Gold as a standard phase reference in complex sum frequency generation measurements

Interfaces consisting of complex mixtures with soft, high-vapor pressure substrates abound in technological applications, the environment, corrosion, and biological systems. Understanding interactions among the several molecules in these complex environments requires a probe that is both molecule and surface specific. At the present time, the vibrational spectroscopy sum frequency generation (SFG)^1–3 is uniquely suited to this task. SFG is surface specific due to the requirement that the environment be non-centrosymmetric on a length scale comparable to the wavelength of the probing beams. SFG involves two input laser beams. If the beams are sufficiently intense, they combine to form a third coherent beam with a frequency that is the sum of the two inputs. Combining is described by the second-order susceptibility, χ⁽²⁾; χ⁽²⁾ vanishes in a centrosymmetric environment leading to surface specificity. The observed intensity is the absolute square of the resulting complex amplitude. The square has practical consequences; separation of the measured intensity into separate resonances due to the surface constituents is nonunique. It is thus challenging to use SFG intensity to measure important surface properties such as constituent concentration, orientation, or lifetime that reveal interactions. In contrast, the complex amplitude is linear in constituent amplitudes and contains all this information.

Measurement of the complex amplitude involves interfering the signal with another. Based on the Euler relationship between a complex quantity and phase, the measurement is often referred to as the phase measurement. Due to the utility of the complex amplitude, several methods have been devised to measure the relative phase.^4–11 The most common generates a signal from a reference to calibrate a surface known as the local oscillator (LO), then exchanges the reference with the sample and measures the sample relative to the LO. Although this method works well to determine the relative sample phase, it has limitations for determining a standard phase, primarily due to uncertainties associated with both swapping the calibration surface for the sample and stabilization of the sample surface during the measurement.¹¹ Swap-stability issues can lead to controversies, particularly for samples with broad resonances.^12–14 One solution to the swap-stability issue is to deposit the film of interest on the phase standard.^8,11,15 Of course, requiring the surface to be a phase standard has clear limitations. In addition, the phase standard must be unaltered by adsorption on the surface.

Note that the selection of a standard is common in chemical physics: Reduction potentials are defined relative to that of the standard hydrogen electrode. Bond energies are defined relative to ionization, defined to be zero. Enthalpy is defined relative to the formation of the elements in their most stable state, etc. A phase standard has not yet been clearly identified for phase measurements. Herein, it is argued that the selection of the right-handed, Z-cut quartz oriented with its +X axis in the beam propagation direction is a robust phase standard. In the specified orientation, the phase of Z-cut quartz can be consistently identified as −90°; this defines a phase standard that is applicable to different polarizations and visible excitation frequencies. Support for this phase standard is presented in the section titled Theory and Experimental Method.

Gold, rather than Z-cut quartz, is often used in complex SFG phase measurements in part because it is an excellent reflector throughout the infrared (IR) and visible regions: nearly 100% compared with less than 10% for quartz. Establishing a standard phase for gold (Au) is more challenging than for Z-cut quartz. There have been two efforts aimed at determining a standard phase for Au.^8,15–17 Both involved adsorbing a long-chain hydrocarbon on the Au surface to use the known response from the terminal CH₃ group to deduce the phase of Au. This method works well for Z-cut quartz¹¹ because the SFG response is due to the chirality of the quartz bulk crystalline structure.^18,19 Because the quartz response is a bulk property, perturbation of the surface electrons by adsorption does not significantly alter the response. In contrast, the Au response is primarily from the surface states.^16,17 How much the response is altered by adsorption remains an open question. Not surprisingly, the two above-mentioned reports on Au do not agree. One⁸ indicates that the phase of Au varies from −45° to +90° as the infrared frequency sweeps from 2800 cm⁻¹ to 3050 cm⁻¹ and is offset relative to that of Z-cut quartz by about +45°. Another report^15–17 indicates that the phase of Au is about 193° for the same 532 nm visible excitation. Given the lack of agreement, it is desirable to determine a standard phase for Au without perturbing the surface.

