- Email: [email protected]
Terahertz time-domain spectroscopy for explosive imaging
ARTICLE IN PRESS
Optik
Optics
Optik 118 (2007) 325–329 www.elsevier.de/ijleo
Terahertz time-domain spectroscopy for explosive imaging Zhengwei Zhang, Yan Zhang, Guozhong Zhao, Cunlin Zhang Department of Physics, Capital Normal University, Xisanhuan Beilu 105, 100037 Beijing, China Received 7 October 2005; received in revised form 17 January 2006; accepted 23 March 2006
Abstract Terahertz (THz) imaging which is a new technology for material classification and nondestructive detection has been extensively investigated in the past decade. The time-domain waveform acquired at each point of the object by using the THz time-domain spectroscopy contains much information about the object. Processing this waveform will present the characters of the object. Several methods are adopted to generate the image of the explosive samples and results are compared and discussed. Experiment results indicate that this new imaging technology can be used for explosive detection. r 2006 Elsevier GmbH. All rights reserved. Keywords: Terahertz; THz imaging; Image processing; Exploder detection
1. Introduction Since Hu and Nuss [1] reported their first image achieved by using the terahertz (THz) time-domain spectroscopy (TDS) in 1995, this technology has been considered as one of the most promising applications of the THz wave science and technology [2–4]. Some data processing methods have been adopted to reconstruct the THz image [5]. For the THz imaging technology, there are three important factors: a powerful and compact THz generator, a high sensitivity and low noise THz detector, and a fast, efficient data processing method. A good data processing method permits the high quality image. Since the samples to be studied are diversiform, developing a single algorithm for all signal processing is impractical. Therefore, discussion and comparison of various imaging approaches are quite important and useful.
Explosive materials are quite dangerous for public safety and national defense. An effective method for explosive detection is quite useful and important. In this paper, three kinds of explosive, including TNT, RDX, and HMX, are used as samples to be imaged by using the THzTDS. Several methods are adapted to process transmitted THz time-domain data for achieving images of samples. The difference of these data processing methods are discussed and compared. It is demonstrated that three explosive materials can be distinguished in the THz image. This paper is arranged as follows: Section 2 describes the experiment setup used for imaging; Section 3 introduces some approaches for data processing; Section 4 presents some images of the explosive samples with discussion, and finally, a conclusion is drawn in Section 5.
2. Experiment setup used for imaging Corresponding author.
E-mail address: [email protected] (Y. Zhang). 0030-4026/$ - see front matter r 2006 Elsevier GmbH. All rights reserved. doi:10.1016/j.ijleo.2006.03.025
The experiment setup used is schematically shown in Fig. 1. A mode-locked Ti: sapphire femtosecond laser is
ARTICLE IN PRESS 326
Z. Zhang et al. / Optik 118 (2007) 325–329 90
P
InAs Emitter
PM4
Chopper
PM1 Si Wafer ZnTe Sensor
CBS
Amplitude (a.u.)
70
QWP
PBS
80
Delay Stage
lock in
60 50 40 30
HWP
20 PM3
Sample x-y Stage
10
PM2 Laser
Fig. 1. Schematic configuration of a THz–TDS transmission imaging setup.
0 7
8
9 Position (mm)
10
11
Fig. 2. Spatial distributing of THz electronic field along the x direction.
