6. Tomographic terahertz imaging 65
6.2. Optical ray tracing simulations
6. Tomographic terahertz imaging
6.2. Optical ray tracing simulations
starting point~sand its directionm, with~ |m~|=1. For each ray a possible intersection with the circle of radius r and with a center position~cis checked using the inequation:
B2−4AC>0 (6.3)
A=m2x+m2y (6.4)
B=2
(sx−cx)·mx + sy−cy
·my
(6.5) C = (sx −cx)2+ sy−cy2
−r2 (6.6)
Further an intersection between each ray and the detector is checked using the three inequations 6.7 and the three Eqs. 6.8 - 6.10:
q¶1 &q¾0 & p>0 (6.7)
q=
sy−d1,y
d2,y−d1,y, d2,y 6=d1,y &|my|=0 &|mx| 6=0
d1,x−sx−mxm y(d1,y−sy) η
(6.8)
η= mx
my d2,y−d1,y
− d2,x−d1,x
(6.9)
p=
d1,x+(d2,x−d1,x)·q−sx
mx , d2,y 6=d1,y &|my|=0 &|mx| 6=0
d1,y−sy+(d2,y−d1,y)·q my
(6.10)
Here, the detector is represented by a line spanned by the two points d~1 and d~2. If a ray intersects with both the detector and the circle, the distance between the starting position~s of the ray and the intersection points~his calculated. In case the ray hits the detector first the ray ends and is marked as detected. Otherwise a new ray is casted, with its starting position at the intersection point and a direction calculated using Snell’s law. The intensity of rays traveling through the circle gets attenuated considering the absorption coefficient of the sample and the length traveled through the object. Also losses due to reflection are taken into account using the Fresnel equations for unpolarized light and non-magnetic media.
Figure 6.7(a) exemplary shows the ray tracing results of four ray bundle and detector arrangements as they are found during the raster scanning of a circular object. First the ray bundle passes the object and is directly measured by the detector. Second a minor part of the rays still passes the object and reaches the detector. But the majority of rays get strongly refracted on the bended surface of the circular object and therefore do not reach the small surface of the detector. By moving the detector and the ray bundle towards the center of the circular object, as shown in the third case, all the rays get refracted away from the detector. In the last case the ray bundle is at the same height as the center of the circle. Here, the rays get focused onto the detector by the bended surface of the object. But the intensity of the rays is
6. Tomographic terahertz imaging
lowered due to the absorption of the sample object. A simulated projection of the circular object is shown in Fig.6.9(a) using the raster scanning method and a small detector area. The object is scanned using 64 start positions of the ray bundle and the detector. Aside from the object a maximum transmission through the surrounding atmosphere can be seen. This suddenly changes when the rays intersect the object and the detected intensity approaches zero very fast. The intensity stays at zero until the ray bundle approaches the center of the circular sample object. Due to the lensing effect, the rays get focused on the detector and an intensity according to the absorption of the object is detected. The achieved simulated projection resembles the projection measured using the HSM technique in the case of a unfilled Teflon cylinder shown in Fig. 6.4.
Next, the small detector is replaced by a detector with a large surface as shown in Fig. 6.7(b). Again the circular sample object is scanned using 64 ray bundles starting from different x-positions, but this time the detector stays at a fixed position. From the ray tracing simulations it can be seen, that even strongly refracted rays reach the detector surface. For each ray bundle the intensity of the rays reaching the detector is summed up and correlated to the x-position of the corresponding bundle. In this way, the simulated projection shown in Fig. 6.9(b) is obtained. Besides two sharp transmission minimums at the sample boarder, the projection looks similar to the projection received using x-ray tomography of a circular shaped object. But the actual values of the intensity at the detector might differ, since the rays get refracted and might travel a longer distance inside the absorbing media in case of terahertz tomography.
This causes a stronger absorption compared to rays propagating on straight lines, which is the case for x-rays.
Figure 6.8.: Ray tracing simulations for the projection imaging of a circular sample object using (a) a multi-pixel detector and (b) the single-pixel imaging approach. In (b) two out of the128 necessary irradiation patterns are selected and shown as example.