The recently reported nonlinear interferometer²⁰ enables the measurement of a standard phase for Au. The nonlinear interferometer naturally separates the sample and the phase standard. Additionally, it directly addresses the swap-stability issue with interferometric alignment and active stabilization using an interferometric signal. The Theory and Experimental Method section includes background on SFG, describes the nonlinear interferometer, and sets out the basis for using Z-cut quartz as the phase standard. The Results section provides data about reproducibility and determining the calibration as well as application to measuring the standard phase of Au. The Discussion section presents the calculation of the standard phase of Au: −222°. This is constant across the infrared band from 2700 cm⁻¹ through 3800 cm⁻¹ relevant to hydrocarbons and water. Finally, the results are summarized and support for this work is acknowledged.

The theory of SFG has been described in numerous articles,^1–3 so only the details essential to complex amplitude measurements are given here. SFG is generated when visible and infrared beams spatially and temporally overlap at the interface. If the intensity is sufficiently high to drive the system into a nonlinear response, the two beams combine as given by the second-order hyperpolarizability, $χS2$⁠, to generate a third beam at the sum frequency: hence, the name Sum Frequency Generation. In vibrational SFG, the response amplitude is resonantly enhanced when one of the input frequencies coincides with a vibrational resonance. Thus, SFG yields a surface vibrational spectrum. With discrete resonances, the surface response is^3,21

where $χNR(2)$ is the non-resonant response, A_q, ω_q, and Γ_q are the oscillator strength, resonant frequency, and damping coefficient of the qth vibrational mode, respectively. If vibrational resonances form a band, the sum in Eq. (1) becomes an integral over the density of states. The SF intensity is proportional to the absolute square of the surface response.

Note that the sign of the imaginary part depends on the sign of iΓ_q in the denominator of Eq. (1). There is some controversy concerning the sign;^4,11,22,23 it has its roots in the Raman amplitude.²⁴ The sign choice is inconsequential as long as it is stated. The negative sign is chosen here because that is the sign used to deduce the phase of the standard: right-handed, Z-cut quartz oriented with its +X axis in the forward beam propagation direction. Equation (1) illustrates the power of measuring the complex response: it is linear in the constituent resonances.

Since the response is complex, it is characterized by an amplitude, A, and a phase, φ, via Euler’s formula

Determination of the phase angle and amplitude allows the determination of the real, Re, and imaginary, Im, parts of the signal via Eqs. (1) and (2)

It is often assumed that the nonresonant response is purely real: $ImχNR(2)=0$⁠. For many substrates, this may be essentially true. However, for films on metals or chiral substrates, this may not be true. Specifically, for Au or Z-cut quartz, a purely real nonresonant response requires—via Eq. (2)—that the phase be nπ. Literature data¹¹ presented below suggest that the phase of the right-handed, Z-cut quartz with the +X axis in the probe plane in the forward beam direction is −90°. A −90° phase is a purely imaginary response; this contradicts the assumption that the nonresonant response is purely real.

Spectra are collected using a 20 Hz, 20 ps Nd:YAG (Ekspla PL-4320A) generating a 1064 nm, 30 mJ pulse. The Nd:YAG pumps an optical parametric generator (OPG)/optical parametric amplifier (OPA) (LaserVision, custom) to generate the 532 nm visible (40 µJ) and tunable infrared from 2000 to 4600 cm⁻¹. The visible angle of incidence is 50° and the infrared is 60°. The visible beam is condensed to approximately 0.5 mm and nearly columnated. The infrared is gently focused with a 250 mm fl CaF₂ lens located just outside the interferometer. The focal spot is just beyond the surfaces. The beam diameter at the surfaces is approximately 0.2 mm. The visible and infrared beams are p polarized. The p polarized SF beam is filtered through a notch filter, a $14$-m monochromator (Newport CS260, 600 µm slits), and detected with a photomultiplier tube (PMT) (Hamamatsu R11540). The PMT pulse signal is routed through a low noise, broad band pre-amplifier and then the pulse is integrated using a gated integrator (Stanford Research SR250). After A-D conversion, the signal is further processed and logged with a custom LabVIEW software application. No high-level mathematical operations (e.g., FFT and smoothing) are performed on the data.

It is well known that the most accurate method for measuring phase is via an interferometer. SFG, however, requires two input frequencies and generates a third. This SFG characteristic makes conventional interferometers (e.g., Michelson and Mach-Zehnder) inappropriate. We have recently demonstrated a nonlinear design²⁰ shown schematically in Fig. 1.

