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J. Opt. Soc. Am. B / Vol. 23, No. 9 / September 2006
Schneider et al.
Generation of terahertz pulses through optical rectification in organic DAST crystals: theory and experiment Arno Schneider, Max Neis, Marcel Stillhart, Blanca Ruiz, Rizwan U. A. Khan,* and Peter Günter Nonlinear Optics Laboratory, Institute of Quantum Electronics, ETH Zurich, 8093 Zurich, Switzerland Received February 3, 2006; revised April 6, 2006; accepted April 7, 2006; posted April 24, 2006 (Doc. ID 67685) We present a combined theoretical and experimental investigation of the generation of few-cycle terahertz (THz) pulses via the nonlinear effect of optical rectification and of their coherent detection via electro-optic sampling. The effects of dispersive velocity matching, absorption of the optical and the THz waves, crystal thickness, pulse diameter, pump pulse duration, and two-photon absorption are discussed. The theoretical calculations are compared with the measured spectra of THz pulses that have been generated and detected in crystals of the highly nonlinear organic salt 4-N , N-dimethylamino-4⬘-N⬘-methyl stilbazolium tosylate (DAST). The results are found to be in agreement with the theory. By the selection of the optical pump wavelength between 700 and 1600 nm, we achieved several maxima of the overall generation and detection efficiency in the spectral range between 0.4 and 6.7 THz, with an optimum at 2 THz generated with 1500 nm laser pulses. © 2006 Optical Society of America OCIS codes: 160.4330, 160.4890, 190.2620, 230.4320, 260.3090.
1. INTRODUCTION Far-infrared pulses of electromagnetic radiation with a spectral content between 0.1 and 10 THz are commonly called terahertz or THz pulses. Several breakthroughs in the past two decades have enabled applications of THz pulses in different areas such as imaging of biological tissue1 and the measurement of fundamental solid-state processes in semiconductor physics.2 Whereas for some applications, namely, THz absorption spectroscopy or spectroscopic imaging, tunable narrowband THz pulses with nanosecond durations may be favorable owing to their better spectral resolution,3 broadband THz pulses on a picosecond time scale offer additional benefits that are unique to this technique. The excellence of such broadband THz pulses is a phase-coherent detection technique that provides the inherent advantage of a time resolution that may be as short as a few tens of femtoseconds.4 Broadband THz pulses may be generated by several methods, which all employ femtosecond laser pulses, namely, using photoconductive switches,5,6 semiconductor surfaces,7 and optical rectification (OR) in nonlinear optical crystals. A frequently used technique for their coherent detection is electro-optic (EO) sampling,8,9 a process that is based on the interaction of an optical pulse with the THz wave in a nonlinear material. For an optimum performance of THz generation via OR and EO sampling, it is important that the THz and the optical pulses propagate through the crystal with the same velocity (velocity matching). This is achieved, e.g., within the inorganic semiconductor ZnTe when one uses laser pulses at a wavelength of 822 nm,10 i.e., within the tuning range of the widely used Ti: sapphire femtosecond lasers. This fac0740-3224/06/091822-14/$15.00
tor made ZnTe the material of choice for the generation of pulses with a broadband spectrum below a frequency of 3 THz. Recently, a number of groups have exploited various experimental schemes to achieve velocity matching, such as angle tuning,11 laser pulses with tilted pulse fronts,12 or periodically poled EO crystals.13 However, for future compact and cost-effective devices, simple experimental geometries (perpendicular incidence, as-grown nonlinear crystals) are desirable. The most straightforward parameter that may be varied to achieve velocity matching in this case is the optical wavelength . Owing to the ongoing progress in the telecommunications industry, stable short-pulsed lasers with ⯝ 1.55 m are becoming readily available. Nagai et al. recently demonstrated a THz generation and detection system using an erbium-doped fiber laser at an optical wavelength of 1.56 m and GaAs as the nonlinear optical and EO material.14 Owing to the relatively low EO coefficient of GaAs—three times lower than that of ZnTe—other materials may yield a higher THz output than that observed in this study. The organic crystalline salt DAST (4-N , Ndimethylamino-4⬘-N⬘-methyl stilbazolium tosylate) exhibits EO and nonlinear optical coefficients that are among the highest of all known materials.15 It was first shown to be an efficient emitter of THz pulses via OR as early as 1992,16 but little progress using this material for THz applications has been documented.17–19 However, the generation of nanosecond pulses of THz radiation by difference-frequency generation has been demonstrated in DAST.20,21 In this paper, we investigate DAST as a source of THz pulses through OR. The paper is organized as follows. In © 2006 Optical Society of America
Schneider et al.
Vol. 23, No. 9 / September 2006 / J. Opt. Soc. Am. B
Section 2, we provide a theoretical overview of OR and subsequent THz emission as well as of EO sampling. We also derive quantities, defined as effective lengths, that may serve as figures of merit as a function of THz frequency and optical wavelength. In Section 3, the linear and nonlinear optical properties of DAST are presented that are relevant for its use in THz applications. Moreover, we calculate the effective lengths for DAST. Section 4 describes the experimental setup. The results are presented and discussed in Section 5, where they are compared with the predictions from the preceding sections.
with 共2兲 and 共3兲 as the second- and third-order nonlinear susceptibilities, which are tensorial response functions. Higher-order terms are neglected. Causality requires that both 共2兲 and 共3兲 vanish if at least one of their arguments is smaller than zero. In an experimental realization, the nonlinear polarization is induced by the electric field of a short laser pulse. The 共3兲 term does not contribute to the THz generation and is therefore neglected. The Fourier transform of Eq. (6) then yields PNL,i共兲 = ⑀0
冕
共2兲 d⬘ijk 共 ; ⬘, − ⬘兲Ej共⬘兲Ek共 − ⬘兲.
R
2. THEORY
共7兲
A. Generation of a Terahertz Pulse 1. Nonlinear Wave Equation In an optically nonlinear medium, the electric displacement vector D in SI units is given by D = ⑀0E + PL + PNL ,
共1兲
with PL and PNL as the linear and nonlinear polarizations, respectively, and E as the electric field. In the time domain, PL共t兲 is coupled to E共t兲 by a response function, the linear susceptibility tensor 共t − 兲 = ⑀共t − 兲 − 1. In the frequency domain this reads as PL共兲 = ⑀0共兲 丢 E共兲
共2兲
ÛD共兲 = ⑀0⑀共兲 丢 E共兲 + PNL共兲,
共3兲
where 丢 denotes the tensor multiplication. The nonlinear polarization PNL will be discussed in Subsection 2.A.2. A nonmagnetic 共 = 1兲 medium with no free charges 共 = 0兲 is assumed. The current density j共兲 may be written in analogy to Eq. (2): j共兲 = 共兲 丢 E共兲,
共4兲
with 共兲 as the frequency-dependent conductivity tensor. From Maxwell’s equations,22 one can derive the nonlinear wave equation in the frequency domain: ⌬E共r, 兲 − ⵜ关ⵜ · E共r, 兲兴 + 20⑀0⑀共兲 丢 E共r, 兲 − i0共兲 丢
E共r, 兲 = − 20PNL共r, 兲.
