Chapter 2. Terahertz Time-domain-spectroscopy (THz-TDS)
2.4. Detection of THz-TDS radiation using PC antenna
For the detection of THz field, which is a Gaussian beam on the detector expressed as [6]:
Fig. 2.4. THz detector.
The total resistance over the detector is:
td switch area between the two electrodes tips, ρ and M ρS are their resistivity,
2 Lm
2 Lm
respectively, and d is the width of electrode and the gap area as shown in Fig. 2.4.
The average resistivity ρS is larger than ρ due to the low duty cycle of the M driving laser (100 fs/10 ns =10−5). The resistivity ρS depends on photogenerated carrier density, which for homogeneous illumination of power Plaser scales as:
laser where ξ is a conversion factor between laser power and number of photogenerated carriers.
The average electric field E across the detector gives rise to a potential difference )
The average electric field across the detector area is
2 )
( is an error function. The peak strength of the
electric field, E , can be expressed in terms of the total power of THz beam: 0
gap at the position given by
−1
= n
R n
h L (2.13)
where RL is the lens radius and n is the index of refraction of the lens material.
Justifying the lens design:
lim 1
By inserting Eq. (2.12), together with Eq. (2.14), into the expression for the detector current, Eq. (2.9), we get
In short, the time domain signal I(ν)relative to the emitted THz electric field E and the response function of the detector. The carrier life time of LT-GaAs, transmission characteristics of the SI-GaAs substrate and LT-GaAs film, resonance characteristics of dipole-type antenna, diffraction effect of the THz beam optics, and pulse widths of probe beam, etc, all contribute to the transient response function of the detector.
2.5. The application of THz-TDS system: extracting the refractive index of a large birefringence LC-DN125262W
The knowledge of the frequency dependence and the magnitude of the refractive indices as well as the birefringence of liquid crystal is a key parameter for practical applications of LCs. Although many groups have investigated the birefringence of LCs in visible or infrared range, we expect a large birefringence with low loss LCs in THz range.
For searching high birefringence LCs, we measure the refractive index of a large birefringence LC_DN125262W, a special mixture from daily polymer corporation, by using THz-TDS to determine it. We prepared two cells: the reference cell is constructed by two fused silica substrates contacting each other; each LC cell is constructed by two fused silica substrates separated by an aluminum spacer and filled with DN125262W as shown in Fig 2.5. The inner surfaces are coated with polyimide films, which are baked and rubbed to give LCs the homogeneous alignment.
LC Alignment Direction
(a) Reference Cell THz beam
LC Cell THz beam
Fused Silica (b)
The Polarization of Incident THz Wave: e‐ray ; o‐ray
If a plane wave pass through the cell normally, the electric field of the THz wave through the reference cell at angular frequency,ω , can be written as
)
where E0(ω)is the electric field of THz wave before passing through the cell, )
(ω
η is a coefficient including all the reflection, transmission and propagation coefficients in fused silica and Pair( dω, )is the propagation coefficient in air, which
dis the thickness of the LC layer. Generally, for a medium over a distance L, the propagation coefficient Pa(ω,L)can be written as exp[(−i~naωL)/c], where n~a (=na −iκ) is the complex refractive index of medium a, which depends on the wave frequency. In η(ω), the echoes of the THz wave or Fabry-Perot effect in fused silica can be neglected since in the thick sample, these echoes are far away from the main signal on the time domain and can be truncated during the spectrum analysis. The electric field of a THz wave passing through the LC cell is given by:
) thickness L. Because of the same thickness of fused silica substrates used in this work, η(ω)is equal to the reference cell. The complex transmission coefficient
) (ω
T of the measured LC as follow:
)
demonstrated the solutions in 1996 for optically thick and thin samples [7]. Here, we make thick sample to simplify the analysis processes such that the echoes of THz waves from multiple reflections of sample are temporally well separated from main signal. So FLC(ω)in Eq. 2.18 can be equal to 1.