When aligned, the SF signal from the sample is generated simultaneously with that from the reference. The signals thus have a well-defined generated phase relationship that also depends on the interaction with the materials of the sample and reference. The two sum frequency beams interfere at the beam combiner revealing the modification of the phase by the sample relative to that of the reference. “When aligned” is a loaded statement. It means that the optical path A is equal to that of C; the path of B is equal to that of D; and the path of E is equal to that of F. This is a seemingly daunting requirement. However, the A-B-C-D paths also constitute a linear Mach-Zehnder (M-Z) interferometer. A balanced M-Z interferometer has the reference path: A + B is equal to that of the sample C + D; this condition is identified via white-light interference. With all beams in the input plane (defined by the surface normal and the incident beam direction), s-oriented, white-light fringes appear. A photograph of typical fringes is shown in Fig. 1 (right). The fringes indicate that the reference is slightly in-plane rotated and the beam combiner does not precisely bisect the B-D angle. These two optics are rotated in the probe plane to attain an excellent overlap at the beam combiner face (reference rotation) and to bisect the B-D angle (BC rotation), resulting in the expansion of the center fringe to encompass the entire footprint ensuring that the sample and reference arms are balanced to nanometer resolution.

The embedded M-Z interferometer also provides a feedback mechanism via the interferogram of a HeNe laser to maintain balance in the nonlinear interferometer. The center portion of the expanded HeNe alignment beam is cropped to a narrow beam—this narrow beam is referred to as the tracker beam. Dichroic beam splitters separate the tracker beams from the signal, sending them to detectors located at I and II. The sum of the intensities at I and II is independent of interference, providing a normalization that accounts for laser fluctuations and other small variations. In balance, the signal to I is constructive and that to II is destructive. The signal at I divided by the sum from I and II thus varies from 0 to 1. A negative feedback proportional-integral-derivative (PID) algorithm drives a piezoelectric actuator (ThorLabs PZS001) to move the BS (in Z) to stabilize the M-Z interferometer. In practice, stabilizing the M-Z interferometer also stabilizes the nonlinear interferometer²⁰ since they have common optics and the one additional nonlinear interferometer optic (the infrared beam splitter) is on the same back plane. Most of the movement is due to thermal fluctuations. Design considerations are discussed further in Ref. 20.

The nonlinear interferometer has two operational modes: a phase-shift and a constant-phase mode. These differ in what is referred to as “frequency-domain fringes”—cycling from SF constructive to destructive interference across the infrared band. Fringes result when the IR (E-F) path-length difference does not compensate for the SF (B-D) path difference. The phase-shift mode is particularly effective for measuring the phase of samples having a broad, non-resonant response such as Au (analysis below), providing contrast to detect phase variation across the infrared band. In addition, it provides a sensitive measurement of position, sample-swap reproducibility. The constant-phase mode operates without frequency-domain fringes. It is the method of choice for samples with narrow features such as CH or OH resonances.

The IR-BS is a compensated plate beam splitter (ISP Optics, 50/50 splitter, and CaF₂ compensator). Putting the BS on top of the CaF₂ plate, splitting face down ensures that the phase of the beam reflected to the reference is the same as that transmitted to the sample. Similarly, the vis-BS (Thorlabs, 25.4 mm 50/50 vis beam splitters; BK7; 400-700 nm) is oriented so that the phase of the beam reflected to the reference matches that of the beam transmitted to the sample; flipping the BS 180° about the splitting face rigorously (via conservation of energy) alters this relationship by 180°. A similar relationship holds for the beams emerging from the BC.

Materials having a broad, nonresonant response such as Au are effectively measured using the phase-shift method. The SFG polarization is given by

where $E1̃$ and $E2̃$ are the harmonic driving fields. The relative phase of the visible (infrared) driving field at the sample or reference is given by its phase at the splitter, $φviso$ (⁠$φIRo$⁠) plus evolution along the optical length ω_vist_vis (ω_IRt_IR), where ω_vis (ω_IR) is the visible (infrared) frequency and t_vis (t_IR) is the time required to reach the surface. The sum frequency polarization is

The sample (reference) phase is contained in the χ⁽²⁾ response. Explicitly,

where δ_m is the material response phase and A_m combines the amplitude of the material response and the optical factors—Fresnel and geometric—for coupling the infrared and visible beam into the material and the generated SF beam out. The resulting SF beam propagates to the BC

At the BC, the sample and reference beams interfere according to their relative phases. To a good approximation, for beam j, ω_jt_j = 2πcd_j/λ_j, where d_j is the path length depicted in Fig. 1. (Any optic in the beam path modifies this according to the refractive index.) The key parameter at the beam combiner is the phase difference, ∆δ, between the generated reference and sample beams

Note: If the sample is the same as the reference material, termed a calibration sample, δ_R − δ_S = 0. The substitution of an unknown sample introduces a phase shift of δ_R − δ_S.

