these plotted simulations that shapes, widths, andmplitude of the beam profiles are generally closeetween the BPM and the CWA simulations. How-ver, some singular differences can be found, for ex-mple, singularity of the profiles �sidelobe, zero . . .�r peak intensity value dispersion. Since there isood agreement between the BPM technique and theWA method when some approximations are taken
nto account, these simulation tools can equivalentlyredict the diffraction behavior of a Gaussian beamhat propagates into a volume grating.
. Theoretical Discussion
aking into consideration a fixed configuration of arating ��, �, B�, we found it interesting to study thevolution of diffracted and transmitted profiles andhe diffraction efficiency as a function of other param-ters such as n1, d, and �0 �and then as a function ofand �. It could, for example, provide a solution to
btain a Gaussian profile with the highest diffractionfficiency or otherwise predict what kind of distortionould be observed for a given configuration.
Some trends of the S- and R-beam characteristicss a function of g and have already been analyzed inhe literature.5–8 The major conclusion is that therofiles remain Gaussian for g �� 1: it means amall value of the grating thickness and�or a largealue of the beam radius. For example, the smallerhe value of grating thickness d �therefore for smallalues of g�, the less significant the interactions andhe less distorted the diffracted and transmittedeams. In this case, diffraction is comparable to thease of the plane wave, and the peak amplitude there-ore follows the sin2� � function.4 The same conclu-ion could be explained in the transfer functionormalism: the impulse response of the grating ap-ears to be close to an impulse and allows us to trans-it the input profile undistorted. Inversely, at a
igher value of the grating thickness, the impulseesponse widens. At a fixed value of g �and g � 1�,he profiles are again Gaussian-like for very largealues of �verifying �g � 8� according to Moharamt al.5�. Physically, this means that, for a smallaussian profile �high value of g�, the distortion is
ompensated by a significant amount of gratingtrength and therefore by a high refractive-indexodulation and�or by a substantial grating thick-ess. As the value of grating strength increases,he concentration of energy in the S beam is closer to��0 � g, and its profile is more and more Gaussian.trade-off exists between conserving a Gaussian pro-
le and obtaining the highest diffraction efficiencyossible.In our analysis we performed a complete study
elated to the shape distortions of diffracted and
ransmitted beams by considering their behavior as aunction of different parameters, taking into accountrating strength and geometry parameter g in theear field.
. S-Beam �First-Order Diffracted Beam�diffracted beam can present different kinds of dis-
ortion that changes it from Gaussian-like because ofifferent values of the grating strength and the ge-metry parameter. Figures 5 and 6 show three-imensional plots of the diffracted beam profile thataries with one of the two parameters �gratingtrength and geometry parameter, respectively�hen the other parameter has a fixed value. We canrst generally observe that these profiles have a dra-atic evolution as a function of these parameters and
hat distortion increases with an increase in the ge-metry parameter value.
ig. 5. Three-dimensional plot of the near-field diffracted S-beamrofile for a Gaussian wave input as a function of gamma �0;5��nd s��0 ��2;8� with a fixed g value of 3. The normalized inten-ity is plotted on the vertical axis.
ig. 6. Three-dimensional plot of the near-field diffracted S-beamrofile 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 islotted on the vertical axis.
For some values of g and �for example, g � 3 and� 2.1��, some sidelobes appear around the princi-
al peak �see Fig. 4�b��. The position of the sidelobesaries as a function of the considered parameters �ashown in Fig. 5 for g � 3�. Another type of distortionas already been observed in the past by Forshaw14
nd predicted by Chu et al.6 As far as we are con-erned, this distortion, called hole burnt into the pro-les,6 resembles two Gaussian shapes that overlapcorresponding to different diffracted contributions�.igure 2�b� shows this effect for � ��4. The burntole is more visible as g increases.For very small values of � �� 1� the diffraction
fficiency is small because of a small refractive-indexodulation and therefore has a uniform diffraction
ver a large length. The profiles have a rectangularhape �Fig. 3�b�� and this effect is more visible as thealue of g increases for a fixed value of . This caseorresponds to a large value of the profile width.ore generally, as the g value increases, the S-beam
rofile width increases; at a fixed value of g, as the alue increases the width decreases. In any case,he S beam �as well as the R beam for the sameeason� is confined to a defined region. The 1�e am-litude width of the S beam �or of the R beam� isounded by a boundary width value of9
d tan B � 2�0 � � gcos B
� 2��0. (8)
owever, for a given value of g, a large value existsor which the S beam full width is much smaller thanhe boundary width value.
