Transformation of Laguerre-Gaussian beam by a ring-lens

Abstract

The propagation property of Laguerre-Gaussian (LG) beams passing through a diffractive ring-lens (RL) was studied, where the RL was generated by a liquid crystal spatial light modulator (LC-SLM). It was found that the LG beam was transformed into a sharp ring at the focal plane first, and then a Bessel-similar beam was formed behind the focal plane but the beam size was enlarged with the increase of propagation distance. With the help of a group of lenses, the beam was further collimated into a Bessel beam. Finally, the “non-diffractive” and self-reconstruction properties of the generated Bessel beams were experimentally verified.

Keywords

ring-lens (RL) / Bessel beam / spatial light modulator (SLM)

1 Introduction

Terahertz pulse imaging and spectroscopy (TPIS) is emerging as a nondestructive-evaluation tool of high growth potential within the fields of art conservation, historical architecture conservation, and archeology [ 1]. In particular with mummies, TPIS has been explored as an alternative—or complement—to X-ray imaging techniques [ 2, 3]. It is the combination of material characterization, time of flight imaging, and the penetration of optically opaque materials that gives rise to applications for subsurface imaging of many culturally significant objects. Moreover, the variety and adaptability of the many electronic, optical, and hybrid terahertz (THz) sources allow for versatile approaches to measurement [ 4– 6]. Spatial resolution for free-space THz measurements can be scaled from tens of micrometers to several millimeters, providing the possibility of taking measurements without sample extraction, in situ, and in the field [ 7]. Lastly, moderate exposure to THz radiation poses significantly less long-term risk [ 8, 9] to the molecular stability of the historical artifact and to humans than X-rays, ultra violet (UV) or visible radiation, because it is non-ionizing and produces low thermal heating. Therefore, THz technology provides a non-ionizing, noninvasive, noncontact, nondestructive toolset [ 10] for the examination of unique and priceless objects [ 11].

The THz region is possibly the least understood and most complicated part of the electromagnetic spectrum. The THz region has been typically defined as being between 30 µm and 3 mm in wavelength, thus putting its scale on the border between the microscopic and macroscopic world. At frequencies between 0.1 and 10 trillion (10¹²) cycles per second, the THz regime overlaps with both the microwave and far infrared regions of the spectrum. The THz regime also corresponds to photon energies between 0.4 and 40 meV. Lower frequency microwave radiation has lower photon energy, therefore the photons cannot be measured directly, only collectively by the electrical bias they induce in a detector. On the other hand, infrared radiation is optical, since its photon energy is large enough that individual photons can be directly measured. Thus, THz radiation uniquely straddles the worlds of electronics and optics. Over the last two decades, means of producing and detecting sub-picosecond, broadband pulses of THz radiation by integrating optoelectronic devices with ultrafast optical lasers have sparked many new forms of research—including time-domain THz spectroscopy and imaging. As the THz technology gap is filled by diverse and practical devices, the number of THz applications constantly increases, including those developed for the chemical-mapping of pharmaceuticals, the non-destructive evaluation of space shuttle foam, people-safe security imaging, and atmospheric-chemical species monitoring [ 12, 13].

Our TPIS technique is based on time-domain spectroscopy (TDS). The THz pulse is measured as a transient electric field, which can be directly transformed into the amplitude and phase spectra of the pulse. When the THz pulse is transmitted through an object of low electrical conductivity (thus relatively low absorption), compositional changes—which include refractive index and optical density—can result in changes of the amplitude, shape and propagation time of the THz pulse. By scanning the object, this can be exploited to image the lateral and axial spatial features buried beneath the visibly opaque surface of the object. While the wide bandwidth of the THz pulses can aid in spectroscopically discriminating between buried materials that exhibit different THz-refractive-index spectra, the short-time-duration nature of THz pulses can also help isolate and distinguish depth information from different interfaces within an object.

In the course of our study, we were permitted access to the conservation laboratory of the Oriental Institute at the University of Chicago to examine several archeological artifacts from their collection. The first is an Egyptian neonate bird mummy (OIM E9164, Fig. 1), which was to be included in a project to image several dozen bird mummies using X-ray computed tomography (XRCT) [ 14]. Challenges of such an endeavor included protecting the fragile objects from physical disturbance and fluctuations in humidity. Special packaging and precise coordination of transport to the medical imaging facilities were essential to the XRCT experiments. In contrast, the availability of portable THz-TDS systems enabled our measurements to be performed in the well-controlled environment of the conservation laboratory.

