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Angular and power characteristics of noncollinear acousto-optic tunable filters
Optics and Lasers in Engineering 31 (1999) 1}12
Angular and power characteristics of noncollinear acousto-optic tunable "lters G. Georgiev *, E. Georgieva, L. Konstantinov Central Laboratory of Mineralogy and Crystallography, Bulgarian Academy of Sciences, Rakovski Str. 92, 1000 Soxa, Bulgaria Institute of Applied Physics, Technical University of Soxa, Soxa, Bulgaria Received 28 August 1998; accepted 7 December 1998
Abstract The angular and the power characteristics of noncollinear acousto-optic tunable "lters are investigated both experimentally and theoretically. The dependencies of the acoustic wave frequency, the bandpass width and the di!raction e$ciency of the "lter are studied as functions of the angle of incidence of the optical wave and of the acoustic signal power density. These dependencies are illustrated and tested on the basis of a realized such a "lter based on tellurium dioxide (TeO ). A good agreement of theory and experiment is demonstrated, which indicates that the approach used can be e!ective in designing such devices. 1999 Elsevier Science Ltd. All rights reserved. Keywords: Acousto-optic tunable "lters; Acousto-optics; Spectral "lters
1. Introduction Acousto-optic tunable "lters (AOTFs) are solid state, RF-signal tunable optical components based on the anisotropic Bragg di!raction of light on acoustic waves, which select a narrow bandpass from a wide incident spectrum. The band central wavelength of such devices can be scanned through an acoustic signal of variable frequency (RF-signal), while the linewidth of the generated acousto-optical grating is equal to the acoustic wavelength. This grating can serve as a dispersive element of electrically tunable spectrometers. The response time of the AOTF is limited by the acoustic transit time across the optical aperture, which is typically on the order of a few microseconds.
* Corresponding author. Tel.: #359-2-872-450: fax: #359-2-884-979; e-mail: [email protected] 0143-8166/99/$ - see front matter 1999 Elsevier Science Ltd. All rights reserved PII: S 01 4 3-8 1 66 ( 9 8) 0 0 04 9 - 9
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G. Georgiev et al. / Optics and Lasers in Engineering 31 (1999) 1}12
The "rst AOTF, reported by Harris and Wallace [1] was based on collinear acousto-optic interaction. On the other hand, the "rst noncollinear AOTF was introduced by Chang [2]. Since then, several di!erent AOTF con"gurations have been reported based on collinear or noncollinear interactions and using various acousto-optic media, which operate from the ultraviolet to the far infrared optical spectrum. AOTFs have recently attracted much attention due to their potential in spectral analysis [3,4], spectropolarimetry [5,6], spectrophotometry [7,8], etc. Amongst all di!erent types of AOTF, noncollinear "lters have been most frequently treated theoretically and experimentally owing to their large angular aperture, simple construction and operation facilities. The spectral characteristics of noncollinear AOTFs have been studied, both theoretically and experimentally, in our previous works [9,10]. In studying these characteristics we have kept the angle between the incident-optical wavevector and the (0 0 1)-axis (the angle of incidence), the power density of the controlling RF-signal and the length of acousto-optic interaction "xed. In this paper we report results on theoretical and experimental investigations of the "lter bandpass width, angular aperture and di!raction e$ciency as functions of the angle of incidence and the RF power density at constant wavelengths of the incident optical wave. For this purpose, we have realized such a "lter based on TeO with parameters close to those of noncollinear AOTFs used in practice, and compared the experimentally measured and the calculated characteristics. The agreement observed between these dependences is discussed and shown to be useful in designing the angular and power characteristics of noncollinear AOTFs.
2. Theoretical background Noncollinear AOTFs operate on the basis of the di!raction of an optical wave on acoustic waves in anisotropic media. The principle of the noncollinear acousto-optic di!raction states, generally, that noncollinear di!raction takes place under the socalled non-critical phase matching condition, where the tangents to the incident and to the di!racted light wavevector loci are parallel, i.e. that their group velocities are collinear. Under this condition, the wavevectors of the incident, k , and the di!racted, k , optical waves and of the acoustic wave, k , are related by k "k $k , where the momenta of incident and di!racted photons and phonon are: 2nn , "k "" j
2nn , "k "" j
2nf "k "" . ? v
For a randomly polarized incident optical wave, containing extraordinarily and ordinarily polarized components, the acousto-optic interaction changes the polarization ratio and the extraordinarily polarized incident optical wave exits the noncollinear AOTF almost ordinarily polarized, and vice versa. For the speci"c case of an extraordinarily polarized incident wave and an ordinarily polarized di!racted wave we have calculated the bandpass width, *j, and the
G. Georgiev et al. / Optics and Lasers in Engineering 31 (1999) 1}12
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di!raction e$ciency, ¹, of a noncollinear AOTF as functions of the angle of incidence, h , and the RF-signal power density, P . As TeO is the most frequently used acousto-optic material in noncollinear AOTFs, especially in the visible and near infrared spectral regions, we have concentrated our analysis on this material. So, all calculations throughout this work were performed for TeO and a "lter was produced and examined to test the dependencies obtained. The geometry of the "lter is shown in Fig. 1a, and the wavevector diagram of the noncollinear acousto-optic interaction in this case is presented in Fig. 1b. The acoustic wave is generated by a transducer of LiNbO , bonded to the crystal of TeO , cut and polished to the appropriate ge ometry. By changing the frequency of the RF-signals applied to the transducer, one can scan the spectral range of interest. If the incident optical wave is unpolarized, there
Fig. 1. (a) Geometry of the noncollinear AOTF of TeO studied. k and k are the incident optical and acoustic wavevectors, respectively. (b) Wavevector diagram of the noncollinear acousto-optic interaction in the "lter.
