Running electric field gratings for detection ofcoherent radiationGERMANO MONTEMEZZANI,1,2,* MASSIMO ALONZO,1,2 VIRGINIE CODA,1,2 MOJCA JAZBINSEK,3 AND

PETER GÜNTER3

1Université de Lorraine, LMOPS, EA 4423, F-57070 Metz, France2CentraleSupélec, LMOPS, EA 4423, F-57070 Metz, France3Rainbow Photonics AG, Farbhofstrasse 21, CH-8048 Zurich, Switzerland*Corresponding author: germano.montemezzani@univ‑lorraine.fr

Received 3 March 2015; revised 7 April 2015; accepted 7 April 2015; posted 10 April 2015 (Doc. ID 235547); published 11 May 2015

A technique for the room temperature electro-optic detection of coherent low-frequency radiation, typically in thefar infrared or terahertz (THz) range, is proposed. It relies on a longitudinally running electric field gratingprobed by phase-matched Raman–Nath diffraction of a higher frequency wave. The expected probe wavediffracted intensity is background-free and is proportional to the desired signal intensity. As an example, wepresent model calculations for the probing of monochromatic THz radiation using the organic crystal 4-N,N-dymethylamino-4 0-N 0-methyl-stilbazolium 2,4,6-trimethylbenzenesulfonate as nonlinear optical sensormaterial, for which new measurements of the refractive index and absorption up to a frequency of 11 THz arepresented. © 2015 Optical Society of America

OCIS codes: (190.2055) Dynamic gratings; (230.2090) Electro-optical devices; (190.4710) Optical nonlinearities in organic

materials; (040.2235) Far infrared or terahertz.

http://dx.doi.org/10.1364/JOSAB.32.001078

1. INTRODUCTION

Approaches based on nonlinear optics are among the most validalternatives to conventional thermal sensors for the detection offar infrared or terahertz (THz) radiation, such as bolometers,Golay cells, or pyroelectric detectors [1], some of which requirecryogenic temperatures. Nonlinear optics brings about a fasterresponse, permits operation at room temperature, and offers theinteresting feature that the signature of the long wavelengthradiation can be transferred to a wave at much shorter wave-length, easily detected by means of standard photodiodes orphotomultipliers. This is true both for nonlinear techniquesbased on frequency conversion [2–5] and for those based onelectro-optics, as used for instance in electro-optic time-domainspectroscopy [6]. Importantly, in the latter case the electro-optic phase shift is proportional to the local electric field ofthe electromagnetic wave under study (signal wave). An inter-esting situation is the one where this wave is spatially inhomo-geneous, e.g., tightly focused. In this case the spatial variationof the cumulated phase shift on a copropagating probe wavecan produce a distortion of its wavefront on top of the averagephase shift. This distortion leads in general to a slight focusingor defocusing of the probe wave. This useful phenomenon wasexploited by Schneider and coworkers [7–9] to develop a sim-ple and effective method to detect ultrashort (nearly singlecycle) broadband THz pulses, also known as terahertz-induced

lensing (TIL). However, extension of this method for the de-tection of radiation with significantly longer pulses and weakerpeak intensity is difficult, an important drawback being the factthat the weak probe wave intensity variation is measured over astrong background that exists also in the absence of the sig-nal wave.

Here we propose and analyze an alternative method thatfaces this drawback, while still exploiting a spatial variationof the signal wave electric field. The proposed technique isbased on the creation of a running electric field gratingand can work for a sufficiently narrow-bandwidth coherentsignal wave able to create few interference fringes. The signalwave grating is probed by a wave being velocity-matchedto the longitudinal speed of the grating and leading to diffrac-tion in angular directions where a potentially background-free measurement is possible. In Section 2 we describe theconcept and the underlying physics and analyze its require-ments. In Section 3 we propose a compact implementationbased on a Lloyd type interferometer with incidence eitherthrough the front surface or an oblique side surface of thenonlinear sample. Finally, in Section 4 we give an applicationexample related to monochromatic THz wave detectionbased on the organic crystal 4-N,N-dymethylamino-4’-N’-methyl-stilbazolium 2,4,6-trimethylbenzenesulfonate (DSTMS).Thanks to new measurements of the THz refractive index

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0740-3224/15/061078-06$15/0$15.00 © 2015 Optical Society of America

of this crystal, we predict the necessary phase-matchingconfiguration and we estimate the achievable performance inthis context.