It has been shown that active stabilization maintains stable path lengths over days,²⁰ so phase stability and frequency-domain fringes are also stable. Furthermore, using white-light balance and the expanded HeNe beam, both frequency domain fringe-spacing and phase are reproduced upon sample removal and replacement (see the Results section and Fig. 4).

As shown below, right-handed, Z-cut quartz oriented with the +X axis in the beam propagation direction is an effective phase standard. Z-cut quartz blanks (University Wafer, 1 in. diameter, 10 mm thick, diamond saw cut) were polished (Advanced Optics, Inc.) to λ/20. These were cleaned to remove organic contaminants (acetone, chloroform, and water) and used in air. The Au surface (ThorLabs, PF10-03-M03, λ/10) was minimally handled, rinsed with methanol, and air dried. All surfaces were held in their optical mount with nylon tipped set screws with minimal pressure to minimize distortion.

Phase is always measured relative to another material. Accordingly, determining the characteristic phase of a material depends on an agreed upon reference standard. There is some controversy about the optimal phase reference.^12–14 Herein, we argue for Z-cut quartz as the primary phase standard. Data to support this choice are in the literature;¹¹ unfortunately, an approximation applied to interpret the data somewhat obscured the merit of Z-cut quartz as a phase reference standard. Herein, we digitize the literature data [Origin 9.0, 64 bit, Figs. 2(a) and 3(a)], re-analyze it eliminating the unnecessary approximation, and show not only that the complex spectra of the film for two orientations of Z-cut quartz are mirror images but also that the deviation from mirror images noted in the original paper is due to the scalar spectrum of the surface film.

Briefly, chemical considerations provide a benchmark for the primary standard: a reasonably ordered long chain hydrocarbon monolayer adsorbed on Z-cut quartz has the terminal CH₃ group dipole pointing out of the surface. The terminal CH₃ nonlinear response depends on the azimuthal average of the molecular nonlinear response: the ⟨β_ijk⟩, where β_ijk denotes the molecular hyperpolarizability tensor. The tensor elements depend on the polarization of the output, visible excitation, and infrared excitation, in that order, as well as the vibrational mode. The result is that the symmetric stretch of the terminal CH₃ group of a hydrocarbon generates a ssp response parallel to the dipole, while the ppp response is antiparallel.^25,26

Attempts to derive a standard phase for nonresonant materials, including Au and Z-cut quartz, use the terminal methyl of a long-chain hydrocarbon.^8,11,15–17 The distinction between Au and Z-cut quartz lies in the source of the nonresonant response. For Au, the nonresonant response derives from the surface electronic states.^16,17 Bonding a monolayer to the surface involves these surface state electrons, thus perturbing them. Particularly, when bonding to Au via a sulfur atom, the surface states and thus the nonresonant response are likely altered. In contrast, the nonresonant response of Z-cut quartz originates within the coherence length: in the bulk within about 20 nm of the surface.^18,19 Surface states thus make a negligible contribution (<0.5%) if the crystal is oriented to generate a maximum SFG response, hence bonding to the surface does not significantly alter the nonresonant phase.

The phase of Z-cut quartz is thus established using a long-chain hydrocarbon: octadecylphosphonic acid (OPA) adsorbed on the surface. Figures 2(a) and 3(a) show that the +X and −X oriented spectra are nearly mirror images. As indicated above, an approximation in Eq. (3) of the referenced paper obscures the significance of deviation from a mirror image. Retaining the dropped term, the interference intensity is

where Re denotes the real part and Im the imaginary part. The last term is the interference

where δ_S (δ_qtz) is the phase of the sample (quartz). Applying the sum angle formula and using |δ_Qtz| = 90°, (10) becomes

