Advanced spectral processing of broadband light using acousto-optic devices with arbitrary transmission functions

In the paper, we developed a dispersive method for transmission function synthesis of collinear and quasi-collinear acousto-optic tunable filters. General theoretical consideration was performed, and modelling was made for broadband and narrowband signals. Experimental results on spectral shaping of femtosecond laser emission were obtained. Binary spectral encoding of broadband emission was demonstrated. © 2014 Optical Society of America OCIS codes: (120.6200) Spectrometers and spectroscopic instrumentation; (230.1040) Acousto-optical devices; (320.7100) Ultrafast measurements. References and links 1. A. Chong, W. H. Renninger, and F. W. Wise, “Properties of normal-dispersion femtosecond fiber lasers,” J. Opt. Soc. Am. B 25, 140–148 (2008). 2. R. Royon, J. Lhermite, L. Sarger, and E. Cormier, “High power, continuous-wave ytterbium-doped fiber laser tunable from 976 to 1120 nm,” Opt. Express 21, 13818–13823 (2013). 3. T. Oksenhendler, D. Kaplan, P. Tournois, G. M. Greetham, and F. Estable, “Intracavity acousto-optic programmable gain control for ultra-wide-band regenerative amplifiers,” Appl. Phys. B-Lasers Opt. 83, 491–494 (2006). 4. V. Ya. Molchanov, S. I. Chizhikov, O. Yu. Makarov, N. P. Solodovnikov, V. N. Ginzburg, E. V. Katin, E. A. Khazanov, V. V. Lozhkarev, and I. V. Yakovlev, “Adaptive acousto-optic technique for femtosecond laser pulse shaping,” Appl. Opt. 48, C118–C124 (2009). 5. S. I. Chizhikov, S. G. Garanin, L. V. Goryachev, V. Ya. Molchanov, V. V. Romanov, N. N. Rukavishnikov, S. V. Sokolovskii, I. N. Voronich, and K. B. Yushkov, “Acousto-optical adaptive correction of chirped laser pulse spectral profile in nd-phosphate glass regenerative amplifier,” Laser Phys. Lett. 10, 015301 (2013). 6. G. Bergner, E. Vater, D. Akimov, S. Schlücker, H. Bartelt, B. Dietzek, and J. Popp, “Tunable narrow band filter for CARS microscopy,” Laser Phys. Lett. 7, 510–516 (2010). 7. M. Kavehrad and D. Zaccarin, “Optical code-division-multiplexed systems based on spectral encoding of noncoherent sources,” J. Lightwave Technol. 13, 534–545 (1995). 8. V. V. Proklov, O. A. Byshevski-Konopko, and V. I. Grigorievski, “On the possibility of developing incoherent fibre-optic data transmission systems based on signal spectral coding with matched acousto-optical filters,” Quantum Electron. 43, 542–545 (2013). 9. L. Cao and C. Gu, “Matched spectral filter based on reflection holograms for analyte identification,” Appl. Opt. 48, 6973–6979 (2009). 10. P. I. Shnitser, I. P. Agurok, S. Sandomirsky, A. A. Avakian, and R. B. Lockwood, “Real-time polarizationsensitive multispectral automatic imaging system for object contrast enhancement and clutter mitigation,” in “Acquisition, Tracking, and Pointing XIV,” , vol. 4025 of Proc. SPIE, M.K. Masten and L.A. Stockum, eds. (SPIE, 2000), vol. 4025 of Proc. SPIE, pp. 229–237. 11. C. Wang and P. Sahay, “Breath analysis using laser spectroscopic techniques: breath biomarkers, spectral fingerprints, and detection limits,” Sensors 9, 8230–8262 (2009). #209997 $15.00 USD Received 11 Apr 2014; revised 6 Jun 2014; accepted 9 Jun 2014; published 19 Jun 2014 (C) 2014 OSA 30 June 2014 | Vol. 22, No. 13 | DOI:10.1364/OE.22.015668 | OPTICS EXPRESS 15668 12. E. Dekemper, N. Loodts, B. Van Opstal, J. Maes, F. Vanhellemont, N. Mateshvili, G. Franssens, D. Pieroux, C. Bingen, C. Robert, L. De Vos, L. Aballea, and D. Fussen, “Tunable acousto-optic spectral imager for atmospheric composition measurements in the visible spectral domain,” Appl. Opt. 51, 6259–6267 (2012). 