2. PHYSICAL CONCEPT

As shown in Fig. 1, the signal wave of angular frequency ωs issplit into two parts and enters a nonlinear optical materialunder the external angle θs;ext, while the auxiliary probe waveat frequency ωa � 2πc∕λa enters the same material under anangle θa;ext. Even though from physical principles there is nolimitation on the values of the two involved frequencies, from apractical point of view ωa should be such that this wave is de-tectable by means of conventional detectors, e.g., a visible orinfrared photodiode.

Let us consider for simplicity that the two signal beams,modeled as plane waves, enter the sample symmetrically withrespect to the longitudinal z-axis. The electric fields of the

corresponding waves are E1�r ; t��E0∕2 e1 exp�i�ωs t − k1 · r��and E2�r ; t� � E0∕2 e2 exp�i�ωst − k2 · r��. Here e1;2 are theunit vectors along the electric field direction of the two waves,e1;2 � �cos θs ; 0;∓ sin θs� for p-polarized waves and e1;2 ��0; 1; 0� for s-polarized waves. The corresponding wave vectors

are k1;2 � k0;s�� sin θs ; 0; cos θs�, where θs is the internal an-gle of propagation with respect to the surface normal (z-axis).The coherent superposition of the two waves E�r ; t� �E1�r; t� � E2�r; t� leads to

E�r; t� � E0ei�ωs t−kz z�e−αs z∕�2 cos θs� ~e�x�; (1)

where we have taken into account the effect of a possibleattenuation of the signal wave upon propagation by meansof an absorption constant αs for its intensity. Here ~e�x� is acomplex vector that depends on the transversal x-coordinateand takes into account the local amplitude and directionof the combined electric field. It is given by ~e�x� ��cos θs cos�kxx�; 0; i sin θs sin�kxx�� for p-polarization and

by ~e�x� � �0; cos�kxx�; 0� for s-polarization, its modulus is al-

ways j ~e�x�j ≤ 1. Equation (1) represents an electric field gratingmodulated along the x-coordinate. The phase of this grating isrunning along the longitudinal z-coordinate at the speedvgr � ωs∕kz � ωs∕�k0;s cos θs� � c∕�ns�θs� cos θs�, where cis the speed of light in vacuum and ns�θs� is therefractive index seen by the signal wave.

If the material where the waves cross is optically nonlinear,the electric field grating can translate into a refractive indexgrating. Here we focus only on materials exhibiting opticalnonlinearities of second order, for which the change in refrac-tive index is linear in the local electric field. In principle alsothird order (Kerr-type) nonlinearities can be used, but thiseffect is generally much weaker and would therefore be usefulonly for detecting very intense signals. By an appropriate linearelectro-optic tensor element the running electric field grating ofEq. (1) gives rise to a running refractive index grating Δn�r ; t�,which is able to influence the propagation of the probe wavethrough the sample. However, a significant cumulation of thelocal phase retardation on the probe wave can occur only if thelongitudinal phase velocity of this wave equals or nearly equalsthe longitudinal velocity of the grating phase. This leads tofollowing phase-matching condition:

ns�θs� cos θs � na�θa� cos θa; (2)

where na and θa are the refractive index and the internal anglefor the auxiliary probe wave, respectively. For later use, wedefine also a phase mismatch parameter as

δn ≡ na�θa� cos θa − ns�θs� cos θs : (3)

As seen in Eq. (2), the phase-matching condition can beadvantageously adjusted to a certain extent by changing theangles θs and θa.