Fig.1 Photograph of ancient Egyptian bird mummy (OIM E9164) [15]

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Among the other selections, we studied corroded copper-alloy (most likely bronze) artifacts from Ancient Sumer (now Iraq), with special attention toward a 5 cm fragment from an unregistered object (Fig. 2) and two cups (Fig. 3). For many archeological artifacts made of bronze or other copper-based alloys, the corrosion process can produce multilayer oxide structures (Fig. 4)—such as copper + cuprite + copper trihydroxy chloride minerals + malachite + azurite—which through time and environmental exposure can entirely convert the metallic material to stratified, composite layers of dielectric minerals [ 16, 17]. This particular application for TPIS is dependent on the ability of THz waves to distinguish dielectric corrosion products from their metallic source material by their transmissivity and reflectivity. Previous work has been done to find defects beneath the coating on metal surfaces [ 18, 19], however not much THz research has been conducted to study the specific corrosion layers themselves. THz pulses have the potential to spectroscopically identify and spatially distinguish each layer.

Fig.2 Photo of corroded copper-alloy fragment cross-section [17]

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Fig.3 Photographs of corroded copper-alloy (a) cup A (OIM A11280A, 12 cm diameter) and cup B (OIM A11399A, 10 cm diameter) [17]

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Fig.4 Diagram of typical copper-based corrosion layers

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2 Experimental methods

Our time-domain THz imaging system consists of computer-controlled, motorized translation stages and the Picometrix T-Ray 4000 (TR4K) commercial THz time-domain spectrometer. The modelocked, two-stage, amplified, Ytterbium fiber laser operates with a center frequency near 1064 nm, a 100 fs pulse width, a 50 MHz repetition rate and a maximum output power of 400 mW.

The THz pulses were generated and then propagated through free-space using a biased, photoconductive switch antenna, consisting of a photosensitive low-temperature grown gallium arsenide semiconductor with two metal electrodes deposited on its surface. The antenna is illuminated at normal incidence by the ultrafast laser pulse, thus generating electron-hole pairs into the semiconductor. A voltage bias is applied to the electrodes to generate a photocurrent. The free-space THz electromagnetic field emanating from the antennas is proportional to the rapid change in the photocurrent—the sub-mechanisms of which determine the THz pulse’s duration and bandwidth. A second photoconductive antenna is used as the THz receiver. The optical pulse generates photocarriers in the receiver by the same photoexcitation mechanism as when the emitter is illuminated. However in this case, the incident electric field of the THz pulse causes a time-varying potential to develop across the receiver, thus serving as an applied voltage bias that induces a transient photocurrent, which is amplified and measured as an electrical signal by using a data acquisition board and computer.

The development of compact, fiber-coupled THz-TDS systems, such as the TR4K, has several benefits which were essential to the advancement of THz applications in art and archeology. As a general rule, cultural heritage conservators aim toward minimizing the disturbance of the objects under investigation. The optical components for the compact spectrometer are contained within an easy to transport (size and weight) box. This allows for lower complication on-site or field measurements. The fiber-coupled THz antennas permit rapid modification of the measurement geometry from transmission (Fig. 5(a)) to reflection (Fig. 5(b)), and vice versa. Additionally, it makes it possible for the object to remain fixed and secure during the scans, and facilitates the measurement of large objects in the transmission geometry.

Fig.5 Photographs of the (a) transmission [15] and (b) reflection setups

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We took into consideration the relatively low THz absorption and refractive indices of polystyrene foam [ 20] and paper [ 21] and utilized a foamcore support (Fig. 5(a)), which could permit us to securely scan the top and bottom of the extremely fragile mummy in transmission with minimal consequence to the signal. Figure 6(a) shows THz time-domain signals through the ambient environment and the foam support. The THz spectra (Fig. 6(b)) are obtained by performing a fast Fourier transform of the time domain signals. The refractive index and absorption of the foamcore (Fig. 6(c)) were ~20% and ~1500% higher than expected, respectively. This may be due to the thickness, type and coating of the board's paper shell. For future experiments, a several millimeter thick sheet of uncoated, un-sheathed polystyrene, or polymethyacrylimide [ 22], would provide equally rigid support with less signal loss.