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G. Georgiev et al. / Optics and Lasers in Engineering 31 (1999) 1}12
are one undi!racted unpolarized and two di!racted and oppositely polarized optical waves at the output of the noncollinear AOTF. The di!racted waves are usually of di!erent intensity and passband width due to the optimal values of h being di!erent for the incident extraordinarily and ordinarily polarized optical waves. The angular separation of the output optical waves in such "lters is usually about 8}93 and, therefore, the use of an output polarizer is not necessary. An important step in designing noncollinear AOTFs is to choose an appropriate con"guration in order to realize a nonzero e!ective elasto-optic coe$cient. The TeO has a large optical rotatory power. The optical activity is a property of a chiral medium. It consists of molecules arranged in a spiral way or its molecules are build up of atoms, arranged in a spiral manner. In such a medium left circularly polarized light and right circularly polarized light propagate with di!erent phase velocities, which results in the rotation of the incident polarization plane. The change of the acousto-optical characteristics of the TeO , when chirality is involved has been studied [11]. The results obtained show that the polarization modes vary for propagation close to the optical axis as the acoustic power is increased. The in#uence of the optical activity of TeO for the geometry and the size of the acousto-optic cell used by us is taken into account in the values of material parameters used in the calculations.
3. Experimental The experimental studies were performed by a noncollinear AOTF of TeO with geometry as that shown in Fig. 1a. The acoustic wave used is a pure transverse mode and propagates in the (!1 1 0) plane, forming an angle a"10.43 with the [1 1 0] axis. A linearly polarized incident optic wave falls normally to the plane, forming an angle h "24.63 with the plane (0 0 1). We used a 1633-cut LiNbO transducer operating at a central frequency of 150 MHz. The dimensions of the transducer are 4;8 mm, the clear aperture of the optical faces is 8;12 mm and the diameter of the incident optical beam used is about 2 mm. By changing the acoustic frequency from 120 to 190 MHz we were able to select optical wavelengths in the spectral range from 450 to 600 nm. The "lter was driven by a RF-signal generator through a power ampli"er supplying RF power of about 3 W at the central frequency. Our primary goal is to show that the theoretical modelling can serve for estimating the spectral parameters of real devices rather than to achieve optimum operating characteristics. The typical speci"cations for commercial noncollinear AOTF's of TeO are given in Table 1. The numerical values of the constants, used by us in the calculation, are given in Table 2. Applying the appropriate RF-signal into the AOTF will enable to di!ract light with corresponding wavelength. In our case it was found that the laser beam of 633 nm was di!racted from the "lter when the RF-signal of 124 MHz was applied into it. For shorter wavelengths is needed higher RF frequency. For example to di!ract wavelengths of 488 and 514 nm the 162 and 175 MHz are needed, respectively. To measure experimentally the dependences given in Section 2, we used the setup shown in Fig. 2. The transmission of the "lter, normalized to unity and corrected to
G. Georgiev et al. / Optics and Lasers in Engineering 31 (1999) 1}12 Table 1 Typical speci"cations for commercial noncollinear AOTF's of TeO [18}20] Parameter
Value
Optical aperture De#ection angle Bandpass width Angular "eld of view Tuning range (single transducer) Di!raction e$ciency RF signal power density RF signal frequency range Transducer material Spectral range
2}6 mm 0}93 1}50 nm 0}73 0.4}0.8 lm; 0.8}1.8 lm; 1.1}2.5 lm; 2.4}5 lm 50}90% (typically '60% over tuning range) 1;10 to 1;10 W m\ 10}250 MHz LiNbO 0.4}5 lm
Table 2 Parameters of TeO used in the calculations [12, 13, 15, 16] Parameter
Value
a b k c c c q (p !p )/2 n n
7.156;10\ 0.1338 262.9;10\ 5.57;10 N m\ 5.12;10 N m\ 2.65;10 N m\ 5.99;10 kg m\ 0.12 2.2597 at 0.6328 lm 2.4119 at 0.6328 lm
Fig. 2. Setup used for measuring the "lter characteristics.
