Piezotronic Nanowire “Taxel” Gives Active/Adaptive Sense of Touch

[From Left to Right] Xiaonan Wen, Zhong Lin Wang, Wenzhuo Wu


Authors: Wenzhuo Wu, Xiaonan Wen and Zhong Lin Wang

Affiliation: 
School of Materials Science and Engineering, Georgia Institute of Technology, Atlanta, USA.


Emulation of human senses via electronic means has long been a grand challenge in research of artificial intelligence as well as prosthetics, and is of pivotal importance for developing intelligently accessible and natural interfaces between human/environment and machine. Unlike other senses (seeing, hearing, smelling and tasting), capability of skin for touch sensing remains stubbornly difficult to be mimicked, which necessitates the development of large-scale pressure sensor arrays with high spatial-resolution, high-sensitivity and fast response.

Implementations of pressure sensor arrays have been previously reported using assembled nanomaterials or microstructured rubber layers based on the change in capacitance or resistance upon pressure, which have been applied to map strain distribution in a matrix format with resolution in the millimeter-scale [1,2]. In those demonstrations, electronic components like traditional field-effect-transistors (FETs) act as read-out elements for detecting pressure-induced property change in the pressure-sensitive media. Intensive efforts have been devoted to minimize the effect of substrate strain on performance of these electronic components while preserving the deformability of the substrate. This scheme of pressure sensing, whereas, not only requires complicated system integration of heterogeneous components but also lacks proficiency in directly interfacing electronics with mechanical actions in an active way that mechanical straining can be utilized to generate new electronic control/feedback. Therefore, new approaches have to be developed for better mimicking or even competing with the tactile sensing capabilities of human skins.

A direct control over the operation of electronic devices by mechanical action could be advantageous for implementing tactile sensing and realizing direct interfacing between machines and human/ambient. Piezoelectric materials are ideal candidates for this purpose, which produce electrical potential upon variations of applied pressure/stress due to the linear electromechanical coupling/interaction between mechanical and electrical state in materials lacking inversion symmetry. The most well-known piezoelectric material is perovskite-structured Pb(Zr_xTi_1-x)O₃ (PZT), which has been widely used for electromechanical sensing, actuating and energy harvesting. PZT, however, is electrically-insulating and hence less useful for building electronic devices. In addition, the extremely brittle nature of ceramic PZT films and incorporation of lead impose issues such as reliability, durability and safety for long term sustainable operations and hinder its applications in areas such as biomedical devices.

Figure 1: [click on image to view higher resolution] Piezopotential in Wurtzite crystal. (a) Atomic model of the Wurtzite-structured ZnO. (b) Distribution of piezopotential along a ZnO NW under axial strain calculated by numerical methods. The growth direction of the NW is along c-axis. The color gradient represents the distribution of piezopotential in which red indicates positive piezopotential and blue indicates negative piezopotential.

Due to the polarization of ions in a crystal that has non-center symmetry, a piezoelectric potential (piezopotential) is created in the crystal by applying a stress [3] (Figure 1). For piezoelectric semiconductor materials such as ZnO, GaN, InN and CdS, the effect of piezopotential on the transport behavior of charge carriers is significant due to their multiple functionalities of piezoelectricity, semiconductor and photon excitation. One of the important effects of piezopotential on p-n junction and metal-semiconductor contact is to tune/control the charge transport across the interface, which is the basis of piezotronic effect (Figure 2) [3,4]. Electronics fabricated by using inner-crystal piezopotential as a “gate” voltage to tune/control the charge transport behavior is called piezotronic device. Our group initiated the research in piezotronics back to 2007 [5]. Since then we have investigated the piezotronic effect for realizing novel applications such as strain-gated piezotronic logic nanodevices [6] and piezotronic strain memory device [7]. In contrast to conventional CMOS units, the strain-gated electronics is driven by mechanical agitation. A brief comparison between piezotronic transistor and field effect transistor (FET) is shown in Figure 3.

Figure 2: [click on image to view higher resolution] Schematic of energy diagram illustrating the effect of piezopotential on modulating the metal-semiconductor contact and p-n junctions. (a) With compressive strain applied, the negative piezoelectric polarization ionic charges induced near the interface (symbols with “-”) increases the local Schottky barrier height (SBH). (b) With tensile strain applied, the positive piezoelectric polarization ionic charges induced near the interface (symbols with “+”) decreases the local SBH. (c) and (d) With strain applied, the piezoelectric polarization ionic charges are induced near the junction interface.

Figure 3: [click on image to view higher resolution] Comparison between traditional FET and Strain-gated piezotronic transistor (SGT).

Designing, fabricating and integrating arrays of nanodevices into a functional system is the key for transferring nano-scale science into applicable nanotechnology. In our latest work [8], we report the first and by far the largest 3D array integration of vertical NW piezotronic transistors circuitry (92 x 92 taxels with 234 taxels per inch) as active/adaptive taxel-addressable pressure-sensor matrix for tactile imaging. Here “taxel” is short for tactile pixel, which is the unit in our array device. We combined the top-down microfabrication processes for fabricating device structure with the bottom-up synthesis of ZnO NWs at low-temperature (85^oC).