Fig.6 Air and foam support reference (a) time domain signal; (b) transmission spectra and (c) refractive index and absorption spectra for the foam support

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3 Discussion

3.1 Terahertz pulse imaging of a bird mummy

Transmission through the thicker, upper region of the mummy was too small to discern a signal through the baseline noise, so we only scanned the lower 80%. The two dimensional scans of the middle and lower regions can be seen in Fig. 7, and were calculated using the amplitude of the minimum peak for each pixel. The spatial resolution, or ability to see detail in the image, was negatively impacted by decreased bandwidth of the signal, as well as the changing spot size of the Gaussian beam through the bird. For the torso region, there is very strong absorption loss where the skeleton and desiccated organ tissue would be (black). For the leg region, the flesh and bone are less dense, resulting in better signal to noise and more distinct features. The shape of what appear to be the claws (although probably tibio-tarsal joints), however, is the most impressive feature of this image.

Fig.7 Terahertz transmission image of bird mummy {color scale: white/yellow = higher transmission, black/violet = lower transmission}

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Figure 8 shows exemplary signals through the middle region, or torso, and lower region, or legs, of the bird mummy, designated by the letters A-H in Fig. 7. The multiple peaks, seen in Figs. 8(a) and 8(b), are a result of internal reflections of the THz pulse as it interacts with the textile wrap, desiccated flesh and bones of the bird. Note the one order of magnitude difference in transmission between the torso region and the leg region. The spectra in Fig. 8(c) also show significant signal loss above 0.2 THz, even in the “empty” region H, which suggests the loss is most likely due to the electric field being scattered by the weave and layers of the cloth wrapping [ 23]. Fortunately, in the range of 0.1 to 0.2 THz, the attenuation of the foam support is comparable to air.

Fig.8 Select (offset) time-domain waveforms from (a) torso region and (b) leg region; (c) spectral waveforms from bird mummy transmission signals

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There are few published comparisons of mummies imaged using both X-rays and THz radiation. Previously, Öhrström et al. also used THz transmission imaging to view the contents of a mummified human hand and fish [ 2]. They demonstrated that with TPIS it was possible to recognize differences in the desiccated flesh and that it adds a temporal aspect that does not exist in other techniques, despite TPIS having coarser resolution than XRCT. Fukunaga et al. also confirmed that THz imaging may complement X-ray images when materials with low radiographic density and contrast are being investigated [ 3]. From the XRCT images taken on this bird mummy (Fig. 9), we can confirm this with significance. It is likely that since the bird was young at the time of its death, its bones were soft, resulting in low-density calcification. As a result, THz provides a much better visualization of the leg region of the bird than X-rays, as applied in this particular CT-scanning session [ 14]. Further comparative measurements using other X-ray energies and more intense THz waves would be desirable.

Fig.9 X-ray computed tomography scan of Egyptian bird mummy E9164 [14] {Courtesy of Charles Pelizzari and Christian Wietholt}

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3.2 Corroded copper-alloy fragment

A copper alloy fragment from an Early Dynastic (~2000 BC) Sumerian unregistered object excavated from the Tell Agrab settlement of the Diyala Valley in Iraq (Fig. 2) was examined to determine whether it is possible to distinguish the metallic and mineralized stratified corrosion layers. Figure 10(a) shows an image of the surface of the fragment, calculated using the peak to peak amplitude of each pixel. The brightest areas are where the surface of the fragment is still metallic, due to the high reflectivity. The pink line indicates the location of the slice, seen as the cross-sectional b-scan in Fig. 10(b). Figure 10(c) shows the time-domain signatures of select point measurements. The outer crust is composed of calcified dirt, possibly tin and copper corrosion products. It is possible, however, to observe some significant reflectivity from the near-surface interfaces.

Fig.10 (a) Peak to peak amplitude terahertz reflection image; (b) cross-sectional b-scan image; and (c) select time-domain signals (offset) [17]

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3.3 Corroded copper-alloy cups

Two Sumerian copper alloy cups excavated from the ancient city of Eshnunna—modern-day Tell Asmar in the Diyala Valley—were also investigated (Fig. 3). Cup A was covered by dark blue corrosion, presumably azurite crystals with some dark green areas, presumably malachite. The transmission through the base of cup A was at the signal noise floor, therefore the subsurface was still entirely metallic (Fig. 11(a)). Cup B was covered in aqua colored corrosion, an admixture of azurite and malachite, with several regions of dark blue and green. Much of the base of cup B is completely mineralized. The black region of the THz transmission measurement in Fig. 11(b) indicates that approximately 13% of the subsurface area has transmission approaching, but not at, the noise floor, unlike with cup A.

