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Abstract
The phase of monochromatic light directly relates to the optical path difference (OPD), but finding this connection for spectrally broadband light is challenging. Due to a missing concept of the compatibility between the phase of randomly fluctuating fields and the OPD, demanding scanning is the only proven way for a highly accurate OPD measurement in white light. Here, we use the self-coherence function (SCF) of the spatially incoherent light to reveal the connection between the white-light phase and the OPD. Our method uses an associated field assigned to the SCF to mimic the intensity oscillation of a correlation pattern. The associated field allows restoring a cumulative OPD integrated into the SCF across all spectral constituents. The method is essential for quantitative phase microscopy, in which the SCF is available even in white light, but its processing beyond the quasi-monochromatic approach is still lacking. Improper assessment of the white-light phase may result in a loss of measurement accuracy, as we demonstrate theoretically and experimentally. Deploying our method in coherence-controlled holographic microscopy, we measured the cumulative OPD in the broadband light with a strongly asymmetric spectrum (bandwidth of 150 nm), achieving accuracy better than 5 nm in the measuring depth range of 2 µm.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
The phase of light is a measurable quantity that has played an indispensable role in developing optics and photonics [1]. The optical phase has a straightforward interpretation for ideal monochromatic light. In this case, the electric field oscillates regularly with the passed optical path changing the phase of light. This so-called dynamic phase differs from the geometric (Pancharatnam-Berry) phase of light [2,3], which varies under the polarization transformation and allows light shaping by flat elements formed by optical metasurfaces or polymer liquid crystals [4–6]. The dynamic phase has a linear dependence on the optical path with the slope being proportional to the oscillation frequency and allows light shaping by traditional volume optical components (lenses, mirrors, spatial light modulators). By processing the dynamic phase, three-dimensional objects are imaged in digital holography [7], and quantitative information on wavefront deformation, surface shape, position change, or time delay caused by alterations in refractive index is obtained using interferometric and topographic methods [7].
More complex approaches are needed to interpret and correctly use the phase of quasi-monochromatic and spectrally broadband light. The phase problem of x-ray diffraction experiments using a spatially coherent quasi-monochromatic field was solved by introducing the associated monochromatic field accessible from the correlation measurement. The amplitude of the field is just the square root of the cross-spectral density and the phase is associated with the spectral degree of coherence; hence this deterministic field represents the statistically average behavior of the actual fluctuating field [8,9]. White-light interferometry measures the optical path difference (OPD) in spatially incoherent white light without the need for the phase restoration. This is possible at the cost of higher technical complexity. The spectrally broadband beams coming from two different interferometric paths create monochromatic interference fringes that are superposed incoherently when producing the resulting interference pattern. The maximum of the interferogram contrast is achieved at zero path difference; hence its location achieved by scanning allows developing measurements in white-light topography and interferometry [10,11]. Another approach used to get information from the white light is based on interferogram decomposition into monochromatic constituents performed electronically or by a spectrograph [12,13].
Quantitative phase imaging (QPI) is another progressive field including a variety of experimental techniques that have evolved from coherent holography [14–17] towards phase imaging benefiting from a low coherence of light [18,19]. The QPI measures variations in the OPD mostly associated with biological samples and allows studying their morphology and dynamics without contrast agents [20,21]. As the QPI has found diverse applications in live and material sciences [22,23], it continues to evolve and gain popularity. In well-established QPI experiments, a quasi-monochromatic spatially incoherent light is used in experimental configurations allowing self-interference. Imaging of biological samples or structures investigated in material research suffers from a low signal-to-noise ratio (SNR); hence, the spectral broadening of light is desirable. The use of spectrally broadband light in QPI improves the SNR, which ensures higher measurement precision. The spectrally broadband QPI is technically feasible and in recent experiments even white-light correlation records have been taken [24–29]. Unfortunately, a full concept of the white-light phase, and in particular the clarification of its connection with the OPD, is still lacking. The broadband correlation records are processed as if they were acquired in monochromatic light of the central wavelength. This approach is not reliable and with its use, the precision improvement achieved by the spectral broadening may be accompanied by the loss of the measurement accuracy. The spectrally broadband light was also used in temporally low-coherent optical diffraction tomography based on angle-scanning Mach-Zehnder interferometry [30].