The taxel area density of SGVPT (strain-gated vertical piezotronic transistor) array is 8464/cm², which is higher than the number of tactile sensors in recent reports (~ 6-27/cm²) [1,2,9] and mechanoreceptors embedded in the human fingertip skins (~ 240/cm²) [10]. The fabricated sensors are capable of mapping spatial profiles of small pressure changes (< 10 kPa). The spatial resolution with taxel dimension (20 x 20 μm²) and pitch size (100 μm) as well as the tactile sensitivity (2.1 μs∙kPa^-1) have been demonstrated, enabling a 15-to-25-fold increase in number of taxels and 300-to-1000-fold increase in taxel area density compared to recent reports.

Each taxel in the array device is made of a strain-gated vertical piezotronic transistor (SGVPT) based on vertically-aligned n-ZnO nanowires (NWs) (Figure 4a, d). SGVPT operates based on modulation of local contact characteristics and charge carrier transport by strain-induced ionic polarization charges at the interface of metal-semiconductor contact, which is the fundamental of piezotronics. Using the piezoelectric polarization charges created at metal-semiconductor interface under strain to gate/modulate transport process of local charge carriers, piezotronic effect has been applied to design independently addressable two-terminal transistor arrays, which convert mechanical stimuli applied on the devices into local electronic controlling signals (Figure 4b-c). Figure 4c shows the equivalent circuit to present the operation scheme of SGVPT array.

Figure 4: [click on image to view higher resolution] (a) Comparison between three-terminal voltage-gated NW FET (left) and two-terminal strain-gated vertical piezotronic transistor (right). (b) Schematic illustration of a 3D SGVPT array with taxel density of 92 × 92 and scheme for spatial profile imaging of local stress. (c) Equivalent circuit diagram of the 3D SGVPT array. (d) SEM image of SGVPT array taken after etching-back the SU 8 layer and exposing top portions (~ 20 μm) of the ZnO NWs. Inset, 30^o-tilt view of the exposed ZnO NWs for single taxel. (e) Optical image of the transparent 3D SGVPT array on flexible substrate. (f) Current responses for single taxel under different pressures, presenting the gate modulation effect of applied pressure on the electrical characteristics of SGVPT.

The elimination of wrap gate offers a new approach for 3D structuring. The local contact profile and carrier transport across the Schottky barriers, formed between ZnO NW and metal electrodes, can be effectively controlled by the polarization-charge-induced potential. Electrical characteristics of the two-terminal SGVPT are therefore directly modulated by external mechanical actions induced strain, which essentially functions as a gate signal for controlling carrier transport in SGVPT. The modulation effect of applied pressure is shown from the plot of current variations against pressure changes (Figure 4f). Cross-bar electrodes have been configured for multiplexed data acquisition and the spatial profiles of applied stress can be recorded and imaged. The output signal is current response so that it is easy to integrate SGVPT array with back-end interface circuits for fast data processing, recording and transmission.

Three unique capabilities of SGVPT array device, which are not readily available in previous reports, have been demonstrated in this work:
1. The capability of using SGVPT array for multi-dimensional signature recording, which not only records the calligraphy or signature patterns, when people write, but also registers the corresponding pressure/force applied at each location/pixel by the person and the speed or writing. This augmented capability can essentially provide means for realizing personal signature recognition with unique identity and enhanced security.
2. The shape-adaptive sensing capability of SGVPT array, which provides means for detecting the shape change of device in situ in real-time and feeding-back the sensed changes in shape for calibration of other functionalities as well as corresponding control/response performed by the system. The real time detection of shape changes caused by stretching or twisting is a desirable feature for sensors embedded in an artificial tissue or prosthetic device. This unique capability enables the potential integration of SGVPT device for applications in artificial/prosthetic skin in smart biomedical treatments and intelligent robotics.
3. Additionally, the SGVPT devices can also function as self-powered active tactile sensors by converting mechanical stimulations into electrical signals utilizing the piezopotential without applied bias, which emulates the physiological operations of mechanoreceptors in biological entities, such as human hair follicles and hair cells in the cochlea.

Table 1: Comparison between piezoresistive effect and piezotronic effect.

Our approach is based on a completely different mechanism from the traditional designs based on piezoresistive effect. A comparison between piezotronic effect and commonly recognized piezoresistive effect is listed in Table 1. This technology eliminates the wrap-gate electrode for fabricating 3D vertical NW based transistor, taking advantage of piezotronic effect. The structural transformation from three-terminal configuration into two-terminal configuration significantly simplifies the layout design and circuitry fabrication while maintains effective control over individual devices. The approach presented has the potential to be integrated with silicon-based CMOS technology for achieving augmented functionalities in smart systems in the era of “More Than Moore”, such as artificial skin, personal electronics and potential integration with compliant energy harvesting modules for self-powered flexible functional systems [11]. The scalability of this platform in integrating in-place synthesized single-crystalline NWs in controllable manners together with its demonstrated compatibility with state-of-the-art microfabrication techniques enables future large-scale implementation of wurtzite NWs for practical applications in human-electronics interfacing, smart skin and micro/nano-electromechanical systems.