13. V. Ya. Molchanov, V. M. Lyuty, V. F. Esipov, S. P. Anikin, O. Yu. Makarov, and N. P. Solodovnikov, “An acoustooptical imaging spectrophotometer for astrophysical observations,” Astron. Lett.-J. Astron. Space Astrophys. 28, 713–720 (2002). 14. O. Korablev, A. Fedorova, E. Villard, L. Joly, A. Kiselev, D. Belyaev, and J.-L. Bertaux, “Characterization of the stray light in a space borne atmospheric aotf spectrometer,” Opt. Express 21, 18354–18360 (2013). 15. A. Goutzoulis and D. Pape, eds., Design and Fabrication of Acousto-Optic Devices (Marcel Dekker, 1994). 16. L. N. Magdich, “Acousto-optic filter apparatus function under the conditions of frequency tuning,” Opt. Spectrosc. 49, 387–390 (1980). 17. L. N. Magdich, V. Ya. Molchanov, and I. P. Ponomareva, “Transmission function of non-collinear acoustooptical filter,” Opt. Spectrosc. 56, 736–739 (1984). 18. V. E. Pozhar and V. I. Pustovoit, “Compression of ultrashort light pulses,” Sov. J. Quantum Electron. 17, 509–511 (1987). 19. M. E. Fermann, V. da Silva, D. A. Smith, Y. Silberberg, and A. M. Weiner, “Shaping of ultrashort optical pulses by using an integrated acousto-optic tunable filter,” Opt. Lett. 18, 1505–1507 (1993). 20. P. Tournois, “Acousto-optic programmable dispersive filter for adaptive compensation of group delay time dispersion in laser systems,” Opt. Commun. 140, 245–249 (1997). 21. F. Verluise, V. Laude, Z. Cheng, Ch. Spielmann, and P. Tournois, “Amplitude and phase control of ultrashort pulses by use of an acousto-optic programmable dispersive filter: pulse compression and shaping,” Opt. Lett. 25, 575–577 (2000). 22. V. I. Pustovoit and V. V. Timoshenko, “An acoustooptical filter with a controlled transmission band,” J. Commun. Technol. Electron. 43, 422–429 (1998). 23. I. C. Chang, “Acousto-optic tunable filters,” Opt. Eng. 20, 824–829 (1981). 24. R. Pieper, A. Korpel, and W. Hereman, “Extension of the acousto-optic bragg regime through hamming apodization of the sound field,” J. Opt. Soc. Am. A 3, 1608–1619 (1986). 25. V. N. Parygin, A. V. Vershoubskiy, and K. A. Kholostov, “Collinear filter controlled by variable ultrasonic pulses,” Opt. Eng. 38, 1149–1153 (1999). 26. E. Yu. Filatova and V. N. Parygin, “Transmission function of acousto-optic cell with apodized piezotransducer,” J. Opt. A-Pure Appl. Opt. 3, S40–S45 (2001). 27. J. Vila-Francés, J. Calpe-Maravilla, J. Muñoz-Mari, L. Gómez-Chova, J. Amorós-López, E. Ribes-Gómez, and V. Durán-Bosch, “Configurable-bandwidth imaging spectrometer based on an acousto-optic tunable filter,” Rev. Sci. Instrum. 77, 073108 (2006). 28. V. I. Balakshy, V. N. Parygin, and L. E. Chirkov, Physical Principles of Acousto-optics [in Russian] (Radio i Svyaz, 1985). 29. A. K. Zaitsev and V. V. Kludzin, “Subcollinear acousto-optic tunable filter based on the medium with a strong acoustic anisotropy,” Opt. Commun. 219, 277–283 (2003). 30. V. Ya. Molchanov, V. B. Voloshinov, and O. Yu. Makarov, “Quasi-collinear tunable acousto-optic filters for systems of wavelength division multiplexing and selection of optical channels,” Quantum Electron. 39, 353–360 (2009). 31. K. B. Yushkov and V. Ya. Molchanov, “Effect of group velocity mismatch on acousto-optic interaction of ultrashort laser pulses,” Quantum Electron. 41, 1119–1120 (2011). 32. K. B. Yushkov and V. Ya. Molchanov, “MTF formalism for measurement of spectral resolution of acousto-optical devices with synthesized transmission function,” Opt. Lett. 38, 3578–3580 (2013).