Rotating the quartz by 180° flips the sign of the last term in Eq. (9), so the sum of the spectrum in +X orientation and that in −X is independent of the interference: it is the intensity spectrum [green spectrum in Figs. 2(b) and 3(b)]. Note that the intensity spectrum: $χS,Im22$⁠, (sum) is weak, hence dropping this term in the original paper is justified. Retaining it indicates both the intensity spectrum and the source of deviation from a mirror image. Conversely, the difference contains only the interference [magenta spectrum in Figs. 2(b) and 3(b)]: it is the Im spectrum of the film. Note that in both polarizations the imaginary spectrum starts and ends at 0 validating choice of ±90° for the phase of the quartz substrate. Furthermore, for ppp excitation, Fig. 2(b) (magenta) indicates a positive peak for the CH₃ symmetric stretch. Since the molecular hyperpolarizability is negative, the quartz phase is −90°. Similarly, for ssp excitation [Fig. 3(b), magenta], the symmetric stretch shows a negative peak. Since the CH₃ symmetric stretch has a positive phase for ssp excitation, the ssp quartz phase must also be −90°. Z-cut quartz is thus an excellent primary phase standard with a phase of −90° with the X axis along the beam propagation direction and +90° if against it.

Accurate phase determination relies on both reproducibility and stability. These two characteristics are evaluated using Au in both the reference and the calibration sample positions because Au generates a strong SFG signal. Since both the sample and the reference are Au, the phase difference is rigorously zero. Figure 4 shows typical results; interferograms are well fitted with a single sine function. This implies that there is no significant shift in phase across the spectrum (see the supplementary material). As indicated in the Theory and Experimental Method section, the frequency-domain interferogram period reflects the optical path length difference. Sample position reproducibility upon sample swap is quantified by variation in fringe period and phase. The data in Fig. 4 show that the phase varies by 1.8° upon repeated sample swap (see the supplementary material); this is an operational phase uncertainty. At the present time, identification of the center white-light fringe and maximizing the central fringe across the footprint are done visually. Replacing the operator’s eye with a sensor and generating a sine-wave fit of the central fringe interferogram are expected to reduce the operational error to subdegree, as expected for an interferometer.

The interferogram amplitude reflects the input beam intensities as well as a beam overlap factor, M. For data reported here, M ∼ 0.75; this is typical.

Since Au generates a stronger SFG intensity than does Z-cut quartz and the IR-BS directs more intensity toward the sample, Z-cut quartz was placed in the sample position. This results in greater contrast in the interferogram (Fig. 5) than putting Au in the sample position and Z-cut quartz in the reference position. The interferogram is fitted as in the Au–Au calibration procedure. Data show that the phase is shifted −132° relative to calibration with Au–Au: so δ_Au − δ_Z-cut = −132°. Since the phase of the right-handed, Z-cut quartz in the +X orientation is −90° for ppp polarization, the standard phase of Au is −222° ± 2° with 532 nm excitation in ppp polarization. Note that from an interaction perspective, all phases are negative; all generated responses are delayed relative to no interaction.

Sum frequency generation (SFG) is uniquely suited for probing soft, high vapor pressure, or buried interfaces with molecular specificity. Molecular specificity occurs when one of the input frequencies coincides with a vibrational resonance; thus, SFG is sometimes referred to as vibrational SFG. Deducing vibrational information—resonant frequencies and lifetimes—can be challenging since SFG is a nonlinear spectroscopy. Fortunately, although the generated intensity is nonlinear, the complex amplitude (both Re and Im parts) is linear. The amplitude also depends on the polarization of the excitation fields, thus containing mode orientation information not usually revealed with infrared spectroscopy. Orientation, frequency shifts, and lifetime can be used to probe interactions among molecules on the surface, so it is highly desirable to determine the complex amplitude. The challenge arises because the complex amplitude cannot be directly measured.

Determining the complex amplitude, often stated as determining the phase, involves optically interfering the surface signal with another, well-defined signal. Because the amplitude is important, several techniques have been devised to determine it.^4–11,27,28 Most generate a signal from what is known as a local oscillator (LO), then interfere the unknown signal with the LO signal. Usually, the LO is physically separated from the unknown surface,^4,6,27 enabling the measurement of a variety of substrates. This procedure produces a relative phase which is often sufficient for the system at hand. Difficulties arise when the system at hand has weak or broad or very narrow resonances.^12–14 Uncertainty arises in part due to sample positioning between calibration and data acquisition.¹¹ Sample swap uncertainty could be reduced if the LO had a standard phase.

