Stability in FDTD Algorithm - Finite-Difference Time-Domain (FDTD) Algorithm

2.3 Finite-Difference Time-Domain (FDTD) Algorithm

2.3.2 Stability in FDTD Algorithm

Now we discuss that how we determine the time step ∆t. An EM wave propagating in free space cannot go faster than the speed of light.

To propagate of one cell requires a minimum time of . When we get to two-dimensional simulation, we have to allow for the propagation in the diagonal direction, which brings the time requirement to

three-dimensional simulation. This is summarized by the well-known

“Courant Condition” [22]

t x

n c

∆ ≤ ∆

⋅

(2.3.3) where n is the dimension of the simulation.

It is convenient to specify the time step for any simulation in (2.3.4).

2 t x

c

∆ = ∆

⋅

(2.3.4)

2.4 The Drude Dispersion Model for Metal

In keeping with IR spectroscopic notation, all frequencies will be expressed in cm⁻¹. The complex dielectric function ε_c and the complex The Drude model dielectric function is [23]

2 imaginary parts yields

In these equations, the plasma frequency is

2 1 the effective mass of the electrons, and

N e m^*

light.

If we have a group of ( , )ε ε₁ ₂ at any frequency, from (2.4.3) we can calculate the value of ω_p and ω_τ by solving the simultaneous equation.

2 1

2 2

(1 )

(1 )( )

p τ

τ2

ω ωε

ε

ω ε ω ω

= −

= − +

(2.4.6) This was done for several values of co to obtain several pairs of co, and cop, which produce the curve with the best eyeball fit to the data.

3 Experiments and Simulated Methods

3.1 THz-TDS using LT-GaAs photoconductive antanna

The THz-TDS system we used was shown in the Fig. 3-1, and transient current was generated by femtosecond laser pulses impinging on a photoconducting material, dipole antenna on a LT-GaAs substrate.

Emission of electromagnetic pulses, THz radiation, of about picosecond duration were produced by this current. The THz pulse was collected and guided by gold-coated parabolic mirrors. Femtosecond laser pulse as a probe beam with time delayed by motor stage .Also using the LT-GaAs antenna to detect the THz wave. Finally, connect the detected voltage signal to the lock-in amplifier. Signal from the lock-in amplifier can be obtained using PC easily. The photo of our THz-TDS system is shown in Fig. 3-2.

Because of its unique properties (high carrier mobility, high dark resistivity and subpicosecond carrier lifetimes) low temperature-(LT) grown GaAs is widely used to be the material for the THz emitters or detectors [24]. Post-growth annealing of LT-GaAs at temperature above 600°C leads to the nucleation of excess arsenic in As precipitates. The influence of As precipitates on the high resistivity of annealed LT-GaAs is related to As clusters, which is also responsible for the fast recombination of the photoinjected carriers. Recently, the trapping time reduction was observed for heavily doped LT-GaAs.

(1)

(5)Mirror coated with Ag (6)Object Lens

(7)Emitter:LT-GaAs Photoconductive Antenna with Silicon Lens

(8)Parabolic Mirror coated with Au (9)Detector:LT-GaAs Photoconductor

Antenna with Silicon Lens (10)Optical Delay Stage

(5)Mirror coated with Ag (6)Object Lens

(7)Emitter:LT-GaAs Photoconductive Antenna with Silicon Lens

(8)Parabolic Mirror coated with Au (9)Detector:LT-GaAs Photoconductor

Antenna with Silicon Lens (10)Optical Delay Stage

(9)

Fig. 3-1 Terahertz Time-Domain Spectroscopy in our lab.

Emitter Detector

THz Emitter

Detector THz

Fig. 3-2 The photo of our THz-TDS system. The photoconductive emitter and detector forms a symmetry type can be seen.

We also use the silicon lens to make the second reflective signal from the substrate far away from the expected time domain waveform.

There are three possible silicon lens designs are depicted in Fig. 3-3 [4].

In the hemispherical design (Fig. 3-3 (a)), the emitted source is located at the center of the silicon lens all rays exit the lens/air interface at normal incidence. The radiation pattern inside the substrate is preserved in the transition to free-space because no refraction occurs. The design in Fig.