Introduction
Methods of adaptive spectroscopy are strongly demanded in various modern applications.The problem of broadband emission spectral processing became especially important with development of ultrashort-pulse lasers.Combined with hollow-fiber supercontinuum generators, they became an efficient source of broadband coherent light.Adaptive optical spectroscopy requires not only random access wavelength tuning but also synthesis of arbitrary spectral transmission functions.Some applications where advanced spectral control of emission is urgently required can be listed.Among them are the problems of designing femtosecond fiber lasers [1,2], spectral gain control in optical amplifiers [3,4,5], CARS spectroscopy [6], etc.Other applications that require synthesis or arbitrary spectral transmission functions are optical matched filtering for telecommunication systems [7,8], recognition of chemical compounds [9], etc. Spectral matching of the receiver's transmission window with the analyzed spectrum can be done to maximize signal-to-noise ratio [10].It can be useful in such applications as identification of gases by their emission or absorption lines in medicine [11], in atmospheric pollution monitoring [12], and in space research and astrophysical observations [13,14].
Acousto-optic (AO) tunable filters (AOTFs) are widely used in photonics for adaptive selection of light spectral components [15].Conventionally, the continuous operation mode is used, when the filter is fed with the tonal radio-frequency (RF) signal.If the driving signal is phase and amplitude modulated in time domain, the traveling acoustic wave converts modulation into space domain.As a result, ultrasonic field distribution in the crystal becomes artificially inhomogeneous that alters the transmission function of the filter.
Different methods of controlling the transmission function of AOTFs were studied before.The transmission function synthesis by means of linear frequency modulation of ultrasonic waves was first proposed by Magdich in early 1980-s for both collinear and noncollinear interaction [16,17].Compression of ultrashort laser pulses based on this phenomenon was proposed by Pozhar and Pustovoit [18].Later, Fermann implied that principle for spectral shaping of femtosecond pulses [19]; Tournois further developed that method for amplitude and phase control of broadband femtosecond laser emission [20,21].Phase modulation of tonal ultrasound can also be used for modifying transmission function of the AOTF [22].The idea of transmission function sidelobes suppression by means of apodization of ultrasonic field came from spectroscopy.Chang [23], Korpel [24], and Parygin [25,26] investigated modification of the transmission function and side-lobes reduction using amplitude-only apodization of the acoustic field in AO interaction media.Multifrequency diffraction can be also used for broadening the bandwidth of AOTFs [10].Slow frequency sweeping during the exposure of the sensor can also be used for effective broadening of the emission spectrum transmitted by the AOTF [27].
In this paper, we propose a novel method for synthesis of arbitrary spectral transmission functions over a full tuning range of the quasi-collinear AOTF.The method is based on both amplitude and phase modulation of the ultrasonic wave.This method enables flexible controlling of light spectrum transmitted through the AOTF.A unified algorithm is deduced for calculation of amplitude and phase modulated ultrasonic waveforms that provide desired transmission functions.
Complex Fourier transform in time domain is used to calculate amplitude and frequency profiles of ultrasonic pulses.Ultrasonic waveforms S(t) are considered as complex time-dependant functions, and correspondent amplitude and frequency profiles are defined as We consider pulsed operation mode of the AOTF.The ultrasonic wave packets are assumed to have finite duration that is smaller than the acoustic wave travel time along the crystal.Temporal distribution of ultrasonic field in the crystal can be considered frozen because the speed of light in the interaction media exceeds the speed of ultrasound at least by 5 orders.Thus, the transmission function of the AOTF remains unchanged while the ultrasonic wave packet is located in the region of acousto-optic interaction.Transient phenomena that take place while the wave packet is entering and exiting the interaction region are not considered.