We are interested here in the diffraction of the auxiliaryprobe wave of vacuum wavelength λa in the Raman–Nathregime [10]. This diffraction is a result of the cumulatedtransverse modulation of its phase upon propagation alongthe running refractive index grating that possesses the transversegrating vector kx � k0;s sin θs � �2π∕λs�ns�θs� sin θs. Providedthat the condition of Eq. (2) can be fulfilled, the ideal situationthat gives the maximum phase modulation is the one wherethe probe wave propagates along the z-axis (θa � 0), perpen-dicularly to the grating vector. Let ϕ�x; t; z � L� �ϕ0�L; t� cos�kxx� be the x-modulated phase shift experiencedby such an incident plane probe wave upon propagationthrough the above running grating of thickness L. This quan-tity can be determined by inserting the running gratingΔn�r ; t� into the wave equation; its amplitude is a solutionof the integral

ϕ0�L; t� �2π

λaΔn0�t�

ZL

0

cos

�2π

λsδnz

�e−αs zdz; (4)

where Δn0�t� is the amplitude of refractive index grating at thetime t corresponding to the entrance of the probe wave intothe sample (at z � 0). The projected amplitude absorptionconstant αs in the above equation is defined as αs≡αs∕�2 cos θs�. Equation (4) leads to

ϕ0�L; t� � �2π∕λa�Δn0�t�∕�α2s � ξ2�· �e−αsL�ξ sin ξL − αs cos ξL� � αs �; (5)

where the quantity ξ is defined as ξ ≡ 2πδn∕λs. If the absorp-tion for the signal wave can be neglected, Eq. (5) reduces to thesimpler expression

ϕ0�L; t� ��2π

λaΔn0�t�L

�sinc

�2π

λsδnL

�; (6)

Signal wave (ωs)

Probe wave (ωa)

Nonlineardetectioncrystal

Conventionaldetector

θ a,ext

θs,ext

x

y z

Fig. 1. Configuration for the detection of the signal wave via anauxiliary probe wave diffracted by a longitudinally running gratingin the nonlinear material. If the waves are pulsed, synchronizationof the pulses is required.

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where the cardinal sine factor disappears at perfect phasematching. The above amplitude is modulated at the frequencyωs of the signal wave according to Eq. (1). In the Raman–Nathdiffraction regime, the transversely phase-modulated probewave leads to several diffraction orders in the transmittedfar-field. The diffraction efficiency into the order n is givenby ηn�t� � J2n�ϕ0�L; t��, where Jn is the Bessel function ofthe first kind of order n. For small values of the phase shiftamplitude (ϕ0 ≪ 1) one obtains for the diffraction efficiencyinto the first order (order �1 or −1)

η�1�t� ≅�ϕ0

2

�2

≅�π

λaΔn0�t�L

�2

sinc2�2π

λsδnL

�: (7)

The second equality in the above expression holds only for neg-ligible absorption (αsL ≪ 1). Since Δn0 is proportional to theelectric field, the above diffraction efficiency is expected to scalelinearly with the intensity of the signal wave. Also, due to thequadratic term in Eq. (7), the diffracted wave intensity is tem-porally modulated in amplitude at twice the frequency ωs of thesignal wave.

Let us now discuss the optimum conditions with respect tothe thickness of the nonlinear sample. First we mention that ifthe absorption of the signal wave is very strong, the interactiondistance will be controlled by its effective absorption lengthLα � 1∕αs. In this case, the use of thick samples is possibleprovided that Lα does not contradict any of the argumentsgiven below. By considering now the case of weak absorption,we see from Eq. (7) that the diffraction efficiency falls to half itsmaximum value if the argument of the sinc2 function in Eq. (7)equals 1.39. Therefore, for a given mismatch δn the gratingthickness should not exceed about Lmax ≅ 0.2215λs∕δn, whichis of the order of the signal wave wavelength, unless the mis-match parameter δn is significantly smaller than 0.2. However,even in the case where the phase-matching condition Eq. (2) iswell satisfied, the thickness L should not be excessively large.This is to avoid that the diffraction process gets into the Braggregime (thick grating). Indeed, satisfying simultaneously theBragg condition and the relation Eq. (2) would be very difficultto achieve. The condition to be satisfied for the Raman–Nathdiffraction regime was given in [11,12] and reads for our case as

