Role of misalignment-induced angular chirp in the electro-optic detection of THz waves

D. A. Walsh; M. J. Cliffe; R. Pan; E. W. Snedden; D. M. Graham; W. A. Gillespie; S. P. Jamison

doi:10.1364/OE.22.012028

1. Introduction

Electro-optic (EO) sampling of THz waves with ultra-fast optical pulses is now a well established technique. It is extensively applied in THz-Time Domain Spectroscopy (THz-TDS), often being cited as the “standard method” [1, 2]. Particular applications include the study of materials and material dynamics [3], chemical recognition [4], non-destructive THz Pulse Imaging [5, 6], and the detection of, and examination of conformational changes in, proteins [7].

The EO sampling process is commonly described as a THz electric-field induced refractive index modification of the EO material, in which an optical probe effectively experiences a THz-field induced change in polarisation. Since its early application the description of the EO detection process has undergone several refinements. In the approach of Nahata et al. [8], the optical group velocity and THz phase velocity are used to describe the propagation of the probe and THz pulses respectively, with the condition that the two be matched. Gallot et al. [9] and Jamison et al. [10] presented an alternative description, with the EO interaction viewed as sum-and difference-frequency generation between optical and THz waves. In the latter, the phase-matching of optical and THz waves naturally reduces to the group/phase velocity matching of Nahata et al. for THz frequencies which are significantly less than the bandwidth of the optical probe.

Theoretical treatments of the EO process such as those detailed above, have constrained their analysis to collinearly propagating wave fronts and do not account for the effect of a finite transverse beam size on phase matching efficiency [1]; in this work we expand on the sum- and difference-frequency description, and unlike previous treatments we allow the optical probe input and generated optical waves to be non-collinear.

It is shown that non-collinear phase-matching gives rise to an angular chirp of the frequencies created through the non-linear three-wave mixing processes that can have a measurable impact on standard EO detection schemes. To provide experimental verification, the χ⁽²⁾ process of Spectral Upconversion is employed to reveal the angular chirp unambiguously. The direct extension of the effect to EO sampling schemes is then discussed.

2. Theory: non-collinear phase-matching in EO detection and spectral upconversion

Following Gallot et al. [9] and Jamison et al. [11], using the optical principal axis frame and applying the small signal limit, an optical probe is modified in an EO interaction according to:

{\tilde{E}}_{out}^{opt} (ω) = {\tilde{E}}_{in}^{opt} (ω) + i α ω χ^{(2)} \int {\tilde{E}}_{in}^{opt} (ω - Ω) ζ (Ω) {\tilde{E}}^{THz} (Ω) d Ω,

where Ẽ represents a complex electric field, χ⁽²⁾ is the nonlinear response function, ω is the optical frequency, Ω represents the frequency components of the THz pulse, and α is an experimental-geometry-dependent constant as defined in [10]. The phase matching and χ⁽²⁾ frequency dependence are described by the function ζ(Ω). In the limiting case of a broadband transform limited optical probe, where we can assume Ẽ^opt (ω − Ω) ≈ Ẽ^opt (ω), and introducing a variable delay τ between THz and optical probe pulses with

ζ (Ω) {\tilde{E}}^{THz} (Ω) \to ζ (Ω) {\tilde{E}}^{THz} (Ω) exp (i Ω τ) \equiv {\tilde{E}}_{eff}^{THz} (Ω) exp (i Ω τ)

, we obtain the expected electro-optically induced phase shift of the optical field:

\begin{array}{l} {\tilde{E}}_{out}^{opt} (ω) & \approx {\tilde{E}}_{in}^{opt} (ω) (1 + i α ω χ^{(2)} E_{eff}^{THz} (τ)) \\ = {\tilde{E}}_{in}^{opt} (ω) exp (i α ω χ^{(2)} E_{eff}^{THz} (τ)) . \end{array}

In an opposing limit, where the optical probe is monochromatic,

{\tilde{E}}_{in}^{opt} (ω) \to E_{in}^{opt} δ (ω - ω_{0})

, we obtain

{\tilde{E}}_{out}^{opt} (ω) = E_{in}^{opt} (δ (ω - ω_{0}) + i α ω χ^{(2)} ζ (ω_{0} - ω) {\tilde{E}}^{THz} (ω_{0} - ω)),

which describes the process of Spectral Upconversion, where the THz spectrum is shifted to optical frequencies and forms sidebands on the input optical probe. Here we consider modifications to Eq. (1) that allow for an angular separation of the χ⁽²⁾-generated field and the input probe, and the subsequent implications for Spectral Upconversion and THz-TDS. In the extreme case where probe and generated waves become spatially separated it is clear that the net effect cannot be considered a phase shift of the optical probe.