References:
[1] Kuniharu Takei, Toshitake Takahashi, Johnny C. Ho, Hyunhyub Ko, Andrew G. Gillies, Paul W. Leu, Ronald S. Fearing & Ali Javey. "Nanowire active-matrix circuitry for low-voltage macroscale artificial skin". Nature Materials, 9: 821-826 (2010). Abstract.
[2] Stefan C. B. Mannsfeld, Benjamin C-K. Tee, Randall M. Stoltenberg, Christopher V. H-H. Chen, Soumendra Barman, Beinn V. O. Muir, Anatoliy N. Sokolov, Colin Reese & Zhenan Bao. "Highly sensitive flexible pressure sensors with microstructured rubber dielectric layers". Nature Materials, 9: 859-864 (2010). Abstract.
[3] Zhong Lin Wang. "Progress in piezotronics and piezo-phototronics". Advanced Materials, 24: 4632-4646 (2012). Abstract.
[4] Zhong Lin Wang. "Piezotronics and piezo-phototronics" (Springer, 2013).
[5] Zhong Lin Wang. "Nanopiezotronics". Advanced Materials, 19: 889-892 (2007). Abstract.
[6] Wenzhuo Wu, Yaguang Wei, Zhong Lin Wang. "Strain-gated piezotronic logic nanodevices". Advanced Materials, 22: 4711-4715 (2010). Abstract.
[7] Wenzhuo Wu and Zhong Lin Wang. "Piezotronic nanowire-based resistive switches as programmable electromechanical memories". Nano Letters, 11: 2779-2785 (2011). Abstract.
[8] Wenzhuo Wu, Xiaonan Wen, Zhong Lin Wang. "Taxel-addressable matrix of vertical-nanowire piezotronic transistors for active/adaptive tactile imaging". Science, 340: 952-957 (2013). Abstract.
[9] Tsuyoshi Sekitani, Tomoyuki Yokota, Ute Zschieschang, Hagen Klauk, Siegfried Bauer, Ken Takeuchi, Makoto Takamiya, Takayasu Sakurai, Takao Someya. "Organic nonvolatile memory transistors for flexible sensor arrays". Science, 326: 1516-1519 (2009). Abstract.
[10] R.S. Johansson, A.B. Vallbo. "Tactile sensibility in the human hand - relative and absolute densities of 4 types of mechanoreceptive units in glabrous skin". Journal of Physiology, 286: 283-300 (1979). Article.
[11] Zhong Lin Wang, Wenzhuo Wu. "Nanotechnology-enabled energy harvesting for self-powered micro-/nanosystems". Angewandte Chemie International Edition, 51: 11700-11721 (2012). Abstract.

Edgy Photonics

[From Left to Right] Mordechai Segev, Julia M. Zeuner, Mikael C. Rechtsman, Yaakov Lumer, Yonatan Plotnik.

Authors: Mikael C. Rechtsman¹, Julia M. Zeuner², Yonatan Plotnik¹, Yaakov Lumer¹, Mordechai Segev¹ and Alexander Szameit²

Affiliation: 
¹Department of Physics and the Solid State Institute, Technion – Israel Institute of Technology, Haifa, Israel
²Institute of Applied Physics, Abbe Center of Photonics, Friedrich-Schiller-Universität Jena, Germany

There is a topological revolution going on in solid-state physics. At its heart are ‘topological insulators’ (TIs), a class of materials that have electrical properties unlike anything else that has been seen before [1–4]. In TIs, electrical current flows only on the edges or surfaces and not through the bulk of the material: this is why the titles of many popular science articles on the topic are variants of “Physics on the edge.” The atoms making up these materials are, for example, Bismuth Selenide (for the three-dimensional case), and Mercury Telluride (for the two-dimensional case).

Alexander Szameit of Friedrich-Schiller-Universität Jena

The two-dimensional TI has perhaps the most bizarre behavior: electrical current flowing on its edge does not get obstructed by imperfections of the lattice. Any kind of defect or disorder cannot scatter the electrons – allowing them to flow in a smooth and seamless way. Essentially because of this robustness, the condensed matter and solid-state physics community has been convinced that these are going to be the materials at the core of futuristic technologies like robust quantum computation and spintronic devices.

Just as electronics uses the flow of electrons to manipulate information, the field of photonics explores the use of light to do the same thing. Examples of photonic technologies are optical fibers (which transmit internet signals across oceans) and DVD players (think “optical drives”), among many, many others. Physicists in the 1980s realized that light could be manipulated with incredible control in “photonic crystals" [5,6], which are lattices made out of some transparent material (for example, glass, polystyrene, or even silicon). Photonic crystals are central in many emerging photonic technologies, and have the ability to manipulate light in much the same way that semiconductors can control and manipulate electrons in computer chips.