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G. Georgiev et al. / Optics and Lasers in Engineering 31 (1999) 1}12
account for the optical system in#uence, was measured as functions of h and P at three "xed wavelengths, j "632.8 nm, j "514 nm and j "488 nm. 4. Results and discussion 4.1. Dependence of the frequency on h The dependence of the frequency of the controlling RF-signal, f, on the angle of incidence h for an extraordinarily polarized incident optical wave, is shown on Fig. 3. The curves in this "gure were calculated from the wavevectors diagram (Fig. 1b): v f (h )" (n#n!2n n cos (h !h ) , j
(1)
where v is the acoustic velocity, h is the de#ection angle and n and n are the refractive indeces for the incident and for the ordinarily polarized optical wave, respectively (see the Appendix). In calculating f (h ) we used values of the indeces of refraction n and n obtained by a two-oscillator Sellmeier model, as proposed by Uchida [13], A A # , 1 1 1 1 ! ! j j j j where the coe$cients A , A , j and j are listed in Table 3. The points in Fig. 3 present the experimentally measured values of the frequency on h , determinated from the "lter transmission at the three incident optical wavelengths j "632.8 nm, j "514 nm and j "488 nm, whereas the theoretical curves are n"1#
Fig. 3. Calculated dependences of the acoustic frequency on h for: j "633 nm, j "514 nm, j "488 nm, ¸"4 mm and h "79.63. The points represent the experimentally measured values.
G. Georgiev et al. / Optics and Lasers in Engineering 31 (1999) 1}12
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Table 3 Indices of refraction of TeO [10] Indices of refraction
A
k (nm)
A
k (nm)
n n
1.4351;10\ 1.5678;10\
134.2 134.2
1.6621;10\ 2.2248;10\
263.8 263.1
Fig. 4. Calculated dependences of the bandpass width on h for: j "633 nm, j "514 nm, j "488 nm, ¸"4 mm and h "79.63. The points represent the experimentally measured values.
calculated using Eq. (1). The good agreement between the theoretical curves and the experimental data con"rms the applicability of the theoretical model used by us at least in the range of angles of incidence from 203 to 303. 4.2. Dependence of the xlter bandpass width on h When a collimated incident optical wave is di!racted by a collimated acoustic wave, the momentum-matching condition is highly selective with respect to the angle of incidence, h . In this case the dependence of the noncollinear AOTF bandpass width, *k (the full width at half maximum) on h , shown in Fig. 4, is given by [14] 1.8nj *j (h )" , B¸ sin h
(2)
where B (j) is the optical dispersive constant and ¸ is the length of acousto-optic interaction (see the Appendix). Fig. 4 illustrates a comparison between the theoretical dependences of the bandpass width on the angle of incidence and experimental data. As usually, the experimental points were speci"ed from the widths of the "lter transmission curves measured at j "632.8 nm, j "514 nm and j "488 nm, while the theoretical curves are cal culated using Eq. (2). It is seen that to attain a narrow "lter bandpass width for the
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G. Georgiev et al. / Optics and Lasers in Engineering 31 (1999) 1}12
given values of ¸ and j one must use angles of incidence exceeding 203. All the observed experimental values of *j are higher than the calculated once, since in real practice one cannot achive a perfect wavevectors' matching. 4.3. Dependencies of the diwraction ezciency on h and P The di!raction e$ciency of a noncollinear AOTF, which is de"ned by the ratio between the intensities of the di!racted and the incident optical waves, are shown in Figs. 5 and 6, as functions of h and P , respectively. The theoretical curves, were
Fig. 5. Calculated dependencies of the di!raction e$ciency on h for ¸"4 mm, h "79.63 and P "9.5;10 W/m. The experimentally measured values are presented by: circle (633 nm), triangle (514 nm) and square (488 nm).
Fig. 6. Calculated dependencies of the di!raction e$ciency on P for ¸"4 mm, h "79.63 and h "24.63. The experimentally measured values are presented by: circle (633 nm), triangle (514 nm) and square (488 nm).
G. Georgiev et al. / Optics and Lasers in Engineering 31 (1999) 1}12
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calculated from the dependence ¹"(ng¸)
sin nX , nX
(3)
where X"¸
g#
*K n ,
1 g" (M P , 2j and the e!ective "gure of merit, relating the di!raction e$ciency to the acoustic power for a given device geometry, np M " , ov where p is the e!ective elasto-optic coe$cient, P is the power density and *K is the wavevector mismatch (see the Appendix). Fig. 5 shows the "lter di!raction e$ciency versus the angle of incidence, h . The theoretical curves are calculated by Eq. (3) for P "9.5;10 W/m using the material parameters of TeO and taking into account the values of the indices of refraction for the corresponding optical wavelengths. The experimental points are determined from the measured "lter transmission at the three wavelengths of the incident optical beam. It is evident that the maximum di!raction e$ciency shifts to greater angles of incidence, thus shortening the corresponding optical wavelength in a sinusoidal squared manner. The "gure reveals a good agreement between theory and experiment. Fig. 6 presents the dependence of the di!raction e$ciency on P for h "24.63, the angle of incidence of the "lter investigated, at the three wavelengths under consideration. The theoretical curves are calculated using Eq. (3) with material parameters of TeO , while the points in the "gure show the corresponding experimental results. Since the calculations by Eq. (3) were performed using the general expression for the di!raction e$ciency rather than that for the maximum e$ciency, the power density in experiments was not optimized for each optical wavelength used. Again, a good agreement of the theoretical curves with the experimental results is evident, which demonstrates the applicability of the model used to describe these kinds of dependence. 4.4. Angular xeld of view This dependence, which is re!ered by Glenar et al. [5], shows the range of external angles of deviation, c, for the incidence beam from the normal to the "lter input surface, where c"0 at h "24.63. The reason to introduce such a parameter is that both the "lter bandpass width and di!raction e$ciency vary when the incident beam does not fall perpendicularly to the surface and losses of intensity occur due to the
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G. Georgiev et al. / Optics and Lasers in Engineering 31 (1999) 1}12
Fig. 7. Experimental dependence of the "lter transmission of the angular "eld of view c for j"633 nm, ¸"4 mm, P "9.5;10 W/m and h "24.63.