If we let the reference cell have an empty gap of the same thickness of LC cell, the ) respectively the calculated and measured complex transmission coefficients, we define an error function, δ(n,κ), and then find (n,κ)by minimizing the error function. The error function is defined as :
2 After several iterations to minimize the error function by computer programs, we get the refractive indices as following results:
We repeated four times of measurements in the LC cell with 4.992 mm cell gap where to get the same result in each one. Fig. 2.6 (a) is the result of extraordinary and ordinary refractive indices, and (b) is the result of imaginary extraordinary and ordinary refractive indices of DN125262W with 4.992mm cell gap. Besides, we also fabricated DN125262W LC cells with 0.565mm cell gap, and Fig. 2.6 (c) is
ordinary refractive indices, shown as a function of frequency from 0.2 to 0.95 THz.
Comparing with the refractive index of E7, Fig. 2.7. (a) and (b) are the refractive indices of E7 with 8mm cell gap, measured by our group before. Therefore, the birefringence of DN125262W is larger than E7, which largely enhances applications to wave plates or filters. Table 2.1 is a comparison of two kinds of LC molecules according to our measurement.
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
Fig. 2.6. The room-temperature (a) extraordinary and ordinary refractive
indices, (b) imaginary extraordinary and ordinary refractive indices of DN125262W with 4.992 mm cell gap; (c) extraordinary and ordinary refractive indices, (d) imaginary extraordinary and ordinary refractive indices of DN125262W with 0.565 mm cell gap, are shown as a function of frequency from
(c)
(d)
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.50
1.55 1.60 1.65 1.70 1.75 1.80
8mm E7 ne 8mm E7 no
Refractive index
Frequency (THz)
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.00
0.01 0.02 0.03 0.04
8mm E7 no 8mm E7 ne
Imaginary Index
Frequency (THz)
Fig. 2.7. The room-temperature (a) extraordinary and ordinary refractive indices, (b) imaginary extraordinary and ordinary refractive indices of E7 with 8mm cell gap. ( Measured by C.-F. Hsieh)
(a)
(b)
E7(8mm) Large Birefringence LC: DN-125262W
(0.565mm)
Large Birefringence LC: DN-125262W
(4.992mm) Ordinary
refractive
1.580 1.544 1.566
Extraordinary refractive
1.713 1.728 1.741
Birefringence 0.133 0.184 0.175
Table 2.1. The measuring results of refractive indices between different thicknesses of DN125262W cells, and one thickness E7 cell.
2.6. References:
1. D. Grischkowsky, S. Keiding, M. van Exter and C. Fattinger: J. Opt. Soc. Am. B 7 (1990) 2006.
2. X.-C. Zhang and D. H. Auston: J. Appl. Phys. 71 (1992) 326.
3. B.B. Hu, X.-C. Zhang, D. H. Auston, and P. R. Smith: Appl. Phys. Lett. 56 (1990) 506.
4. H. G. Roskos, M. C. Nuss, J. Shah, K. Leo, D. A. B. Miller, A. M. Fox, S.
Schmitt-Rink and K. Kohler: Phys. Rev. Lett. 68 (1994) 2216.
5. M. Born and E. Wolf: Principles of Optics ( Pergamon, New York, 1959)
6. P. Uhd. Jepsen, R. H. Jacobsen, and S. R. Keiding, “Generation and detection of terahertz pulses from biased semiconductor antennas,” J. Opt. Soc. Am. B, Vol. 13, no. 11, pp. 2424-2436, Nov. 1996
7. L. Duvillaret, F. Garet, and J. Coutaz, “A Reliable Method for Extraction of Material Parameters in Terahertz Time-Domain Spectroscopy,” IEEE J. Sel. Top.
Quantum Electronics 2, 739-746 (1996).