To formalize a description of the interaction and generation of non-collinear waves via the χ⁽²⁾ non-linearity, we define the non-linear source term, P̃(ω₃, ṟ), where ṟ = (x, y, z) is a vector in cartesian coordinates, for a pair of input single frequency components ω₁ and ω₂ as

\tilde{P} (ω_{3}, \underline{r}) = α χ^{(2)} {\tilde{E}}^{opt} (ω_{1}, \underline{r}) {\tilde{E}}^{THz} (ω_{2}, \underline{r}) .

The complex fields Ẽ are separated into a slowly varying amplitude Ã and rapidly varying phase term, yielding:

\tilde{P} (ω_{3}, \underline{r}) = α χ^{(2)} {\tilde{A}}_{1} (ω_{1}, \underline{r}) {\tilde{A}}_{2} (ω_{2}, \underline{r}) exp (i ({\underline{k}}_{1} + {\underline{k}}_{2}) \cdot \underline{r})) .

The non-collinear geometry of the interaction is shown in Fig. 1. We have chosen to define the z-axis of the coordinate frame as the propagation direction of the probe, which in the case of ZnTe is also set parallel to the [110] direction; however, the conclusions are not dependent on this choice. Solving the paraxial wave equation along z for a crystal of thickness L, with no depletion of the fields at ω₁ and ω₂, yields the field Ã₃ (ω₃, θ, ϕ) at the exit of the crystal:

{\tilde{A}}_{3} (ω_{3}, θ, ϕ) = α ω_{3} χ^{(2)} \frac{exp (i Δ k_{z} L) - 1}{Δ k_{z}} \iint {\tilde{A}}_{1} (ω_{1}, x, y) {\tilde{A}}_{2} (ω_{2}, x, y) exp (i (Δ k_{x} x + Δ k_{y} y)) d x d y,

where Δḵ ≡ ḵ₁ + ḵ₂ − ḵ₃ which is a function of ω₃, θ and ϕ. It is assumed that the transverse beam profiles are large enough that transverse walk-off of the probe and THz beams within the typically millimetre-scale thickness of the EO material can be neglected. If Ã₁ (ω₁, x, y) and Ã₂ (ω₂, x, y) now have Gaussian transverse profiles with e⁻¹ E-field radii σ_x and σ_y, and amplitudes Ã₁ (ω₁) and Ã (ω₂) we obtain:

{\tilde{A}}_{3} (ω_{3}, θ, ϕ) = α ω_{3} χ^{(2)} exp (- \frac{1}{2} (σ_{x}^{2} Δ k_{x}^{2} + σ_{y}^{2} Δ k_{y}^{2})) \frac{exp (i Δ k_{z} L) - 1}{Δ k_{z} L} {\tilde{A}}_{1} (ω_{1}) {\tilde{A}}_{2} (ω_{2}),

which is consistent with the phase matching efficiency of Boeuf et al. [12]. The relative contributions of the transverse profile dependent Gaussian, and (Δk_zL) dependent terms of Eq. (7) are shown in Fig. 2 for a 4 mm thick (110) ZnTe crystal. In calculating Δḵ the THz and optical refractive index are evaluated from the parametrization of Gallot et al. [13].

Fig. 1 k-vector diagram illustrating the modelled non-collinear phase matching condition. The angles and k-vectors are not drawn to scale to increase clarity.

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Fig. 2 Phase-matching efficiency plots calculated for a 4 mm (110) ZnTe crystal. Here the external angle (ϕ_ext) refers to the angle of incidence of the THz wave on the ZnTe surface, which is related to the internal angle (ϕ) by Snell’s law. a) Plot showing contribution of the z-dependent component. The vertical green lines indicate the 50% efficiency width from plot b, b) ϕ_ext = 0°, L = 50μm, c) ϕ_ext = 0°, L = 500μm, d) ϕ_ext = 12°, L = 50μm, e) ϕ_ext = 12°, L = 500μm.

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As the angles of the waves external to the crystal are of more practical interest we now define the output upconverted wave angle as θ_ext and incident THz angle as ϕ_ext, which are related to θ and ϕ through Snell’s law. In Fig. 2 the phase matching efficiency is shown for a monochromatic optical probe, with ω₁ = 375THz (800nm); the extension to a broadband pulse is discussed in Section 4. Figure 2(a) shows the magnitude of the sinc²(Δk_zL) term as a function of the external generation angle θ_ext and THz frequency with collinear THz and optical probe beams (ϕ_ext = 0°). The 50% contour of the Δk_x and Δk_y dependent Gaussian for a transverse width of σ_x = σ_y = 50μm is superimposed. The total phase-matching efficiency is displayed for a number of configurations in Figs. 2(b)–2(e). The THz wave is taken to be incident either collinearly with the optical probe or at an external THz incident angle (ϕ_ext) of 12°. It can be seen that the transverse phase matching dominates and determines the angular spread of generated waves, while the Δk_z matching dominates the frequency response of the χ⁽²⁾ mixing process.