This paper contributes to a general phase concept of low coherence light developed in the time domain. The white-light phase is introduced using the self-coherence function (SCF), whose normalized form is represented by the complex degree of temporal coherence (CDTC). Our aim is to strengthen phase-sensitive techniques conducted in white light and represented here by the QPI realized in coherence-controlled holographic microscopy (CCHM) [31]. The results obtained show a connection between the white-light phase and the cumulative OPD integrated into the SCF throughout the light spectrum, validate this connection experimentally, and demonstrate its use for achieving a high-accuracy OPD restoration in the QPI using spectrally broadband light with different shapes of the frequency spectrum.
2. Materials and methods
2.1 White-light phase and its connection to the optical path
When a monochromatic light wave travels through a geometric path $g$ in a phase object with a refractive index $n$ not varying along $g$, the phase of the electric field oscillation changes as $\mathrm{\Phi } ={-} 2\pi \nu d/c$, where $\mathrm{\Phi },\nu $ and $c$ denote the dynamic phase, the frequency and the phase velocity of light in vacuum, respectively, and $d = ng$ represents the optical path (wavelength of light in vacuum is $\lambda = c/\nu $). The spectrally broadband (white) light is composed of monochromatic components whose initial phases are mutually uncorrelated. Hence, the broadband optical field passing through the phase object (signal field) does not provide information about variations in the optical path when processed separately (phase of the entire field fluctuates rapidly and randomly). In the measurement, the signal field correlates with an optical field not affected by the phase object (reference field) coming from the same light source. The monochromatic constituents of the signal and reference fields create interferograms that are superposed incoherently forming a white-light correlation pattern. Proven methods for the determination of the OPD use the processing of individual monochromatic interferograms [12,13] or locate the amplitude maximum of the resulting correlation pattern by scanning [10] or modulation of the refractive index [11].
Beyond these widely used principles, the phase of white light is also accessible to the OPD measurement. The white-light phase is introduced as the phase of intensity oscillation observable in the correlation pattern when the OPD changes, and is directly available from the CDTC. The CDTC is separated by the temporal or spatial phase shifting [32,33] from a white-light interference pattern created in the self-correlation experiments [18,19,23]. In the QPI, the spatial variations in the OPD caused by a phase sample are encoded into the CDTC that is mapped simultaneously in the entire field of view. Several phase-sensitive experiments demonstrated recording of white-light correlation patterns for various phase samples, but the OPD was restored using techniques justified only for quasi-monochromatic light. The aim of this paper is to reveal the connection of the white-light phase with the OPD and to investigate effects influencing the OPD restoration in light with a low temporal coherence. The proposed theoretical treatment is valid for white light with an asymmetric frequency spectrum and it considers the dispersion of the phase object.