This begs the question: can photons be topologically protected from disorder the same way that electrons can in TIs? Can topological robustness be realized photonic crystals just as it was in solids? As Raghu and Haldane showed in a theoretical paper in 2008 [7], the answer is yes – all you need to do is make a photonic crystal out of a gyromagnetic material (like, for example, Yttrium Iron Garnet - YIG), and apply an external constant magnetic field. The external field acts to break “time-reversal symmetry,” meaning that if light is flowing forward, it could be prevented from being scattered (due to the unavailability of a backwards-propagating wave).

 Raghu and Haldane’s theory was proven correct by an experiment carried out in Marin Soljacic’s group in 2009 [8,9]. But there was a catch: because of the weakness of the gyromagnetic effect at optical wavelengths, their experiment could only be done at microwave frequencies. That meant that the photonic crystals involved could be (roughly) no smaller than the metallic grating you can find on the front cover of your microwave oven. Another scheme – in other words, different physics – would have to be found in order to scale down to micron scales and beyond.

There were quite a number of ideas put forward on how to exactly to scale down in order to make photonic topological insulators with light [10–12]. Our team of researchers (a collaboration between the Segev group at the Technion – Israel Institute of Technology in Haifa and the Szameit group at Friedrich-Schiller University in Jena, Germany) conceived of a new way of doing exactly this, and then implemented it in the lab [13]. The group used an array of waveguides made of fused silica – i.e., glass – to create a honeycomb lattice. The waveguides – which are essentially wires for light – were made to propagate in a helical fashion, rather than straight (see Fig. 1). The helicity is the key ingredient – it allows the light to differentiate between diffracting off to the left or to the right, and only thus can topological protection be achieved. An animation of a simulation in which light misses a defect and then goes around a corner is included below. Full experimental results are available in a recent publication in the journal Nature [13].

Figure 1: This is a schematic diagram of a photonic topological insulator realized in Ref. 13. Each helix represents a waveguide that guides light just as a wire guides electrical current. Each waveguide is about 6μm in diameter and the spacing between waveguides is 15μm.

Caption for Animation: This animation is a simulation of how light propagates through a photonic topological insulator. As light travels down the waveguides, it encounters a defect in the form of a missing waveguide and simply bypasses it without getting scattered. Then, light encounters the corner of the array and simply goes up along the edge because it cannot go back. The reason that the whole array is spinning in a circular way is due to the helicity of the waveguides.

The realization of topological protection of light will impact photonic crystal science and beyond – especially by making photonic devices more robust to disorder, and hence – hopefully – cheaper to manufacture. The optical setting will also enable probing new topological physics that may never have been accessed otherwise. Perhaps the most beautiful aspect of photonic TIs is that they are another example of “emergent physics” [14] – the idea that in complex systems, physics can be much more than the sum of its parts.

References:
[1] C. L. Kane and E. J. Mele. "Quantum Spin Hall Effect in Graphene". Physical Review Letters, 95, 226801 (2005). Abstract.
[2] Markus König, Steffen Wiedmann, Christoph Brüne, Andreas Roth, Hartmut Buhmann. "Quantum Spin Hall Insulator State in HgTe Quantum Wells". Science, 318, 766–770 (2007). Abstract.
[3] D. Hsieh, D. Qian, L. Wray, Y. Xia, Y. S. Hor, R. J. Cava, M. Z. Hasan. "A topological Dirac insulator in a quantum spin Hall phase". Nature, 452, 970–974 (2008). Abstract.
[4] M.Z. Hasan, C.L. Kane. "Colloquium: Topological insulators". Review of Modern Physics, 82, 3045–3067 (2010). Abstract.
[5] Eli Yablonovitch. "Inhibited Spontaneous Emission in Solid-State Physics and Electronics". Physical Review Letters, 58, 2059–2062 (1987). Abstract.
[6] Sajeev John. "Strong localization of photons in certain disordered dielectric superlattices". Physical Review Letters, 58, 2486–2489 (1987). Abstract.
[7] F.D.M. Haldane and S. Raghu. "Possible Realization of Directional Optical Waveguides in Photonic Crystals with Broken Time-Reversal Symmetry". Physical Review Letters, 100, 013904 (2008). Abstract.
[8] Zheng Wang, Y. D. Chong, John D. Joannopoulos, Marin Soljačić. "Reflection-Free One-Way Edge Modes in a Gyromagnetic Photonic Crystal". Physical Review Letters, 100, 013905 (2008). Abstract.
[9] Zheng Wang, Yidong Chong, J. D. Joannopoulos, Marin Soljačić. "Observation of unidirectional backscattering-immune topological electromagnetic states". Nature, 461, 772–775 (2009). Abstract.
[10] Mohammad Hafezi, Eugene A. Demler, Mikhail D. Lukin, Jacob M. Taylor. "Robust optical delay lines with topological protection". Nature Physics, 7, 907–912 (2011). Abstract.
[11] Kejie Fang, Zongfu Yu, Shanhui Fan. "Realizing effective magnetic field for photons by controlling the phase of dynamic modulation". Nature Photonics, 6, 782–787 (2012). Abstract.
[12] Alexander B. Khanikaev, S. Hossein Mousavi, Wang-Kong Tse, Mehdi Kargarian, Allan H. MacDonald, Gennady Shvets. "Photonic topological insulators". Nature Materials, 12, 233–239 (2013). Abstract.
[13] Mikael C. Rechtsman, Julia M. Zeuner, Yonatan Plotnik, Yaakov Lumer, Daniel Podolsky, Felix Dreisow, Stefan Nolte, Mordechai Segev & Alexander Szameit. "Photonic Floquet topological insulators". Nature, 496, 196–200 (2013). Abstract.
[14] P.W. Anderson. "More Is Different". Science 177, 393–396 (1972). Abstract.