Chapter3. Magnetically Controlled Liquid-Crystal-Based Terahertz Tunable
Solc Filter
3.1. Introduction
Spectral filters can be based on the interference of polarized light. These filters play an important role in many optical systems where filters of extremely narrow bandwidth with wide angular fields or tuning capability are required. In Solar Physics, for example, the distribution of hydrogen may be measured by photographing the solar corona in light of the Ha (λ=6563Ao ) line. In view of the large amount of light present at neighboring wavelengths, a filter of extremely narrow bandwidth (~ 1λ =Ao ) is required if reasonable discrimination is to be attained. Such filters consist of birefringence crystal plates (wave plates) [1]. The two basic versions of these birefringence filters are Lyot-Ohman filters [2-5] and Solc filters [6, 7]. The Solc filter a type of birefringence filters widely employed in the visible and near infrared, consists of a stack of identical birefringence plates with folded azimuth angles between crossed polarizers or fanned azimuth angles between parallel polarizers. Solc filters can be made tunable using active birefringent retarders such as electro-optic crystals or LC cells. For example, electro-optic Solc-type wavelength filter have been demonstrated in near infrared tuning from 1532.67 to 1529.35nm [8]. In this chapter, we construct and characterize a
controlled retardation in LC.
3.2. Theory of Solc filter
The folded Solc filter as Fig. 3.1 is composed of a series of even half-wave plates between crossed polarizers and each of wave plates must be oriented at the azimuth angle ρ and – ρ alternatively with respect to the polarization of incident electromagnetic wave.
Fig. 3.1. A six-stage folded Solc filter
Calculating by Jones matrix formula, the transmittance can be simplified into ρ
N Sin
T = 22 .The transmissivity is 100% if the azimuth angle is equal to 4N ρ = π .
The polarization mechanism of light after passing six-stage folded Solc filter can be understood as the picture shown in Fig.3.2.
ρ
Fig. 3.2. The light path mechanism after passing six half wave plates.
According to Fig. 3.2, we can deduce that the final azimuth angle of polarized light after N plates is 2Nρ. Furthermore, if the final azimuth angle of polarization is equal to 90°at central frequencies, the light passes through the rear polarizer without any loss of intensity but light at other frequencies will have losses. The central frequency fc is given by:
...,
where no and ne are refractive indices of ordinary and extraordinary wave, respectively, d is the thickness of each wave plate, and c is the speed of light in vacuum, m is the order of the half-wave plates. On the other hand, the transmittance can be transferred into
) ]
where
ν
m is the central frequency respecting to the order of half wave plates The large birefringence of LCs is well known and has been employed successfully for phase shifting of microwave and millimeter wave signals previously. In THz frequency range, we have recently determined the complex refractive index of a NLC E7 at room temperature by THz time-domain spectroscopy (THz-TDS) [9]. The birefringence of NLC:where θ is the angle between optical axis of LC and direction of propagation of light, therefore, if we vary θ, according to eq. (3.4), birefringence will be changed and then according to eq. (3.1), the central frequency could be tuned.
3.3. Design and measurement
Technologically, we apply two orientations of magnetic field to the same LC cells to avoid magnets blocking THz wave. Therefore, the large tunable range of two folded THz filter is obtained. One of two setups consists of two LC cells and two annular permanent magnets, which are fixed on rotary stages to re-orientate the molecules of LC as shown in the inset (a) of Fig. 3.3. The homeotropic LC cells are used in our filter. The cells are coated with N,N-dimethyl-n-octadecyl- 3-aminopropyltrimethoxysilyl chloride (DMOAP) [10], to align the LC molecules perpendicular to the surface of fused silica plates. Each cell is constructed with two fused silica plates with aluminum spacers and filled with E7 (Merck) whose ne and no
are about 1.71 and 1.57 in the THz frequency range. The LC-based Solc filter has two folded retarders which are placed between a pair of parallel wire-grid polarizers (Specac, No. GS57204 with an extinction of > 1000). The LC cells as tunable retarders (TR) are at the center of the rotatable magnet. TRs are oriented at 22.5° and - 22.5° with respect to the polarization of input light used to achieve the desired variable phase retardation, ΔΓ (Fig. 3.3).
The thickness of LC layers is 5.7 mm. The threshold field required to reorient LC molecules in the LC cell when the magnetic field is perpendicular to the alignment direction is less than 0.001 Tesla [11]. The maximum magnetic field at cell position in the rotary permanent magnets (sintered Nd-Fe-B) is 0.427 Tesla ensuring that all LC molecules are parallel to the magnetic field. The retardation provided by homeotropic cells-TRs is zero when the direction of LC molecules is parallel to the z-axis and changes with the reorientation of the LC molecules by rotating magnets.