As has been mentioned in Subsection 2.A, even ifhe S-beam profiles are distorted they are constantlyymmetrical around the s��0 � g axis. This positions the center of the S-beam pattern for small values of. At a high fixed value of g, the central peak tendso reach this position when the value increases.onsidering that the medium ends at z � d at theame position as the grating, the beam position out-ide the medium �z � d� can be written as9
x � d tan B � � z � d�tan for the R beam, (9)
x � �d tan B � � z � d�tan for the S beam,(10)
here is the Bragg angle of the reading beam out-ide the medium �obtained from the Bragg angle inhe medium by Snell’s law�.
Even if optimization of the signal coupling at thend face of the volume grating is not important forar-field behavior, it would be interesting to considert. Usually from near-field considerations we caneduce some properties for the far-field profiles andnversely as the intensity in the far field is thequared amplitude of the Fourier transform of themplitude in the near field. In general, for the far-eld distance we consider the S-beam profiles as aentral peak with sidelobe ripples.14 These ripplesccur in all the far-field profiles and are symmetrical
round the Bragg angle position � B. Themaller the values of grating thickness d �therefore,or small values of g�, the less distorted the diffractedeam, the smaller the sidelobe ripples, and the lessroadened is the S-beam profile width�. In fact,ost of the energy is associated with a plane wave,hich respects the Bragg condition. At a high valuef the grating thickness, the impulse response wid-ns, the number of components of the incident beamngular spectrum with respect to the Bragg conditionecreases, and the sidelobe ripples are more impor-ant �the diffracted beam is more distorted�.
. R-Beam �Transmitted Beam�he major distortion of the R beam consists of theresence of sidelobes but generally it is not too dis-orted �as could be the case for the S beam� and aaussian profile is recognizable �see Figs. 2�a� and�a��. In fact, the increased value required to ob-ain a Gaussian-like R beam �starting with a dis-orted configuration� is lower than the increased alue required for the S beam. Inversely, we alsoave a Gaussian-like R beam for high values of g and
or �� 1, which corresponds to a rectangular shapef the S beam �see Fig. 3�a��.In some cases, the R beam can be split into two
eams, an ordinary transmitted and a forward dif-racted, an effect that can be observed, for example, inig. 8. The more the g value increases, the moreistorted is the R-beam profile and the less it resem-les a Gaussian profile; this evolution can be seen inig. 8. The distortion is due to the coupling of en-rgy between the R and the S beams during theragg scattering process.7 The R-beam profileidth is greater than the incident beam width be-
ause, even if the transfer function of the grating ismpulselike, it always has a broadening. As valuesf increase �see Fig. 7� or values of g decrease, therofile width values increase. As well as what has
ig. 7. Three-dimensional plot of the transmitted R-beam profileor a Gaussian wave input as a function of �0; 5�� and r��0 ��2;� with a fixed g value of 3. The normalized intensity is plotted onhe vertical axis.
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een explained for the S beam, the full width of the Ream is bounded.The peak of the beam profile shifts spatially as the
rating strength increases. At a fixed value of g, ashe value of increases, the energy concentratesround the r��0 � g axis and the shift increases �Fig.�. However for g �� 1, the R beam is symmetricalround the r � 0 axis whatever the value of . The-beam far-field pattern is confined to a small angu-
ar range and centered at the negative Bragg angle of� �B. The major distortion of the R beam con-
ists in a dip or even a deep null in the middle of the-beam profile.6 A deep null appears in the centerosition when * � ���2� � m�, where m is an inte-er. The central portion of the Gaussian spectrumf the transmitted beam has completely converted itsnergy into a Bragg-scattered wave, which results indepletion of energy from its beam-center position
nd corresponds to the split of the R beam into twoeams in the near field �see Fig. 7�. For some dif-erent values of , we could expect only a small dip.