used as light source. It has 100 fs pulse duration, 810 nm central wavelength, 88 kW peak power, and 82 MHz repetition rate. The laser beam is split into pump light and probe light by a cubic beam splitter (CBS). The pump pulse passes through a variable time-delay stage and illuminates the /1 0 0S InAs THz emitter for generating THz wave. Two pairs of gold-coated parabolic mirrors are used to collimate and focus the generated THz beam. A high resistance silicon wafer is placed after the fourth parabolic mirror to combine the probe beam with the THz beam. This allows the probe beam to travel collinearly with the THz beam inside the 1-mm-thick /1 1 0S ZnTe crystal, where the probing beam is modulated by the electric field of the THz radiation via the electro-optic effect. A quarter-wave plate (QWP), a Wollaston prism (PBS), and a pair of photodiodes are assembled for the balanced detection of the probe beam. The signal is read by a lock-in amplifier (LIA) from the detector at 1 kHz rate and is feed into a computer. The spectrum width of the system used is between 0.2 and 2.5 THz with resolution better than 10 GHz. The dynamic range is greater than 3000 and the SNR is about 600. The sample is placed on the focal point of the second parabolic mirror. Two crossed motorized translation stages are used to move the object in the x–y plane perpendicular to the beam. At each x–y position, a complete THz time-domain spectrum is recorded by scanning the delay line. These time-domain spectra contain the information of the sample, which can be used to exhibit different characters of the object. The time used for imaging depends on the time constant of the LIA, the number of the spatial points, and the number of sampling points in the time domain. The spatial resolution of the system is determined by the size of the focal point of the THz radiation focused on the sample. Assuming that the THz beam is a Gaussian distribution, the spatial resolution of the imaging can be expressed by pffiffiffi R ¼ 2ð4l=pÞðf =dÞ, where d is the diameter of the
collimated THz beam before the second parabolic mirror, f is the focal length of the second parabolic mirror, and l is the peak wavelength of pffiffithe ffi involved broad-band THz radiation [6]. The factor 2 is introduced due to the fact that the THz electronic field rather than the intensity is measured in this system. The parameters used in our experiment are: d ¼ 2:8 mm, f ¼ 50:8 mm, and l ¼ 0:31 mm. Thus the resolution R can be calculated as1.1 mm. The size of the focal point has also been experimentally measured by using the blade scanning method. The differential of the experimentally measured data is shown in Fig. 2. It can be found that the THz beams has a very good Gaussian distribution. The full width at half maximum is about 1.1 mm, which corresponds to the theoretical expectation very well.
3. Data processing methods The TDS imaging technology collects a waveform in the time domain; therefore, it can present much information about the sample at each x–y position. One can extract the information of a discretionary point form the recorded time-domain waveform or its corresponding Fourier transform spectrum to construct an image. Fig. 3 presents a typical THz time-domain signal at a discretionary point on the sample and its corresponding Fourier transform spectrum. One can use many parameters to construct the image for expressing different characters of the object. Here, we introduce several feasible data processing methods for THz TDS imaging.
3.1. Time-domain mode Some information of the sample can be directly extracted from its THz time-domain waveform. As
ARTICLE IN PRESS Z. Zhang et al. / Optik 118 (2007) 325–329
sample. It should be pointed out that the time-domain signal reflects the synthesized effect of all frequencies contained in the THz pulse. Therefore, it exhibits the average effect of the object of the different frequencies; thus, the imaging methods in the time-domain mode have good quality but small contrast.
Amplitude (a.u.)
2
1
0
3.2. Frequency-domain mode
-1 0
4
8
12
Time (ps)
(a) 8 7
Amplitude (a.u.)
6 5 4 3 2 1 0 0.0
(b)
327
0.5
1.0
1.5
2.0
2.5
3.0
Frequency (THz)
Fig. 3. (a) THz time-domain signal and (b) its corresponding frequency-domain spectrum.
shown in Fig. 3(a), many parameters can be used to describe the characters of the object, such as the maximum amplitude, the minimum amplitude, the peak-to-peak value (the difference of the maximum amplitude and the minimum amplitude), the energy in a Rt time interval (the energy D ¼ t12 E 2 ðtÞ dt, where t1 and t2 limit a duration of time, which are usually the time of the zero amplitude before and behind the peak value), the time of the peak value, the time of the minimum value, and the time of the zero amplitude between the peak value and the minimum value. These methods present different images of an object and can be classed as two modes, i.e., the amplitude mode and the phase mode. The first four methods used time-domain amplitude information to describe the object; they can reflect the thickness and the absorption characteristic of the sample. Other methods used phase information to construct the image, which can present the thickness and refractive index distribution of the sample. Different THz images have different contrast due to the fact that they have been constructed by using different processing methods, which represent different information of the
Information of the sample can also be extracted from the frequency-domain spectrum as shown in Fig. 3(b). The maximum amplitude in the frequency domain, the amplitude of a arbitrary frequency, the phase of a discretionary frequency, the absorption at a arbitrary frequency, the refractive index at a discretionary frequency, andR the energy contained in a frequency o interval (D ¼ o12 E 2 ðoÞ do, where o1 and o2 are the limitation of the frequency interval) can be used to construct the image. These approaches present the characters of the sample at different frequencies. They can describe the absorption, thickness, and the refraction index of the object for a single frequency. Since only a single frequency is selected, the contrast of the constructed image has been obvious improved. However, the resolution has been decrease due to the diffraction limitation of the long wavelength (more than 300 mm). This shortcoming can be overcome by choosing higher frequency since the THz spectrum exceeds over 2 THz.