共5兲
In a linear medium, i.e., in which PNL = 0, Eq. (5) describes the propagation, absorption, and diffraction of an electromagnetic wave. All operators are linear; thus any superposition of solutions will solve the equation. 2. Nonlinear Polarization The nonlinear polarization PNL共t兲 is induced by the electric field and can be expressed as the quadratic and higher terms of a Taylor expansion of the total polarization in the electric field: PNL共t兲 = ⑀0
冋冕冕
册
dt1dt2共2兲共t1,t2兲 丢 E共t − t1兲E共t − t2兲 关
R2
+
1823
冕冕冕
1
E共t兲 = 2 关E0共t兲exp共iot兲 + c.c.兴,
dt1dt2dt3共3兲共t1,t2,t3兲 丢 E共t − t1兲 共6兲
共8兲
where E0共t兲 is the time-varying complex amplitude and o is the optical carrier frequency. The complex conjugate is denoted by c.c. A Fourier transform yields the frequency spectrum 1
E共⬘兲 = 2 关E0共⬘ − o兲 + E0*共⬘ + o兲兴,
共9兲
which has to be inserted into Eq. (7). For a laser pulse length T that is long compared with one optical cycle, T 共2 / o兲, the bandwidth ⌬ of E0共兲 is smaller than o, and the integration in Eq. (7) runs only over 共−o − ⌬ , −o + ⌬兲 and 共o − ⌬ , o + ⌬兲. Therefore PNL共兲 is nonzero only in three distinct spectral ranges. Two of them are centered at ±2o and lead to second-harmonic generation (SHG). The third lying within ±2⌬ causes OR and is the source of the THz radiation. If o is not near a resonance frequency of the material under consideration, the dispersion of 共2兲共 ; ⬘ , − ⬘兲 is generally small; i.e., it does not change significantly when ⬘ is varied within ±共o − ⌬ , o + ⌬兲. Thus we introduce an effective susceptibility OR that is relevant for OR and, consequently, for THz generation:
OR共 ; o兲 ª 共2兲共 ; ⬘, − ⬘兲,
兩o − ⬘兩 o . 共10兲
With this definition we can evaluate the OR term in Eq. (7): 1
POR共兲 = 2 ⑀0OR共 ; o兲共E0 ⴱ E0*兲共兲.
共11兲
The asterisk denotes the convolution. Using the Fourier transform of the laser intensity I共兲 = 关⑀0n共o兲c兴 / 2共E0 * E0*兲共兲, we get POR共兲 =
R3
⫻E共t − t2兲E共t − t3兲 + ¯ 兴,
For simplicity, we assume a laser polarization along a main axis of the crystal (say, m) and only one element 共2兲 lmm of the nonlinear susceptibility tensor to be active, where l need not be different from m. In this case, the nonlinear polarization P is along the l axis, and Eq. (7) reduces to a scalar equation. The field of the laser pulse may be written as
OR共 ; o兲 n共o兲c
I共兲.
共12兲
Note that this description is equivalent to a calculation of difference-frequency generation within the spectral content of a finite laser pulse.
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Schneider et al.
I共,z兲 = I0共兲exp关i共ng/c兲z兴exp共− ␣oz兲.
3. Plane-Wave Approximation We consider a pump pulse propagating along the z direction, perpendicular to the optical and the nonlinear polarization, in the plane-wave approximation, i.e., I / x = I / y = 0. The nonlinear polarization in Eq. (5) gives rise to a rectified field at THz frequencies as well as to a field at the doubled optical frequency. The latter can safely be neglected, since the phase-matching conditions for SHG are generally not fulfilled.23 Furthermore, we assume that the generated THz field is weak compared with the optical pump field and does therefore not lead to cascaded nonlinear effects. Thus we take the nonlinear source term on the right-hand side of Eq. (5) to be zero for the fundamental optical frequency o. The propagation of the pump pulse is then not affected by nonlinear effects, and it can be treated as in a linear medium: I共t,z兲 = I0共t − z/vg兲exp共− ␣oz兲,
Group-velocity dispersion does not play a significant role in I共 , z兲 if pump pulses longer than 100 fs are used as in this paper. With Eqs. (12) and (15), the nonlinear wave equation (5) becomes
2E共兲 z
冏
n
冏
0
E共,z兲 = n共o兲
再冋
c ␣ T共 兲
2
冋
exp − i
册
冎
c2
册
− i 0 共 兲 E
OR共, o兲 n共o兲c
exp关− i共ng/c兲z兴exp共− ␣oz兲I0共兲,
0c n共兲
共兲.
共17兲
In the solution of Eq. (16) we consider only forwardtraveling waves, i.e., waves in the same direction as the pump pulse. With the boundary condition E共 , 0兲 = 0 and low THz absorption 关␣T共兲 nx共兲 / c兴, we find the solution:
共14兲
+ ␣o + i关n共兲 + ng兴
2n 2共 兲
␣ T共 兲 =
o
共 ; o兲 I 0共 兲
冋
where we used c2 = 共0⑀0兲−1 and n2共兲 = ⑀共兲. The THz absorption coefficient ␣T共兲 is related to the conductivity 共兲 by
The intensity spectrum I共兲 in Eq. (12) follows from Eq. (13):
OR
+
共16兲
共13兲
.
2
= − 2 0
with I0共t兲 as the temporal shape of the input pulse that effectively entered the crystal at z = 0, i.e., after Fresnel reflections at the front surface. ␣o denotes the absorption coefficient at the frequency o, and vg共o兲 = c / ng共o兲 is the optical group velocity with ng共o兲 as the optical group index: ng共o兲 = n共o兲 + o
共15兲
n共兲 c
册 冋
z exp −
␣ T共 兲 2
␣ T共 兲 2
册 冉
z − exp − i
ng c
冊
z exp共− ␣oz兲 .
共18兲
− ␣o + i 关n共兲 − ng兴 c
The second fraction on the right-hand side of Eq. (18) has the unit of a length. Its maximum absolute value of z is reached for zero absorption 关␣T共兲 = ␣o = 0兴 and equal propagation velocities of the THz wave and probe pulse [velocity matching, n共兲 = ng]. Therefore we define its absolute value as the effective generation length Lgen共 , z兲:
Lgen共,z兲 =
冢
再冋
exp关− ␣T共兲z兴 + exp共− 2␣oz兲 − 2 exp −
冋
␣ T共 兲 2
册 冉冊 2
− ␣o
The z dependence of the electric field in Eq. (18) is then exclusively given by Lgen共 , z兲, which depends strongly on the material’s dispersion through the 关n共兲 − ng兴 terms. A study of the length and pump wavelength dependence of THz generation in ZnTe was recently presented by van der Valk et al.24 The denominator in the first fraction of Eq. (18) varies more slowly with the dispersion than Lgen, since it con-
␣ T共 兲
+
c
2
册冎 再
+ ␣o z cos
2
关n共兲 − ng兴2
c
关n共兲 − ng兴z
冎
冣
1/2
.
共19兲
tains the sum n共兲 + ng instead of the difference; the contribution of ␣T共兲 and ␣o is negligible for moderate absorption. Hence the dependence of the emitted field E共 , z兲 on the linear properties (refractive indices, absorption coefficients) in both THz and optical spectral ranges is essentially contained in Lgen共 , z兲. The function I0共兲 of a Gaussian pulse with the intensity profile I0共t兲 is given by
Schneider et al.