General consideration
It is well known that the transmission function of collinear AOTFs under the plane-wave approximation is directly determined by the spectrum of ultrasound in the crystal, i.e. the Fourier transform of the ultrasonic field distribution along the direction of light propagation [28].For example, for a stationary tonal ultrasonic wave that has a circular frequency Ω 0 , the spectrum is where T 0 is the wave travel time along the crystal and S 0 is a constant amplitude coefficient.Equation (2) determines the bandshape of typical AOTFs [23].For quasi-collinear AOTFs under the plane-wave approximation, the transmission function is also determined by the Fourier transform of the ultrasonic field distribution, providing the same bandshape as for collinear AOTFs [29].In practice, divergence of optical and acoustic beams can alter the transmission function of AOTFs.Spectral transmission function of the AOTF becomes efficiently wider when the divergency of the input light exceeds the angular aperture of the AOTF [30].Acoustic anisotropy and diffraction of ultrasonic beams in the crystals can result in asymmetry of the transmission function, however for most practical cases the FWHM single-frequency bandwidth is not notably influenced by ultrasonic diffraction [29].
In this Section, we propose a dispersive method for AOTF transmission function synthesis.The approach that is developed here concerns principal changes of AOTF transmission function that take place due to amplitude and phase modulation of RF driving waveforms.Meanwhile, the phenomena of transmission function asymmetry [29] are more or less manifested depending on the parameters of optical and acoustic beams.The results of this Section can be applied both for collinear and quasi-collinear AOTFs.
First, we consider a real rectangular spectrum A trivial Fourier transform leads to the following ultrasonic waveform: The waveform S rect (t) in Eq. ( 4) determines signal with a constant frequency F(t) = Ω 0 /(2π).Thus, the amplitude-only modulation can produce a rectangular signal spectrum; the maximum of the signal amplitude grows with the bandwidth, maxU(t) ∝ ∆Ω 0 , and the width of the peak is inversely proportional to the signal bandwidth.Linear frequency modulation of ultrasound is another method for obtaining broadband signals, and it can be applied for synthesis of transmission functions of the AOTFs.The width of the AOTF transmission window can be controlled by means of frequency-only linear modulation [16].That made possible to control broadening of the filtered window, but the transmission function had oscillations.
Our method is based in solving the inverse problem: to find a complex ultrasonic waveform that provides the desired optical transmission function.Proper phasing of spectral components is necessary to equalize the peaks of ultrasonic waveforms.For this purpose, we add a definite spectral phase to the real transmission function.It is well known that the first-order dispersion results only in a group delay of a wave packet without changes in its envelope.Thus we must provide quadratic dispersion in order to provide desired changes in the waveform.
In the case of rectangular spectral window S rect (Ω) that results in a complex spectrum S chirp (Ω): where Ψ 2 is the second-order dispersion coefficient, i.e. the second coefficient of Taylor series expansion of the ultrasonic phase in the vicinity of the central frequency Ω 0 .One can show that the complex waveform that provides this spectrum is where B chirp (t) is determined by the Fresnel integrals and can be expressed using complex error function: Unlike Eq. ( 4), the waveform S chirp (t) defined by Eq. ( 6) is both amplitude and frequency modulated.
The function B chirp (t) is a result of shift, scaling and addition of two complex terms erf (−1) 3/4 t .Modelling of signal waveforms defined by Eq. ( 7) is shown in Fig. 1 for different parameters of spectral width ∆Ω 0 and second-order dispersion Ψ 2 .The examples show dramatic changes of signal shape in time-domain, from quasi-rectangular to quasi-sinc.Timebandwidth parameter of the waveforms, T chirp ∆Ω 0 = 2∆Ω 2 0 Ψ 2 , varies from 5000 (curve 1) to 2 (curve 4).
The waveform |B chirp (t)| has a characteristic width The signal phase Φ(t) = arg S chirp (t) equals to where constant phase offset is neglected.The first two terms in Eq. ( 9) determine pure linear frequency modulation of ultrasound.The third term contains higher-order phase derivatives, i.e. higher-order dispersions, that makes difference of the proposed approach with the previous method [16].
The functions S rect (t) in Eq. ( 4) and S chirp (t) in Eq. ( 6) are supported in (−∞, ∞), however in practice only finite waveforms can be used.For collinear AOTFs, maximum waveform duration is equal to the acoustic wave travel time through the crystal T 0 .Otherwise, some part of the waveform will be out of the interaction region.Thus, we can accept that t ∈ [−T 0 /2, T 0 /2].