changed re#ection conditions. In addition, for non-perpendicular incidence of input beam, the noncritical phase matching condition is not perfectly satis"ed (*K+0), which also leads to reduction in the di!raction e$ciency. Fig. 7 shows the experimentally measured values of c at j"633 nm ( f"124 MHz). The "gure indicates that for deviations of the incidence optical beam with $1.53 from the normal to the input surface, the transmission of the "lter remains above 50% of that at c"03 (h "24.63). The geometry size of our "lter makes it impossible to change experimentally the values of the angle of deviation, c, out of the aforesaid range. We tried to model this dependence and found it similar in character to the experimentally observed one. Unfortunatelly, the intensity losses at the front and the exit surfaces of the cell when the light is incident non perpendicularly as well as the parasitic re#ected optical waves into the cell in this case makes it di$cult to discuss quantitatively the correlation between theory and experiment.
5. Conclusions The presented results clearly demonstrate that the bandpass width and the di!raction e$ciency of the noncollinear AOTF based on TeO can be successfully modelled in the framework of a simple theoretical approach used. It is important to point out that in this study the "xed independent parameters of the "lter are: j * the optical wavelength, the angle of incidence of optical wave * h and the length of acousto optic interaction * ¸. Since the angle of acceptance of the "lter is rather large it is expected that the features of the noncollinear AOTFs as all solid state nonmoving parts dispersive device, are particularly valuable for spectroscopic techniques and other technological applications such as imaging and remote sensing.
G. Georgiev et al. / Optics and Lasers in Engineering 31 (1999) 1}12
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Appendix A In Eq. (1) v is calculated from the characteristic sound velocity modes [12]: ( (C !C ) cos a#C sin a v" , o where a"903!h is the acoustical tilt angle (the angle between the direction of ? incidence of the acoustic wave and the (1 1 0) axis), C , C and C are the elastic sti!ness constants, o is the mass density of TeO and n sin h !n sin h h "!arctan ? n cos h !n cos h is the angle between the incident acoustic wavevector and the (0 0 1) axis. The angle between the di!racted optical wave and the (0 0 1) axis, h , was calculated from the indicatrix relations:
n tan h , n with the index of refraction for an extraordinary incident optical wave given by h (h )"arctan
1 n (h )" . cos h sin h # n n In Eq. (2) the optical dispersive constant [12] is,
B (j)"2n dn!j
*dn , *j
and a dn (j)" #b j!j A is the birefringence (a, b and j are constants, characterizing the optical properties of the medium, Table 2). The length of acousto-optic interaction, given by [5] is = ¸" , cos (h !a) = is the transducer width (in our case ="4 mm). In Eq. (3) the momentum mismatch of the noncollinear interaction [20] is
sin h b*j *K" #ndn (F (*h)#F (* )) , F ( j j
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G. Georgiev et al. / Optics and Lasers in Engineering 31 (1999) 1}12
where F "2 cos h !sin h , F F "2 cos h #sin h , ( j *h" , ¸B (j) "F " F j * " . ¸B (j) "F " (
References [1] Harris SE, Wallace RW. Acousto-optic tunable "lter. J Opt Soc Am 1969;59:744. [2] Chang IC. Noncollinear acousto-optic "lter with large angular aperture. Appl Phys Lett 1974;25:370. [3] Kurtz I, Dwelle R, Katzka P. Rapid scanning #uorescence spectroscopy using an acousto-optic tunable "lter. Rev Sci Instrum 1987;58:1996. [4] Hueber DM, Stevenson CL, Vo-Dinh Tuan. Fast scaning synchronous luminescence spectrometer based on acousto-optic tunable "lters. Appl Spectrosc 1995;49:1624. [5] Glenar DA, Hillman JJ, Saif B, Bergstralh J. Acousto-optic imaging spectropolarimetry for remote sensing, Appl Opt 1994;33:7412. [6] Glenar DA, Hillman JJ, LeLouarn M, Fugate RQ, Drummond JD. Multispectral imagery of Jupiter and Saturn using adaptive optics and acousto-optic tuning. Publ Astr Soc Paci"c 1997;109:326. [7] Baptista MS, Tran CD, Gao GH, Near-infrared detection of #ow injection analysis by acousto-optic tunable "lter based spectrophotometry, Anal Chem 1996;68:971. [8] Pasquini C, Lu J, Tran CD, Smirnov S. Detection of #ow injection analysis with pH gradient by acousto-optic tunable "lter based spectrophotometry. Anal Chem Acta 1996;319:315. [9] Georgiev G, Konstantinov L. Design of the bandpass width, the spectral resolution and the di!raction e$ciency of a noncollinear acousto-optic tunable "lter of TeO . Opt and Laser Technol 1995;27:383. [10] Georgiev G, Konstantinov L. Spectral characteristics of noncollinear acousto-optic tunable "lters. Opt Laser Technol 1997;29:267. [11] Seymour RS. Acoustooptic Bragg di!raction in anisotropic optically active media. Appl Opt 1990;29:822. [12] Yano T, Watanabe A. Acousto-optic TeO tunable "lter using far-o!-axis anisotropic Bragg di!rac tion. Appl Opt 1976;15:2250. [13] Uchida N, Optical properties of single-crystal paratellurite (TeO ). Phys Rev B 1971;4:3736. [14] Chang IC. Acousto-optic tunable "lters. Opt Eng 1981;20:824. [15] Catalog of Infrared Fiber Systems, Inc. 1995. [16] Catalog of A.A Opto-electronique, 1996. [17] Catalog of NEOS Technologies, Inc., 1996. [18] Ohmachi N, Uchida N, Niizeki N. Acoustic wave propagation in TeO single crystal, J Acoust Soc Am 1972;51:164. [19] Yano T, Watanabe A, Acousto-optic "gure of merit of TeO for circularly polarized light. J Appl Phys 1974;45:1243 [20] Chang IC. Analysis of the noncollinear acousto-optic "lter. Electron Lett 1975;11:617.