Vol. 23, No. 9 / September 2006 / J. Opt. Soc. Am. B
冉 冊
I0共t兲 = Im exp −
t2
2
2
冉 冊
Û I0共兲 = Im exp −
2 2 2
E共,l兲 =
,
冕
l
E 0共 兲 z
0
共20兲 with Im as the peak amplitude. The bandwidth ⌬ in the angular frequency, where I0共兲 drops to 1 / e2 of its maximum value, is given by 2 / . In terms of the frequency = / 共2兲, we receive for a typical value of = 64 fs—equivalent to a full width at half-maximum (FWHM) pulse length of 150 fs—a bandwidth of ⌬ = 5.0 THz. 4. Finite Pump-Pulse Diameter In the following, we discuss the change in the emitted spectrum that arises for a finite pump pulse diameter due to diffraction. Its profile is assumed to be radially symmetric with a Gaussian profile:
冋
I共,t兲 = Ic共t兲exp −
2 2s2共z兲
册
W0 W共兲
冋
exp −
共21兲
,
2 W 2共 兲
册
,
冑
1+
2 z02共兲
.
2c
W02 =
c
s2 .
c
共23兲
共24兲
In the case of velocity matching and negligible absorption in both optical and THz spectral ranges, the total field component at the THz frequency on the beam axis 共 = 0兲 at the end of the crystal at z = l is composed of the contributions emitted at all positions z within the crystal:
1 1+
c2共l − z兲2
dz
2s 4
冉 冊 cl
s2 arcsinh
s 2
共25兲
.
The factor E0 / z contains the material parameter OR and the pump intensity Ic共兲. In the limit of a plane wave, or s2 → ⬁, we get E共,l兲 →
E 0共 兲 z
共26兲
l,
which has to be identical to the result in Eq. (18) with Lgen = l; thus
冏 冏 E 0共 兲
=
0OR共 ; o兲 2n共o兲ng
I 0共 兲 .
共27兲
Combining Eq. (25) with Eq. (27), we get the spectrum of a THz pulse generated by a Gaussian pump beam for zero absorption and velocity mismatch: 兩E共,l兲兩 =
0OR共 ; o兲 2cn共o兲ng
冉 冊
s2I0共兲2 arcsinh
cl
s 2
. 共28兲
The difference between Eq. (28) and the plane-wave solution of Eq. (18) is significant only for low frequencies 共 ⬍ cl / s2兲. Assuming an unfocused optical beam 共s ⱖ 0.25 mm兲 and a typical crystal length 共l ⱕ 1 mm兲, one finds that the plane-wave result represents a valid approximation for frequencies above 1 THz. 5. Influence of Two-Photon Absorption For a high conversion efficiency and to prevent damage to the nonlinear crystal, one usually chooses the pump wavelength within the transparency range of the material, i.e., ␣o ⯝ 0. However, the energy of the pump pulse may be decreased through two-photon absorption (TPA). In the absence of linear absorption, the intensity is I共t,z兲 =
The frequency-dependent Rayleigh length is given by
z 0共 兲 =
z
共22兲
with W0 = 冑2s as the initial beam waist and the momentary beam size as
W共兲 = W0
E 0共 兲
z
with Ic共t兲 as the time-dependent intensity in the center of the beam, as the radial coordinate, and s共z兲 as the beam size parameter. For a crystal length l s2共z兲 / , i.e., much shorter than the Rayleigh length of the optical beam, its diffraction is negligible, and s is approximately constant within the crystal. Typical values 共 ⬃ 1 m , l ⬃ 0.5 mm兲 lead to s 23 m; i.e., the pump beam must not be focused onto the nonlinear crystal. However, diffraction must not be neglected for the THz radiation due to its wavelength that is 2 orders of magnitude longer than that of the optical pulse. The initial Gaussian distribution of the THz field, which is proportional to the pump intensity distribution in Eq. (21), will remain Gaussian after a propagation distance with a change in beam size and peak amplitude:
E共, 兲 = E0
=
冑
1825
Ii共t − z/vg兲 1 + ␣2Ii共t − z/vg兲z
,
共29兲
where Ii共t兲 is the intensity at z = 0 and ␣2 is the TPA coefficient defined by I / z = −␣2I2. The inhomogeneous relative intensity change in Eq. (29) leads to a flattening of the pump pulse intensity and, consequently, to a frequency-dependent change in I共兲. Thus the THz spectrum in the presence of TPA will be different from that with linear absorption. For a quantitative estimation of this effect, we make a Taylor expansion of Eq. (29) in ␣2Iiz. Assuming a Gaussian pulse as in Eq. (20), the first terms of the intensity change ⌬I共t , z兲 = Ii共t兲 − I共t , z兲 are
冋
⌬I共t,z兲 = ␣2I02 exp −
冋
⫻exp −
共t − z/vg兲2
2
3共t − z/vg兲2 22
册
册
z − ␣22I03
z2 .
共30兲
Both terms in Eq. (30) have a Gaussian shape with dura-
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Schneider et al.
tions that are shorter than the original pulse Ii共t兲 by factors of 冑2 and 冑3. In the frequency domain, their bandwidths are consequently widened by the same factor compared with Eq. (20). Hence the flattening in time rather suppresses high frequencies. Since the emission of the THz field E共兲 is proportional to the pump intensity I共兲, the total emitted THz spectrum is effectively shifted toward lower frequencies by TPA. From Eq. (30), one can see that this frequency shift increases with I0 and thus with the energy of the pump pulse. TPA also leads to a spatial flattening of the pump pulse intensity. In analogy to the above, the first two correction terms have diameters of s / 冑2 and s / 冑3 if I共兲 is a Gaussian as defined in Eq. (21). This affects the THz spectrum through changes in diffraction as in Eq. (28), mainly for low frequencies. However, the estimation there is still valid; thus we expect no significant influence of the spatial pump pulse flattening on the emitted THz spectrum, provided that the pump beam is unfocused. We will show in Section 5 that in DAST crystals the intensity-dependent changes in the emitted THz spectrum due to TPA are small (see Subsection 5.D).
冕冕 L
⌬共td兲 = ko
0
⌬n共z,t兲A关t − 共ngz/c兲 − td兴dtdz, 共31兲
R
where t is the time at which the pulse entered the crystal, ko = o / c is the wavenumber of the probe beam with frequency o, and ⌬n共z,t兲 = −
n3o 2
rETHz共z,t兲,
共32兲
with r as the active Pockels coefficient for the given polarizations of ETHz and the probe pulse. In the case that more than one element of the r tensor is active, the individual phase shifts have to be added. The spectrum ⌬共兲 of the induced phase shift is ⌬共兲 = − ko
n3o 2
冕 冕冕 L
r
ETHz共z,t兲A关t − 共ngz/c兲 − td兴
R2
0
⫻exp共itd兲dtdtddz = − ko
n3o 2
rA共兲
冕
冉
L
ETHz共z, 兲exp − i
0
ng c
冊
z dz. 共33兲
Since the THz pulse undergoes linear absorption and dispersion within the crystal, its spectrum at the position z is given by
B. Propagation Effects during Detection For the coherent detection of the THz transients, we employed conventional EO sampling9,10 and THz-induced lensing.25 In both methods, the THz electric field induces a change ⌬ in the phase of a copropagating optical probe pulse (or one of its polarization components) through the linear EO effect (Pockels effect). A measurement of ⌬ as a function of the time delay td between the THz and the probe pulse allows the determination of ETHz共t兲. The impact of dispersive THz propagation within the detection crystal on the phase shift has been described elsewhere.26,27 We resume the results by using a comprehensive, slightly modified approach, with an emphasis on the dependence of ⌬ on the crystal length. Consider a probe pulse with the normalized amplitude A共t兲 propagating through the crystal in the same way as in Eq. (13). The group velocity vg is taken to be unaffected by the THz field, since the induced relative index change ⌬n / no is small. The total phase shift ⌬ that is averaged over the probe pulse duration and accumulated over the propagation through the EO detection crystal, which ranges from z = 0 to z = L, is given by
Ldet共兲 =
冢
冋
ETHz共,z兲 = Ei共兲exp −
⌬共兲 = ko
n3o 2
rA共兲
冋
1 − exp − ⫻
−
n共兲 c
册
z , 共34兲
册 再
␣共兲
z exp i 关n共兲 − ng兴z 2 c
␣共兲 2
冎
Ei共兲.