Δn02n

k2xL2 � 2π2Δn0n

L2

Λ2 ≤ 1; (8)

where n is the average refractive index seen by the probe waveand Λ � 2π∕kx is the grating periodicity. It is unlikely that theratio Δn0∕n will much exceed a value of the order of 10−5, sothat the ratio L∕Λ can be up to an order of about 100 accordingto this criterion.

The above limitations on the thickness of the grating holdsfor perpendicular incidence of the probe wave (θa � 0). In thecase of oblique incidence, even under phase matching, the am-plitude of phase modulation ϕ0�L; t� is decreased due to thefact that the phase corrugation accumulated by the probe waveis spatially averaged as a result of propagation in a region withdifferent (and eventually opposite) refractive index contrasts.This issue was already recognized in the original work byRaman and Nath [10]. A reasonable condition to avoid thiseffect may be given by requiring that the side drift of the probewave should not exceed about half a grating period over the

propagation distance through the sample, that is tan θa <Λ∕2L � π∕�kxL� or, equivalently, L < Λ∕�2 tan θa�. Underphase matching, this gives the strongest limitation on the usablegrating thickness already for an internal angle θa of a few tenthsof a degree. Note, however, that secondary (but weaker)maxima of the cumulated phase corrugation are found for spe-cific angles θa exceeding the above limit and corresponding tothe conditions j tan θaj � �3� 2p�Λ∕2L, where p is a positiveinteger. This relaxes somehow the last inequality if the use oflarger probe wave angles is needed for fulfilling Eq. (2). Theamplitude of the phase modulation ϕ0�L; t� is reduced byroughly a factor of 5 for p � 0 with respect to the case ofnormal incidence (θa � 0), and by nearly a factor of 10for p � 1.

3. SINGLE SIGNAL BEAM CONFIGURATION

In certain wavelength ranges, the use of the beam splitter for thesignal wave depicted in Fig. 1 may be impractical. A more com-pact alternative approach is the use of a Lloyd-type interferom-eter, as shown in Fig. 2. In this case only one signal wave isdirected to the nonlinear optical detection sample and the sec-ond wave is created by a mirror in contact with the sample and/or by total reflection at its side surface. The probe wave prop-agates parallel (or nearly parallel) and very close to this side sur-face. If the signal wave enters through the front surface[Fig. 2(a)], the phase-matching condition Eq. (2) can be ful-filled for θa � 0 provided that the refractive index ns of thesignal wave is in the range

na ≤ ns ≤ffiffiffiffiffiffiffiffiffiffiffiffiffin2a � 1

p; (9)

as limited by the refraction at the entrance surface. This rangeholds also for the case of the two-wave interference of Fig. 1 andis depicted in Fig. 3 as the violet shadowed area. Due to thelimitation of the range of ns in Eq. (9), the front surface con-figuration of Fig. 2(a) is suitable for materials for which the re-fractive indices at the frequencies ωa and ωs do not differ toomuch, which may be the case for several organic crystals evenifωs is in the THz range. If the difference of the refractive indicesns and na is larger, one may still fulfill Eq. (2) for θa � 0 if thesignal wave is incident through a wedged side surface, as shownin Fig. 2(b). In this case, the allowed values for the refractiveindex ns depend on the wedge angle β and we haveffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

n2a − 2na cos β� 1p

sin β≤ ns ≤

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffin2a � 2na cos β� 1

psin β

: (10)

Fig. 2. Compact Lloyd-type interferometer design with a single in-cident signal wave. In (a) the signal wave enters through the front sur-face of the nonlinear detection crystal (NDC); in (b) it enters througha wedged side surface. M, optional mirror; Det, conventional detector.