Angular and power characteristics of noncollinear acousto-optic tunable "lters G. Georgiev *, E. Georgieva, L. Konstantinov Central Laboratory of Mineralogy and Crystallography, Bulgarian Academy of Sciences, Rakovski Str. 92, 1000 Soxa, Bulgaria Institute of Applied Physics, Technical University of Soxa, Soxa, Bulgaria Received 28 August 1998; accepted 7 December 1998
Abstract The angular and the power characteristics of noncollinear acousto-optic tunable "lters are investigated both experimentally and theoretically. The dependencies of the acoustic wave frequency, the bandpass width and the di!raction e$ciency of the "lter are studied as functions of the angle of incidence of the optical wave and of the acoustic signal power density. These dependencies are illustrated and tested on the basis of a realized such a "lter based on tellurium dioxide (TeO ). A good agreement of theory and experiment is demonstrated, which indicates that the approach used can be e!ective in designing such devices. 1999 Elsevier Science Ltd. All rights reserved. Keywords: Acousto-optic tunable "lters; Acousto-optics; Spectral "lters
1. Introduction Acousto-optic tunable "lters (AOTFs) are solid state, RF-signal tunable optical components based on the anisotropic Bragg di!raction of light on acoustic waves, which select a narrow bandpass from a wide incident spectrum. The band central wavelength of such devices can be scanned through an acoustic signal of variable frequency (RF-signal), while the linewidth of the generated acousto-optical grating is equal to the acoustic wavelength. This grating can serve as a dispersive element of electrically tunable spectrometers. The response time of the AOTF is limited by the acoustic transit time across the optical aperture, which is typically on the order of a few microseconds.
* Corresponding author. Tel.: #359-2-872-450: fax: #359-2-884-979; e-mail: [email protected] 0143-8166/99/$ - see front matter 1999 Elsevier Science Ltd. All rights reserved PII: S 01 4 3-8 1 66 ( 9 8) 0 0 04 9 - 9
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G. Georgiev et al. / Optics and Lasers in Engineering 31 (1999) 1}12
The "rst AOTF, reported by Harris and Wallace [1] was based on collinear acousto-optic interaction. On the other hand, the "rst noncollinear AOTF was introduced by Chang [2]. Since then, several di!erent AOTF con"gurations have been reported based on collinear or noncollinear interactions and using various acousto-optic media, which operate from the ultraviolet to the far infrared optical spectrum. AOTFs have recently attracted much attention due to their potential in spectral analysis [3,4], spectropolarimetry [5,6], spectrophotometry [7,8], etc. Amongst all di!erent types of AOTF, noncollinear "lters have been most frequently treated theoretically and experimentally owing to their large angular aperture, simple construction and operation facilities. The spectral characteristics of noncollinear AOTFs have been studied, both theoretically and experimentally, in our previous works [9,10]. In studying these characteristics we have kept the angle between the incident-optical wavevector and the (0 0 1)-axis (the angle of incidence), the power density of the controlling RF-signal and the length of acousto-optic interaction "xed. In this paper we report results on theoretical and experimental investigations of the "lter bandpass width, angular aperture and di!raction e$ciency as functions of the angle of incidence and the RF power density at constant wavelengths of the incident optical wave. For this purpose, we have realized such a "lter based on TeO with parameters close to those of noncollinear AOTFs used in practice, and compared the experimentally measured and the calculated characteristics. The agreement observed between these dependences is discussed and shown to be useful in designing the angular and power characteristics of noncollinear AOTFs.
