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Line scans of a model structure with machined air gaps ranging in thickness from 12.5 to 100 m. The inset shows a schematic of this sample. The line scans are shown as relative changes in the peak-to-peak amplitude of the wave forms as a function of position across the sample. Without interferometry, ͑ dashed curve ͒ only the largest air gap can be discerned. However, the interferometric case ͑ solid ͒ shows a dramatic increase in contrast, demonstrating the ability to detect the smallest ͑ 12.5 m ͒ gap. These measurements were performed with fairly long THz pulses, with a coherence length of approximately 0.8 mm.
Source publication
We describe an imaging technique for few-cycle optical pulses. An object to be imaged is placed at the focus of a lens in one arm of a Michaelson interferometer. This introduces a phase shift of approximately π between the two arms of the interferometer, via the Gouy phase shift. The resulting destructive interference provides a nearly background-f...
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Context 1
... between these two pulses resulting in a signal reduced in amplitude by more than 90%. The dashed curves in Figs. 2 ͑ b ͒ and 2 ͑ c ͒ illustrate the effect of introducing a small modification to the mirror in the sample arm. Here, a thin ͑ 45 m ͒ layer of adhesive tape has been affixed to the mirror, at the position where the THz beam is focused. The dashed curve in Fig. 2 ͑ b ͒ shows the sample arm wave form with this perturbation in place. It indicates, primarily, a shift in arrival delay but also a small decrease in amplitude due to the absorption and index of the thin polymer layer. The dashed curve in Fig. 2 ͑ c ͒ shows the equivalent result, with the reference arm un- blocked so that the two pulses can interfere. The relative change in amplitude resulting from the perturbation is much larger in this case, since the change in arrival delay substan- tially disrupts the destructive interference. We note that the interfered signal, Fig. 2 c , is not quite zero even when the sample and reference reflectors are identical. We can understand this incomplete destructive interference as a manifestation of the frequency dependence of the Gouy phase. The additional phase shift acquired by the fo- cusing beam is given by ⌽ ͑ v ͒ ϭ Ϫ 2 tan Ϫ 1 2 v c , where v c f c / w 0 , f is the focal length of the lens and w 0 is the beam waist before the lens. 5 For frequencies larger than the critical frequency v c the Gouy phase shift approaches , but for lower frequencies the confocal parameter of the fo- cusing Gaussian beam approaches the focal length of the lens, and the phase shift rapidly drops to zero. For our experimental configuration, with a lens with f ϭ 13.2 cm, we estimate a beam waist of 1.5 cm and thus a critical frequency of v c ϭ 176 GHz. Thus, a phase shift of is not expected for all the wavelengths in the THz pulse. As a result, the interference between the sample and reference arms is not complete, and a small low-frequency wave form is measured. We can confirm that this incomplete cancellation is re- sponsible for the observed wave forms by measuring the interfered wave forms as a function of the delay between the sample and reference arms. In this case, both arms are reflected with identical metal mirrors. In Fig. 3, we compare these measured wave forms ͑ solid lines ͒ with simulated data. For these simulations, we measure the reference arm pulse, E ref ( ), at each delay position. We then compute the sum of this reference pulse and a delayed, phase-shifted replica of itself. That is, we plot the Fourier transform of E ref ( ) ͓ 1 ϩ e i • e i ⌽ ( v ) ͔ for each position of the delay stage. This simulates the coherent superposition of the reference and sample arms, using only the retroreflected reference arm as an input. The excellent agreement between the measurements and simulations indicates that the Gouy phase is suf- ficient to explain the observed wave forms. Figure 3 emphasizes the distinction between our imaging technique and conventional optical coherence tomography. In OCT, the measurement variable is the delay of the reference arm. It is necessary to obtain data at many different reference arm positions in order to determine the electric field of the pulse train reflected from the sample. 1 In our measurements, a complete wave form can be measured at any reference arm delay since the THz electric field is measured directly. This permits us to exploit the destructive interference in a manner which is not currently possible at optical frequencies. For the measurements shown in Fig. 3, the coherence length of the THz pulse is about 0.8 mm. We note that, for delays less than half of the coherence length of the pulse, it is difficult to discern two pulses in the wave forms. The resulting wave forms are quite complex, exhibiting evidence of both constructive and destructive interference. This results in a large range of peak-to-peak amplitudes, from nearly zero to almost twice the peak-to-peak amplitude of the original pulse. We note that previous THz imaging experiments have demonstrated a high degree of sensitivity to small shifts in the delay of the measured pulse. 8 The interferometric technique converts these small delay shifts into relatively large amplitude shifts, with a corresponding increase in sensitivity. It also permits us to detect two closely spaced surfaces, which would ordinarily generate little or no reflection signal due to destructive interference. 