4. Explosive imaging Several explosive, including TNT, RDX, and HMX, have been imaged by using the THz–TDS, they were wrapped by a piece of lens paper in experiment. The white light photo of the samples is presented in Fig. 4. The lens paper is removed for directly showing. It can be seen that samples have different shape. Some THz images of the samples, which are constructed by using the time-domain mode, are presented in Fig. 5. The maximum amplitude, the minimum amplitude, and the peak-to-peak value are adopted to construct images in Fig. 5(a)–(c), respectively. These images reflect the absorption characteristic of the samples for whole THz radiation. From these images, the distribution of the spatial density and thickness of the sample can be obtained. Moreover, these images are very similar with each other because the variation of the time-domain signal reflects an average effect of the sample to all of frequencies contained in the THz pulse. It can also be found that the intensity at the object’s edge is lower than that at the inner area of the HMX and TNT. This is because that the THz radiation was scattered at the sample’s edge. Fig. 5(d) adopts the phase information to
ARTICLE IN PRESS 328
Z. Zhang et al. / Optik 118 (2007) 325–329
Fig. 4. White light photo of the samples.
Fig. 6. Frequency-domain imaging of the exploders: Amplitude distribution at (a) 0.27 THz and (b) 1.09 THz and phase distribution at (c) 0.27 THz and (d) 1.09 THz.
effect is more distinct for the lower frequency than that for the higher frequency. Fig. 6(c) and (d) adopt the phase information at 0.27 and 1.09 THz for image generating. They can show the distribution of the sample’s refractive index at different frequency. Moreover, it can be found that at the lower frequency, the phase modes have better image quality than that at the higher frequency.
5. Conclusion Fig. 5. Time-domain imaging of the exploders: (a) timedomain maximum amplitude imaging, (b) time-domain minimum amplitude imaging, (c) time-domain peak-to-peak value imaging, and (d) time delay imaging.
generate image. The gray scale expresses the relative time delay of the peak value, which can reflect the thickness and refractive index information of the sample. The resolution of this method is limited by the smallest step of time delay in the system. In this experiment, the smallest time delay is 0.1 ps, the corresponding optical distance is 30 um. Some THz images constructed by using the frequencydomain mode are given in Fig. 6. Fig. 6(a) and (b) adopt the amplitude information at 0.27 and 1.09 THz to construct images. These images can reflect the absorption coefficient of the sample at corresponding frequency. Moreover, it can be drawn that the scattering
The THz time-domain spectroscopy imaging is an important THz imaging technology. Some exploders have been imaged by using the THz–TDS. The shape of the explosive materials can be effectively distinguished. Because the samples to be investigated are various, different display modes should be investigated for different proposes. The results obtained here indicate that the imaging quality is different for different approaches. The time-domain modes can present images with relatively good quality, but the phase mode can eliminate the edge scattering in the image. One should select a suitable method to construct an image for a special propose in order to achieve the best effect.
Acknowledgments This work was financially supported by the National Science Foundation (10390160) and the Beijing Science Nova Program (2004B35).
ARTICLE IN PRESS Z. Zhang et al. / Optik 118 (2007) 325–329
References [1] B.B. Hu, M.C. Nuss, Imaging with terahertz waves, Opt. Lett. 20 (1995) 1716–1718. [2] D.M. Mittleman, R.H. Jacobsen, M.C. Nuss, T-ray imaging, IEEE J. Sel. Top. Quant. Electron. 2 (1996) 679–692. [3] B. Ferguson, X.C. Zhang, Materials for terahertz science and technology, Nat. Mater. 1 (2002) 26–33.
329
[4] J. Xu, C. Zhang, X.C. Zhang, Recent progress in terahertz science and technology, Progr. Nat. Sci. 12 (2002) 0. [5] M. Herrmann, M. Tani, K. Saki, Display modes in timeresolved Terahertz imaging, Jpn. J. Appl. Phys. 39 (2000) 6254–6258. [6] Q. Chen, Terahertz wave imaging and its application, Doctoral Thesis of the Rensselaer Polytechnic Institute, Troy, New York, 2002.