The connection between the phase and the OPD is discussed using Fig. 1, comparing the monochromatic and broadband light situation. Figure 1 shows the optical paths ${d_S}$ and ${d_R}$ of the signal and reference fields as they pass through the object plane. With the geometry used, the OPD introduced between the signal and reference wave $\mathrm{\Delta } = {d_S} - {d_R}$ can be expressed as $\mathrm{\Delta } = {g_S}({{n_S} - {n_R}} )+ {n_R}\mathrm{\Delta }{g_S}$, where ${g_S}$ is the geometrical path of light in a phase sample with the refractive index ${n_S}$, and ${n_R}$ and $\mathrm{\Delta }{g_S}$ denote the refractive index of the surrounding medium and a spatially varying geometric path difference. We consider the sample in the air with $\Delta = {g_S}({{n_S} - 1} )+ \mathrm{\Delta }{g_S}$ to simplify the discussion. The OPD $\mathrm{\Delta }$ varies across the sample plane. In QPI, spatial changes of $\mathrm{\Delta }$ are measured simultaneously in the entire field of view. Well-established QPI methods use light with a spectral bandwidth of few nanometers, which can be considered monochromatic. The phase of the intensity oscillation in the interference pattern then changes linearly with the OPD introduced for the frequency ${\nu _E}$ referred to as the evaluation frequency. The related OPD is ${\Delta _E} = {g_S}({{n_{SE}} - 1} )+ \mathrm{\Delta }{g_S}$, where ${\Delta _E} \equiv {\Delta _E}({{\nu_E}} )$ and $\; {n_{SE}} \equiv {n_S}({{\nu_E}} )$. When a broadband light illuminates the same sample, the OPD differs for individual spectral constituents, and the CDTC phase $\mathrm{\Phi }$ changes non-linearly depending on ${\Delta _E}$ (Fig. 1(B)). To clarify our approach, we write $\mathrm{\Phi }$ in a polynomial form $\mathrm{\Phi } = \alpha {\mathrm{\Delta }_E} + \mathop \sum \limits_m {\beta _m}\mathrm{\Delta }_E^m$, where $\alpha $ and ${\beta _m}$ are unknown constant coefficients. We ask whether getting the OPD from phase $\mathrm{\Phi }$ is possible and, if so, how. We strive to restore the OPD $\mathrm{\Delta }$ cumulated in $\mathrm{\Phi }$ across all spectral components to be the best estimate of the OPD obtained for monochromatic light with the evaluation frequency $\; \mathrm{\Delta } \cong {\mathrm{\Delta }_E}$. In our approach, the white-light phase $\mathrm{\Phi }$ is connected with $\mathrm{\Delta }$ by formally the same relation as in monochromatic light $\mathrm{\Phi } ={-} 2\pi {\nu _A}\Delta /c$, in which ${\nu _A}$ is frequency to be determined (${\nu _A}$ referred to as the frequency of the associated field mimicking the oscillation of the white-light interference pattern). The cumulative OPD $\mathrm{\Delta }$ represents the best approximation of ${\mathrm{\Delta }_E}$ just when ${\nu _A}$ is given by ${\nu _A} ={-} ({c/2\pi } )\left[ {\alpha + \mathop \sum \limits_m {\beta_m}\Delta _E^{m - 1}} \right]$. In the linear approach, ${\nu _A}$ is constant and determined by the coefficient $\alpha $, which will be examined in dependence on the light power spectrum and the sample dispersion. The OPD range, in which the linearity holds, will be investigated theoretically and experimentally. When the OPD is determined from the phase $\mathrm{\Phi }$ measured in the white light using ${\nu _E}$, the dependence shown by the black solid line in Fig. 1(B) is obtained. With ${\nu _A}$ correctly determined and used in the OPD reconstruction, the slopes of $\mathrm{\Delta }$ to $\mathrm{\Phi }$ dependence are the same for white light (red solid line in Fig. 1(B)) and monochromatic light (black solid line in Fig. 1(A) and black dashed line in Fig. 1(B)). When measuring large OPDs, ${\nu _A}$ varies throughout the measurement range to maintain consistency between $\mathrm{\Delta }$ and ${\mathrm{\Delta }_E}$.
Fig. 1. Illustration of the interference in monochromatic and white light. (A) Intensity pattern and linear phase dependence on the OPD ${\Delta _E}$ for monochromatic light. (B) White-light intensity pattern (black line) with an associated field (red line) mimicking its oscillation. Nonlinear change of the cumulative OPD $\Delta $ determined from the white-light phase $\mathrm{\Phi }$ by the frequency ${\nu _E}$ (black line). Our method uses the associated field frequency ${\nu _A}$ to restore $\Delta $ (red line) as the best estimate of ${\Delta _E}$ measured in monochromatic light (black dashed line).