− i 关n共兲 − ng兴 c 共35兲
Defining an effective detection length Ldet共兲 in analogy to the effective generation length Lgen共兲 in Eq. (19),
冋
␣ T共 兲 2
冋 册 冉冊 2
2
册 冋
z exp i
where Ei共兲 is the complex spectrum of the THz electric field at the crystal entrance after Fresnel reflection. The phase shift is thus
exp关− ␣T共兲z兴 + 1 − 2 exp −
␣ T共 兲
␣共兲
2
+
c
2
册 再
z cos
c
关n共兲 − ng兴
2
关n共兲 − ng兴z
冎
冣
1/2
,
共36兲
Schneider et al.
Vol. 23, No. 9 / September 2006 / J. Opt. Soc. Am. B
we may write the absolute value of the phase shift ⌬共兲 as 兩⌬共兲兩 = ko
n3o 2
rA共兲Ldet共兲Ei共兲.
共37兲
The spectrum that is measured by EO sampling is thus proportional to that of the THz pulse, filtered by two multiplicative factors A共兲 and Ldet共兲. The function A共兲 is solely determined by the temporal shape and the length of the probe pulse. For a Gaussian pulse with A共t兲 ⬀ I共t兲 as in Eq. (20) with = 64 fs, corresponding to a pulse length of 150 fs (FWHM), the function A共兲 reduces the detection sensitivity by a factor of 0.13 for a frequency = / 2 = 5 THz, by 0.02 for = 7 THz, and by 3 ⫻ 10−4 for = 10 THz. The second filtering function is the effective detection length Ldet. It is identical to the effective generation length Lgen共兲 in the case of vanishing optical absorption ␣o. Hence the detection sensitivity does not depend on ␣o, provided that the transmitted probe pulse energy remains high enough to be measured. In the time domain, the filtering effect of Ldet共兲 corresponds to a distortion of the THz waveform.26 C. Optimization of the Crystal Length In this subsection we will derive an expression for the calculation of the optimum crystal length lo for the generation and detection of a THz pulse in the presence of THz absorption. The value Lmax共兲 ª Lgen共 , z = lo兲 is then a measure for the highest possible conversion efficiency from optical intensity I0共兲 to the THz electric field E共兲. In the absence of absorption, the conversion efficiency of a nonlinear process reaches a maximum after the coherence length lc = / ⌬k, where ⌬k is the difference of the wave vectors that are involved in the nonlinear process. In collinear geometries, ⌬k is proportional to the index difference.23 Likewise, the effective lengths Lgen and Ldet defined in the previous subsections are at maximum after l c, lo = lc =
c 关n共兲 − ng兴
共␣o = ␣T = 0兲,
共38兲
and we find Lmax = 共2 / 兲lo. Note that in this case the coherence length depends on the optical group index ng [Eq. (14)] rather than on the optical refractive index n.28 For nonvanishing THz absorption, the optimum length lo cannot be calculated analytically. However, an approximative solution may be found by a Taylor expansion of Lgen共z兲 up to second order around z = lc:
lo共␣T,lc兲 = lc −
2
␣T
冢冦
␣2T 4
1+2
冉冊 lc
冧冣 1/2
关1 + exp共− ␣Tlc/2兲兴 2
−
␣2T 4
−1 exp共− ␣Tlc/2兲
.
共39兲
We assumed ␣o = 0; the approximation is thus valid for both Lgen and Ldet. We evaluated Eq. (19) also numerically to find lo and Lmax, and we compared these values with those from Eq. (39). Although the deviation in lo ranges
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up to 10% for ␣T 1 / lc [i.e., if the limitation of the effective length Lgen in Eq. (19) is due to absorption rather than due to velocity mismatch], Lmax is correct within 0.1%, since the function Lgen共z兲 is flat in the proximity of the extremum. Hence Lmax may be found by one’s inserting z = lo from Eq. (39) into Eq. (19). For velocity matching 关ng = n共兲兴, lo becomes infinite; however, Lgen converges to 2 / ␣ T. For determining the best thickness of a generation or detection crystal or both, Lgen ought to be large over the range of frequencies that are emitted locally [see, e.g., Eq. (20)]. Therefore we calculate for which crystal lengths z the effective generation length Lgen共z兲 is close to its maximum value Lmax. In the case of velocity matching, Lgen共z兲 is above 0.8Lmax for z ⬎ 1.61Lmax. For ␣ = 0, the 80% limit is reached within the interval 0.93Lmax ⬍ z ⬍ 2.21Lmax, with the maximum at 共 / 2兲Lmax. Generally, i.e., for velocity mismatch and nonvanishing absorption, one sees that generation and detection efficiency of more than 80% of its theoretical maximum is achieved if the crystal length lies between 1.6 and 2.2 times Lmax, which is calculated from absorption and refractive index data with Eqs. (19) and (39).
3. ORGANIC NONLINEAR OPTICAL CRYSTAL DAST A. Crystal Preparation The molecular crystal DAST (4-N , N-dimethylamino-4⬘ -N⬘-methyl stilbazolium tosylate) is known for its large nonlinear and EO coefficients.15,29 For our experiments, single crystals were grown in-house from a supersaturated solution in methanol. After a growth period of one to three days at a constant temperature in the metastable zone,30 we obtained c plates that were between 0.1 and 1 mm in width and possessed an area of between 4 and 10 mm2. The crystals were removed from the solution and polished on both sides, i.e., on the ab planes, using standard techniques to a surface roughness of less than 100 nm root-mean-square, measured by atomic force microscopy. Optical microscopy between crossed polarizers showed the crystals to be single crystalline. B. Linear Optical Properties The refractive indices of DAST single crystals in the near infrared15,31 show extraordinarily high birefringence (n1 − n2 = 0.8 at 800 nm, where n1 and n2 correspond to polarizations along the dielectric x1 and x2 axes, respectively) as can be expected from the highly aligned arrangement of the chromophores. The dispersion in the transparency range (700 to 1650 nm29), where the absorption coefficient ␣o is below 2 cm−1 for all three polarizations, is governed by single electronic resonances between 500 and 540 nm, whose absorption has also been observed experimentally.32 This dispersion leads to a large variation of the optical group index ng,1 from 3.38 at 800 nm to 2.24 at 1600 nm, which is relevant for the strong wavelength dependence of THz generation. By THz time-domain spectroscopy (THz-TDS), Walther et al. observed a strong resonance at 1.1 THz, which is attributed to a transverse optical (TO) phonon, the anion– cation vibration. It dominates the linear properties in the
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␣共兲 = 2K2
Fig. 1. Refractive index n1 of DAST between 1.3 and 4.1 THz. Circles, measured data; solid curve, calculation of n1 based on the harmonic-oscillator model with two oscillators at 1.10 and 3.05 THz.