3-3 (b) is the aplanatic hyper-hemispherical substrate lens. Like the hemispherical design, it has no spherical aberration or coma, and when using silicon as a lens material, no chromatic dispersion. The design specification for the aplanatic design is

r n

ρ =

(3.1.1) where ρ is the distance from the center of the lens to the focal point, r is the radius of the lens, and n is the refractive index. This design provides a slight collimation of the beam. The design in Fig. 3-3 (c) locates the dipole source at the focal point of the lens so that geometrical optics predicts a fully collimated output beam. However, diffraction is important because of the long wavelength of the THz wave compared to the lateral dimension of the beam exiting the lens, and, unless substrate lenses with a diameter much larger than the wavelength are used, the beam propagates with a larger divergence compared to the other designs.

(a) (b) (c)

Fig. 3-3 Three different silicon design. (a) the non-focusing hemispherical design. (b) the hyper-hemispherical lens, and (c) , the collimating substrate lens.

The type of our silicon lens is hyper-hemispherical lens, the radius is 6.75mm and the total thickness is 8.35mm. When using the hyper-hemispherical silicon lens, the focal length of the first collective Au-coated parabolic mirror is changed due to the silicon lens, and we can use the equation in geometric optics. For example, in Fig. 3-4, this is the case of our setup, the thickness of the GaAs substrate is 0.355mm. If we want to know where is the imaging point of the dipole antenna through the hyper-hemispherical silicon lens, we can use the well-known equation in geometric optics.

1 ( 1) '

n n

l r

= − −

l

(3.1.2)

where is the object distance, is the image distance and r is the radius of the silicon lens. n is the refractive index of silicon in THz region, the value is about 3.5. Filling the values into the equation, we can get

l l'

1 2.5 3.5

' 31.548 ' 6.75 8.705 l m

l = − ⇒ = − m

Therefore, the imaging point is located in the left side of GaAs substrate and the distance L is 31.548-8.705 = 22.843mm. If the focal length of the first parabolic mirror is 7.5cm, we have to make it short to 7.5-2.284 = 5.216mm.

0.355mm 1.6mm

6.75mm L

0.355mm 1.6mm

6.75mm L

Fig. 3-4 The schematic diagram of silicon lens and GaAs substrate. The radius of silicon lens is 6.75mm and the total thickness is 8.35mm. The thickness of the wafer is 0.355mm. The imaging point of the dipole antenna through the silicon lens is located in the left side with L distance.

The effective focal length of the first parabolic mirror is very critical, so we strongly recommend that combining the emitter with the object lens in one stage shown in Fig. 3-5. That will be convenient for avoiding missing the focus of femtosecond laser on the photoconductive antenna when aligning the effective focal length of the first parabolic mirror.

Fig. 3-5 Photo of the setup of the THz emitter and the object lens. The emitter is combined with the object lens in one stage.

In our system, THz wave is parallel between two parabolic mirrors, the MHA is place on the middle of the optical path. We put a pinhole beyond the sample in order to control the THz beam size passing through the MHA. Assuming that the THz beam was spatial Gaussian distribution, the THz radiation can be considered as a plane wave, i.e. normal incident to the sample, as shown in Fig. 3-6. For convenience, we define the transmission direction is z-axis and the periodical structures is distributed in x-y plane.

MHA

Incident THz wave

Transmitted THz wave

x

y

MHA

Incident THz wave

Transmitted THz wave

x

y

x

y

Fig. 3-6 Schematic diagram of the transmission properties of THz wave through MHA. THz wave is parallel between two parabolic mirrors and normal incident to the sample.

We measure THz radiation propagating through MHA or not as a signal or a reference, respectively. Two time-domain waveforms can be used to obtain frequency-domain spectra using numerical fast Fourier transform, and then the MHA signal divided by the reference in frequency domain should be the transmittance of this sample. The transmission properties of certain sample with a board band frequency spectrum can be observed. From the above steps, we can obtain the basic transmission properties of MHAs.

3.2 The Structure of MHAs and the sample fabrication

Our sample is a aluminum film perforated periodically with hexagonal lattice of circular holes shown in Fig. 3-7, and the photo of the real sample is shown in Fig. 3-8. The spacing between holes ,s is 995µm and the diameter of holes ,d is 565µm. The thickness of MHA ,t is about 500µm. These parameters can be controlled to change the transmission properties of MHA.