3. Experiment

To observe the angular chirp of the upconverted waves, a test system based around a large-area semi-insulating GaAs photoconductive antenna (PCA) THz source [14] (excited by 50 fs FWHM, 805 nm pulses) and 4 mm thick ZnTe crystal was constructed, a schematic of which is shown in Fig. 3(a). The time and frequency representations of the THz source, as measured using THz-TDS and spectral upconversion, are presented in Figs. 3(b) and 3(c).

Fig. 3 a) Schematic layout of the experimental system illustrating the gimballed mirror and conjugate parabolic mirror pair arrangement for stable, angle-only, tuning of the THz pulse incident on the ZnTe. Also shown is the implementation of a 90° rotating periscope in the optical transfer line before a lens which mapped the upconverted waves onto the imaging spectrometer. The quarter-wave plate (QWP) either compensated for residual birefringence, or provided an optical bias for balanced detection. b) Example temporal profile recovered via TDS measurements with a 50 fs probe and c) Example THz spectrum obtained via Spectral Upconversion with a 10 ps probe compared to a spectrum found via the FFT of the THz TDS trace (vertically offset for clarity). The Spectral Upconversion trace in this plot was limited by the resolution of the spectrometer.

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The THz radiation emitted from a large-area PCA preserves the wavefront curvature of the exitation pulse [15]. This effect was exploited by first diverging and then refocussing the 805 nm excitation pulse through the PCA such the generated THz pulse came to a focus on the center of a gimbal-mounted mirror. The distance between PCA and gimbal-mounted mirror was 61 cm, being set such that the PCA was fully illuminated. A conjugate pair of identical gold-coated parabolic mirrors then imaged this mirror surface on to a ZnTe detection crystal. This enabled, through the adjustment of the gimbal-mounted mirror angle alone, the tuning of the angle of incidence of the THz pulse (ϕ_ext) on an EO crystal without altering the THz focus position, spot size, or time of flight. The optical probe and THz pulses were combined using an ITO coated fused silica substrate placed 6 cm before the ZnTe crystal.

The probe pulse was picked off prior to the beam exciting the PCA and passed through a zero-dispersion 4-f spectral filter. This allowed the pulse to be tuned from a 50 fs (for THz-TDS) to a 10 ps long pulse (for this set-up the best bandwidth/power compromise for Spectral Upconversion). After the spectral filter an aperture was placed to reduce the size of the beam to a diameter of 1 mm in order to sample the center portion of the several-millimetre diameter THz beam.

After the ZnTe crystal, and quarter-wave plate used for tuning/compensating for the optical polarisation state, a flip-mirror allowed switching between a balanced detection system or a spatially resolved Spectral Upconversion arrangement. The adjustment of the THz incidence angle (ϕ_ext) was in the horizontal plane, giving rise to an angular chirp of the upconverted wave that was also in the horizontal plane. A 90° turning periscope was used to rotate the chirp to be parallel to the vertical plane, and together with focussing onto the entrance slit of an iHR550 imaging spectrometer an initially horizontal angular chirp was transformed into a vertical displacement. At the imaging plane of the spectrometer single-shot spectrograms mapping the angular dependence of the spectrum of the upconverted waves were produced. These spectrograms were recorded using an intensified CCD camera (DICAM pro, PCO).

In Fig. 4 we show the experimental observation of the angular chirp introduced by an angular offset of the THz beam from a collinear geometry with the input probe. For well aligned collinear THz and optical probe beams the THz spectrum, as observed through the optical sid-bands produced via the nonlinear processes, is also collinearly propagating (with some small, approximately frequency independent, angular spread) for all THz frequencies. In contrast, with an external THz angle of incidence (ϕ_ext) of 12° (internal angle ϕ ≈ 4°), the sidebands display a clear angular chirp; which in this case is 0.032degrees/THz. In the corresponding calculations the bandwidth of the source has not been included, enabling the full extent of the response function and angular chirp to be displayed. The experimental angular chirp was evaluated by fitting a Gaussian to the angular distribution of each spectral component, and then performing a linear fit to the data, as shown in Fig. 4(c). The measured and predicted angular chirp for a range of THz angles is shown in Fig. 4(f). The ≈ 7% difference in gradient is attributed to errors in the focusing lens and the positioning of the camera in the spectrometer exit plane. Additionally, a range of values for the THz refractive index of ZnTe can be found in the literature, indicating that the refractive index may vary from crystal to crystal, perhaps due to growing conditions, which could also be a contributing factor.