Fig. 2. Absorption coefficient ␣1 of DAST. Circles, measured data; solid curve, sum of two Lorentzian functions centered at 1.1 and 3.05 THz.
THz range for x1 polarization,33 resulting in a dispersion of n1 from 2.3 to 3.0 below the resonance frequency and from 1.8 to 2.3 above it and in an absorption constant ␣1 above 100 cm−1 between 0.9 and 1.3 THz. However, the published data are limited to below 3.0 THz. Therefore we also carried out a THz-TDS experiment and measured n1 and ␣1 up to 4.1 THz (Figs. 1 and 2). Below the resonance frequency of 1.1 THz, our data agree well with those previously published. However, between 1.4 and 2.8 THz, we measured values n1 that were about 0.03 higher than those of Walther et al. The systematic experimental error of ⌬n = 0.01 is the same in both measurements and is due to the uncertainty in the sample thickness (250± 2 m in our experiment); thus the discrepancy between our and the published data cannot be fully explained. An additional feature that manifested itself as a kink in n1 at 3 THz was observed. We believe that this is due to a second phonon resonance that also leads to the absorption line that Taniuchi et al.21 reported at 3.1 THz. It is indirectly confirmed by the decrease in the amplitude in a typical THz spectrum from DAST (see Subsection 5.A). The curve plotted in Fig. 1 is the sum of two Lorentzian line shapes as it follows from the harmonic-oscillator model:
n共兲 = n⬁ +
兺
2 i=1,2 共i
ai共i2 − 2兲 − 兲 + 2 2
4␥i22
.
2 ␥ ia i
2 i=1,2 共i
− 2兲2 + 4␥i22
.
共41兲
The parameters of the two oscillators were fitted simultaneously to the measured values of n共兲 (Fig. 1) and ␣共兲 (Fig. 2); their values are given in Table 1. The empirical conversion factor K = 17.9⫻ 1012 s−1 mm−1 was fitted to the measured index and absorption data. The effect of an additional resonance at 5.2 THz18,21 on the refractive index could not yet be determined experimentally. However, we will show evidence in Section 5 that the value of n1 at 6 THz is still close to 2.3 (see, e.g., Fig. 11). Figure 2 indicates that the absorption coefficient of our samples is about 30% lower than previously published data in the frequency range between 1.6 and 2.6 THz.33 The reason for this deviation might be a better quality of the crystal samples that were investigated in this paper. The THz pulses used for our THz-TDS measurements (Figs. 1 and 2) were generated and detected in DAST crystals. It is observed in a typical spectrum of such a THz pulse (see, e.g., Fig. 10) that the measured THz field decreases significantly above 2.7 THz owing to the previously mentioned absorption line at 3.05 THz. The low spectral amplitude leads to a low resolution in the measured absorption data, which explains the large errors in Fig. 2 above 2.7 THz. A third absorption line at a frequency near 5 THz18,21 might be responsible that the measured values of the absorption coefficient ␣ above 3.5 THz are higher than the plotted function. C. Nonlinear and Electro-Optic Properties The crystal symmetry of DAST (point group m, see Subsection 3.A) allows the following EO coefficients rijk to be nonzero: r111, r221, r331, r131 = r311, r232 = r322, r122 = r212, r113, r223, r333, and r313 = r331. Out of these, four were reported15,34 to have values above 1 pm/ V, namely, r111, r221, r113, and r122. OR The nonlinear optical susceptibility ijk defined in Eq. (10) that is responsible for OR is related to the EO tensor rjki by35 1
OR ijk 共 ; o兲 = − 2 nj2共o兲nk2共o兲rjki共o ; 兲,
共42兲
where nj and nk denote the refractive indices along the dielectric j and k axes, respectively, denotes the THz modulation frequency, and o denotes the frequency of the optical pump or the probe beam. The dispersion of rjik with the optical frequency o was discussed and tabulated by Pan et al.15 Depending on the modulation frequency , Table 1. Parameters for the Refractive Index n1 of DAST in the Harmonic-Oscillator Modela,b
Oscillator
Amplitude ai
Frequency i = i / 共2兲 (THz)
Damping Parameter ␥i 共s−1兲
1 2
6.9 3.4
1.10 3.05
0.40⫻ 1012 3.1⫻ 1012
共40兲
n⬁ = 2.316 is the limit of the refractive index for frequencies much higher than the highest-lying oscillator frequency. The plotted curve of the absorption coefficient ␣共兲 in Fig. 2 is composed of the contributions of the same oscillators:
兺
a b
See Eq. 共40兲.
The amplitude ai contains the oscillator strength and the number density of the material.
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Table 2. Largest Electro-Optic Coefficients rijk and OR of DASTa Nonlinear Susceptibilities, kij Wavelength 共nm兲 800 1535 800 1535
r111
r221
r113
r122b
77± 8 47± 8 OR 111 1230± 130 490± 90
42± 4 21± 4 OR 122 166± 16 69± 13
15± 2 5±1 OR 311 239± 32 52± 10
17± 1.5 — OR 212 135± 12 —
Values of r from Pan et al. and those of OR calculated with Eq. 共42兲. Absolute values in units of picometers per volt. a
b
15
From Spreiter.34
several contributions distinguished36:
to
the
r
tensor
r共兲 = ra共兲 + ro共兲 + re共兲.