Another possibility to measure the projection of an object is the use of a detector array as shown in Fig. 6.8(a). Here, the red line indicates 64 equally sized and spaced detectors, each summing up the intensity of rays hitting the corresponding pixel. In that way the intensity for each pixel is recorded and
6.2. Optical ray tracing simulations
an image of the object is take. A total of 104equally spaced rays are casted and their traces to the detector are calculated. For clarity, Fig. 6.8(a) only shows the trace of every hundredth ray. The evaluation of the intensities measured at each detector pixel is shown in Fig. 6.9(c). The intensities on the left (0 to 5 mm) and right (20 to 25 mm) side of the projection mostly represent the rays passing the object. However, coming from the boarders of the projection, the intensity increases slowly when approaching the object boarder. This increase in intensity is caused by rays, which intersect with the top and bottom part of the object and as a result get strongly refracted. The simulated projection shows an abrupt decline of the intensity at the positions 5 and 20 mm, which corresponds to the top and bottom border of the object.
Followed by these two positions towards the center of the projection, the intensity rises and reaches a small plateau of a maximum transmission through the object. This maximum reaches an intensity level close to the transmission level through the surrounding atmosphere. This can be understood by regarding the circular object as a lens, which focus the rays onto the two central pixels of the detector. Thus, a displacement of the detector in z-direction would lead to a different appearance of sample projection.
The last imaging approach investigated here is the so-called single-pixel imaging already introduced in
Position / mm10 15 20 25
Intensity / arb. U.
0 20 40 60 80 100
5 0 (a)
Position / mm10 15 20 25
Intensity / arb. U.
10 30 50 70 90
5 0 (b)
Position / mm10 15 20 25
Intensity / arb. U.
20 60 100 140 180
5 0 (c)
Position / mm10 15 20 25
Transmission
0.1 0.3 0.5 0.7 0.9 1.1
Losses due to reflection
5 0 (d)
Figure 6.9.: Simulated projections of a cylindrical sample object with a refractive index of1.5and an absorption coefficient of1cm−1using ray tracing of the different imaging concepts. The object projections are obtained by raster scanning of the object using (a) a small and (b) a large detector, (c) taking an image using a multi-pixel detector and (d) calculating the image using the single-pixel imaging technique.
6. Tomographic terahertz imaging
section 2.3.2. The utilized detector is the same bucket detector as in the case of the previously described raster scanning using a large surface detector. But instead of raster scanning the object with 64 ray bunches, rays forming different masks defined by a 64×64 Hadamard matrix are casted towards the object. The results of the ray tracing simulations are shown exemplary for the 3r d and 20th pattern mask in Fig. 6.8(b). The detector measures the total intensity for each radiation pattern to obtain the measurement vectorM~, from which afterwards the projection of the object is calculated using Eq. 2.23.
The reconstructed projection is shown in Fig. 6.9(d). It is noted, that the projection employing single-pixel imaging looks the same as the projection received using raster scanning in combination with a large area detector (Fig. 6.9(b)). As already discussed before, the projection resembles a projection using x-rays.
The only differences are two absorption spikes near the positions 5 and 20 mm, which are the result of strongly refracted rays. Further, the rays at these positions incident on the object under large angles, which leads to high losses due to reflection.
Since the investigated object possesses a rotational symmetry, the reconstruction of the two-dimensional
Position / mm 10 15 20
Position / mm
0 5 10 15
20 10
20 30 40 50 60
5 0
Absorption / arb. u.
Position / mm 10 15 20
Position / mm
0 5 10 15
20 10
20 30 40 50 60
0 5
Absorption / arb. u.
Position / mm 10 15 20
Position / mm
0 5 10 15
20 10
20 30 40 50 60
0 5
Absorption / arb. u.
Position / mm 10 15 20
Position / mm
0 5 10 15
20 10
20 30 40 50 60
Absorption / arb. u.
0 5
(a) (b)
(c) (d)
Figure 6.10.: Resulting reconstructions using the simulated projection employing raster scanning us-ing (a) a small and (b) a large detector surface, (c) a multi-pixel detector and (d) the single-pixel approach.