Determining the LO standard phase requires a phase standard to measure it against. This is very similar to the problem of defining reduction potentials; a problem solved by defining the standard hydrogen electrode to have a potential equal to zero. Or the problem of defining electron energies relative to ionization being zero: bound electrons then have negative energies and free ones positive. Herein, it is suggested that adopting a phase standard of Z-cut quartz aligned with the +X axis in the forward beam direction, as having −90° phase is a good phase standard. It has the virtue that Z-cut quartz is relatively easy to obtain, easy to clean, and the phase is essentially unaffected by any surface contaminants since the signal originates in the near-surface bulk. Establishing the quartz phase-standard uses this property—the literature data are cited¹¹ and analyzed in the Theory and Experimental Method section.

It might be tempting to establish a standard phase for other materials using the adsorption procedure. However, adsorbing a long-chain hydrocarbon on the substrate limits the substrates that can be used. Furthermore, this method depends on not perturbing the substrate response by adsorption. While non-perturbation is a good assumption for Z-cut quartz, it may not be for Au or other reference materials. Indeed, two reports in the literature for Au report quite different phases.^8,15–17

Similarly, using the LO method to establish a standard phase for Au using Z-cut quartz requires a stable, well-defined spatial relationship between the two surfaces. It is not clear how this could be accomplished.

Using an interferometer circumvents limitations of both the LO and adsorption methods. The sample and the reference are separated, and the positioning-stability issues are small. As shown in Fig. 4, currently the linear M-Z interferometer has an operational sample repositioning accuracy of 1.8°. Stability within a measurement depends on maintaining stable path lengths; the negative feedback control provides this stability. The Au–Au data (not shown) demonstrate that this stability is better than 1/10⁹.

The interferometer was used to measure the phase of Au against the proposed phase standard, Z-cut quartz. It is found that the phase shift is −132°. Since the Z-cut quartz phase standard is −90°; the standard phase of Au is −222° (where all phase delays are negative). Note that this result applies for ppp polarization and 532 nm visible excitation. Since the phase is altered by visible resonances,^15–17,29 it is expected that the phase for systems based on 800 nm excitation differs from this value. The measurement technique, however, can be applied to any visible excitation.³⁰ 

Note that a phase of −222° means that the nonresonant response from Au is neither purely real nor purely imaginary. This is not surprising given the color of the Au surface, which indicates nearby visible resonances.²⁹ Constancy of the Au phase across the infrared band is consistent with the phase being due to the visible excitation and the absence of resonances in the infrared. In contrast, quartz resonances are far into the ultraviolet; this phase standard is thus essentially independent of the visible excitation frequency—another justification for adoption of Z-cut quartz as the phase standard. Note that the purely imaginary phase of Z-cut quartz is due to the chirality of the quartz crystal structure.¹⁹ 

In addition to measuring the phase of Au relative to Z-cut quartz, the data answer questions about the nonlinear interferometer. The interferometric method currently results in reproducible sample swap to ±1.8°. Although optics undoubtedly move when the interferometer box is opened for sample swap and the active control loop is necessarily disengaged, recovering balance in the linear M-Z interferometer brings the nonlinear interferometer back to its previous position with nanometer accuracy. These data also show that materials with very different reflectivity can be measured with the interferometer.

Finally, a new operational mode, a phase-shift mode, has been demonstrated. This mode is particularly effective for samples with broad resonances. Application to such surfaces is a study in progress.

A phase standard, Z-cut quartz oriented with the +X axis in the beam propagation direction is suggested. Using this phase standard, the standard phase of Au is measured to be −222° for 532 nm visible excitation and ppp polarization.

This measurement uses a nonlinear interferometer. This work demonstrates the stability and reproducibility of phase measurements using this interferometer. Based on repeated sample replacements, operational phase reproducibility is currently ±1.8°. A high degree of instrument stability and reproducibility enables measuring the standard phase of a wide variety of materials with numerous visible excitation frequencies.

See supplementary material for procedure to determine phase reproducibility and method to probe phase variation across the IR band.

Partial support from the United States National Science Foundation (Grant Nos. CHE1306933 and CHE1565772) is gratefully acknowledged. Special thanks to Professor Denis Hore, Professor Mischa Bonn, and Professor Tahei Tahara for valuable discussions about various aspects of phase measurements.