In the following Subsection, we observe two important particular cases: the broad spectral window ∆Ω 0 T 0 ≫ 1, and the narrow spectral window ∆Ω 0 T 0 ∼ 1.

Case studies: broad and narrow spectral windows
Broad bandwidth of input emission is typical for many applications of AOTFs in the field of AO spectral processing, e.g.ultrashort laser pulse shaping, matched optical filtering, multispectral imaging, etc. [4,5,10].For these applications, the processed spectral window is much wider than the passband of the filter, ∆Ω 0 ≫ 1/T 0 .Before now, the value of the second-order dispersion Ψ 2 was considered as a free variable.Hereinafter, the value of Ψ 2 should be fixed for given waveform parameters.The chirped waveform S chirp (t) is defined in Eq. ( 6).Given bandwidth ∆Ω 0 is fixed, the waveform duration T chirp is proportional to the second order dispersion Ψ 2 .The waveform duration can not exceed the acoustic wave travel time in the crystal T 0 .Otherwise, sufficient truncation of the waveform will take place and transmission function of the AOTF will become non-stationary.Thus, maximum value of Ψ 2 that can be used equals to Eq. ( 10) corresponds to the group delay dispersion of the optical signal that takes place in acousto-optic light dispersive delay lines [31].We use Eq. ( 10) as a definition of Ψ 2 for further calculations.
From Eq. ( 9) one can estimate the frequency span in the waveform (6).It equals to ∆F ≈ T 0 /(4πΨ 2 ) = ∆Ω 0 /(2π) ≫ 1/T 0 .Therefore, the desired spectral shape of transmission function for broad spectral windows is mostly obtained due to frequency modulation within the ultrasonic waveform.
Another application field for dispersive transmission synthesis is improvement of transmission function of the filter.In this problem, it is necessary to synthesize a narrow transmission function with correspondent ultrasonic spectral width ∆Ω 1 ∼ 1/T 0 , while the central frequency Ω 1 is slowly tuned like in a stationary operation mode of the AOTF.
If ∆Ω 1 were used in Eq. ( 10) instead of ∆Ω 0 it would follow that Ψ 2 ∼ T 2 0 .The pulse envelope S chirp (t) in this case has a quasi-sinc shape with slowly decreasing wings at t > T chirp , and expanding the temporal window T 0 > T chirp is not efficient in maintaining a uniform flat-top transmission.
To solve the problem of the transmission window distortions for chirped signals with narrow spectrum, we propose to use broadening of the spectral interval.When it is necessary to synthesize a narrow rectangular spectral window with the width ∆Ω 1 , we use a broader spectral interval ∆Ω 0 ≫ ∆Ω 1 to support S chirp (Ω).The exact value of ∆Ω 0 can be made equal to either emission bandwidth, or the tuning range of the AOTF.Then, the complex spectrum is defined as where the spectral window ∆Ω 1 centered at the variable frequency Ω 1 is a subset of the bandwidth ∆Ω 0 centered at Ω 0 .
In general, the Fourier transform of (11) will be similar to Eqs. ( 6) and ( 7), but the following inequalities will take place: From analytical point of view it means that the scaling of complex error functions in the expression for B chirp (t) is the same as for broad passband ∆Ω 0 , and the shift is defined by the window width ∆Ω 1 .Under these conditions, the envelope |B chirp (t)| will be close to | sinc(∆Ω 1 t/2π)|, and linear frequency chirp will be much smaller than the frequency bandwidth of the AOTF, i.e.F(t) ≈ 2πΩ 1 .

Arbitrary transmission function
Rectangular bandshape synthesis that was considered in Subsection 2.2 is a particular case of arbitrary transmission function.In general, it is assumed that some unrestricted spectral transmission function is defined in the wavelength interval λ ∈ [λ min , λ max ].The problem of finding the ultrasonic waveform S arb (t) for implementation of that transmission function can be solved in several steps: • accurate scaling from optical wavelength to ultrasonic frequencies; • calculation of the ultrasonic waveform; • estimation of the resulting transmission function with respect to single-frequency bandwidth of the AOTF.