995 s = µ m

565 d = µ m t = 500 µ m

995 s = µ m

565 d = µ m t = 500 µ m

Fig. 3-7 The structure of MHA. hole diameter d=0.56 mm, lattice constant s=0.99 mm, and thickness of the metal plate t=0.5 mm

Fig. 3-8 The photo of the real MHA sample.

In order to investigate the effect of hole material on MHA, we filled UV-gel into the holes. The UV-gel we used is produced by Norland company and the number of this product is 68. The reason we chose UV-gel is due to the negligible absorption and the flat curve of dielectric function. The procedure of filling UV-gel into the holes of the MHA is shown below and drawn in Fig. 3-9.

1. We let the MHA lay on the UV-gel from top to bottom in order to avoid producing the bubble within the UV-gel.

UV-gel (MHA) UV-gel

(MHA)

MHA filled with UV-gel MHA filled with UV-gel

Fig. 3-9 (a) Procedure 1 of fabricating the MHA which holes filled with UV-gel. Let the MHA lay on the UV-gel from top to bottom, and then a MHA filled with UV-gel can be obtained.

2. Devising a mold for fixing the MHA to benefit thinning the thickness of the MHA.

The mold The mold

Fig. 3-9 (b) Procedure 2 of fabricating the MHA which holes filled with UV-gel. Devising a mold for fixing the MHA.

3. Using the miller to drill the MHA, and to obtain a thickness t of the MHA which holes filled UV-gel we want by controlling the miller.

The miller The miller

ig. 3-9 (c) Procedure 2 of fabricating the MHA which holes filled with

tt

UV-gel. Using the miller to drill the MHA, and to obtain a thickness t we want.

The photo of the real sample is shown below.

ig. 3-10 The photo of the MHA with expected thickness which holes F

filled with UV-gel.

3.3 Parameters of FDTD Algorithm

r is P4 2.4GHz with 2G RAM. For

sed to numerically simulate the THz The equipment of our compute

convenience, we use the commercial software “FULLWAVE” invented by the RSOFT company to proceed the FDTD simulation. The parameters of the simulation are listed below.

The FDTD method is u

propagation problem. In this approach, the 2D-MHA occupies the space from z = 0 to t. Terahertz wave is normally incident in the z-direction from z<0 and polarized in the x-direction. The grid spacing for FDTD calculation are 30µm in x and y dimensions and 8µm in the z-direction.

For the periodic triangular arrays with period of 990µm, the total computational grid is about 7.5mm × 8.3mm × 0.9mm. The beam size is 6mm which is large enough to avoid the finite size effect. The system is defined by specifying the relative dielectric constant, εR(x, y, z, ω). At first, the hole material and the adjacent medium are air, so we set εR = εAir = 1, and for Aluminum we set εR = εAl(ω) using the Drude model shown in Fig. 3-11 (a). For example, it gives εAl = -3.3820×104 +2.1882×106i at 0.3THz. Complex refractive index is also shown below in Fig. 3-11 (b). The grids are truncated with uniaxial perfect matching layers to simulate absorption of field components approaching grid edges where appropriate [20].

1 10 100 1000

0.03 0.3 3 30

10³ 10⁴ 10⁵ 10⁶ 10⁷ 10⁸

frequency (THz)

dielectric function (ε 1+iε 2) in Al

frequency (cm^-1)

- ε

ε

Fig. 3-11 (a) Dielectric function of Aluminum versus frequency.

10 100 1000

0.3 3 30

10 100 1000 10000

frequency (THz)

refractive index (n-iκ) in Al

frequency (cm^-1)

n - κ

Fig. 3-11 (b) Refractive index of Aluminum versus frequency.

4 Results and Discussion

4.1 Free Space THz-TDS Waveform and Spectrum

THz time domain waveform is shown in Fig. 4-1. The amplitude of oscillations after main peaks are due to the humidity can be seen in the tail of the waveform. It can speculate that the oscillation is caused by water vapor absorption.

0 20 40 60 80 100 120 140

-0.00010 -0.00005 0.00000 0.00005 0.00010 0.00015 0.00020

electric field (a.u.)

time delay (psec)

Reference

Fig. 4-1 Free space THz time domain waveform

Because of the silicon lens contacted with the substrate of antenna, we can eliminate the reflective signal from the waveform. Then, using numerical fast Fourier transform can obtain the frequency spectrum shown in Fig. 4-2. Some deep dips at 0.556, 0.754, 0.988, 1.113, 1.164, 1.208, 1.230, 1.413 THz are cause by water absorption [25].