1080 Vol. 32, No. 6 / June 2015 / Journal of the Optical Society of America B Research Article

The corresponding range for ns is depicted in Fig. 3 as theorange shadowed area. This approach may allow one to employnonlinear materials for which the dispersion of the refractiveindices between the frequencies ωa and ωs is rather large, asexpected for most inorganic crystals. Note that the inequalityEqs. (9) and (10) assume ordinary (angle-independent)refractive indices for the signal wave. An extension to the caseof an extraordinary polarization is straightforward. Notealso that Snell’s law prevents the possibility to fulfill Eq. (2)under side illumination of a rectangular nonwedgedsample (β � 0).

The wedged configuration of Fig. 2(b) requires a minimumpropagation distance of the signal wave in the sample beforereaching the interaction region. If the wedge angle β is nottoo large, this distance, corresponding to the average samplewidth, may be kept below 1 mm. Nevertheless, in the casewhere the absorption constant αs is such that the signal waveis already strongly depleted over such a distance, the two-beamconfiguration of Fig. 1 or the front surface one-beam configu-ration of Fig. 2(a) have to be preferred.

4. APPLICATION EXAMPLE

Even though the theoretical treatment does not depend on thevalues of the frequencies ωs and ωa, we illustrate the concept bya possible application related to detection of monochromaticTHz radiation, in a spectral range where room temperature de-tection is generally difficult. As a nonlinear material we choosethe example of the crystalline organic salt DSTMS [13], whichhas attracted attention recently as one of the best organic THzwave generators [14,15]. Generally, due to their strong electro-optic response which is predominantly electronic in nature,noncentrosymmetric organic crystals are specially suited fordetecting fast-varying electric fields.

The linear and nonlinear optical properties of DSTMS inthe visible and near infrared range were described in [16].In the THz range the refractive index n1 for polarization direc-tion along the polar axis of the crystal was measured in [14]between 0.4 and 3.5 THz. Figure 4 presents an extensionof these measurements up to a frequency of 11 THz, as ob-tained by THz time-domain spectroscopy [17] using a broad-band THz spectrometer (TeraKit, Rainbow Photonics) basedon optical rectification of femtosecond pulses. The frequencydown-conversion from the near infrared to the THz range isphase-matched if the group velocity in the near infrared equalsthe phase velocity at the THz frequency. This is true both foroptical rectification of ultrashort (femtosecond) pulses as well as(in the first approximation) for difference frequency generationof longer quasi-monochromatic infrared pulses. In the case ofDSTMS, as seen in Fig. 4, the group refractive index ng at thewavelength of approximately 1350 nm (dotted curve in theupper graph) nearly equals the refractive index at 4 THz, wherealso a local minimum of absorption is observed. Therefore, forour example we consider phase-matched monochromatic radi-ation at about 4 THz frequency generated by difference fre-quency generation in DSTMS out of two waves near the1350 nm wavelength. Such a process may be realized eitherby using two distinct lasers as pumps, dual wavelength opticalparametric oscillators (OPO) [19], or compact dual frequencysolid-state or fiber laser sources [20–23]. We will now check

Fig. 3. Accessible range of values of the refractive index ns that cansatisfy the phase-matching condition Eq. (2). The orange-shaded areashows the allowed ns values versus the wedge angle β for the case ofFig. 2(b) and na � 2, θa � 0. The violet-shaded area shows theallowed values for the same parameters when entering through thefront surface of the crystal, as in Figs. 1 and 2(a).