The tuning curve of the filter F(λ ) is used to correlate the scale of the optical wavelengths with ultrasonic frequencies.From the calibration routine (see Sec. 3.1) we define the spectral window of ultrasound so that the frequency Ω 0 + ∆Ω 0 /2 provides Bragg phase matching at the wavelength λ min , and Ω 0 − ∆Ω 0 /2 corresponds to phase matching at λ max .Thus, the real function H arb (Ω) that determines transmission of the AOTF is defined by the user in the bandwidth ∆Ω 0 .
Then, the chirped rectangular window S chirp (Ω) is determined from Eq. ( 6), and the secondorder dispersion coefficient Ψ 2 satisfies Eq. ( 10).The complex spectrum of ultrasound S arb (Ω) equals to The ultrasonic waveform S arb (t) is defined as a Fourier transform of S arb (Ω).Contrast characteristics of the resulted transmission function depend on the single-frequency bandwidth of the AOTF.The fact that the waveform S arb (t) is defined in the finite interval t ∈ [−T 0 /2, T 0 /2] is equivalent to convolution of S arb (Ω) with the stationary transmission function S stat (Ω).However, for accurate estimation of contrast at the output of the AOTF using Eq. ( 2) can be not sufficient.Theoretical analysis of the transmission function of the AOTF can be made using the angular spectrum of ultrasound in the AOTF [29].In the experiment, the effect of the single-frequency AOTF bandwidth on the spectral contrast of diffracted light can be experimentally measured using spectral modulation transfer function [32].

Calibration procedure
For a quasicollinear filter, Bragg angle θ B is related to the ultrasonic energy flow angle ψ [30]: In general, the polar light incidence angle of light θ can be slightly varied near the value of θ B + α, and the tuning relationship of the filter can be expressed as where V is the phase velocity of ultrasound, α is the tilt angle of ultrasonic wave vector, n o is the ordinary refractive index, and n e (θ ) is the refractive index of the extraordinary wave versus the polar angle.The tilt angle α is angle between the ultrasonic wavefront and the Z axis of the crystal.It is a constructive parameter of the AOTF which determines the ultrasonic phase velocity V and the energy flow angle ψ.
Before the experiments, it is necessary to calibrate the AOTF because orientation of the AOTF crystal relatively the crystallographic axes has a tolerance, material constants V , n o , and n e are temperature-dependent, etc.It is a common assumption that the phase matching condition can be approximated as λ F = const.This approximation is handy for computations because it determines simple linear scaling of optical spectra to ultrasonic spectra.Nevertheless, experimental studies revealed this approximation gives rise to noticeable calibration errors: the wavelength scale at the output of the AOTF does not correspond to the wavelength scale of the initial spectral transmission function even in relatively narrow (∆λ 10 nm) spectral bands.For that reason, we use full expression for ultrasonic frequency with respect to refractive index dispersion in Eq (15).
The first step of the calibration routine is to find an actual value of the incidence angle θ cal .For that purpose a spectrum with a marker (e.g. a narrow magnitude dip) at a given radio frequency F mark is used.The wavelength λ mark that corresponds to the marked frequency is meas- ured experimentally.Then, the value of θ cal is determined by numerical solution of Eq. ( 15) with λ = λ mark and F = F mark .
After that, scaling of the transmission function from the optical wavelength domain to the ultrasonic frequency domain can be made.Unambiquous correspondence of optical wavelengths to ultrasonic frequencies F(λ ) is determined by Eq. ( 15) where the incidence angle of light θ = θ cal is obtained from preceding calibration.

Experimental setup description and performance
Experimental setup is schematically shown in Fig. 2. Femtosecond mode-locked laser Femtolasers Femtosource Synergy with the FWHM emission bandwidth 95 nm was used as a light source.Arbitrary waveform generator (AWG) Agilent N8241A was controlled by the computer.Periodical RF output of the AWG was amplified and supplied to the AOTF.Diffracted light spectrum was measured by the optical spectrum analyzer Agilent 86142B.The output of the AWG was controlled by the RF spectrum analyser Rohde&Schwarz FSH3.