0.0 0.5 1.0 1.5 2.0 2.5 3.0 1E-23

1E-22 1E-21 1E-20 1E-19 1E-18 1E-17 1E-16 1E-15 1E-14

power (a.u.)

frequency (THz)

Reference

Fig. 4-2 THz frequency domain spectrum

From the frequency spectrum, we can estimate the signal-to-noise ratio in our system is larger than and the bandwidth approaches 2THz.

Because this spectrum was measured in free space without passing any sample, we call this signal is reference signal.

106

4.2 Characteristics of MHAs

MHA in THz region is like a band-pass filter, and the normalized transmittance at peak frequency is about 260% to the porosity of the sample. The porosity means the area of holes divided by the total area of illuminating, the value is about 0.29 in our sample. Theory of this extraordinary phenomenon is still unclear; the widespread explanation is

the coupling between the incident electromagnetic wave and surface plasmon polariton. Recall the equation (2.2.26) about the resonant peaks in triangular structure,

2 2 2

max 0

[ ( 4 )]

3 a i ij j

λ = + +

⁻

ε

(2.2.26)

the maximum transmittance only concern with the spacing and the adjacent material. This phenomenon has been verified by experiments, for instance, the peak can be tuned when the adjacent dielectric material is changed from air to tape. However, in this formula, the max.

transmittance peak seems to be independent of the thickness of the MHA and the material stuffed with the holes. We want to check this point ,and then some brand-new phenomena happened in our experiments that the formal experiences can’t to expound.

4.2.1 Basic Transmission properties of MHAs

First, we measure the THz waveform in free space as the reference, and then we measure THz propagating through MHA as the siganl. Two time-domain waveforms can be used to obtain frequency-domain spectra using numerical fast Fourier transform, and dividing MHA signal by the reference in frequency domain will be the transmittance of this sample.

The transmission properties of certain sample with a board band frequency spectrum can be observed. From the above steps, we can obtain the basic transmission properties of MHAs.

From the above steps, we can obtain the basic transmittance properties of MHAs shown in Fig. 4-3 (a) is the time domain signal

FFT from (a). (c) is the power transmittance, we can observe a obvious characteristic of band-pass filter, and the magnitude of the peak almost get up to 100%. The little peaks at 0.55 and 0.75THz are the inaccuracies owing to the water absorption.

The finite time extent of the terahertz pulse scans, 136 ps (in 1024 steps), limits the frequency resolution of the numerical Fourier transforms in THz-TDS. To perform a numerical interpolation between the measured frequency points, the measured pulses in the time domain were extended with zeros (zero padding) [11] to a total time duration of 1500 ps.

0 20 40 60 80 100 120 140

-0.0002 -0.0001 0.0000 0.0001 0.0002 0.0003

electric field (a.u.)

time delay (psec) Ref.

MHA signal amplified to 300%

Fig. 4-3 (a) The time domain signal compared with the reference

0.0 0.5 1.0 1.5 1E-25

1E-24 1E-23 1E-22 1E-21 1E-20 1E-19 1E-18 1E-17 1E-16

2.0

power (a.u.)

Frequency (THz)

Ref.

MHA

Fig 4-3. (b) The frequency domain signals via the FFT from (a)

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

0.0 0.2 0.4 0.6 0.8 1.0

water absorption power transmittance of the MHA

power transmittance

frequency (THz)

Fig. 4-3 (c) The power transmittance of the MHA. A obvious characteristic of band-pass filter, and the magnitude of the peak almost

In order to grasp this phenomena more, we use the FDTD algorithm to simulate this structure. It is successful that we got almost the same result compared with the experiments. Owing to the lack of our RAM, the spectral linewidth of the peak is wider slightly than the experiment, but the peak frequency is also located at 0.3THz nearby. So we can affirm that this simulated tool is accurate enough to express the real situation.

The simulated results shown Fig. 4-4 (a) is the time domain signal compared with the reference. (b) is the frequency domain signals via the FFT from (a). (c) is the power transmittance.

0 20 40 60 80 100 120 140

-0.25 -0.20 -0.15 -0.10 -0.05 0.00 0.05 0.10 0.15 0.20 0.25

electric field (a.u.)

time delay (psec) Ref.