3 To demonstrate this latter ability, we have constructed a model sample containing a series of thin well-controlled features. The inset in Fig. 4 shows a schematic of this teflon– metal This model, work with has air been gaps supported between in the part two by pieces the National ranging Science from 12.5 Foundation to 100 m in and width. the We Environmental image a line scan Protection across Agency. this sample, and compare the results with and without the interferometric cancellation. The results are shown in Fig. 4 as the percent change in peak-to-peak amplitude of the measured wave forms, relative to a wave form measured on a spot that does not contain an air gap. The contrast of the interferometric signal is enhanced by an order of magnitude over the noninterferometric signal. Without the reference arm almost no change in amplitude is observed and only the largest air gaps can be detected. In the interferometric mode the areas with no air gap show strong destructive interference. The change in the cancellation when an air gap is en- countered results in a large increase in contrast. As a result, it is possible to detect the smallest air gap using the interference effect. This 12.5 mm gap is roughly 80 times smaller than the coherence length of the terahertz pulses used to collect the data of Fig. 4. 12 In conclusion, we have described an imaging technique which uses the Gouy phase shift to provide a destructive interference between two arms of an interferometer. This, in turn, permits a nearly background-free method for imaging with a corresponding dramatic contrast enhancement for sub- coherence length features in a sample. This phase-shift interferometry permits imaging well below the conventional Rayleigh bandwidth limit. Since the Gouy phase is a geometric phase, and is a very general phenomenon, this technique is not limited to THz imaging. In any situation where few-cycle pulses are available, this phase-shift method can provide sub- stantial improvements in depth resolution. With recent ad- vances in femtosecond pulse techniques, 13 it could find important applications in optical imaging methods such as coherence tomography. This work has been supported in part by the National Science Foundation and the Environmental Protection ...
Context 2
... larger in this case, since the change in arrival delay substan- tially disrupts the destructive interference. We note that the interfered signal, Fig. 2 c , is not quite zero even when the sample and reference reflectors are identical. We can understand this incomplete destructive interference as a manifestation of the frequency dependence of the Gouy phase. The additional phase shift acquired by the fo- cusing beam is given by ⌽ ͑ v ͒ ϭ Ϫ 2 tan Ϫ 1 2 v c , where v c f c / w 0 , f is the focal length of the lens and w 0 is the beam waist before the lens. 5 For frequencies larger than the critical frequency v c the Gouy phase shift approaches , but for lower frequencies the confocal parameter of the fo- cusing Gaussian beam approaches the focal length of the lens, and the phase shift rapidly drops to zero. For our experimental configuration, with a lens with f ϭ 13.2 cm, we estimate a beam waist of 1.5 cm and thus a critical frequency of v c ϭ 176 GHz. Thus, a phase shift of is not expected for all the wavelengths in the THz pulse. As a result, the interference between the sample and reference arms is not complete, and a small low-frequency wave form is measured. We can confirm that this incomplete cancellation is re- sponsible for the observed wave forms by measuring the interfered wave forms as a function of the delay between the sample and reference arms. In this case, both arms are reflected with identical metal mirrors. In Fig. 3, we compare these measured wave forms ͑ solid lines ͒ with simulated data. For these simulations, we measure the reference arm pulse, E ref ( ), at each delay position. We then compute the sum of this reference pulse and a delayed, phase-shifted replica of itself. That is, we plot the Fourier transform of E ref ( ) ͓ 1 ϩ e i • e i ⌽ ( v ) ͔ for each position of the delay stage. This simulates the coherent superposition of the reference and sample arms, using only the retroreflected reference arm as an input. The excellent agreement between the measurements and simulations indicates that the Gouy phase is suf- ficient to explain the observed wave forms. Figure 3 emphasizes the distinction between our imaging technique and conventional optical coherence tomography. In OCT, the measurement variable is the delay of the reference arm. It is necessary to obtain data at many different reference arm positions in order to determine the electric field of the pulse train reflected from the sample. 1 In our measurements, a complete wave form can be measured at any reference arm delay since the THz electric field is measured directly. This permits us to exploit the destructive interference in a manner which is not currently possible at optical frequencies. For the measurements shown in Fig. 3, the coherence length of the THz pulse is about 0.8 mm. We note that, for delays less than half of the coherence length of the pulse, it is difficult to discern two pulses in the wave forms. The resulting wave forms are quite complex, exhibiting evidence of both constructive and destructive interference. This results in a large range of peak-to-peak amplitudes, from nearly zero to almost twice the peak-to-peak amplitude of the original pulse. We note that previous THz imaging experiments have demonstrated a high degree of sensitivity to small shifts in the delay of the measured pulse. 