Fig. 4 Comparison of the calculated chirp in the phase matching efficiency (a and b) with the experimentally observed chirp in the Spectral Upconversion (d and e). Plot c shows the shift of the peak of the detected spectra (Gaussian fit centers) vs the propagation angle for a subset of measured THz incident angles. Plot f compares how the calculated and experimental chirp of the upconversion varies with THz incident angle, including linear fits, showing a small systematic error but good overall agreement.

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As the measured angular chirp is numerically small, we emphasize that nothing other than the gimbal mirror in the optical arrangement is altered during these successive measurements, therefore only the THz angle at the crystal was changed. Indeed, the gimbal mirror angle was tuned with the system running and the variation of the chirp on turning a single optical actuator was observable in real time.

4. Implications for EO THz measurements

The angular chirp introduced by non-collinear phase matching has a number of consequences for THz waveform measurements that utilize the EO effect in ZnTe and other nonlinear mixing materials. The Spectral Upconversion experiment detailed in Section 3 is structurally identical to those in the literature [16, 17], except for the deliberate resolution of the angular chirp. In Spectral Upconversion the spectral content of a THz waveform is translated into the optical regime, where the spectral and temporal content of the waveform can be simultaneously characterized using a number of methods, including Frequency Resolved Optical Gating (FROG) [18] and Spectral Interferometry for Direct Electric-field Reconstruction (SPIDER) [19]. Any aspect of these measurements which distorts the spectral profile of the upconversion will prevent an accurate retrieval of the input THz waveform/spectrum. Most obviously, the experiments explicitly show that THz measurements via Spectral Upconversion that are subject to a finite angular acceptance in the collection optics will be affected. For example, if the chirp were not rotated in the experiment described in Section 3 then the waves generated by higher frequency THz components would be attenuated by the slit of the spectrometer.

Figure 5 shows the results of a Spectral Upconversion measurement where the input THz beam has been deliberately misaligned to an incidence angle of 10°. Here the same THz and narrowband optical probe (1 mm diameter) pulses as described previously were mixed in a 4 mm ZnTe crystal. A 150μm slit was placed 50 cm after the ZnTe crystal, just in front of a 10 mm diameter fibre coupler connecting to a multi mode fibre. This arrangement was then scanned laterally across the detection plane whilst the output of the fibre was characterised using an iHR550 spectrometer. This can approximately be thought of as sampling a vertical slice though Fig. 4(e). A clear shift in spectral intensity detected from almost exclusively sum-frequency, to almost exclusively difference-frequency, components of the mixing occurs as the slit is scanned from +0.6mm to −0.6mm. This example serves to demonstrate the potential for the small angular chirp in the mixing processes, coupled to the angular acceptance of real optical collection and detection systems, to give rise to distortions in measured THz spectra. This shift is in agreement with expectations, and compares well with our calculations that include the finite transverse sizes of the upconverted waves.

Fig. 5 a) Schematic representation of the optics used to investigate the effect of non-collinear phasematching (10° THz angle of incidence) combined with a finite aperture (a 150 μm slit placed in front of a fiber-coupler) on the THz spectrum recorded via Spectral Upconversion. b) Plot showing the modification to the recorded spectrum as the position of the slit was adjusted.

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In Fig. 6 we plot the calculated spectral intensity of the upconverted radiation from ZnTe as a function of frequency for a range of output angles and two different THz beam radii (Fig. 6(A): 100μm; Fig. 6(C): 1.0 mm). The integrated bandwidth within a range of output angles are plotted in Figs. 6(B) and 6(D). The measured bandwidth of the THz waveform is found to decrease with the range of measured output angles over which a signal is detected; for example, in the case of a 1mm @ e⁻¹ radius beam in 100μm thick ZnTe, the detection bandwidth decreases from 3.8 to 1.3 THz as the acceptance cone is reduced to θ = 0.05°. A decrease from 3.8 to 1.9 THz is predicted for a 100μm @ e⁻¹ radius beam in the same crystal.

Fig. 6 Phase-matching efficiency for the non-linear mixing processes (Spectral Upconversion) in 100 μm ZnTe for A 100 μm (e⁻¹ radius) and B 1.0 mm THz spot size. The corresponding total bandwidths for a given range of output angles (0.05, 0.10, 0.15 and 0.20 degrees, see lines) are shown in B and D.