2. Theoretical background Noncollinear AOTFs operate on the basis of the di!raction of an optical wave on acoustic waves in anisotropic media. The principle of the noncollinear acousto-optic di!raction states, generally, that noncollinear di!raction takes place under the socalled non-critical phase matching condition, where the tangents to the incident and to the di!racted light wavevector loci are parallel, i.e. that their group velocities are collinear. Under this condition, the wavevectors of the incident, k , and the di!racted, k , optical waves and of the acoustic wave, k , are related by k "k $k , where the momenta of incident and di!racted photons and phonon are: 2nn , "k "" j
2nn , "k "" j
2nf "k "" . ? v
For a randomly polarized incident optical wave, containing extraordinarily and ordinarily polarized components, the acousto-optic interaction changes the polarization ratio and the extraordinarily polarized incident optical wave exits the noncollinear AOTF almost ordinarily polarized, and vice versa. For the speci"c case of an extraordinarily polarized incident wave and an ordinarily polarized di!racted wave we have calculated the bandpass width, *j, and the
G. Georgiev et al. / Optics and Lasers in Engineering 31 (1999) 1}12
3
di!raction e$ciency, ¹, of a noncollinear AOTF as functions of the angle of incidence, h , and the RF-signal power density, P . As TeO is the most frequently used acousto-optic material in noncollinear AOTFs, especially in the visible and near infrared spectral regions, we have concentrated our analysis on this material. So, all calculations throughout this work were performed for TeO and a "lter was produced and examined to test the dependencies obtained. The geometry of the "lter is shown in Fig. 1a, and the wavevector diagram of the noncollinear acousto-optic interaction in this case is presented in Fig. 1b. The acoustic wave is generated by a transducer of LiNbO , bonded to the crystal of TeO , cut and polished to the appropriate ge ometry. By changing the frequency of the RF-signals applied to the transducer, one can scan the spectral range of interest. If the incident optical wave is unpolarized, there
Fig. 1. (a) Geometry of the noncollinear AOTF of TeO studied. k and k are the incident optical and acoustic wavevectors, respectively. (b) Wavevector diagram of the noncollinear acousto-optic interaction in the "lter.
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G. Georgiev et al. / Optics and Lasers in Engineering 31 (1999) 1}12
are one undi!racted unpolarized and two di!racted and oppositely polarized optical waves at the output of the noncollinear AOTF. The di!racted waves are usually of di!erent intensity and passband width due to the optimal values of h being di!erent for the incident extraordinarily and ordinarily polarized optical waves. The angular separation of the output optical waves in such "lters is usually about 8}93 and, therefore, the use of an output polarizer is not necessary. An important step in designing noncollinear AOTFs is to choose an appropriate con"guration in order to realize a nonzero e!ective elasto-optic coe$cient. The TeO has a large optical rotatory power. The optical activity is a property of a chiral medium. It consists of molecules arranged in a spiral way or its molecules are build up of atoms, arranged in a spiral manner. In such a medium left circularly polarized light and right circularly polarized light propagate with di!erent phase velocities, which results in the rotation of the incident polarization plane. The change of the acousto-optical characteristics of the TeO , when chirality is involved has been studied [11]. The results obtained show that the polarization modes vary for propagation close to the optical axis as the acoustic power is increased. The in#uence of the optical activity of TeO for the geometry and the size of the acousto-optic cell used by us is taken into account in the values of material parameters used in the calculations.
3. Experimental The experimental studies were performed by a noncollinear AOTF of TeO with geometry as that shown in Fig. 1a. The acoustic wave used is a pure transverse mode and propagates in the (!1 1 0) plane, forming an angle a"10.43 with the [1 1 0] axis. A linearly polarized incident optic wave falls normally to the plane, forming an angle h "24.63 with the plane (0 0 1). We used a 1633-cut LiNbO transducer operating at a central frequency of 150 MHz. The dimensions of the transducer are 4;8 mm, the clear aperture of the optical faces is 8;12 mm and the diameter of the incident optical beam used is about 2 mm. By changing the acoustic frequency from 120 to 190 MHz we were able to select optical wavelengths in the spectral range from 450 to 600 nm. The "lter was driven by a RF-signal generator through a power ampli"er supplying RF power of about 3 W at the central frequency. Our primary goal is to show that the theoretical modelling can serve for estimating the spectral parameters of real devices rather than to achieve optimum operating characteristics. The typical speci"cations for commercial noncollinear AOTF's of TeO are given in Table 1. The numerical values of the constants, used by us in the calculation, are given in Table 2. Applying the appropriate RF-signal into the AOTF will enable to di!ract light with corresponding wavelength. In our case it was found that the laser beam of 633 nm was di!racted from the "lter when the RF-signal of 124 MHz was applied into it. For shorter wavelengths is needed higher RF frequency. For example to di!ract wavelengths of 488 and 514 nm the 162 and 175 MHz are needed, respectively. To measure experimentally the dependences given in Section 2, we used the setup shown in Fig. 2. The transmission of the "lter, normalized to unity and corrected to
G. Georgiev et al. / Optics and Lasers in Engineering 31 (1999) 1}12 Table 1 Typical speci"cations for commercial noncollinear AOTF's of TeO [18}20] Parameter
Value
Optical aperture De#ection angle Bandpass width Angular "eld of view Tuning range (single transducer) Di!raction e$ciency RF signal power density RF signal frequency range Transducer material Spectral range
2}6 mm 0}93 1}50 nm 0}73 0.4}0.8 lm; 0.8}1.8 lm; 1.1}2.5 lm; 2.4}5 lm 50}90% (typically '60% over tuning range) 1;10 to 1;10 W m\ 10}250 MHz LiNbO 0.4}5 lm
Table 2 Parameters of TeO used in the calculations [12, 13, 15, 16] Parameter
Value
a b k c c c q (p !p )/2 n n
7.156;10\ 0.1338 262.9;10\ 5.57;10 N m\ 5.12;10 N m\ 2.65;10 N m\ 5.99;10 kg m\ 0.12 2.2597 at 0.6328 lm 2.4119 at 0.6328 lm
Fig. 2. Setup used for measuring the "lter characteristics.