Fig.11 Transmission images calculated using THz pulse minimum peak amplitude for (a) cup A and (b) cup B {color scale, same: white/yellow = high transmission, black/violet = low transmission}

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First, we analyzed the minimum peak time-of-flight of both cups. Since no THz signal transmitted through cup A, the image of the minimum peak time was random, as seen in Fig. 12(a). On the other hand, for cup B (Fig. 12(b)), the minimum peak times were locally ordered and fall within a 10 ps time window. This helps to discount the possibility that any dominant pulse in that temporal region is an artificial product of the denoising process, as we would expect it to be random as well. Interestingly, there is not a direct correlation between the peak amplitude and time-of-flight; both regions with the highest and lowest peak amplitudes also have the longest time-of-flights. Additionally, the rim shape of the cup base has no impact on the transmitted peak time, suggesting no change in optical density with deformation for all material components.

Fig.12 Minimum peak time-of-flight for transmission signals through (a) cup A and (b) cup B {color scale, same: white/yellow = shortest time-of-flight [132 ps], black/violet = longest time-of-flight [142 ps]}

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Fig.13 (a) Transmission factor, h, map of cup B; (b) and (c) select frequency- and time-domain (inset) waveforms

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Figure 13(a) shows a discrete color map of cup B’s transmission factor, defined as

(1) $$\eta =\mathrm{100}\left[{\displaystyle \frac{\log \left(E_j/E_{\min }\right)}{\log \left(E_{\max }/E_{\min }\right)}}\right],$$

where E_j is the peak to peak amplitude of the time signal for an individual pixel, normalized by E_min, the peak to peak amplitude of the smallest pixel signal, and E_max is the peak to peak amplitude of the largest pixel signal (specifically, an air reference). For η greater than 50%, the pattern is consistent with the minimum peak amplitude image in Fig. 11(b). However below 50%, the erstwhile black region now has four levels of intensity with the least transmissive region being reduced to an area approximately 9 mm wide and 36 mm long. The compositional nature of this region is not obvious, although it is probable that this smaller black region describes the location of the only remaining metallic subsurface.

We used Coif4 wavelets with 4 levels to denoise the time-domain waveforms and extract the transmitted signal, shown in Figs. 13(b) and 13(c). Despite an order of magnitude difference in transmission between the wholly-mineralized and partially-mineralized region, there are at least two plausible explanations for the presence of THz signal in the black region of Fig. 13(a). The first possibility is that at least part of that subsurface region is non-transmissive metal, with the full-width, Gaussian-beam diameter exceeding that surface area, resulting in lower frequencies leaking through the surrounding dielectric areas. This is consistent with the “concentric” shape of the blue, red and green regions around the black area, and with the narrow bandwidth, low-central frequency spectra selected as D-H in Fig. 13(c).

On the other hand, it is also plausible that the black region comprises a remnant metallic layer of total thickness less than the skin depth of copper through which the THz pulse is propagating. If the skin depth of copper is given by [ 24]

(2) $$\sqrt{\mathrm{4300}/\nu },$$

where $\nu$ is the THz frequency, then the total thickness of any remnant metal must be less than 75 nm, since there is signal up to 0.6 THz. The likelihood of such a fine layer of copper is worth considering and intriguing, given the totality of the cup’s oxidation, and the frequency independent attenuation of metal.

The bases of both cups (Fig. 14) were also measured in reflection. Figure 15(a) shows the image of cup A, calculated using the minimum peak amplitude of the reflected THz pulse. There was significant loss around the edges, due to the increased curvature; however, the reflectivity of the surface is very sensitive to the thickness and number of corrosive layers. Figure 15(b) shows the b-scan—or cross-sectional signal amplitude image, with vertical axis corresponding to time and the horizontal axis corresponding to the space—of the slice denoted by the green line in Fig. 15(a). This cross-section shows that in the violet areas, there are only two large reflections—one on the mineral surface and one on the metal surface. In the orange area, extra lines appear in the cross-section between the two strongest reflections, suggesting one or two different corrosion layers, whereas, in the yellow area, the reflections are the weakest and the cross-section shows that the yellow and white area is highly stratified.

Fig.14 Photographs of the bottom sides of (a) cup A and (b) cup B

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Fig.15 (a) Terahertz pulse reflection minimum peak image of cup A {color scale: white/yellow = low reflection, black/violet = high reflection}; and (b) select cross-sectional b-scan, sliced at green line in (a)

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Figure 16 shows the minimum peak image and a cross-sectional b-scan for cup B. Similarly to cup A, the shape of the base had an impact on the image patterns. A few clusters of higher reflectivity are visible in Fig. 16(a), but none provide insight into the nature of the low transmission area in Fig. 13. The b-scan in Fig. 16(b) is a slice from this area. Interestingly, it shows that between pixels 85 and 35 (left), there are four widely spaced interfaces, separated by 5-10 picoseconds; while between pixels 35 and 15 (right), there are thinner, intermediate layers and the reflections lose intensity at a depth equivalent to the second of the left side’s layers. This may be the depth of the metallic layer.