. Diffraction Efficiency and Conversion of Energy
or a plane wave, the Kogelnik theory4 predicts aomplete conversion of the input beam into a dif-racted beam for values of grating strength * � ���2�
m�, where m is an integer. By using finite beamse can observe that R beams still contain an amountf energy for these values of the grating strength.onsidering the diffraction efficiency that has beenefined in Subsection 2.A as � � SS* �where S is theutput signal at z � d for an incident plane wave ofnit amplitude�, the � value of a Gaussian beam islways less than the � value of a plane wave.7 Aotal conversion cannot therefore occur with a Gauss-an beam. It has been interesting and innovative fors to plot in three dimensions diffraction efficiency �efined previously �see Fig. 6� as a function of g and�see Fig. 9�. From Fig. 9 we can observe that, as g
ig. 8. Three-dimensional plot of the near-field transmitted-beam profile for a Gaussian wave input as a function of g �0,6�nd r��0 ��2,12� with a fixed value of 9��4. The normalizedntensity is plotted on the vertical axis.
ncreases, the diffraction efficiency decreases, for axed value of g, as increases, the diffraction in-reases and tends to reach the plane-wave diffractionfficiency4 sin2� �. To obtain a maximum diffractionfficiency value, the grating strength value had to bequal to a * value. Even though that is sufficientor g �� 1, for larger values of g a large value of * islso necessary.
. Theoretical Conclusion
rom the above analyses we can conclude that, for g� 1 or �� 1 and � 8�g, R and S beams areaussian-like. For intermediate values of g and ,
he R and S intensity profiles are no longer Gaussiannd contain some distortion. The distortion in-reases when the g values increase �also for largeralues of the grating thickness and�or for smalleralues of the beam radius� and the diffraction effi-iency decreases. There is also a loss in the effi-iency of converting energy from the input beam intohe S beam. We now turn our attention to experi-entation for the purpose of comparing experimental
esults with theoretical predictions.
. Experimentation of Bragg Diffraction of Gaussianeams by a Transmission Unslanted Thick Grating
orshaw first reported a study of the diffraction of aarrow laser beam by a thick hologram,14 but thisind of analysis has never been carried out experi-entally. In other published papers, the principal
xperimental interest focused on grating selectivitynd diffraction efficiency results. Here we presentur experimentation of Gaussian beam diffractionith a thick holographic grating by taking into ac-
ount the diffracted beam distortion with regard tober coupling.
. Experimental Setup
he thick gratings that we checked in our experimen-ation were holographic, recorded by a fairly commonetup, and could be used for optical storage experi-
ig. 9. Three-dimensional plot of the diffraction efficiency of aaussian wave as a function of g �0,6� and �0,5��.
Fig. 10. Experimental setup of the grating recording.
3
ents. The recording material we used was a 1m � 1 cm � 2 cm photorefractive crystal ofiNbO3:Fe �0.05 mol% of Fe doping� produced by Del-
ronics. The crystal was a 0°-cut and the beamsere horizontally polarized to achieve maximal mod-lation of the refractive-index recording. The exper-
mental setup is shown in Fig. 10. The 488-nmmission of an Ar-ion laser was expanded by a lensnd filtered by a spatial filter �pinhole�. This firstart of the setup was used to obtain a plane wave.he horizontal polarized light was obtained by a ��2ave plate. The incident beam was then divided
nto two beams of equal intensity by a 50:50 beamplitter. Together they produced an interferenceattern in the recording material. We produced therating by exposing the LiNbO3:Fe crystal to a spa-ially varying pattern of light intensity. The posi-ion of the mirrors induced an equal path length andn equal incident angle of 15° for both recordingeams. This geometric arrangement led to a fringeeriod � of 942.7 � 0.6 nm, if we consider an error of0.01° on the incident recording angles. Refractive-
ndex modulation n1 was controlled by the amount ofxposure time.14
ith a goniometer and a rotating stage with a mea-urement accuracy of 0.001°. The lighted areas onhe material were 1 cm � 1 cm squares. By mea-uring the total area of illumination on the input facef the crystal, we deduced the geometric form of therating. To obtain a grating with a mostly rectan-ular shape, we canceled the back part of the originalexagon shape of the grating by means of an incidentr beam of incoherent light with a suitable spatialindow. This technique was also used to controlrating thickness.Read out is based on the so-called two-� method,ith a Gaussian beam as readout radiation at 1550m, the well-known third window � of optical com-unications, to analyze the experimental behavior of
nteresting cases for their future use in the field ofptical communications.The crystal was placed on a rotating stage oriented
t 55.2° �angle in air corresponding to the Bragg an-le in the medium� from the laser reading source ashown in Fig. 11. The light source was a semicon-uctor laser �1550 nm� pigtailed to a fiber whose ends placed on an x–y scanning device, which allowed uso analyze the entire surface of the recorded volumerating. Moreover its position in combination with
Fig. 11. Experimental setup for the grating analysis.