5
6
G. Georgiev et al. / Optics and Lasers in Engineering 31 (1999) 1}12
account for the optical system in#uence, was measured as functions of h and P at three "xed wavelengths, j "632.8 nm, j "514 nm and j "488 nm. 4. Results and discussion 4.1. Dependence of the frequency on h The dependence of the frequency of the controlling RF-signal, f, on the angle of incidence h for an extraordinarily polarized incident optical wave, is shown on Fig. 3. The curves in this "gure were calculated from the wavevectors diagram (Fig. 1b): v f (h )" (n#n!2n n cos (h !h ) , j
(1)
where v is the acoustic velocity, h is the de#ection angle and n and n are the refractive indeces for the incident and for the ordinarily polarized optical wave, respectively (see the Appendix). In calculating f (h ) we used values of the indeces of refraction n and n obtained by a two-oscillator Sellmeier model, as proposed by Uchida [13], A A # , 1 1 1 1 ! ! j j j j where the coe$cients A , A , j and j are listed in Table 3. The points in Fig. 3 present the experimentally measured values of the frequency on h , determinated from the "lter transmission at the three incident optical wavelengths j "632.8 nm, j "514 nm and j "488 nm, whereas the theoretical curves are n"1#
Fig. 3. Calculated dependences of the acoustic frequency on h for: j "633 nm, j "514 nm, j "488 nm, ¸"4 mm and h "79.63. The points represent the experimentally measured values.
G. Georgiev et al. / Optics and Lasers in Engineering 31 (1999) 1}12
7
Table 3 Indices of refraction of TeO [10] Indices of refraction
A
k (nm)
A
k (nm)
n n
1.4351;10\ 1.5678;10\
134.2 134.2
1.6621;10\ 2.2248;10\
263.8 263.1
Fig. 4. Calculated dependences of the bandpass width on h for: j "633 nm, j "514 nm, j "488 nm, ¸"4 mm and h "79.63. The points represent the experimentally measured values.
calculated using Eq. (1). The good agreement between the theoretical curves and the experimental data con"rms the applicability of the theoretical model used by us at least in the range of angles of incidence from 203 to 303. 4.2. Dependence of the xlter bandpass width on h When a collimated incident optical wave is di!racted by a collimated acoustic wave, the momentum-matching condition is highly selective with respect to the angle of incidence, h . In this case the dependence of the noncollinear AOTF bandpass width, *k (the full width at half maximum) on h , shown in Fig. 4, is given by [14] 1.8nj *j (h )" , B¸ sin h
(2)
where B (j) is the optical dispersive constant and ¸ is the length of acousto-optic interaction (see the Appendix). Fig. 4 illustrates a comparison between the theoretical dependences of the bandpass width on the angle of incidence and experimental data. As usually, the experimental points were speci"ed from the widths of the "lter transmission curves measured at j "632.8 nm, j "514 nm and j "488 nm, while the theoretical curves are cal culated using Eq. (2). It is seen that to attain a narrow "lter bandpass width for the
8
G. Georgiev et al. / Optics and Lasers in Engineering 31 (1999) 1}12
given values of ¸ and j one must use angles of incidence exceeding 203. All the observed experimental values of *j are higher than the calculated once, since in real practice one cannot achive a perfect wavevectors' matching. 4.3. Dependencies of the diwraction ezciency on h and P The di!raction e$ciency of a noncollinear AOTF, which is de"ned by the ratio between the intensities of the di!racted and the incident optical waves, are shown in Figs. 5 and 6, as functions of h and P , respectively. The theoretical curves, were
Fig. 5. Calculated dependencies of the di!raction e$ciency on h for ¸"4 mm, h "79.63 and P "9.5;10 W/m. The experimentally measured values are presented by: circle (633 nm), triangle (514 nm) and square (488 nm).
Fig. 6. Calculated dependencies of the di!raction e$ciency on P for ¸"4 mm, h "79.63 and h "24.63. The experimentally measured values are presented by: circle (633 nm), triangle (514 nm) and square (488 nm).
