Optics Communications 285 (2012) 715–719

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Optics Communications

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Mid-infrared high-frequency high-resolution reflective acousto-optic filters inmercury halides

Raman Maksimenka ⁎, Pierre TournoisFASTLITE, Centre Scientifique d'Orsay, Bât. 503, 91401 Orsay, France

⁎ Corresponding author.E-mail address: raman@fastlite.com (R. Maksimenka

0030-4018/$ – see front matter © 2011 Elsevier B.V. Alldoi:10.1016/j.optcom.2011.10.078

a b s t r a c t

a r t i c l e i n f o

Article history:Received 22 July 2011Received in revised form 28 September 2011Accepted 27 October 2011Available online 12 November 2011

Keywords:Acousto-optic filtersMercury halidesMultispectral imagingLaser spectroscopyPulse shaping

Due to the long optical wavelengths of the mid-infrared region and to the low acoustic shear wave velocitiesof mercury halides, the high acoustic frequency solution of the acousto-optic wave-vector diagram can beused to design new high resolution acousto-optic filters, such as reflective AOTFs and AOPDFs, which aretechnically impossible to realize in the UV, visible and near-infrared regions.

© 2011 Elsevier B.V. All rights reserved.

1. Introduction

Two main categories of acousto-optic filters have found numerousapplications: the non-collinear Acousto-Optic Tunable Filters (AOTF)[1] showing wide optical angular aperture for one filtered acousticfrequency and the Acousto-Optic Programmable Dispersive Filters(AOPDF) [2,3] utilizing a collinear beam interaction between the inputoptical wave and the acoustic wave energies [4,5].

Due to their wide angular aperture, the AOTF filters can extract,from white large sources, many different images of different spectralcontents (multispectral imaging). The spectral resolution and diffrac-tion efficiency of these filters are limited by the size of the acoustictransducer which determines the acousto-optic interaction length.

The AOPDF filters, which show a weak angular aperture for eachapplied acoustic frequency, are mainly dedicated to ultra-short laserpulse shaping in amplitude and phase. Their spectral resolution anddiffraction efficiency are no more limited by the transducer size butonly by the crystal length.

In this paper, the designs of newhigh-resolution acousto-optic filters,referred to as “reflective AOTF” and “reflective AOPDF”, respectively, areproposed. As opposed to the classical AOTFs and AOPDFs, which use asmall deviation of the diffracted wave from the incident wave, thesefilters use a large deviation of the diffracted wave from the incidentwave, after coherent reflections on the acoustic wave planes.

).

rights reserved.

2. High-frequency solution in mercury halides crystals

In the visible and near-infrared acousto-optic filters the mostfrequently used crystal is the Tellurium dioxide (TeO2) belongingto the tetragonal crystal class. However, its transparency window(0.35–4.5 μm) does not allow using it in the mid-infrared opticalregion. In that region the most suitable crystals are mercury(I)halides such as Calomel (mercury(I) chloride, Hg2Cl2), mercury(I)bromide (Hg2Br2) and mercury(I) iodide (Hg2I2), which belong alsoto the tetragonal crystal class [6–8]. Their optical ordinary and extraor-dinary indices, in the mid-infrared, are respectively: no=1.898, 2.05and 2.26, and ne=2.445, 2.75 and 3.21. In the (1�10) plane of the crystaland along an angle θd from the [110] axis, the extraordinary index ndwrites:

nd θdð Þ ¼ none=

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffin2o cos

2θd þ n2e sin

2θdq

: ð1Þ

In the (1�10) plane of a tetragonal crystal, a pure shear acousticwave propagates, along an angle θA from the [110] axis, with aphase velocity given by:

V θAð Þ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiV2110 cos

2θA þ V2001 sin

2θAq

; ð2Þ

with V110=347, 282, 253 m/s and V001=1084, 1008, 871 m/s beingthe shear wave velocities of the Calomel, mercury bromide andmercury iodide along the [110] and [001] axis.