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he free-space propagation until the input into therystal and the choice of collimator enabled us toontrol the beam radius values in the crystal. Welso measured the powers and diffraction efficiencyith some powermeters, and we observed the beam
pots with an infrared camera. Our observationsere made for both the near field and the far field.It is important to stipulate that these experimental
etups enabled us to change different parametersasily to achieve a large range of g and values thatere useful for our analyses.
. Experimental Results
any representative cases related to particular setsf g and values have been experimentally tested.ere we present only a few examples. As in theogelnik theory, we consider a zeroth-order beamnd one first-order beam whose profiles have beenegistered in near-field and far-field conditions. Wepplied the Kogelnik CWA and BPM simulations toach set of parameter values �both near-field andar-field solutions�, which we then compared with thexperimental results. Here we present only theomparison with the CWA simulations because thePM simulations do not provide additional informa-
ion.For each output beam we consider the direction
long which the beam has been diffracted �which haseen defined in the Kogelnik theory as the s axis�.he experimental profile intensities are normalizedver the CWA profiles. The beam radius parametert the input face of the crystal is equal to �0 � 323 �mn all the experiments, and the far-field condition iseached for z � zRayleigh where zRayleigh � 180 mm forhis configuration; the observation distance in the fareld was always chosen to be 350 mm.
. Sidelobe Example: g � 6, � 2.1�n addition to the incident beam radius being preset,he other experimental parameters are an index mod-lation of n1 � 6 � 10�4 and a grating thickness of� 5000 �m. These parameters lead to a relatively
ig. 12. Comparison of the CWA simulated S-beam intensity pro.1� in �a� the near field and �b� the far field.
igh value of g when the value is not high enougho obtain Gaussian profiles, which is the case for theon-Gaussian shape of the S-beam intensity profile �aentral peak accompanied by sidelobe ripples�.
Figure 12�a� shows the kind of distortion that af-ects a diffracted beam. In the horizontal direction,he profile of the S beam is not Gaussian but is com-osed of a central peak and two sidelobes, in agree-ent with the theoretical shape �both CWA or BPM�.owever, we could observe some differences in therofiles with theory: the repartition of intensity inhe three different lobes is not the same, and thexperimental profile is not symmetrical around the��0 � g � 6 axis. Besides, the two profiles do notave the same dimensions and the experimental pro-le is not as broad as the simulated profile. Thisajor difference could be explained by the supposed
oor quality of the grating recording. To obtain aeep refractive-index modulation such as the pre-icted 6 � 10�4, an exposure time of several minutess necessary. During such a long exposure, prob-ems of stability and perturbations become criticalnd can affect the grating uniformity.By considering the same diffracted beam but in
ar-field observation, we can see in Fig. 12�b� thatistortion is also visible in that position �an undis-orted profile in the near field would also present aaussian shape in the far field�. The experimentalrofile respects globally the predicted model: thehape is nearly the same and the same broadeningccurs. But not all the small collateral sidelobes arebserved in the experimental profile and the depthalue of the hole in the middle of the central peak isore marked in the simulated profile.If we consider the transmitted beam, we can see
hat it is also distorted. In Fig. 13�a� we observehat the R-beam profile is theoretically composed of aentral peak, a major sidelobe, and a small sideloben the other side. The first two components can bebserved in the experimental profile with less inten-ity in the sidelobe. This lack of intensity is coun-erbalanced by a minor deep hole. The global
ith the experimental S-beam intensity profile for g � 6 and �
xperimental R-beam profile is narrower than theheoretical profile. This R-beam profile in the hori-ontal direction is not symmetrical either in the sim-lation or in the experiment. In the far field, thehape of the central peak and its width are main-ained as the experimental result, whereas all theidelobe peaks are reduced �see Fig. 13�b��.We have also taken into consideration that in our
xperimentation the grating recording is not homo-eneous along the z axis because the power of theecording beams decreases when they progress insidehe crystal �from absorption losses�. The gratingtrength presents the same evolution because theefractive-index modulation decreases, and, hence,he achieved profiles are not symmetrical and not aerfect match with respect to the theoretical profiles.Finally, even with these differences, the measured
iffraction efficiency equals the value predicted byheory ����1.55 �m � 0.28�. Apart from this rela-ively small value of the diffraction efficiency, a fiberoupling would be largely compromised in this con-guration because of the significant loss of power that
ig. 13. Comparison of the CWA simulated R-beam intensity pro.1� in �a� the near field and �b� the far field.