Optik
Optics
Optik 118 (2007) 325–329 www.elsevier.de/ijleo
Terahertz time-domain spectroscopy for explosive imaging Zhengwei Zhang, Yan Zhang, Guozhong Zhao, Cunlin Zhang Department of Physics, Capital Normal University, Xisanhuan Beilu 105, 100037 Beijing, China Received 7 October 2005; received in revised form 17 January 2006; accepted 23 March 2006
Abstract Terahertz (THz) imaging which is a new technology for material classification and nondestructive detection has been extensively investigated in the past decade. The time-domain waveform acquired at each point of the object by using the THz time-domain spectroscopy contains much information about the object. Processing this waveform will present the characters of the object. Several methods are adopted to generate the image of the explosive samples and results are compared and discussed. Experiment results indicate that this new imaging technology can be used for explosive detection. r 2006 Elsevier GmbH. All rights reserved. Keywords: Terahertz; THz imaging; Image processing; Exploder detection
1. Introduction Since Hu and Nuss [1] reported their first image achieved by using the terahertz (THz) time-domain spectroscopy (TDS) in 1995, this technology has been considered as one of the most promising applications of the THz wave science and technology [2–4]. Some data processing methods have been adopted to reconstruct the THz image [5]. For the THz imaging technology, there are three important factors: a powerful and compact THz generator, a high sensitivity and low noise THz detector, and a fast, efficient data processing method. A good data processing method permits the high quality image. Since the samples to be studied are diversiform, developing a single algorithm for all signal processing is impractical. Therefore, discussion and comparison of various imaging approaches are quite important and useful.
Explosive materials are quite dangerous for public safety and national defense. An effective method for explosive detection is quite useful and important. In this paper, three kinds of explosive, including TNT, RDX, and HMX, are used as samples to be imaged by using the THzTDS. Several methods are adapted to process transmitted THz time-domain data for achieving images of samples. The difference of these data processing methods are discussed and compared. It is demonstrated that three explosive materials can be distinguished in the THz image. This paper is arranged as follows: Section 2 describes the experiment setup used for imaging; Section 3 introduces some approaches for data processing; Section 4 presents some images of the explosive samples with discussion, and finally, a conclusion is drawn in Section 5.
2. Experiment setup used for imaging Corresponding author.
E-mail address: [email protected] (Y. Zhang). 0030-4026/$ - see front matter r 2006 Elsevier GmbH. All rights reserved. doi:10.1016/j.ijleo.2006.03.025
The experiment setup used is schematically shown in Fig. 1. A mode-locked Ti: sapphire femtosecond laser is
ARTICLE IN PRESS 326
Z. Zhang et al. / Optik 118 (2007) 325–329 90
P
InAs Emitter
PM4
Chopper
PM1 Si Wafer ZnTe Sensor
CBS
Amplitude (a.u.)
70
QWP
PBS
80
Delay Stage
lock in
60 50 40 30
HWP
20 PM3
Sample x-y Stage
10
PM2 Laser
Fig. 1. Schematic configuration of a THz–TDS transmission imaging setup.
0 7
8
9 Position (mm)
10
11
Fig. 2. Spatial distributing of THz electronic field along the x direction.