716 R. Maksimenka, P. Tournois / Optics Communications 285 (2012) 715–719

The direction βA of the acoustic energy flow from the [110] axis, isgiven by:

tan βA ¼ V001=V110ð Þ2· tan θA: ð3Þ

When an incident optical ordinary wave, polarized along the [1�10]direction, with a wave vector ko which makes an angle θo with the[110] axis, interacts with the shear acoustic wave of wave vector K,an optical extraordinary wave, polarized in the (1�10) plane, is dif-fracted with a wave vector kd which makes an angle θd from the[110] axis.

The wave vectors matching condition allows writing:

ko cos θoð Þ þ K cos θAð Þ ¼ kd cos θdð Þko sin θoð Þ þ K sin θAð Þ ¼ kd sin θdð Þ: ð4Þ

Taking into account Eq. (1) and with: 2δ=(ne2−no2)/no2, the fol-

lowing relations can be written:

1þ 2δ sin2 θAð Þh i K

ko

� �2þ cos θo−θAð Þ þ 2δ sin θoð Þ sin θAð Þ½ � 2K

ko

� �−2δ cos2 θoð Þ ¼ 0 ð5Þ

K1;2

ko¼

∓ cos θo−θAð Þ þ 2δ sin θoð Þ sin θAð Þ½ � þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ 2δð Þ· cos2 θo−θAð Þ þ 2δ sin2 θAð Þ� �q

1þ 2δ sin2 θAð Þ� � :

ð6Þ

Approximations of these solutions, to first order in δ, are:

K1

ko¼ δ⋅ cos2θo

cos θo−θAð Þ andK2

ko¼ 2 cos θo−θAð Þ þ δ⋅ cos

2 θo−2θAð Þcos θo−θAð Þ : ð7Þ

The first solution K1/ko of the second order Eq. (5) correspondsto a small vector K1 and a weak deviation of the optical diffractedwave from the incident wave, whereas the second solution K2/ko cor-responds to a long vector K2 and to a large deviation of the optical dif-fracted wave from the incident wave, after coherent reflections on theacousticwave planes.With λ being the optical wavelength in a vacuum,the low frequency f1 associated to vector K1 and the high frequency f2associated to vector K2 are given by:

f 1;2 ¼ noV θAð Þλ

K1;2

ko: ð8Þ

Fig. 1. AOTF wave-vector diagram for Calomel. Acoustic wave-vectors K1 and K2 as well adenoted by red color; incidence wave-vector ko and ordinary index curve (internal circle) aindex curve (small ellipse) are denoted in green. Tangents to the ordinary and extraordina

In the visible and near-infrared region, the large acoustic absorp-tion in TeO2 prevents the use of the high-frequency solution. But, inthe mid-infrared, where the wavelengths are longer, and, in the mer-cury halide crystals, where the acoustic shear velocities are small, thehigh-frequency solution can be very useful to design high spectralresolution filters following the large number of acoustic wavelengthsin the interaction length.

3. High-resolution AOTF for wide angle spectroscopy and multi-spectral imaging

The wide angle aperture AOTF filters are based on the conceptof parallel tangents to the optical slowness curves for the incidentand diffracted waves (Fig. 1). Then, the angles θd1 and θd2 of thediffracted waves need to be related to the angle θo of the incidentwave by:

tan θdð Þ1;2 ¼ no=neð Þ2· tan θo: ð9Þ

In the case of the long vector K2, θd2 must be taken as: θd2=θd1+πas shown in Fig. 1. Then, the ratio K2/ko, which can be written as:

K2

ko¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ nd

no

� �2þ 2

nd

nocos θ0−θd1ð Þ

sð10Þ

is shown, using Eq. (9), on Fig. 2a, versus angle θo for the threemercury halides.

When introducing the acoustic angle θA2 given by:

tan θA2ð Þ ¼ −no sin θo−nd sin θd1−no cos θo−nd cos θd1

ð11Þ

in expressions (2) and (8), the parameter f2λ can be drawn versus θo(Fig. 2b).