For this research we specially designed and fabricated high-resolution paratellurite-based quasicollinear AOTF.The tilt angle of the ultrasonic phase velocity was equal to α = 5 • .Theoretical minimum passband for this filter design calculated from Eq. ( 2) was equal to 0.12 nm, however the single-frequency measurement of the transmission function resulted in the minimum passband δ λ min = 0.18 nm FWHM because of divergency of laser emission.In the experiments, ultrasonic waveform duration was equal to T 0 = 51.2µs with the duty cycle 2:1.Experimental spectral transmission functions were normalized by subtraction of reference emission spectrum from the spectra of signals.
We performed a series of experimental measurements of spectral transmission windows S wind (Ω) with variable width for characterization of the dispersive method and experimental apparatus for transmission function synthesis.In Fig. 3, rectangular transmission window was defined according to Eq. ( 11).The results were compared to the single-frequency transmission function that was obtained in the stationary operation mode.The spectral window width was varied from ∆Ω 1 = δ Ω min to ∆Ω 1 = 12δ Ω min , where the single-frequency bandwidth of the filter δ Ω min corresponds to the optical passband δ λ min = 0.18 nm.
For the narrowest transmission windows (plots 1 and 2 in Fig. 3b), convolution of the defined transmission function with the intrinsic bandshape of the AOTF shown in Fig. 3a reduces diffraction efficiency.Calculation shows that for ∆Ω 1 = δ Ω min the relative intensity of 0.6 can be reached in the center of the transmission window, and for ∆Ω 1 = 2δ Ω min the maximum is 0.9.When the transmission window broadens, the flat top is observed, as seen on the plots 4, 8, and 12. Intensity oscillations with the period about 0.4 nm on the top of the transmission function were caused by the slow drifts of laser emission spectral power density which were about 0.5 dB in magnitude.

Broadband arbitrary bandshape synthesis
Synthesis of arbitrary transmission functions was performed in a broadband regime with the optical emission bandwidth ∆λ = 120 nm FWHM.That corresponds to RF spectral window bandwidth ∆Ω 0 = 84 • 10 6 s −1 .The defined spectral magnitude (left), calculated amplitude and frequency profiles of RF waveforms (center), and measured normalized transmission (right) are shown in Figs. 4, 5, 6 for three different user-defined functions H arb (Ω).
First, the flat-top transmission function was implied (Fig. 4).The second experiment demonstrates synthesis of a dual transmission window (Fig. 5).Two rectangular peaks and the gap between them have the width of 4.8 nm.Finally, we simulated spectral encoding of broadband emission (Fig. 6).The spectrum was divided to a number of sub-bands, each had a width of 2.4 nm.These sub-bands were well resolved with peak-to-saddle contrast better than 17 dB, and sub-band spacing could be reduced even more.Total number of usable sub-bands in the experiment was equal to 50.The sequence of sub-bands was coded with a binary quasirandom sequence.Such modulation of optical spectra is typical for OCDMA communication   systems [7,8].The RF signal providing binary encoded spectrum can be characterized as quasinoise though the trend for frequency modulation and the slope are the same as for a rectangular window without modulation.

Summary
In the paper, dispersive method of arbitrary transmission function synthesis of AOTFs was proposed.The essence of the method is to add a certain quadratic spectral phase to the given transmission spectrum and then to calculate the RF waveform that provides the desired spectral composition.Such RF waveforms are characterized with higher-order dispersions and appropriate amplitude modulation.Rigorous and unified mathematical approach was developed for generating user-defined arbitrary optical transmission spectra with finite bandwidth.This method proved to be effective for side-lobe suppression as well.
Traditionally, different approaches were used in design of bandpass and notch AOTFs.A tunable bandpass filter uses the Bragg order output beam, while for notch filtering zero-order output beam is used.Dispersive synthesis of the spectral transmission functions gives rise to multipurpose applications of the AOTFs, when the same device can be used either for bandpass of for notch filtering at the Bragg output.Such feature is uncommon for acousto-optic devices.
The calibration procedure ensured good correspondence of experimentally measured spectra with initially defined transmission functions in a series of test experiments.There was demonstrated that adaptive spectral encoding of femtosecond laser emission can be easily obtained.Adaptive high-performance methods of tailoring spectral transmission functions are also extremely important in acousto-optic devices for controlling spectral gain of optical amplifiers in ultra-high-intensity laser systems as well as pulse shaping.