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4.1. Electro-optic sampling

While the discussion has so far been principally on Spectral Upconversion using a monochromatic probe, the conclusions on optical angular acceptance and the distortion of measured THz spectra can be carried over directly to the broadband optical sampling of THz signals such as THz-TDS. Following Eq. (2), the extension to a broad-bandwidth probe is incorporated in an integral over the probe bandwidth. It is found that for a typical configuration using ZnTe or GaP, with probe wavelengths centred around 800 or 1064 nm, the angular chirp of the upconverted beam is, to a first approximation, independent of the probe wavelength over a bandwidth commensurate with typical ultrashort pulse probes. The χ⁽²⁾ generated signal from the individual frequencies of the probe will add up to form a continuum of frequency shifted (as indicated by the vertical arrows) angularly chirped waves in Fig. 6(C). For a given angular acceptance in the optical system, the bandwidth limiting truncation of the upconverted beam is therefore approximately constant over the full bandwidth of the optical probe, and so it can be concluded that the frequency response of the detection system will be the same for monochromatic and broadband sampling detection.

For EO sampling an additional constraint arises in the optical detection system. As highlighted by Eq. (3) the effective phase shift in the probe relies on an interference between input probe and the χ⁽²⁾ generated optical wave. If there is a spatial walk-off between these beams it is no longer appropriate to consider the EO interaction as a polarisation rotation.

For a given EO or Spectral Upconversion detection system, the impact of the angular effects on the measured THz signal will depend on the specific optical collection system. For Spectral Upconversion a sufficiently large angular acceptance will ensure that all THz frequencies are detected faithfully. For EO polarisation rotation systems, both angular acceptance and imaging/spatial overlap of beams play a role. To provide a generally applicable quantitative guide it is however possible to specify the angular chirp of the χ⁽²⁾ generated beam. The angular chirp, defined in degrees of optical angular offset per THz, is itself dependent on the THz misalignment from collinear. As this dependence is largely linear (see Fig. 4 and associated discussion), we specify an angular chirp rate, defined as the angular chirp per degree of THz misalignment from collinear. Values of chirp rate for common crystal and probe wavelength combinations are tabulated in Table 1. The angular spread of the generated beam, as determined by the phase matching Gaussian in δk_⊥σ_⊥, is also relevant for the complete THz detection. We therefore also specify the opening angle parameter, which when divided by the beam width yields the angular half-width at which the phase matching efficiency drops to e⁻¹. The angular spread is found to be largely independent of the THz frequency, and so values are also presented in Table 1.

Table 1. Chirp rates and opening angle parameters for several commonly used crystals and probe wavelengths.

View Table

5. Conclusions

We have presented a description of the EO effect that accounts for a non-collinear phase-matching between, and the finite transverse profiles of, the optical and THz waves. For non-collinear phase-matching using a narrow bandwidth probe an angular chirp on the optical waves generated through non-linear mixing was predicted, and has been confirmed through experiment. This chirp, which spectrally maps the content of THz pulses to the output angles of optical waves from the EO crystal, can lead to significant experimental errors in Spectral Upconversion measurements (effectively a spatial aperture induced bandwidth filtering), and therefore by extension also to THz-TDS measurements. This analysis is general, and can be applied to other experimental geometries and non-linear mixing materials for which the refractive index is known. In cases where collinear EO detection is not possible, care should be taken in the design of the experiment to minimise these distortions in the measured THz pulse profile.

Acknowledgments

The authors would like to acknowledge that this work was partly supported by CERN through CLIC Contract Number KE1865/DG/CLIC.

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Material	Probe Wavelength nm	Chirp Rate $\frac{degrees (θ_{ext}) / THz}{degrees (ϕ_{ext})}$	Opening Angle Parameter (degrees θ_ext).m

ZnTe	800	2.7 × 10⁻³	1.0 × 10⁻⁵
GaP	800	2.4 × 10⁻³	9.3 × 10⁻⁶
ZnTe	1064	3.8 × 10⁻³	1.4 × 10⁻⁵
GaP	1064	3.3 × 10⁻³	1.3 × 10⁻⁵

Role of misalignment-induced angular chirp in the electro-optic detection of THz waves

Abstract

1. Introduction

2. Theory: non-collinear phase-matching in EO detection and spectral upconversion

3. Experiment

4. Implications for EO THz measurements

4.1. Electro-optic sampling

5. Conclusions

Acknowledgments

References and links

Cited By

Figures (6)

Tables (1)

Equations (7)

Optics Express