may
be 共43兲
re共兲 is the electronic response with resonance frequencies in the optical spectral range; it remains constant throughout the THz range. The contribution of the acoustic phonons, ra, is limited to kilohertz and megahertz frequencies; thus it does not play a role in the THz regime. ro is due to optical phonons that are both infrared active and Raman active. Since there is no absorption line observed in the measurements described in Subsection 3.B below the one at 1.1 THz, it may be assumed that there is no phonon contributing to ro共兲 below 1.1 THz, and that ro共兲 has the same value in the sub-THz range as at radio frequencies. We conclude that the so-called clamped EO coefficient rs共兲 = ro共兲 + re共兲 has to be used for the calculation of OR in Eq. (42). Above 1.1 THz it may deviate by the contribution of the phonon at this frequency, but we estimate that this deviation is at most of the order of 20% of ro共兲 [or 5% of rs共兲], since there are at least four internal vibration modes of the stilbazolium chromophore between 1160 and 1580 cm−1 (35 to 48 THz兲 that are infrared and Raman active and constitute the main part of ro共兲.36 In Table 2 we present the relevant EO coefficients rijk OR and nonlinear susceptibilities kij for two selected wavelengths. D. Velocity-Matching Configurations In Section 2, we demonstrated from theory that a small deviation ⌬n between the optical group index ng共兲 and the THz refractive index n共兲 has a large impact on the amplitude of the emitted THz field (velocity matching). It is important to note that the requirement for ⌬n is less strict than for other nonlinear conversion processes such as second-harmonic generation (SHG) because the coherence length is proportional to the wavelength of the generated radiation, which is thus 2 orders of magnitude larger for THz generation than for SHG with equal index differences. In the following, we investigate which configurations of the crystal orientation and the polarization direction may be used for the velocity-matched generation of THz pulses in DAST, and we compute and plot the maximum effective generation length Lmax defined in Subsection 2.C. The material parameters that determine Lmax are the optical group index ng共兲, the refractive in-
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dex n共兲 in the THz spectral range, and the THz absorption ␣共兲; ␣o is negligibly small. The values for ng共兲 were calculated analytically from the Sellmeier functions for n1共兲 and n2共兲15; those of the THz refractive indices are the two-oscillator function shown in Fig. 1 for n1共兲 and an analogous three-oscillator function fitted to the data of Walther et al.33 for n2共兲. The absorption function ␣1共兲 is the one plotted in Fig. 2. 1. c Plates DAST surfaces with optical quality are most easily produced in the ab plane (see Subsection 3.A). The eigenpolarizations of both optical and THz beams are therefore along the a or the b axis, which allows the use of two of the four elements of the OR tensor in Table 2, namely, OR OR 111 and 122 , for a beam direction perpendicular to the crystal surface. (The small nonzero angle between the crystallographic a and the dielectric x1 axes, respectively, the c and the x3 axes, may be neglected.) The quantity Lmax comprises the effect of THz absorption and velocity mismatch on the optimum THz generation efficiency, and it also applies to the EO detection of THz pulses. We computed Lmax, i.e., the effective generation length Lgen共z兲 from Eq. (19) with the optimum z = lo from Eq. (39), for the polarization configurations using OR OR 111 (or r111 for EO detection) and 122 共r221兲, respectively (see Figs. 3–6), as a function of optical wavelength and THz frequency . The results are shown in two figures per polarization configuration, since the THz refractive index shows two clearly distinct branches, separated by the TO phonon frequency of 1.1 THz. Figure 3 shows Lmax in the OR 111 configuration for frequencies below the TO phonon resonance at 1.1 THz. In this regime, an optical wavelength in the range between 1000 and 1200 nm is favorable. Varying within this range offers the opportunity of tuning the central frequency of the THz pulse between 0.3 and 0.7 THz. This was observed experimentally in Ref. 19 where a local maximum of the conversion efficiency was found for an optical wavelength of 1050 nm. The corresponding data for frequencies above 1.1 THz are presented in Fig. 4. The regime with the highest values of Lmax lies between 1400 and 1700 nm and covers frequencies between 1.7 and 2.3 THz, which also corresponds to the regime with the lowest THz absorption (see Fig. 2). Above 1700 nm, the optical absorption coefficient ␣o increases and therefore inhibits efficient THz generation. For ⬃ 1400 nm, Lmax reaches a second maximum above 3 THz. The absolute value of Lmax in this area is
Fig. 3. Contour plot of the maximum effective length Lmax for OR DAST using 111 or r111.
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efficient THz generation is from 700 to 740 nm; i.e., it lies within the tuning range of common Ti: sapphire lasers, which still represent the most widely used source of femtosecond pulses. Together with the resonantly enhanced OR values of 122 and r221 (see Table 2), the availability of well-suited lasers makes the use of b-polarized pump pulses a serious alternative in DAST-based THz systems despite of the potential disadvantages mentioned above.
Fig. 4. Contour plot of the maximum effective length Lmax for OR or r111. The thick curve represents 0.5 mm, and DAST using 111 the line spacing equals 0.1 mm throughout.
Fig. 5. Contour plot of the maximum effective length Lmax for OR DAST using 122 or r221.
Fig. 6. Contour plot of the maximum effective length Lmax for OR or r221. The thick curve represents 0.5 mm, and DAST using 122 the line spacing equals 0.1 mm throughout.
mainly given by 1 / ␣T, since the velocity-mismatch parameter n共兲 − ng共兲 is small in this regime. The absorption line centered at 3.05 THz (see Fig. 2) causes the valley between these two maxima, where Lmax does not increase above 0.2 mm. Figures 5 and 6 present the results analogous to Figs. 3 and 4 but for optical light polarized along the b axis. It is first observed that the usable wavelength range is much OR owing to the stronger optical disnarrower than for 111 persion of DAST at the velocity-matching wavelength .15 For shorter wavelengths within this range, DAST already shows increased optical absorption 共␣ = 1.5 cm−1 for = 700 nm, ␣ = 5 cm−1 for = 650 nm for b-polarized light29), which results in less efficient THz generation and a higher risk of damaging the crystal owing to heating. Moreover, the strong dispersion leads to nonnegligible group-velocity dispersion and, consequently, to lower generation efficiency and detection sensitivity. However, it is observed in Fig. 6 that the optimum wavelength range for
2. Other Crystal Orientations OR OR Apart from 111 , 311 is the next largest component of the nonlinear susceptibility tensor. It can be used in b plates with the pump beam polarized along the a axis, generating a rectified polarization along the c axis. (The simultaOR neous polarization along a due to 111 is identical to the one described in Subsection 3.D.1 and is independent OR contribution.) from the 311 Unfortunately, there are no data available on the refractive index n3共兲 at THz frequencies. However, one may set the requirement for n3 to be velocity matched to ng,1. Both the long-wavelength limit of the Sellmeier dispersion function for n1共兲15 and the mid-infrared value above 2000 cm−1 are at 2.1± 0.05. Hence n3共兲 has to attain a value of at least 2.0 to achieve nearly velocitymatched THz generation. On the other hand, one may estimate from the average mid-infrared value36 (500 to 2500 cm−1, 15 to 75 THz兲 of n3 ⬃ 1.45 and the dielectric constant in the kilohertz frequency range,15 冑⑀3 = n3 = 1.73± 0.09, that the THz refractive index will not exceed 2.0 except in the proximity of an eventual resonance, where absorption inhibits an efficient conversion from optical power to the THz field. We therefore conclude that OR the component 311 cannot be used for velocity-matched THz generation. All four elements in Table 2 require optical pump polarizations in the ab plane. To make use of unreduced values of OR, the pump beam should possess an eigenpolarization in this plane. This is fulfilled for a propagation direction k in the ab plane, where the o polarization is along the c axis and the e polarization is perpendicular to k in the ab plane under an angle to the a axis. The efOR 共兲 is then given by fective nonlinear susceptibility eff OR OR OR OR eff 共兲 = cos3共兲111 + sin共兲sin共2兲共 2 122 + 212 兲, 1
共44兲 where both optical and pump beams are e polarized. The corresponding effective refractive indices neff共兲 for both beams are23
Fig. 7. Velocity-matched wavelengths as a function of propagation angle (see text for details). Solid curve, = 2.0 THz; dotted curve, = 2.5 THz. Error bars are calculated with an uncertainty of the index difference ⌬n of 0.03.
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1 2 neff 共兲
=
cos2共兲 n12
+
sin2共兲 n22
.