The merit factor M2, which characterizes the strength of theacousto-optic interaction and determines the diffraction efficiency,is given by:

M2 ¼ n3o nd θdð Þ½ �3p2ρ V θAð Þ½ �3 ð12Þ

s acoustic energy direction and f2λ-scaled slowness curve (large external ellipse) arere denoted by blue color; diffracted wave-vectors kd1 and kd2 as well as extraordinaryry index curves are shown at the ends of the vectors ko, K1 and K2.

a

d

b

c

Fig. 2. Acoustic-to-optic wave-number ratios (a), frequency-wavelength products (b) and merit factors M2 (c) as functions of incidence angle in mercury halides reflective AOTFs:Calomel (solid red line), mercury bromide (dashed magenta line) and mercury iodide (dotted green line); diffraction-efficiency optimized design of the high-resolution reflectiveAOTF in Calomel (d): optical beams are shown in dark gray color, transducer — in black and acoustic beam — in light gray. A blue line in panels a–c denotes the maximum of meritfactor M2 in Calomel (θo=25°).

a

b

Fig. 3. θA2 as a function of θo in the condition of K2 derivative nulling (a), frequency-wavelength products versus θo for fixed θA2 (b); see Fig. 2 for curve assignments.

717R. Maksimenka, P. Tournois / Optics Communications 285 (2012) 715–719

where: ρ=7.19, 7.307 and 7.70 g/cm3 are the densities of Calomel,mercury bromide and mercury iodide and p is the effective elasto-optic coefficient given by:

p ¼ − p11−p12ð Þ=2½ � sin θo cos θA þ p44 cos θo sin θA ð13Þ

with: p11=0.551, p12=0.44 and p44=0.In Fig. 2c, it is shown that the curves of M2 versus θo passes, for

θo≈25°, through maxima of 53, 148 and 484 mm2/GW for three mer-cury halides. For this particular angle θo of 25°, (θd2−π)=15.70°,14.53°, 13.01°, θA2=19.81°, 19.06°, 18.03°, βA2=74.12°, 77.24°,75.47°, V=491.5, 423.5, 361.4 m/s and f2λ=2100, 1997, 1938 m/sfor Calomel, mercury bromide and mercury iodide respectively.

The large independency of the acoustic frequency from the inci-dence angle, based on the parallel tangent concept, is precisely obtainedwhen the derivative of K2 versus θo, in Eq. (7), is nulled. This nullingrelies θA2 to θo as shown in Fig. 3a for the three halides. To demon-strate a large independency of the acoustic frequency from the inci-dence angle, f2λ is plotted on Fig. 3b versus θo by introducing a fixedacoustic phase angle θA2 (Eq. (11)), calculated for θo=25°, in rela-tions (6) and (8).

As an example, Fig. 2d shows a high-resolution reflective AOTFfilter, built in Calomel crystal. For the optical wavelengths of 10 and5 μm, the acoustic frequencies are: 210 and 420 MHz respectively. Foran acousto-optic interaction length of 1 cm, the power densities tobe applied to the acoustic transducer, to totally diffract each opticalwavelength, are about 2.5 W/mm2 at 5 μm and 10 W/mm2 at10 μm. The spectral resolutions are: 0.5 nm at 5 μm and 2 nm at10 μm, i.e.: 0.2 cm−1, and, with these resolutions, the angular aper-tures are 1.5° at 5 μm and 3° at 10 μm.

4. High-resolution AOPDF for laser spectroscopy and pulse shaping

In the AOPDF filters, to maximize the acousto-optic interactionlength and, thus, the spectral resolution and diffraction efficiency,the acoustic energy beam and the optic incident beam are aligned(Fig. 4).

Fig. 4. AOPDF wave-vector diagram for Calomel. Acoustic wave-vectors K1 and K2 as well as acoustic energy direction and f2λ-scaled slowness curve (large half-ellipse) are denotedby red color; incidence wave-vector ko and ordinary index curve (internal half-circle) are denoted by blue color; diffracted wave-vectors kd1 and kd2 as well as extraordinary indexcurve (small half-ellipse) are denoted in green.