ig. 14. Comparison of the CWA simulated S-beam intensity pro�2 in �a� the near field and �b� the far field.
s due to the presence of the sidelobes, and the centraleak of the S-beam intensity profile is also wider thanhe Gaussian profile. We can define an estimatedalue of the coupling efficiency as the central value ofhe cross-correlation function between the diffractedntensity function and the incident Gaussian func-ion. If we make the approximation of plane phaseronts, we obtain an estimate of 9.3% of the couplingoefficient.
. Hole Burnt Example: g � 2 � ��2he diffraction of a Gaussian beam by a volume ho-
ographic grating has been extensively studied theo-etically but only a few experiments have beeneported. An experiment carried out by Forshaw14
as made with a thick holographic transmissionrating and the diffraction of a narrow laser beamas studied. He observed a hole burnt into the
eroth-order transmitted �R� far-field patterns and aentral maximum �twice the width of the hole in theero-order beam� for the first-order diffracted �S� far-eld pattern. The position of the hole in the far-field
ith the experimental R-beam intensity profile for g � 6 and �
ith the experimental S-beam intensity profile for g � 2 and �
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eroth-order beam varies as the incident anglehanges. These conclusions are in qualitativegreement with the Chu et al. theory and computa-ion.6 The same kind of distortion of the transmit-ed intensity profile is experimentally obtained by usFig. 14�a��. The experimental parameters areefractive-index modulation n1 � 4.2 � 10�4 andrating thickness d � 1700 �m. Figure 14�a� showshat the experimental intensity profile of the S beamn the near field is in good agreement with the theo-etical CWA and BPM intensity profile. The slightifferences could be due to a lack of homogeneity ofhe recorded crystal, which is confirmed by the facthat they are not symmetrical around the s*��0 �� 2 axis as they would be theoretically.Compared with the case g � 6 and � 2.1�, theajor difference is that the value of the grating thick-
ess is smaller here. The smaller the values of grat-ng thickness d �therefore for small values of g�, theess important are the interactions and the less dis-ortion emerges from the diffracted beam. For far-eld observations we can see that the sidelobe ripplesre smaller here than in the previous case �Fig.4�b��. As was determined by Forshaw,14 we alsobserved a distortion on the R-beam profile composedf a central peak separated from a sidelobe by a zero.The cross-correlation plot between the incidentaussian beam and the profile of the S beam in theear field gives an estimate of 20% for coupling coef-cient, with respect to a diffraction efficiency of 46%
or a Gaussian wave and 100% for a plane wave.he same conclusions are valid if we consider therofiles in the vertical direction for both the R and thebeams.
. Conclusion
he effects of volume grating diffraction on a finiteaussian beam have been investigated experimen-
ally and by simulations. The good agreement be-ween the simulations and the experimentaleasurements allowed us to conclude that the
oupled-wave theory and the BPM are accurate toolso simulate the propagation of Gaussian beams in ahick holographic grating. The two methods giveimilar results except for some minor differences inhe profile intensities. The methods can be ex-loited to predict some distortions of transmitted oriffracted beams. The results indicate that, for spe-ific values of grating parameters g and , the dif-racted and transmitted beams remain Gaussian-ike. On the other hand, hard distortions andfficiency loss can occur. The slight differences be-
ween simulated and experimental beam profiles ob-ained in our analysis have been explained by severalxperimental nonideal conditions, such as temporalriting beam instability, optical absorption loss of
he grating material, and inhomogeneity of the grat-ng strength distribution.
By means of such predictions, it is possible to de-ign optical devices based on volume holography op-imized in terms of fiber coupling, which would makehem quite attractive for use in the field of opticalber communications.
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