The other setup consists of the same LC cells but two pairs of cylindrical magnets, which are also fixed on rotary stages to re-orientate the molecules of LC as shown in the inset (b) of Fig. 3.3. The LC cell is put in the middle of two magnets and LC cells are reoriented at the same mechanism to achieve a desired variable phase retardation, ΔΓ. Fig. 3.4 also shows experimental pictures of the LC-based tunable THz Solc filter , relative to the schematic diagram in Fig. 3.3 (a) and (b), respectively.
Fig. 3.3. Schematic diagram of the LC-based tunable THz Solc filter. The inset (a) and (b) show two different ways to apply magnetic field on the same tunable LC THz retarders, respectively.
Polarizer
Y
X
ρ
Z
Analyzer
Tunable retarder (b)
Fig. 3.4. Experimental setup of the LC-based tunable THz Solc filter. The Fig. (a) and (b) refer to the schematic diagram in Fig. 3.3 (a) and (b), respectively.
(a)
(b)
However, the magnetic field at LC cell position in this setup is 0.19 Tesla which is large enough to align all LC molecules. The retardation provided by TRs is the maximum when the direction of LC molecules is perpendicular to the z-axis and decreases with changing the rotating angle of magnets. Besides filling E7 in the homeotropic cell, we filled LC-DN125262W (daily polymer corporation) in homogeneous cells, which are coated with polyimide following by rubbing to align the LC molecules parallel to the surface of fused silica plates, and with aluminum spacers. Therefore, the high birefringence (~ 0.18) LC-DN125262W made the central pass-band frequency lower. The characteristics of DN125262 such as refractive indexes have been discussed in Sec.2.5.
The filter was characterized by using a photoconductive-antenna-based THz time-domain spectrometer (THz-TDS) [12]. In short, the pump beam from a femtosecond mode-locked Ti:sapphire laser illuminated the antenna fabricated on low-temperature-grown GaAs to radiate the broad band THz pulse, which was collimated and propagated through the filter. The transmitted THz pulse was monitored by the probe beam from the same laser and the same kind of antenna. The measurements are done at room temperature (~25ºC).
3.4. Results
Fig. 3.5 shows the spectral transmittance of the filter filled with E7, converted by taking into account the parallel-polarization configuration, at the 90± between magnetic field and z axis. The transmitted peak frequencies are 0.176 THz, 0.556 THz, 0.892THz, periodically, which is consistent with the relation in formula (3.1).
The theoretical curves (solid curves) are also shown and calculated by considering the absorption of LC molecules [9], which is in good agreement with the experimental data.
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
Experimental data Theoreticalcurve
Transmittance
Frequency (THz)
Fig. 3.5. An example of the transmittance of the broadband THz pulse through LC Solc filter filled with E7. The circles are experimental data and the solid line is theoretical prediction.
Furthermore, with the rotation angle from 90° to other smaller angles, transmitted THz spectrum of the filter is shifted to higher frequencies; the band width
rotation angle from 90° to 70°, transmitted THz spectrum and the band width are both in accord with the formula (3.2) and (3.3) respectively. On the other hand, although the imaginary parts of ordinary and extraordinary refractive indexes (κo,
κe) of NLC are different, result in filtering effect imperfect, central pass-band frequencies are almost the same as prediction. This phenomenon can be explained by the coupled mode theory [1]. Two energy modes are coupled in this quasi-period filter. The incident light is converted from one mode to the other, as the phase matching condition is satisfied. While considering the difference betweenκoand κo, the phase matching condition will be not satisfied and energy can’t be transformed completely. Therefore, it confines the filtering effect of a THz Solc filter.
Fig. 3.6. An example of the transmitted spectrum of the broadband THz pulse which is tuned using a LC (E7) THz Solc filter. The insert is time domain profiles. The circles and squares are experimental data and the solid lines are the theoretical prediction.