111098765432100

10

20

30

400 1 2 3 4 5 6 7 8 9 10 11

1200 1300 1400 1500 1600

2.0

2.1

2.2

2.3

2.4

2.5

2.6

Ref

ract

ive

inde

x n

1

Frequency [THz]

Abs

orpt

ion

cons

tant

[mm

−1]

Wavelength [nm]

ngRef. [14]200 µm520 µmLorentz

Fig. 4. Terahertz refractive index and absorption of DSTMS crystalsfor polarization parallel to the polar a-axis. The red crosses representthe measurements up to 3.5 THz from Ref. [14], while the opencircles and the full squares are the new measurements up to 11 THzfor a 200 μm and a 520 μm thick crystal sample, respectively. Onlydata within the dynamic range of our system are included. The solidcurve is according to the Lorenz model like in Ref. [18], where sevenLorentz oscillators are considered. For comparison, the thick dottedcurve in the upper graph gives the group refractive index ng at nearinfrared wavelengths (upper scale), as given in Ref. [16].

Research Article Vol. 32, No. 6 / June 2015 / Journal of the Optical Society of America B 1081

whether the THz radiation can in principle be detected with theapproach presented in this work by recycling part of one of thenear infrared pump waves as the auxiliary probe wave at the de-tection stage, which gives a potentially compact ensemble.

Let us therefore take νs � ωs∕2π � 4 THz for the signalfrequency and νa � ωa∕2π � 222 THz (λa � 1350 nm)for the probe frequency. Inspection of the situation forDSTMS leads to an optimum configuration if both signaland probe waves are polarized along the (vertically oriented)polar a-axis of the crystal, which permits us to activate the larg-est high-frequency electro-optic coefficient r111 ≈ 37 pm∕V ofDSTMS that was evaluated for the wavelength of 1.9 μm [16].The refractive indices needed to determine the optimum con-figuration are therefore n1�ωa� � 2.091 [16] and n1�ωs� ≅2.24 (Fig. 4). Since n1�ωs� > n1�ωa� and the inequality Eq. (9)holds, the phase-matching condition of Eq. (2) can be fulfilledhere for the ideal case where the IR probe wave is normal to thegrating vector and all waves enter through the front surface.This is shown in Fig. 5(a), which shows the expected normal-

ized diffraction efficiency as a function of the internal angle θsof the incident THz waves, under the condition θa � 0. It isseen that the choice of the phase-matching angle θs is not toocritical, since the curves show a rather broad angular aperture(about 5 deg full width at half-maximum for a sample withL � 500 μm). The thick red and blue curves in Fig. 5(a) aredetermined by explicitly considering the effect of the signalwave absorption (αs � 5 mm−1) on the output phase corruga-tion amplitude in Eq. (5). For comparison, the case of no ab-sorption is also shown (thin green curve) and gives the referencefor the diffraction efficiency normalization. Figure 5(b) com-pares the angular rocking curve for L � 250 μm and νs �4 THz of case (a) with the corresponding curves for the sameprobe wavelength but other THz frequencies, i.e., νs � 2 THz(n1 ≅ 2.22, αs � 4 mm−1) and νs � 6 THz (n1 ≅ 2.22, αs �15 mm−1). This shows that a nearly identical geometricalconfiguration of the detector (moderate change of θs) can beemployed to detect the coherent radiation in a rather broadspectral range, provided that the dispersion of the relevant re-fractive indices in this range is not too strong, which is the casefor the example of DSTMS.

It is useful to get some order of magnitudes for the expectedeffects. For the example depicted in Fig. 5(a), the optimumgrating period is Λ � 93 μm and the external diffraction angleof the �1 order with respect to the transmitted zeroth order isλa∕Λ ≈ 14 mrad. If we assume that the signal wave has 1 Wpeak power (e.g., 10 ns long pulses with roughly 10 nJ energy)concentrated over a surface of 1 mm2 we obtain an electric fieldof the order of 180 V/cm and a maximum refractive indexmodulation Δn0 ≈ 3 · 10−6 in the case of DSTMS. For L �500 μm and considering the THz absorption this leads to ϕ0 ≈0.004 in Eq. (5) and a diffraction efficiency of η�1 ≈ 0.0004%into the first Raman–Nath order. Even though this value mayseem small, the diffraction spots appear over an otherwise darkbackground in the absence of scattering and should therefore beeasy to measure by appropriate conventional detectors respond-ing at the probe wave frequency. It appears reasonable that adiffraction efficiency at least two orders of magnitude lowerwould be still detectable, corresponding to electric fields of afew tens of V/cm or a peak intensity of a few mW∕mm2. Notealso that for the above parameters we have 2π2�Δn0∕n�L2∕Λ2 ≈ 0.001 ≤ 1 and we are therefore well inside the Raman–Nath diffraction regime. In general, the major source of noisefor the detection is likely to be given by linearly scattered lightat the probe wavelength λa, which magnitude depends on thecrystal and surface quality, as well as on the wavelength λa andthe expected diffraction angle. If necessary, an additional slowexternal modulation of the signal wave might be used to im-prove the discrimination against this scattering using lock-intechniques.