Fig.16 (a) Terahertz pulse reflection minimum peak image of cup B {color scale: white/yellow = low reflection, black/violet = high reflection}; and (b) select cross-sectional b-scan, sliced at pink line in (a)

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Figure 17 shows a reflection image of cup B, calculated by integrating the spectrum of each pixel from 1.0 to 2.0 THz. In this spectral range, there is not much variation in the reflectivity of the mineral clusters. However, what is most noticeable is the resemblance of the violet (i.e., the most reflective) region shape to that of the black area in Fig. 13(a). This reinforces the notion that the material beneath the surface is not predominantly absorbing, but highly reflective and probably metallic in nature. The dimensions are slightly narrower and better defined in the reflection image, which could be attributed to the smaller beam spot size attributable to the shorter focal length lens used.

Fig.17 Frequency-domain power integration between 1.0 and 2.0 THz image of cup B {color scale: white/yellow = low reflection, black/violet = high reflection}

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4 Conclusions

In this paper, we have shown that THz pulsed imaging may be useful to archaeologists and conservation scientists for the non-destructive study of artifacts. Compared to other imaging modalities, portable THz systems can provide more flexibility in the measurement geometry and scale, and convenience by permitting the system to be taken to the object’s location. While the spatial resolution is not as detailed as in X-ray imaging, and there is significant signal loss as the object scale increases, we have demonstrated that THz imaging is better able to resolve contrast in the low density components of an ancient Egyptian bird mummy. Additionally, through measurements of ancient Sumerian copper alloy artifacts, we have shown that TPIS can be used for the identification of copper corrosion layers. It can differentiate between metallic and mineralized layers and can qualitatively distinguish different corrosion products.

References

[1] Schmid S, Thalhammer G, Winkler K, Lang F, Denschlag J H. Long distance transport of ultracold atoms using a 1D optical lattice. New Journal Physics, 2006, 8(1):73–197

[2] Arlt J, Garcés-Chávez V, Sibbett W, Dholakia K. Optical micromanipulation using a Bessel light beam. Optics Communications, 2001, 197(4–6): 239–245

[3] Matsuoka Y, Kizuka Y, Inoue T.The characteristics of laser micro drilling using a Bessel beam. Application Physics A: Materials Science and Processing, 2006, 84(4): 423–430

[4] Shen X, Zhang H,Hao H, Li D, Li Q, Yan P, Gong M. High energy, single-polarized, single-transverse-mode, nanosecond pulses generated by a multi-stage Yb-doped photonic crystal fiber amplifier . Optics Communication , 2015, 345: 168–172

[5] Zhang H, Shen X, Chen D, Zheng C, Yan P, Gong M. High energy and high peak power nanosecond pulses generated by fiber amplifier. IEEE Photonics Technology Letter, 2014, 26(22): 2295–2298

[6] Durnin J, Miceli J Jr, Eberly J H. Diffraction-free beams. Physics Review Letters, 1987, 58(15): 1499–1501PMID:10034453

[7] Arlt J, Dholakia K.Generation of high-order Bessel beams by use of an axicon. Optics Communications,2000, 177(1–6): 297–301

[8] Tiwari S K, Mishra S R, Ram S P, Rawat H S.Generation of a Bessel beam of variable spot size. Applied Optics, 2012, 51(17): 3718–3725PMID:22695648

[9] Vasara A, Turunen J, Friberg A T. Realization of general nondiffracting beams with computer-generated holograms. Journal of the Optical Society America A,1989,6(11):1748–1754PMID:2585173

[10] Goodell J B, Eccentric lenses for producing ring images. Applied Optics, 1969,8(12):2566PMID:20076084

[11] Descour M R, Simon D I, Yeh W H. Ring-toric lens for focus-error sensing in optical data storage. Applied Optics, 1999, 38(8): 1388–1392PMID:18305758

[12] Gao C, Qi X, Liu Y, Weber H.Superposition of helical beams by using a Michelson interferometer. Optics Express, 2010, 18(1): 72–78PMID:20173824

Acknowledgements

This work was supported by the Program for ChangJiang Scholars and Innovative Research Team in University, PCSIRT(IRT1212), the National Natural Science Foundation of China (Grant Nos. 61475021 and 61108047).

RIGHTS & PERMISSIONS

2016 Higher Education Press and Springer-Verlag Berlin Heidelberg