Precision Interferometry in a New Shape: Higher-order Laguerre-Gauss Modes for Gravitational Wave Detection

[Paul Fulda is the recipient of the 2012 GWIC (Gravitational Wave International Committee) Thesis Prize for his PhD thesis “Precision Interferometry in a New Shape: Higher-order Laguerre-Gauss Modes for Gravitational Wave Detection” (PDF). -- 2Physics.com]

Author: Paul Fulda

Affiliation:

Currently at Department of Physics, University of Florida, Gainesville, USA

PhD research performed at School of Physics and Astronomy, University of Birmingham, UK.


With the approach of advanced detector science runs, the field of gravitational wave detection stands on the brink of finally making the much sought after first direct detection of gravitational waves. Never a bunch to rest on our laurels, however, we are always trying to think of ways to push the sensitivity of these kilometer-scale interferometers ever higher. A single detection, with its ramifications in the areas of astrophysics, cosmology and General Relativity, would be an incredible achievement for a scientific field long in the making.

However, since the observable astrophysical event rate scales as the cube of the observable event distance (and hence roughly as the cube of the broadband sensitivity), any sensitivity increase we can make will bring with it an ever greater number of detected gravitational wave events. The more events we have, the better statistics we can do with them, and the better we can refine our model of the universe itself. With that in mind, in this article I will focus on one of many ideas for improving the sensitivity of gravitational wave detectors beyond the 'advanced' generation; the idea of using higher-order Laguerre-Gauss modes to reduce the thermal noise of the test-masses of the interferometers.

2Physics articles by past winners of the GWIC Thesis Prize:

Rutger van Haasteren (2011): "Pulsar Timing Arrays: Gravitational-wave detectors as big as the Galaxy"
Haixing Miao (2010): "Exploring Macroscopic Quantum Mechanics with Gravitational-wave Detectors"
Holger J. Pletsch (2009): "Deepest All-Sky Surveys for Continuous Gravitational Waves"
Henning Vahlbruch (2008): "Squeezed Light – the first real application starts now"
Keisuke Goda (2007): "Beating the Quantum Limit in Gravitational Wave Detectors"
Yoichi Aso (2006): "Novel Low-Frequency Vibration Isolation Technique for Interferometric Gravitational Wave Detectors"
Rana Adhikari (2003-5)*: "Interferometric Detection of Gravitational Waves : 5 Needed Breakthroughs"
*Note, the gravitational wave thesis prize was started initially by LIGO as a biannual prize, limited to students of the LIGO Scientific Collaboration (LSC). The first award covered the period from 1 July 2003 to 30 June 2005. In 2006, the thesis prize was adopted by GWIC, renamed, converted to an annual prize, and opened to the broader international community.

The rest of this article won't make a lot of sense if I don't first explain what a higher-order laser mode is, but in order to understand this concept we also need to know a little bit about optical cavities. An optical cavity exists wherever there is an arrangement of mirrors that form a closed optical path. Perhaps the simplest case is a linear cavity, which consists of two partially transmissive mirrors, one flat and one curved, orientated with the reflective surfaces parallel to each other (see Fig. 1). When the length of the cavity is tuned just right relative to the frequency of the light, light entering through one partially transmissive mirror creates a standing wave inside the cavity. This condition is known as a resonance of the cavity, and it creates a higher circulating light power inside the cavity than the light power entering. The light is 'stored' inside the cavity for some time before it leaks out again.

Fig 1: A cartoon picture of an optical cavity. Light enters the cavity through the input mirror from the left, and resonates inside the cavity if the cavity length is divisible by an integer number of half wavelengths.