8 The interferometric technique converts these small delay shifts into relatively large amplitude shifts, with a corresponding increase in sensitivity. It also permits us to detect two closely spaced surfaces, which would ordinarily generate little or no reflection signal due to destructive interference. 3 To demonstrate this latter ability, we have constructed a model sample containing a series of thin well-controlled features. The inset in Fig. 4 shows a schematic of this teflon– metal This model, work with has air been gaps supported between in the part two by pieces the National ranging Science from 12.5 Foundation to 100 m in and width. the We Environmental image a line scan Protection across Agency. this sample, and compare the results with and without the interferometric cancellation. The results are shown in Fig. 4 as the percent change in peak-to-peak amplitude of the measured wave forms, relative to a wave form measured on a spot that does not contain an air gap. The contrast of the interferometric signal is enhanced by an order of magnitude over the noninterferometric signal. Without the reference arm almost no change in amplitude is observed and only the largest air gaps can be detected. In the interferometric mode the areas with no air gap show strong destructive interference. The change in the cancellation when an air gap is en- countered results in a large increase in contrast. As a result, it is possible to detect the smallest air gap using the interference effect. This 12.5 mm gap is roughly 80 times smaller than the coherence length of the terahertz pulses used to collect the data of Fig. 4. 12 In conclusion, we have described an imaging technique which uses the Gouy phase shift to provide a destructive interference between two arms of an interferometer. This, in turn, permits a nearly background-free method for imaging with a corresponding dramatic contrast enhancement for sub- coherence length features in a sample. This phase-shift interferometry permits imaging well below the conventional Rayleigh bandwidth limit. Since the Gouy phase is a geometric phase, and is a very general phenomenon, this technique is not limited to THz imaging. In any situation where few-cycle pulses are available, this phase-shift method can provide sub- stantial improvements in depth resolution. With recent ad- vances in femtosecond pulse techniques, 13 it could find important applications in optical imaging methods such as coherence tomography. This work has been supported in part by the National Science Foundation and the Environmental Protection ...
Context 3
... wave forms which illustrate this technique. Figures 2 ͑ a ͒ and 2 ͑ b ͒ show wave forms from the reference arm and sample arm, respectively, with a metal mirror placed in the focus of the lens. These wave forms illustrate the nearly phase shift acquired by the sample arm, relative to the reference arm. The wave form in Fig. 2 ͑ c ͒ shows the strong destructive interference between these two pulses resulting in a signal reduced in amplitude by more than 90%. The dashed curves in Figs. 2 ͑ b ͒ and 2 ͑ c ͒ illustrate the effect of introducing a small modification to the mirror in the sample arm. Here, a thin ͑ 45 m ͒ layer of adhesive tape has been affixed to the mirror, at the position where the THz beam is focused. The dashed curve in Fig. 2 ͑ b ͒ shows the sample arm wave form with this perturbation in place. It indicates, primarily, a shift in arrival delay but also a small decrease in amplitude due to the absorption and index of the thin polymer layer. The dashed curve in Fig. 2 ͑ c ͒ shows the equivalent result, with the reference arm un- blocked so that the two pulses can interfere. The relative change in amplitude resulting from the perturbation is much larger in this case, since the change in arrival delay substan- tially disrupts the destructive interference. We note that the interfered signal, Fig. 2 c , is not quite zero even when the sample and reference reflectors are identical. We can understand this incomplete destructive interference as a manifestation of the frequency dependence of the Gouy phase. The additional phase shift acquired by the fo- cusing beam is given by ⌽ ͑ v ͒ ϭ Ϫ 2 tan Ϫ 1 2 v c , where v c f c / w 0 , f is the focal length of the lens and w 0 is the beam waist before the lens. 5 For frequencies larger than the critical frequency v c the Gouy phase shift approaches , but for lower frequencies the confocal parameter of the fo- cusing Gaussian beam approaches the focal length of the lens, and the phase shift rapidly drops to zero. For our experimental configuration, with a lens with f ϭ 13.2 cm, we estimate a beam waist of 1.5 cm and thus a critical frequency of v c ϭ 176 GHz. Thus, a phase shift of is not expected for all the wavelengths in the THz pulse. As a result, the interference between the sample and reference arms is not complete, and a small low-frequency wave form is measured. We can confirm that this incomplete cancellation is re- sponsible for the observed wave forms by measuring the interfered wave forms as a function of the delay between