718 R. Maksimenka, P. Tournois / Optics Communications 285 (2012) 715–719

This condition writes:

tan θA ¼ V110=V001ð Þ2· tan θo: ð14Þ

When introducing this acoustic angle in expressions (2), (6) and(8), the products f2λ are only function of θo shown on Fig. 5a. Itcan be seen, on this figure, that f2λ passes through a minimum of1157.27 m/s at θo=60.6° for Calomel, 957.19 m/s at θo=64.0° formercury bromide and 1021.4 m/s at θo=62.3° for mercury iodide.

From Fig. 4 the diffracted angles (θd)1,2 are given by:

tan θdð Þ1;2 ¼ ko sin θo � K1;2 sin θAko cos θo � K1;2 cos θA

: ð15Þ

a

b

c

Fig. 5. Frequency-wavelength products (a), diffracted beam deviation angles (b) and merit fred line), mercury bromide (dashed magenta line) and mercury iodide (dotted green line)

Fig. 5b shows the deviations angles (θd2−θo) in mercury halidesAOPDFs versus θo. For the high frequency solutions, these deviationsare positive and equal 90° for θo=60.2° in Calomel and θo=60.1° inmercury iodide, close to the frequency minima at 60.6° and 62.3°respectively. In mercury bromide deviation angle attains 90° forθo=57.3° with almost 7° offset from the frequency minimum. Inspite of this, f2λ stays the lowest of all three mercury halides at998.9 m/s.

Fig. 5c shows that for the high frequency solutions the M2 factorpasses, for θo=57°, through maxima of 420, 1350 and 3670 mm2/GWrespectively.

A very interesting remark follows the examination of Fig. 5a–c: forthe deviation angle (θd−θo)=90°, the frequency factors are close totheir minima whereas the merit factors are close to their maxima.

d

actors M2 (c) as functions of incidence angle in mercury halides AOPDFs: Calomel (solid.

719R. Maksimenka, P. Tournois / Optics Communications 285 (2012) 715–719

This is an excellent situation to design an efficient perpendicular-diffraction AOPDF filter as the one shown on Fig. 5d.

In this design, diffraction by the filter can be applied twice to thesignal by a total reflection to return the light back in the crystal, sothat the global transfer function of the new device is a square of theone-way transfer function. This is possible owing to the high diffrac-tion efficiencies discussed above. For the same interaction length,same optical wavelength and same incident angle θo, the spectralresolution, which is related to the number of acoustic wavelengthsin the crystal length, is higher for the high frequency AOPDF thanfor the low frequency AOPDF by the ratio of the acoustic frequencies,i.e. one order of magnitude higher.

As an example, for the device shown in Fig. 5d built in a Calomelcrystal with an acousto-optic interaction length of 1 cm, the acousticfrequencies are: 116 and 232 MHz. for the optical wavelengths of 10and 5 μm. The power densities to be applied to the acoustic transduc-er, to totally diffract each optical wavelength, are about 0.7 W/mm2 at5 μm and 3 W/mm2 at 10 μm. The spectral resolutions are: 1 nm at5 μm and 4 nm at 10 μm, i.e.: 0.4 cm−1, and, with these resolutions,the angular apertures are 0.25 mrad at 5 μm and 0.5 mrad at 10 μm.

5. Conclusion

It has been shown that, due to the long wavelengths in the mid-infrared and very low acoustic velocities in mercury halide crystals,a new class of high-frequency high-resolution reflective acousto-optic

filters can be designed in addition to the usual AOTFs and AOPDFs.Resolutions as fine as 0.2 cm−1 for the reflective AOTF and 0.4 cm−1

for the reflective AOPDF could be obtained in the currently availableCalomel. Reflective AOTF is preferred when higher resolution andwider aperture are required, for instance, in the multispectral imaging.On the other hand, reflective AOPDF is preferred when higher diffrac-tion efficiencies are desired, for example in the ultrafast pulse shap-ing. Experimental confirmation of the theoretical performances ofthe new designs is currently underway in the laboratory, as comple-ment to the already realized demonstration of the low-frequencyAOPDF in Calomel [9].

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