There is another condition on top of the length/frequency condition in order for light to resonate in an optical cavity; the amplitude cross-section of the light (i.e. the beam shape) must be unchanged after several reflections from the mirrors that make up the cavity. If this is not the case, the light will not properly add up after each reflection to make a standing wave. For a cavity made up of spherically curved mirrors, this condition is satisfied by sets of so-called "spatial modes". There are two main families of these spatial modes - Hermite-Gauss (HG) modes and Laguerre-Gauss (LG) modes (see Fig. 2). In a well designed optical cavity, each higher mode order (the order being n+m for HG modes and 2p+l for LG modes) is resonant at a different cavity tuning. These modes are mathematical solutions to the wave equation in the beam-like or 'paraxial' limit. However, far from being purely abstract mathematical constructs, they really exist and show up in interferometers all the time, as we shall soon see!

Fig 2: [click on the image to see better resolution] Intensity patterns for ideal higher-order HG (top left) and LG (top right) modes. The lower half of the image shows the higher-order LG modes generated in the lab using an SLM (in negative to show detail).

Advanced interferometric gravitational wave detectors make use of optical cavities in many places. Advanced LIGO for example has a pre- mode-cleaner cavity, an input mode cleaner cavity and an output mode cleaner cavity, in addition to the 4 coupled cavities of the central interferometer (see Fig. 3) [1]. These optical cavities are used outside the central interferometer as 'mode cleaners' for their ability to filter the frequency and spatial properties of light, and inside the central interferometer in order to increase the circulating light power and the interaction time between the light and passing gravitational waves. LIGO would not have been able to achieve the unprecedented strain sensitivity it did without using these crucial interferometric tools.

Fig 3: [click on the image to see better resolution] The optical layout of an Advanced LIGO interferometer, showing the input and output mode cleaner cavities, the two 4km long arm cavities and the two recycling cavities.

Usually one aims to use only the lowest order 'fundamental' mode (the HG00/LG00 mode), for precision interferometric metrology. The presence of any of the higher-order modes commonly signifies an imperfection in the optical setup, for example a misalignment or a discrepancy in the characteristic size of two interfering beams. As a result, there are several activities directed towards reducing the presence of these modes in interferometers [2, 3].

Recently, however, it was shown that by using one of the higher-order Laguerre-Gauss modes instead of the funamental mode it may be possible to reduce the effects of mirror thermal noise, which is expected to be one of the limiting noise sources in gravitational wave detectors [4]. Mirror thermal noise appears as random fluctations of the reflecting surfaces, which can mask any gravitational wave passing through the interferometer. Higher-order LG modes are able to reduce the effect of this noise by averaging better over the mirror surface, effectively smoothing out the fluctuations. Higher-order LG modes also have the interesting property of having spiral phasefronts, thereby carrying angular momentum - a property which has made them useful in the fields of cold atoms and bio-photonics [5].

Could it then be a case of "from villain to hero" for higher-order modes? This is the question we sought to answer with a program of research into the application of LG modes in precision interferometry. Although the thermal noise benefits of using LG modes made them seem like an attractive option, it was crucial to test their compatibility with all of the other technologies that are used in gravitational wave detectors. At the University of Birmingham we started this investigation using simulations of advanced interferometer where the main fundamental mode beam was exchanged for a LG33 beam (top right in each panel of Fig. 1) [6]. The LG33 beam performed very favorably in the simulations, giving us the motivation to move forward to table-top lab experiments.

We used a device called a spatial light modulator (SLM) to generate the higher-order LG modes in the lab. This device works somewhat like an LCD screen, except that each pixel can change the phase of the light reflected from it instead of the intensity. By displaying a specific phase pattern on the SLM, we were able to convert a fundamental mode beam into almost any higher-order LG mode we wanted (see the lower half of Fig. 2). We confirmed the spiral phase front of the beams by interfering them with a fundamental mode beam, creating the spiral intensity patterns shown in Fig. 4. However, the mode purity we could achieve with the SLM alone was not high enough for use with gravitational wave detectors.

Fig. 4: Images of the interference pattern between various SLM generated higher-order LG beams and a fundamental mode beam. The spiral pattern indicates the presence of a spiral phasefront in the LG beams. From left: LG22, LG33, LG44, LG55.

The positive aspect of this was that it gave us the perfect opportunity to make one of the most crucial compatibility tests for the LG33 mode - if the LG33 mode was to be effective in gravitational wave detectors, it would have to be compatible with mode cleaner optical cavities and the control schemes that are commonly used with them. If it worked, we would have the added bonus of a `cleaned' LG33 mode after the mode cleaner. As predicted by the earlier simulations, the LG33 mode was compatible with a mode cleaner cavity, and we achieved mode purity in excess of 99%. Figure 5 shows images of two kinds of LG33 mode before and after the mode cleaner cavity [7].

Fig. 5: Images of the SLM generated LG33 mode before (top) and after (bottom) transmission through a linear optical cavity.

With these positive results we decided to move on to the next level of compatibility testing: using LG modes with a suspended 10m prototype interferometer in Glasgow. The specific aim here was to address a tricky problem with LG modes that was identified through simulations; with larger beam sizes, the imperfections in the mirrors that form an optical cavity can cause a significant reduction in the mode purity [8,9].