the sample and reference arms. In this case, both arms are reflected with identical metal mirrors. In Fig. 3, we compare these measured wave forms ͑ solid lines ͒ with simulated data. For these simulations, we measure the reference arm pulse, E ref ( ), at each delay position. We then compute the sum of this reference pulse and a delayed, phase-shifted replica of itself. That is, we plot the Fourier transform of E ref ( ) ͓ 1 ϩ e i • e i ⌽ ( v ) ͔ for each position of the delay stage. This simulates the coherent superposition of the reference and sample arms, using only the retroreflected reference arm as an input. The excellent agreement between the measurements and simulations indicates that the Gouy phase is suf- ficient to explain the observed wave forms. Figure 3 emphasizes the distinction between our imaging technique and conventional optical coherence tomography. In OCT, the measurement variable is the delay of the reference arm. It is necessary to obtain data at many different reference arm positions in order to determine the electric field of the pulse train reflected from the sample. 1 In our measurements, a complete wave form can be measured at any reference arm delay since the THz electric field is measured directly. This permits us to exploit the destructive interference in a manner which is not currently possible at optical frequencies. For the measurements shown in Fig. 3, the coherence length of the THz pulse is about 0.8 mm. We note that, for delays less than half of the coherence length of the pulse, it is difficult to discern two pulses in the wave forms. The resulting wave forms are quite complex, exhibiting evidence of both constructive and destructive interference. This results in a large range of peak-to-peak amplitudes, from nearly zero to almost twice the peak-to-peak amplitude of the original pulse. We note that previous THz imaging experiments have demonstrated a high degree of sensitivity to small shifts in the delay of the measured pulse. 8 The interferometric technique converts these small delay shifts into relatively large amplitude shifts, with a corresponding increase in sensitivity. It also permits us to detect two closely spaced surfaces, which would ordinarily generate little or no reflection signal due to destructive interference. 3 To demonstrate this latter ability, we have constructed a model sample containing a series of thin well-controlled features. The inset in Fig. 4 shows a schematic of this teflon– metal This model, work with has air been gaps supported between in the part two by pieces the National ranging Science from 12.5 Foundation to 100 m in and width. the We Environmental image a line scan Protection across Agency. this sample, and compare the results with and without the interferometric cancellation. The results are shown in Fig. 4 as the percent change in peak-to-peak amplitude of the measured wave forms, relative to a wave form measured on a spot that does not contain an air gap. The contrast of the interferometric signal is enhanced by an order of magnitude over the noninterferometric signal. Without the reference arm almost no change in amplitude is observed and only the largest air gaps can be detected. In the interferometric mode the areas with no air gap show strong destructive interference. The change in the cancellation when an air gap is en- countered results in a large increase in contrast. As a result, it is possible to detect the smallest air gap using the interference effect. This 12.5 mm gap is roughly 80 times smaller than the coherence length of the terahertz pulses used to collect the data of Fig. 4. 12 In conclusion, we have described an imaging technique which uses the Gouy phase shift to provide a destructive interference between two arms of an interferometer. This, in turn, permits a nearly background-free method for imaging with a corresponding dramatic contrast enhancement for sub- coherence length features in a sample. This phase-shift interferometry permits imaging well below the conventional Rayleigh bandwidth limit. Since the Gouy phase is a geometric phase, and is a very general phenomenon, this technique is not limited to THz imaging. In any situation where few-cycle pulses are available, this phase-shift method can provide sub- stantial improvements in depth resolution. With recent ad- vances in femtosecond pulse techniques, 13 it could find important applications in optical imaging methods such as coherence tomography. This work has been supported in part by the National Science Foundation and the Environmental Protection ...
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... After the rapid development of THz technology in recent decades, THz imaging has become more mature. It has been developed for different scenarios, such as THz timedomain spectral imaging [5,6], THz near-field imaging [7][8][9][10][11][12][13][14], THz tomography imaging [12][13][14], THz digital holographic imaging [15,16], THz compressed sensing imaging [17][18][19][20], etc. These THz imaging techniques have been widely used or have shown potential applications in security [21], medical science [22], industrial non-destructive testing [23] and other fields [24,25]. ...
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This article is part of the theme issue ‘Advanced electromagnetic non-destructive evaluation and smart monitoring’.
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