MPC signal magnified to 300%

Fig. 4-4 (a) Simulated result: The time domain signal compared with the reference

0.0 0.3 0.6 0.9 1.2 1.5 1E-20

1E-19 1E-18 1E-17 1E-16 1E-15 1E-14 1E-13 1E-12 1E-11 1E-10

power (a.u.)

frequency (THz)

Ref.

MHA

Fig. 4-4 (b) Simulated result: The frequency domain signals via the FFT from (a)

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

0.0 0.2 0.4 0.6 0.8 1.0

power transmittance of the MHA

power transmittance

frequency (THz)

Fig. 4-4 (c) Simulated result: The power transmittance of the MHA. The spectral linewidth of the peak is wider slightly than the experiment, but the peak frequency is also located at 0.3THz nearby.

From both experimental and simulated results above, we can observe a almost monochromatic waveform in time domain that can response to the peak at 0.3 THz in the power transmittance.

4.2.2 Using SPP Model to explain the extraordinary transmission peak The MHA we used is an Aluminum plate perforated with an array of triangular holes (hole diameter d=0.56 mm, lattice constant s=0.99 mm, and thickness of the metal plate t=0.5 mm). For this sample, the cut-off frequency is νcutoff=0.311 THz. In Fig. 4-3 (c), we can observe a obvious peak at 0.301THz which is left to the cutoff frequency. The enhanced transmission is commonly believed to be due to the resonant coupling of incident light with SPPs.

Nevertheless, the peak frequency calculated from eq.(2.2.26) is 0.348THz which is not accurate enough to the observed peak at 0.301THz shown in Fig. 4-5 (a). Furthermore, we checked the MHA with cubic arrays with the same parameters (d=0.56 mm, s=0.99 mm, and t=0.5 mm) to the triangular one using FDTD algorithm, the theoretical peaks also laid on the right side to the simulated peaks, as shown in Fig 4-5 (b).

Although the peak positions are not the same with SPP model, they are still very close to each other. And the spacing between the two peaks in Fig. 4-5 (b) show almost identical range. For this reason, we can presume SPPs play an important role in the extraordinary transmissions.

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.0

0.2 0.4 0.6 0.8 1.0

0.301THz 0.348THz

Power transmittance

Frequency (THz)

Fig. 4-5 (a) Using SPP model to estimate the transmission peak in the triangular MHA. The peak frequency calculated from eq.(2.2.26) is 0.348THz right to the observed peak at 0.301THz

0.0 0.2 0.4 0.6 0.8 1

0.0 0.1 0.2 0.3 0.4 0.5

0.426THz 0.301THz

0.381THz 0.271THz

power transmittance

frequency (THz)

Fig. 4-5 (b) Using SPP model to estimate the transmission peak in the cubic MHA. The peak frequencies calculated both are right to the the

4.2.3 Altering the thickness of MHAs

In order to make sure that the maximum transmittance peak frequency won’t change with the thickness of MPC. We use two different samples to do experiments. The structure of the first sample is s (spacing)

=1.13mm, d (diameter) =0.68mm and the second sample is s =0.99mm, d

=0.56mm. From the experimental results shown in Fig. 4-6, we find only the cutoff frequency shifts to left, but the peak frequency is almost invariable. This agrees with the prediction from the equation (2.2.26).

0.1 0.2 0.3 0.4 0.5

0.0 0.2 0.4 0.6 0.8 1.0

d=0.68mm,s=1.13mm 0.20 mm

0.25 mm 0.30 mm 0.50 mm 1.00 mm 2.00 mm

power transmittance

frequency (THz)

Fig. 4-6 (a) Altering the thickness of the first sample. The structure of the first sample is s =1.13mm, d =0.68mm.

0.2 0.3 0.4 0.5 0.0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

power transmittance

frequency (THz)

d=0.56mm,s=0.99mm 500µm

400µm 300µm

Fig. 4-6 (b) Altering the thickness of the second sample. The structure of the second sample is s =0.99mm, d =0.56mm.

4.3 Features of MHAs when their holes filled up with UV-gel

We recently demonstrated a THz tunable filter by controlling the index of refraction of nematic liquid crystal filling the holes and adjacent to the MHA on one side. New phenomena appeared when holes of the MHA are filled with dielectric material. The effect of filling dielectric material into the holes cannot simply be explained by increased effective hole diameter of the 2D-MHA and equation (2.2.26) also cannot predict the peak frequencies. Therefore, we want to fill some dielectric material

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