G. Georgiev et al. / Optics and Lasers in Engineering 31 (1999) 1}12
9
calculated from the dependence ¹"(ng¸)
sin nX , nX
(3)
where X"¸
g#
*K n ,
1 g" (M P , 2j and the e!ective "gure of merit, relating the di!raction e$ciency to the acoustic power for a given device geometry, np M " , ov where p is the e!ective elasto-optic coe$cient, P is the power density and *K is the wavevector mismatch (see the Appendix). Fig. 5 shows the "lter di!raction e$ciency versus the angle of incidence, h . The theoretical curves are calculated by Eq. (3) for P "9.5;10 W/m using the material parameters of TeO and taking into account the values of the indices of refraction for the corresponding optical wavelengths. The experimental points are determined from the measured "lter transmission at the three wavelengths of the incident optical beam. It is evident that the maximum di!raction e$ciency shifts to greater angles of incidence, thus shortening the corresponding optical wavelength in a sinusoidal squared manner. The "gure reveals a good agreement between theory and experiment. Fig. 6 presents the dependence of the di!raction e$ciency on P for h "24.63, the angle of incidence of the "lter investigated, at the three wavelengths under consideration. The theoretical curves are calculated using Eq. (3) with material parameters of TeO , while the points in the "gure show the corresponding experimental results. Since the calculations by Eq. (3) were performed using the general expression for the di!raction e$ciency rather than that for the maximum e$ciency, the power density in experiments was not optimized for each optical wavelength used. Again, a good agreement of the theoretical curves with the experimental results is evident, which demonstrates the applicability of the model used to describe these kinds of dependence. 4.4. Angular xeld of view This dependence, which is re!ered by Glenar et al. [5], shows the range of external angles of deviation, c, for the incidence beam from the normal to the "lter input surface, where c"0 at h "24.63. The reason to introduce such a parameter is that both the "lter bandpass width and di!raction e$ciency vary when the incident beam does not fall perpendicularly to the surface and losses of intensity occur due to the
10
G. Georgiev et al. / Optics and Lasers in Engineering 31 (1999) 1}12
Fig. 7. Experimental dependence of the "lter transmission of the angular "eld of view c for j"633 nm, ¸"4 mm, P "9.5;10 W/m and h "24.63.
changed re#ection conditions. In addition, for non-perpendicular incidence of input beam, the noncritical phase matching condition is not perfectly satis"ed (*K+0), which also leads to reduction in the di!raction e$ciency. Fig. 7 shows the experimentally measured values of c at j"633 nm ( f"124 MHz). The "gure indicates that for deviations of the incidence optical beam with $1.53 from the normal to the input surface, the transmission of the "lter remains above 50% of that at c"03 (h "24.63). The geometry size of our "lter makes it impossible to change experimentally the values of the angle of deviation, c, out of the aforesaid range. We tried to model this dependence and found it similar in character to the experimentally observed one. Unfortunatelly, the intensity losses at the front and the exit surfaces of the cell when the light is incident non perpendicularly as well as the parasitic re#ected optical waves into the cell in this case makes it di$cult to discuss quantitatively the correlation between theory and experiment.
5. Conclusions The presented results clearly demonstrate that the bandpass width and the di!raction e$ciency of the noncollinear AOTF based on TeO can be successfully modelled in the framework of a simple theoretical approach used. It is important to point out that in this study the "xed independent parameters of the "lter are: j * the optical wavelength, the angle of incidence of optical wave * h and the length of acousto optic interaction * ¸. Since the angle of acceptance of the "lter is rather large it is expected that the features of the noncollinear AOTFs as all solid state nonmoving parts dispersive device, are particularly valuable for spectroscopic techniques and other technological applications such as imaging and remote sensing.
G. Georgiev et al. / Optics and Lasers in Engineering 31 (1999) 1}12
11
Appendix A In Eq. (1) v is calculated from the characteristic sound velocity modes [12]: ( (C !C ) cos a#C sin a v" , o where a"903!h is the acoustical tilt angle (the angle between the direction of ? incidence of the acoustic wave and the (1 1 0) axis), C , C and C are the elastic sti!ness constants, o is the mass density of TeO and n sin h !n sin h h "!arctan ? n cos h !n cos h is the angle between the incident acoustic wavevector and the (0 0 1) axis. The angle between the di!racted optical wave and the (0 0 1) axis, h , was calculated from the indicatrix relations:
n tan h , n with the index of refraction for an extraordinary incident optical wave given by h (h )"arctan
1 n (h )" . cos h sin h # n n In Eq. (2) the optical dispersive constant [12] is,
B (j)"2n dn!j
*dn , *j
and a dn (j)" #b j!j A is the birefringence (a, b and j are constants, characterizing the optical properties of the medium, Table 2). The length of acousto-optic interaction, given by [5] is = ¸" , cos (h !a) = is the transducer width (in our case ="4 mm). In Eq. (3) the momentum mismatch of the noncollinear interaction [20] is
sin h b*j *K" #ndn (F (*h)#F (* )) , F ( j j
12
G. Georgiev et al. / Optics and Lasers in Engineering 31 (1999) 1}12
where F "2 cos h !sin h , F F "2 cos h #sin h , ( j *h" , ¸B (j) "F " F j * " . ¸B (j) "F " (
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