used as light source. It has 100 fs pulse duration, 810 nm central wavelength, 88 kW peak power, and 82 MHz repetition rate. The laser beam is split into pump light and probe light by a cubic beam splitter (CBS). The pump pulse passes through a variable time-delay stage and illuminates the /1 0 0S InAs THz emitter for generating THz wave. Two pairs of gold-coated parabolic mirrors are used to collimate and focus the generated THz beam. A high resistance silicon wafer is placed after the fourth parabolic mirror to combine the probe beam with the THz beam. This allows the probe beam to travel collinearly with the THz beam inside the 1-mm-thick /1 1 0S ZnTe crystal, where the probing beam is modulated by the electric field of the THz radiation via the electro-optic effect. A quarter-wave plate (QWP), a Wollaston prism (PBS), and a pair of photodiodes are assembled for the balanced detection of the probe beam. The signal is read by a lock-in amplifier (LIA) from the detector at 1 kHz rate and is feed into a computer. The spectrum width of the system used is between 0.2 and 2.5 THz with resolution better than 10 GHz. The dynamic range is greater than 3000 and the SNR is about 600. The sample is placed on the focal point of the second parabolic mirror. Two crossed motorized translation stages are used to move the object in the x–y plane perpendicular to the beam. At each x–y position, a complete THz time-domain spectrum is recorded by scanning the delay line. These time-domain spectra contain the information of the sample, which can be used to exhibit different characters of the object. The time used for imaging depends on the time constant of the LIA, the number of the spatial points, and the number of sampling points in the time domain. The spatial resolution of the system is determined by the size of the focal point of the THz radiation focused on the sample. Assuming that the THz beam is a Gaussian distribution, the spatial resolution of the imaging can be expressed by pffiffiffi R ¼ 2ð4l=pÞðf =dÞ, where d is the diameter of the
collimated THz beam before the second parabolic mirror, f is the focal length of the second parabolic mirror, and l is the peak wavelength of pffiffithe ffi involved broad-band THz radiation [6]. The factor 2 is introduced due to the fact that the THz electronic field rather than the intensity is measured in this system. The parameters used in our experiment are: d ¼ 2:8 mm, f ¼ 50:8 mm, and l ¼ 0:31 mm. Thus the resolution R can be calculated as1.1 mm. The size of the focal point has also been experimentally measured by using the blade scanning method. The differential of the experimentally measured data is shown in Fig. 2. It can be found that the THz beams has a very good Gaussian distribution. The full width at half maximum is about 1.1 mm, which corresponds to the theoretical expectation very well.
3. Data processing methods The TDS imaging technology collects a waveform in the time domain; therefore, it can present much information about the sample at each x–y position. One can extract the information of a discretionary point form the recorded time-domain waveform or its corresponding Fourier transform spectrum to construct an image. Fig. 3 presents a typical THz time-domain signal at a discretionary point on the sample and its corresponding Fourier transform spectrum. One can use many parameters to construct the image for expressing different characters of the object. Here, we introduce several feasible data processing methods for THz TDS imaging.
3.1. Time-domain mode Some information of the sample can be directly extracted from its THz time-domain waveform. As
ARTICLE IN PRESS Z. Zhang et al. / Optik 118 (2007) 325–329
sample. It should be pointed out that the time-domain signal reflects the synthesized effect of all frequencies contained in the THz pulse. Therefore, it exhibits the average effect of the object of the different frequencies; thus, the imaging methods in the time-domain mode have good quality but small contrast.
Amplitude (a.u.)
2
1
0
3.2. Frequency-domain mode
-1 0
4
8
12
Time (ps)
(a) 8 7
Amplitude (a.u.)
6 5 4 3 2 1 0 0.0
(b)
327
0.5
1.0
1.5
2.0
2.5
3.0
Frequency (THz)
Fig. 3. (a) THz time-domain signal and (b) its corresponding frequency-domain spectrum.
shown in Fig. 3(a), many parameters can be used to describe the characters of the object, such as the maximum amplitude, the minimum amplitude, the peak-to-peak value (the difference of the maximum amplitude and the minimum amplitude), the energy in a Rt time interval (the energy D ¼ t12 E 2 ðtÞ dt, where t1 and t2 limit a duration of time, which are usually the time of the zero amplitude before and behind the peak value), the time of the peak value, the time of the minimum value, and the time of the zero amplitude between the peak value and the minimum value. These methods present different images of an object and can be classed as two modes, i.e., the amplitude mode and the phase mode. The first four methods used time-domain amplitude information to describe the object; they can reflect the thickness and the absorption characteristic of the sample. Other methods used phase information to construct the image, which can present the thickness and refractive index distribution of the sample. Different THz images have different contrast due to the fact that they have been constructed by using different processing methods, which represent different information of the
Information of the sample can also be extracted from the frequency-domain spectrum as shown in Fig. 3(b). The maximum amplitude in the frequency domain, the amplitude of a arbitrary frequency, the phase of a discretionary frequency, the absorption at a arbitrary frequency, the refractive index at a discretionary frequency, andR the energy contained in a frequency o interval (D ¼ o12 E 2 ðoÞ do, where o1 and o2 are the limitation of the frequency interval) can be used to construct the image. These approaches present the characters of the sample at different frequencies. They can describe the absorption, thickness, and the refraction index of the object for a single frequency. Since only a single frequency is selected, the contrast of the constructed image has been obvious improved. However, the resolution has been decrease due to the diffraction limitation of the long wavelength (more than 300 mm). This shortcoming can be overcome by choosing higher frequency since the THz spectrum exceeds over 2 THz.