We set up an LG mode conversion path on the Glasgow prototype input bench (see Fig. 6), and directed the beam into the vacuum tank towards a 10m long optical cavity. We employed a sophisticated method to precisely tune the length of the cavity, while observing the transmitted light power with a high-speed infra-red camera. Since different modes are resonant in the cavity at different tunings, the high-speed camera measurements gave us a good picture of the higher-order mode content in the cavity.

Fig. 6: [click on the image to see better resolution] The LG mode conversion path for the Glasgow 10m prototype interferometer experiment. The red line shows the LG33 conversion path, the green line shows the fundamental mode 'control' path, and the purple line shows the original laser path.

We saw that even when we injected a LG33 mode, we could only observe the rectangular shaped HG modes resonating in the cavity (see Fig. 7). The work reported in [8] and [9] suggested that this behavior might be expected in a cavity where the mirrors are astigmatically curved, rather than the spherically curved ideal case. This was a setback for the use of LG modes in gravitational wave interferometers, as it was the first experimental evidence that mirror surface imperfections might have a stronger impact on mode purity for the LG33 mode than for the fundamental mode. The cavity mirrors were taken out of the prototype and independent measurements of the surface figures were made. The measured astigmatism values were used as inputs for a simulation of the 10m cavity, which reproduced the resonance structures seen in Fig. 7 very well. This was the confirmation that was needed to show that the LG33 mode really was more sensitive to mirror surface imperfections that the fundamental mode [10].

Fig. 7: [click on the image to see better resolution] A trace showing reflected power from the 10m cavity as a function of time as the cavity is swept across a resonance with the LG33 input beam. The resonance is split into several peaks, as predicted in [8] and [9]. Images from the high-speed camera in transmission of the cavity show the resonant mode shape around several of the separate resonance peaks.

From here it was possible to derive tolerances on the mirror imperfections that would be allowable if the LG33 mode was to be used as the readout beam in a real gravitational wave detector. The requirements that the LG33 mode puts on the quality of the mirror surfaces are beyond what is achievable with the current technology, and as such they do look very tough to meet. However, coating and polishing technology is always moving forward with the demands of ever more precise experiments, so at some time in the future we may yet see the LG33 mode flashing on the projector screens in a gravitational wave detector installation.

Those of us who work in the business of laser interferometry become quite accustomed to the sight of higher-order spatial laser modes. It can be easy to take their aesthetic qualities for granted, especially when a lot of our efforts are directed towards minimizing their presence. I still remember the first time I aligned an optical cavity though, and how I was mesmerized by the small CCTV camera picture showing the strange and beautiful patterns in the cavity transmitted light. I'm very glad that my PhD research took me in the direction of studying a particular one of these higher-order laser modes in great detail. By the end of my studies, the higher-order modes weren't just obscure patterns of light to me any more; rather they seemed like cryptic hieroglyphs which when translated could provide a wealth of knowledge about the state of an optical system.

References:
[1] Gregory M. Harry and the LIGO Scientific Collaboration. "Advanced LIGO: the next generation of gravitational wave detectors". Classical and Quantum Gravity, 27(8):084006 (2010). Abstract.
[2] Euan Morrison, Brian J. Meers, David I. Robertson, and Henry Ward. "Automatic alignment of optical interferometers". Applied Optics, 33:5041–5049 (1994). Abstract.
[3] Guido Mueller, Qi-ze Shu, Rana Adhikari, D. B. Tanner, David Reitze, Daniel Sigg, Nergis Mavalvala, and Jordan Camp. "Determination and optimization of mode matching into optical cavities by heterodyne detection". Optics letters, 25(4):266-268 (2000). Abstract.
[4] Benoît Mours, Edwige Tournefier and Jean-Yves Vinet. "Thermal noise reduction in interferometric gravitational wave antennas: using high order TEM modes". Classical and Quantum Gravity, 23:5777-5784 (2006). Abstract.
[5] H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop. "Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity". Physical Review Letters, 75(5):826–829 (1995). Abstract.
[6] Simon Chelkowski, Stefan Hild, and Andreas Freise. "Prospects of higher- order Laguerre-Gauss modes in future gravitational wave detectors". Physical Review D, 79(12):122002 (2009). Abstract.
[7] Paul Fulda, Keiko Kokeyama, Simon Chelkowski, and Andreas Freise. "Experimental demonstration of higher-order Laguerre-Gauss mode interferometry". Physical Review D, 82(1):012002 (2010). Abstract.
[8] Charlotte Bond, Paul Fulda, Ludovico Carbone, Keiko Kokeyama, and Andreas Freise. "Higher order Laguerre-Gauss mode degeneracy in realistic, high finesse cavities". Physical Review D, 84(10):102002 (2011). Abstract.
[9] T. Hong, J. Miller, H. Yamamoto, Y.Chen, R. Adhikari. "Effects of mirror aberrations on Laguerre-Gaussian beams in interferometric gravitational wave detectors". Physical Review D, 84(10):102001 (2011). Abstract.
[10] B. Sorazu, P. Fulda, B.W. Barr, A.S. Bell, C. Bond, L. Carbone, A. Freise, S.H. Huttner, J. Macarthur, K.A. Strain. "Experimental test of higher-order Laguerre–Gauss modes in the 10m Glasgow prototype interferometer". Classical and Quantum Gravity, 30(3):035004 (2013). Abstract.