The above application example can be extended to variousother nonlinear materials as detectors. For instance, onemight use instead of DSTMS the well-known and closelyrelated DAST crystals, 4-N,N-dymethylamino-4 0-N 0-methyl-stilbazolium tosylate [24–26]. Let us consider the THz fre-quency νs � 2 THz and the probe wavelength λa �1.5 μm, which in DAST represent a nearly phase-matched pairat the generation stage [25]. With the IR and THz refractive

5 10 15 20 25 30 35

0.00

0.25

0.50

0.75

1.00

Internal THz angle θs [deg]

Nor

mal

ized

diff

ract

ion

effic

ienc

y

L = 500 µmL = 250 µm

(a)

5 10 15 20 25 30 35

0.00

0.05

0.10

0.15

Internal THz angle θs [deg]

Nor

mal

ized

diff

ract

ion

effic

ienc

y L = 250 µm (b)

2 THz

4 THz

6 THz

L = 500 µm(no absorption)

Fig. 5. (a) Expected dependence of the normalized diffractionefficiency η�1�t� � �ϕ0∕2�2 into the first Raman–Nath order on theinteraction angle θs . The cases of a DSTMS detection crystal withνs � 4 THz, λa � 1350 nm, and grating thickness of 500 μm (solidlines) or 250 μm (dotted line) are shown. The thick lines are for anabsorption αs � 5 mm−1, while the green thin line gives the case withno signal absorption. (b) Corresponding angular dependences for thefrequencies of 2 THz (dashed line), 4 THz (dotted line), and 6 THz(dashed–dotted line) and 250 μm thickness (λa � 1350 nm, θa � 0).

1082 Vol. 32, No. 6 / June 2015 / Journal of the Optical Society of America B Research Article

indices given in [24] and [26] one obtains an optimum con-figuration of the type of Figs 1 and 2(a) with an internal angleθs � 19.7 deg . For the same intensity as in the example aboveone would expect here a diffraction efficiency η�1 ≈ 0.0009%in a 500 μm thick sample, which is of the same order of mag-nitude as the one in the DSTMS example.

5. CONCLUSIONS

We have proposed and analyzed a method to detect coherentradiation by means of diffraction on the induced refractive in-dex grating whose phase is velocity-matched to the one of aneasily detectable auxiliary probe wave. The electric field gratingthat drives the effect may be obtained by a standard two-waveinterferometer or by a compact Lloyd-type interferometer. Theexpected diffracted probe wave is potentially obtained over ablack background. Due to the nearly instantaneous electro-optic effect, the response time of the measurement will belimited only by the speed of the conventional detector usedto observe the diffraction. Conversely, the technique proposedhere may be used for material characterization, for instance forthe determination of the refractive indices at the signal fre-quency ωs under knowledge of those at the frequency ωa.As an illustration, an application example related to THz wavedetection using electro-optic organic crystals was given. In thiscontext it is expected that the proposed technique could be ableto detect radiation electric fields as low as a few tens of V/cm orpeak intensities as low as a few mW∕mm2 in a room temper-ature environment.

European Commission (EC) (284792).

We thank M. Zgonik and R. Cudney for very usefuldiscussions.

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Research Article Vol. 32, No. 6 / June 2015 / Journal of the Optical Society of America B 1083