4. Explosive imaging Several explosive, including TNT, RDX, and HMX, have been imaged by using the THz–TDS, they were wrapped by a piece of lens paper in experiment. The white light photo of the samples is presented in Fig. 4. The lens paper is removed for directly showing. It can be seen that samples have different shape. Some THz images of the samples, which are constructed by using the time-domain mode, are presented in Fig. 5. The maximum amplitude, the minimum amplitude, and the peak-to-peak value are adopted to construct images in Fig. 5(a)–(c), respectively. These images reflect the absorption characteristic of the samples for whole THz radiation. From these images, the distribution of the spatial density and thickness of the sample can be obtained. Moreover, these images are very similar with each other because the variation of the time-domain signal reflects an average effect of the sample to all of frequencies contained in the THz pulse. It can also be found that the intensity at the object’s edge is lower than that at the inner area of the HMX and TNT. This is because that the THz radiation was scattered at the sample’s edge. Fig. 5(d) adopts the phase information to
ARTICLE IN PRESS 328
Z. Zhang et al. / Optik 118 (2007) 325–329
Fig. 4. White light photo of the samples.
Fig. 6. Frequency-domain imaging of the exploders: Amplitude distribution at (a) 0.27 THz and (b) 1.09 THz and phase distribution at (c) 0.27 THz and (d) 1.09 THz.
effect is more distinct for the lower frequency than that for the higher frequency. Fig. 6(c) and (d) adopt the phase information at 0.27 and 1.09 THz for image generating. They can show the distribution of the sample’s refractive index at different frequency. Moreover, it can be found that at the lower frequency, the phase modes have better image quality than that at the higher frequency.
5. Conclusion Fig. 5. Time-domain imaging of the exploders: (a) timedomain maximum amplitude imaging, (b) time-domain minimum amplitude imaging, (c) time-domain peak-to-peak value imaging, and (d) time delay imaging.
generate image. The gray scale expresses the relative time delay of the peak value, which can reflect the thickness and refractive index information of the sample. The resolution of this method is limited by the smallest step of time delay in the system. In this experiment, the smallest time delay is 0.1 ps, the corresponding optical distance is 30 um. Some THz images constructed by using the frequencydomain mode are given in Fig. 6. Fig. 6(a) and (b) adopt the amplitude information at 0.27 and 1.09 THz to construct images. These images can reflect the absorption coefficient of the sample at corresponding frequency. Moreover, it can be drawn that the scattering
The THz time-domain spectroscopy imaging is an important THz imaging technology. Some exploders have been imaged by using the THz–TDS. The shape of the explosive materials can be effectively distinguished. Because the samples to be investigated are various, different display modes should be investigated for different proposes. The results obtained here indicate that the imaging quality is different for different approaches. The time-domain modes can present images with relatively good quality, but the phase mode can eliminate the edge scattering in the image. One should select a suitable method to construct an image for a special propose in order to achieve the best effect.
Acknowledgments This work was financially supported by the National Science Foundation (10390160) and the Beijing Science Nova Program (2004B35).
ARTICLE IN PRESS Z. Zhang et al. / Optik 118 (2007) 325–329
References [1] B.B. Hu, M.C. Nuss, Imaging with terahertz waves, Opt. Lett. 20 (1995) 1716–1718. [2] D.M. Mittleman, R.H. Jacobsen, M.C. Nuss, T-ray imaging, IEEE J. Sel. Top. Quant. Electron. 2 (1996) 679–692. [3] B. Ferguson, X.C. Zhang, Materials for terahertz science and technology, Nat. Mater. 1 (2002) 26–33.
329
[4] J. Xu, C. Zhang, X.C. Zhang, Recent progress in terahertz science and technology, Progr. Nat. Sci. 12 (2002) 0. [5] M. Herrmann, M. Tani, K. Saki, Display modes in timeresolved Terahertz imaging, Jpn. J. Appl. Phys. 39 (2000) 6254–6258. [6] Q. Chen, Terahertz wave imaging and its application, Doctoral Thesis of the Rensselaer Polytechnic Institute, Troy, New York, 2002.