Losing Energy with Hamilton’s Principle of Least Action

Chad Galley

Author: Chad Galley

Affiliation: Theoretical Astrophysics (TAPIR), California Institute of Technology, Pasadena, USA

Classical mechanics is the foundation of physics and of all students’ course work in engineering and the physical sciences. The bricks of this foundation were first laid by Galileo then by Newton and finally by the likes of D’Alembert, Hamilton, Lagrange, Poisson, and Jacobi in the 18th and 19th centuries. What resulted was a framework of physical laws and formalisms for virtually any problem one wishes to study in fluid mechanics [1], electromagnetism [2], statistical mechanics [3], and even quantum theory later on, to give just a few examples.

One important and pervasive formulation of classical mechanics is due to Hamilton, who showed that a physical system evolves to either minimize or maximize a quantity called the action which, loosely speaking, is the accumulation in time of the difference between the kinetic and potential energies [4]. This important result, called Hamilton’s variational principle of stationary action or Hamilton’s principle for short, is the primary way to derive equations of motion for many systems, from the ubiquitous simple harmonic oscillator to supersymmetric string theories. Unfortunately, Hamilton’s principle has a well-known shortcoming: it generically cannot account for the irreversible effects of energy loss that are always present in any real world application, experiment, or problem. But why is that?

The answer has to do with the very formulation of Hamilton’s principle: “The physical configuration of a system is the one that evolves from the given state A at the initial time to the given state B at the final time such that the action is stationary.” This raises the question: how can one know the final state, especially when the system is losing energy? Isn’t the point to determine the final state from initial conditions? That’s how the real world works after all, through cause then effect. Remarkably, answering these questions correctly leads to a natural way to describe generic systems with a variational principle, even those that do not conserve energy [5].

The questions above are usually addressed, if at all, using a somewhat circular reasoning as follows. In practice, one applies Hamilton’s principle to derive equations of motion that are then solved with initial data. The fixed final state used in Hamilton’s principle is argued then as being the one associated with that specific solution. However, that specific final state is only determined after applying Hamilton’s principle to get the equations of motion in the first place. Perhaps this is a passable explanation but it doesn’t seem completely satisfactory because we usually do not have access to the environment that a system loses energy to so we cannot freely adjust the final states of those inaccessible degrees of freedom to accommodate the above explanation.

For these reasons and others, it is important to generalize Hamilton’s variational principle in a way that does not require fixing the final state of the system but is determined instead from the initial state only. The details of how this is achieved are reported in [5]. The take-home result is that eliminating dependence on the final state requires a formal doubling of the degrees of freedom in the problem. These doubled variables are fictitious but their average values are the physical ones of interest whereas their difference does not contribute to the physical evolution of the system. Figure 1 one shows a cartoon of the usual Hamilton’s principle on the left and of Hamilton’s principle on the right generalized to accommodate for energy losses (or gains). The arrows in Figure 1 indicate the direction in time to integrate the Lagrangian of the system along that path.

Figure 1: Left: A cartoon of Hamilton’s principle. Dashed lines denote the virtual displacements and the solid line denotes the stationary path. Right: A cartoon of Hamilton’s principle compatible with initial data (i.e., the final state is not fixed). In both cartoons, the arrows on the paths indicate the integration direction for the line integral of the Lagrangian.

Doubling the variables in this formal way has an interesting natural consequence. Just as the potential V in classical mechanics is an arbitrary function for conservative systems, we now have the freedom to introduce an additional arbitrary function K that couples together the doubled variables. In many ways K is analogous to V in classical mechanics because K generates the forces and interactions that account for energy loss or gain in a similar way that V generates forces and interactions that conserve energy.

To summarize, the seemingly innocuous problem with specifying the final state in Hamilton’s principle leads to a generalization based solely on the initial state. Achieving this requires formally doubling the degrees of freedom that, in turn, allows for an extra arbitrary function K to be introduced that generically accounts for the dynamical forces and interactions that cause energy loss or gain in the system. This new variational principle may have broad applicability in a wide range of practical and theoretical problems across multiple disciplines.

References
[1] G. K. Batchelor, "An Introduction to Fluid Dynamics" (Cambridge University Press, Cambridge, England, 1967).
[2] J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1999), 3rd ed.
[3] K. Huang, Statistical Mechanics (Wiley, New York, 1963).
[4] H. Goldstein, Classical Mechanics (Addison-Wesley, Reading MA, 1980), 2nd ed.
[5] C. R. Galley, “Classical mechanics of nonconservative systems”, Physical Review Letters, 110, 174301 (2013). Abstract.