According to the dexterous tunable retarders in Fig. 3.3, the first order central frequency of a Solc filter is tuning from 0.176 to 0.793 THz (Fig. 3.7) by rotating the magnetic field from 90± to 30± referring to z axis. Therefore, tunable range of a THz filter is largely improved.
30 35 40 45 50 55 60 65 70 75 80 85 90
The first order central frequency (THz)
Rotational angle (degree)
Fig. 3.7. The first-order central frequencies of the filter filled with E7 versus rotation angle θ between magnetic field and z axis. The circles are experimental data. The curve is theoretical calculation.
Besides infusing E7 into the 5.7mm LC cells, we also infused DN125262W-high birefringence LC into 3.15mm LC cells to reduce thickness of the filter. Although this kind of LC has larger birefringence, its absorption increases with frequency apparently, so that it is hard to use in higher frequencies. The insertion loss is ~3.7 dB at 0.3THz. If we compare with results of E7, it reduces the insertion to~
1.5dB. Fig. 3.8 is another example of LC solc filter filled DN125262W; Fig. 3.9 is the relation between rotation angle and the first order central frequency. Therefore, no matter which kind of LCs is applied, the experimental results both consistent with
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2
Fig. 3.8. An example of the transmitted spectrum of the broadband THz pulse through LC Solc filter filled DN125262W. The circles are experimental data and the solid line is theoretical prediction.
55 60 65 70 75 80 85 90
first order central frequency (THz)
rotational degree
Fig. 3.9. The central transmitted frequencies of the filter filled with DN125262W versus rotation angle θ between magnetic field and z axis. The circles are experimental data. The curve is theoretical.
3.5. Discussion
Fig. 3.10 (a) and (b) shows the profiles in time domain of the Solc filter and the reference cell whose THz polarization is parallel to the direction of LC molecules.
Theoretically, a THz wave through the filter is separated into three main peaks, which can be understood as following: the THz wave is separated into o-ray and e-ray after passing the first LC cell and further these two waves are separated into o-o ray, o-e ray, e-o ray, and e-e ray after passing the second one. Moreover, two LC cells have same retardations so that e-o ray and o-e ray overlap with each other;
again e-e ray with the largest retardation overlaps with the main signal of the reference (ne) due to the same total thickness between filter and reference cells; o-o ray with the smallest retardation is the early signal, but due to the larger absorption according to κo comparing with κe [9], the o-o ray is not revealed apparently.
Therefore, in Fig. 3.6 only two main peaks are appeared in the time domain profile of this Solc filter. Besides, we also calculate the time delay between each peak. In E7 based filter, the time delay between e-e ray and o-e (or e-o) ray is 2.755 ps theoretically, which is closed to 2.735 ps in the measurement; in DN125262 based filter, the time delay between e-e ray and o-e (or e-o) ray is 1.932 ps theoretically, which is closed to 2.001 ps in the measurement.
0 4 8 12 16 20 24
Fig. 3.10. The temporal profiles of a THz pulse through (a) E7 based Solc filter and reference LC cells (b) DN125262W based Solc filter and reference ones.
(a)
(b)
3.6. Comparison with our previous work: a THz Lyot filter
Comparing with our previous work, a THz tunable Lyot filter [13] whose tuning range is from 0.388 to 0.564 THz and insertion loss is 8 dB. However, the tunable range of THz E7 based Solc filter is largely increased from 0.176 to 0.793 THz and the insertion loss is about 5 dB. Moreover, two elements utilized in the Solc filter instead of four elements used in the Lyot filter, we promote practicability and convenience of a THz filter. Furthermore, a Lyot filter has intrinsic confinement of different thicknesses in each retarder whereas a Solc filter can have a uniform thickness easily operated in various designs.
Lyot Solc
Setting 4 components 2 components
Tunable Range
0.388~0.564THz (0.18THz)
0.176~0.794THz (0.61782THz)
Bandwidth 0.1THz 0.117~0.465THz
Insertion Loss
~8dB ~5dB
Natural Confinement
Non-filtering low frequency waves fixed at zero
Easily used in any frequency range
Table 3.1. The compassion of Lyot and Solc filter in THz range.