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共45兲
With these values, one can calculate the index difference ⌬n as a function of optical wavelength , THz frequency , and angle . We focus on the frequency range between the THz absorption lines at 1.1 and 3.05 THz. The dependence of the velocity-matching wavelength on the angle for two selected THz frequencies is plotted in Fig. 7. For small angles , the velocity-matching wavelength does not deviate significantly from the one described OR . For close to 90°, the effecabove for c plates using 111 OR 共兲 drops to zero [Eq. tive nonlinear susceptibility eff (44)]. Thus the area of potential benefit is for angles around 45° that are matched or nearly matched to wavelengths between 1000 and 1200 nm. This implies the possibility of using neodymium-doped lasers with ⬃ 1064 nm. Although the exact dispersion of the OR coefficients is not known, one may estimate from Table 2 and OR 共45° 兲 is of the order of 300 pm/ V, which Eq. (44) that eff results in a strong rectified polarization. However, the fact that the THz absorption for b polarization is stronger than for a polarization may lead to a severe attenuation of the emitted THz radiation. Owing to the lack of crystals that were oriented under 45° to the a and b axes, we were not able to verify the above calculations experimentally.
4. EXPERIMENTS For the generation of the THz pulses, we used a series of DAST crystals with different thicknesses between 0.16 and 0.69 mm, grown in our laboratory. The laser source was an amplified Ti: sapphire laser with a central wavelength of 776 nm and a repetition rate of 1 kHz (ClarkMXR, CPA 2001). Typical pulse parameters were an energy of 0.8 mJ and duration of 160 fs FWHM. The tunability of the wavelength was achieved by means of an optical parametric generator–amplifier (Quantronix, TOPAS). The parametric signal wave was tunable from 1100 to 1600 nm with pulse energies of the order of 50 J, varying with wavelength. With optional frequency doubling of either signal or idler wave, the tuning range was extended down to 550 nm. The FWHM length of the laser pulses in the experiment was measured by autocorrelation to be between 130 and 140 fs. The setup for the generation and detection of the THz pulses is shown in Fig. 8. All experiments were carried out at room temperature and under ambient air. The THz pulses were detected by two different methods. Most measurements were done with THz-induced lensing25 as indicated in Fig. 8, a technique that is inherently linear for moderate THz electric fields. The detection crystal was a DAST crystal oriented such that both THz and probe pulses were polarized along the crystal’s a axis, thus making use of the EO coefficient r111. In the wavelength range below 950 nm, however, in which the velocity mismatch disfavors DAST crystals in the above-mentioned configuration, we employed EO sampling using a 0.5 mm thick 具110典 ZnTe crystal.28 The slightly modified setup has been published elsewhere.19
Fig. 8. Experimental setup for the generation of THz pulses by optical rectification and their detection by THz-induced lensing. EM, ellipsoidal mirror; IR block, a sheet of paper that blocks the pump beam and its second harmonic but that is transparent to THz radiation (indicated by a shade of gray).
OR Fig. 9. Typical THz transient from DAST 111 at a wavelength of 1500 nm. Left y scale, modulation ⌬I / I in THz-induced lensing; i.e., the intensity in the center of the probe beam changed by up to 50%. Right y scale, electric field of the THz pulse. The oscillations for t ⬎ 0.5 ps are due to ambient water vapor absorption.
5. RESULTS AND DISCUSSION OR A. Generation Using 111 We measured the THz transients with the setup depicted in Fig. 8 for various combinations of source crystal thickness, orientation, and laser wavelength. From the pulse duration of 140 fs in our system, one can calculate with Eqs. (18) and (37) that the maximum of the measured spectrum in the limit of thin crystals lies near 2 THz. Figure 9 presents a THz waveform that was generated with a pump pulse at a wavelength of 1500 nm, which is close to the optimum wavelength for the generation of 2 THz (see Fig. 4). The lengths were 0.40 mm for the source crystal and 0.69 mm for the detection crystal. The absorption of water vapor in the ambient air manifests itself in the time domain by the oscillating tail that persists for t ⬎ 0.5 ps. We calculated the electric field from the induced modulation ⌬I / I as described in Ref. 25, taking Ldet关 / 共2兲 = 2 THz, z = 0.69 mm兴 = 0.43 mm as the effective crystal length [see Eq. (36)]. Note that the maximum induced modulation of 0.5 (at t = −0.9 ps兲 exceeds the linear regime of THz-induced lensing, which results in a slightly distorted waveform; however, the maximum deviation from the real waveform is estimated to be smaller than 10%.25 The spectrum of this THz pulse as a function of the frequency = / 共2兲 is shown in Fig. 10. All the numerous narrow dips in the spectrum can be identified with ab-
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sorption lines of water vapor,37 such that one has to consider the envelope as the effectively emitted spectrum. For comparison, the theoretical spectrum [multiplication of the emitted spectrum from Eq. (18) with the detection sensitivity from Eq. (35)] is included in the graph. Fresnel reflections at the output surface of the generation crystal and the input surface of the detection crystal have been taken into account. In the central frequency range between 0.7 and 3.3 THz, the plotted spectrum agrees well with the measured data. This shows that the dispersion of the nonlinear susceptibility OR in the THz frequency —which was taken as a constant in the calculation—does not play a significant role in molecular crystals, in contrast to inorganic semiconductors.38 The deviation for lower frequencies is partially due to the effect of the finite beam diameter [Eq. (28)]. Additionally, low frequencies can be focused down only to a diameter of about their wavelength. Indeed, 0.7 THz corresponds to a vacuum wavelength of 0.5 mm, which is also a typical diameter of the probe beam. In the frequency range above 3.3 THz, the discrepancy may be due to an uncertainty in the THz absorption coefficient ␣共兲. The calculated spectrum uses the two-oscillator function plotted in Fig. 2, which does not include the DAST resonance at 5 THz that has been observed indirectly by a number of groups18,21 and confirmed by us (see Fig. 11). Apart from leading to an increased absorption, this line might also have an impact on the THz refractive index and, consequently, on the velocity-matching parameter ⌬n = n共兲 − ng共兲. In Fig. 11, spectra from pulses generated with different pump wavelengths are presented. The Fourier transform was in this case evaluated for the first three cycles of the oscillating THz field only, thus suppressing the influ-
Fig. 10. Solid curve, spectrum of the THz transient in Fig. 9. The numerous narrow absorption lines are from ambient water vapor (compare, e.g., Ref. 37). The dashed curve represents the theoretical spectrum.
Fig. 11. Normalized THz spectra generated at different pump wavelengths. Solid curve, 1350 nm; dashed curve, 1400 nm; dotted curve, 1500 nm. Crystal thickness of 0.25 mm for generation and 0.69 mm for detection.
Schneider et al.
Fig. 12. Spectra of THz pulses generated in a 0.69 mm thick DAST crystal. Pump light polarized along the crystal b axis. Solid curve, = 740 nm; dashed curve, = 710 nm.
ence of water vapor absorption. The spectra were normalized to their maximum value. Their wavelength dependence may well be explained by Lmax共 , 兲 as plotted in Figs. 3 and 4. Below 1 THz, the amplitude drops with increasing wavelength. In the spectral range between the two absorption frequencies 1.1 and 3.05 THz, the spectra are similar, with a slight shift toward lower frequencies for increasing . For ⬎ 3 THz, the highest THz output is achieved with ⬃ 1350 nm, especially above the resonance at 5 THz up to the measured cutoff frequency of 6.7 THz. The latter is given by the duration of the pump pulse, which limits the emitted bandwidth of E共兲 by the factor I0共兲 [Eq. (20)] and the detection sensitivity with the equivalent factor A共兲 in Eq. (37). One can conclude that the area with high values of Lmax in Fig. 4 above 3 THz extends to higher frequencies with only small variations in the related optical wavelength. This is equivalent to a THz refractive index that remains constant to within 0.02 between 3.5 and 6.7 THz with a possible exception near the absorption frequency at 5 THz.
OR B. Generation Using 122 We described in Subsection 3.D that efficient THz generaOR may be expected only in the wavelength tion using 122 range between 700 and 750 nm (Figs. 5 and 6). Figure 12 shows spectra of two pulses generated with b-polarized pump light at 740 and 710 nm, calculated from the first 5 ps of the THz pulse only, again in order to suppress the influence of water vapor absorption. The pulses were detected with EO sampling in a 0.5 mm thick ZnTe 具110典 crystal using the same wavelength. Both spectra have two distinct maxima. The first one below 1 THz is in good agreement with the theoretical expectation and also the shift toward a higher frequency for the lower wavelength. The second peak, however, is expected from Fig. 6 to be at 1.6 THz for 740 nm and at 1.9 THz for 710 nm. The shift toward 1.4 THz for both wavelengths is caused by the material properties of the ZnTe detection crystal. With experimental data for the refractive index n共兲 and absorption function ␣共兲 of ZnTe in the THz spectral range,39 we calculated also for ZnTe the frequency-dependent effective detection length Ldet [Eq. (36)], which decreases monotonously with wavelength for below its velocitymatching value of 822 nm.10 The calculation with Lgen of DAST and Ldet of ZnTe reproduces the measured spectra well.
Schneider et al.
C. Length Dependence The dependence of the THz waveforms on the crystal length changes with frequency. Thus not only the amplitude but also the shape of the pulses varies with crystal size. This is shown in Fig. 13 where the THz radiation from four different crystals is compared under otherwise identical conditions. The pump wavelength was 1500 nm, and the detector crystal thickness was 0.69 mm. We evaluated the Fourier transform of these pulses at two frequencies, namely, 2.0 and 3.4 THz, both of which are at a local maximum of the measured spectra (see Fig. 10). The normalized amplitudes are plotted in Fig. 14 as a function of the source crystal length. In our theory, the length dependence is solely given by the effective genera-
Fig. 13. THz pulses generated in DAST crystals with a thickness d of 0.16, 0.33, 0.40, and 0.60 mm (from bottom to top) with a laser wavelength of 1500 nm. The signals are vertically offset for clarity and shifted in time by ngd / c to compensate for the different propagation times of the pump pulse.
Fig. 14. Spectral amplitude of the THz waveforms in Fig. 13 for two different frequencies. The data were normalized such that the slope is equal to 1 mm−1 in the limit of zero thickness (dotted line). The other curves represent the function Lgen共d兲 for the same frequencies. Filled circles and solid curve, 2.0 THz; open squares and dashed curve, 3.4 THz.
Vol. 23, No. 9 / September 2006 / J. Opt. Soc. Am. B
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tion length Lgen共z兲, which is plotted for comparison. The result is in good agreement with the predictions. Whereas the amplitude at 2.0 THz shows only a small deviation from a linear increase, that at 3.4 THz is already close to saturation after 0.4 mm. These data may also be interpreted easily using Fig. 4. As discussed in Subsection 2.C, the amplitude at a given frequency reaches a maximum for a crystal length d ⬃ 2Lmax, i.e., for ⬃1.2 mm at 2 THz and ⬃0.65 mm at 3.4 THz, in agreement with experiments. D. Pump Energy Dependence In Eq. (18), the spectral amplitude of the THz electric field E共兲 was calculated to be proportional to the intensity spectrum I0共兲 of the pump pulse and, consequently, to the total pump pulse energy Wopt. However, two-photon absorption (TPA) may affect the shape of the spectrum through two different mechanisms (see Subsection 2.A.5). On the one hand, the spatial flattening of the pump intensity modifies the diffraction of the THz pulse, which influences mainly low frequencies (typically below 0.5 THz). More importantly, the reduction of I0共兲 due to the TPAinduced change in the temporal pulse profile I共t兲 increases with frequency; hence the spectrum E共兲 is shifted toward lower frequencies. Figure 15 shows the spectra of five THz pulses normalized with their respective pump pulse energy Wopt. The wavelength of 1300 nm was chosen such that the wavelength / 2 of the two-photon process lies within the absorption range for a-polarized light. At this wavelength, we measured the relevant TPA coefficient ␣2 = 0.7 cm/ GW with the method described by Bechtel and Smith.40 However, this method does not take the self-focusing of the optical pulse into account, such that the actual value of ␣2 will effectively be lower. Nevertheless, the leading term in Eq. (30) with typical experimental parameters (I0 = 20 GW/ cm2, z = 0.5 mm) may lead to a relative intensity change ⌬I / I0 in the center of the beam at the end of the crystal of some tens of percent, which means that TPA may not be neglected. Indeed, small variations are observed above 3 THz, where the amplitude is highest for the lowest pump energy, in agreement with the above discussion, whereas in the central part between 1 and 3 THz, the normalized spectra are essentially identical.
6. CONCLUSIONS
Fig. 15. Spectra of THz pulses generated and detected in DAST crystals with a thickness of 0.40 and 0.69 mm, normalized with the pump pulse energy; = 1300 nm. Pump pulse energies, from bottom to top: 32.7, 27.0, 18.4, 12.9, and 6.2 J. The traces are vertically offset for clarity. The dips in the spectra near 1.7, 2.2, and 2.7 THz are due to ambient water vapor.37 The spectra were calculated from the first 4 ps of the THz pulse.
We have determined the optimum conditions for the generation of terahertz pulses in organic DAST crystals through optical rectification of femtosecond laser pulses and for their detection through electro-optic sampling. Our experimental results have confirmed the theoretical expectations that took into account velocity matching, THz and optical absorption, crystal thickness, and laser pulse duration. We have identified several combinations of the optical wavelength and THz frequency with enhanced THz emission. Within the tuning range of Ti: sapphire lasers, velocity-matched THz generation is achieved using the 共2兲 nonlinear coefficient 122 . Of primary technological importance is that telecom wavelengths around 1500 nm are velocity matched to frequencies between 1.5 and 2.7 THz
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共2兲 if the largest nonlinear coefficient 111 is used. Using DAST crystals of less than 0.7 mm thickness for generation and detection, we have demonstrated a THz-induced modulation of 50% in a nominally linear regime. With the advent of compact and stable femtosecond lasers at 1.5 m (e.g., erbium-doped fiber lasers), highly efficient and cost-effective THz systems may be developed using DAST as the THz emitter. Higher frequencies (3.3 to 6.7 THz) are most efficiently generated using near 1350 nm. These results demonstrate the considerable potential of DAST for THz applications and how it can be fully exploited.
ACKNOWLEDGMENTS The authors thank J. Hajfler for his expert sample preparation and Ch. Bosshard for helpful discussions. This work has been supported by the Swiss National Science Foundation. A. Schneider, the corresponding author, can be reached by e-mail at
[email protected].
*Current address, Advanced Technology Institute, University of Surrey, Guildford GU2 7XH, United Kingdom.
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