3. Results
3.1 Theoretical concept for getting the OPD from the phase of the CDTC
Calculations showing how to restore the cumulative OPD from the phase of the CDTC are performed using signal and reference fields ${U_S}$ and ${U_R}$ created in the self-correlation quantitative phase imaging (QPI) [18,19,23]. In the model used, a phase object with a complex transmittance function ${T_S}$ to be determined is placed in the object plane of the signal imaging path. The object plane ${\vec{R}_{R \bot }} = ({{X_R},{Y_R}} )$ of the reference path has a complex transmittance ${T_R}$, which is known and provides a reference phase. The object planes of the signal and reference paths are illuminated by spectrally broadband light originating from the same source. To achieve this illumination, a grating is used in our CCHM experiment (Fig. 2). The monochromatic constituents u of the broadband light have randomly fluctuating phase. The object planes are imaged by the signal and reference imaging systems with the impulse response functions ${h_S}$ and ${h_R}$, respectively. The created images coincide in the image plane ${\vec{r}_ \bot } = ({x,y} )$ and their complex amplitude is given by
Fig. 2. (A) Experimental scheme of CCHM operating in a reflection mode: IL1-IL3 illumination lenses, G linear binary phase grating, M1, M2 mirrors, BS1, BS2 beam splitters, MO1, MO2 identical microscope objectives, SM, RM signal and reference objects (plane mirrors in calibration experiment), CL camera lens, CCD charged coupled device, FP field planes of the microscope. (B) Achromatic off-axis hologram recorded in broadband light (spectral width 150 nm). Hologram corresponds to surface evaluated in Fig. 7. (C) Hologram Fourier spectrum making the CDTC accessible for processing.
3.2 Approach I
3.2.1 Associated field for symmetric frequency spectrum of light
In this approach, we consider broadband light with a symmetric frequency spectrum $S(\nu )$, whose maximum is at the frequency ${\nu _E}$ and drops to zero at the frequencies ${\nu _E} - \mathrm{\Delta }\nu $ and ${\nu _E} + \mathrm{\Delta }\nu $, respectively. Introducing the frequency $\bar{\nu } = \nu - {\nu _E}$, the integration in (9) is performed from $- \mathrm{\Delta }\nu $ to $\mathrm{\Delta }\nu $, and $\mathrm{\Omega } = 2\pi \nu {\Delta _E}/c$ is used for assumed dispersion-free phase object placed in the air [$\mathrm{\Delta }{n_S}(\nu )= 0$]. As the spectrum $S({\bar{\nu }} )$ is symmetric, the functions $S({\bar{\nu }} )\bar{\nu }\cos \Omega $ and $S({\bar{\nu }} )\sin \Omega $ are odd and their integration results in zero within the given integration limits. The associated field mimicking the intensity oscillation of the correlation record has constant frequency which is given by (9). In this case it is just equal to the center frequency of the spectrum, ${\nu _A} = {\nu _E}$. This result agrees with [8], where a spatially coherent quasi-monochromatic radiation was examined using the cross-spectral density function. When the OPD introduced by a dispersion-free object is restored from the phase $\mathrm{\Phi }$ of the CDTC acquired in a broadband light with the symmetric frequency spectrum, the same relation is applicable as if the monochromatic light with the frequency ${\nu _E}$ was used, $\mathrm{\Delta } = {\Delta _E} ={-} c\Phi /({2\pi {\nu_E}} )$.
The influence of the sample dispersion on the frequency of the associated field (9) can be assessed according to the magnitude of the dispersion term of the cosine argument $\mathrm{\Omega }$,
3.3 Approach II
3.3.1 Associated field for a small-stroke phase object and asymmetric frequency spectrum of light
In this approach, a dispersion-free object placed in the air is again considered [$\mathrm{\Delta }{n_S}(\nu )= 0$], but the asymmetric frequency spectrum is used. The approach is applicable to objects with a small phase stroke introducing the OPD not exceeding the value of ${\Delta _{{E_{max}}}}$. The largest acceptable value follows from the condition $2\pi {\nu _{\textrm{max}}}{\Delta _{{E_{max}}}}/c < q$, where ${\nu _{\textrm{max}}}$ is the maximum frequency of the broadband light used, and q is chosen to justify $\cos q \cong 1$ and $\sin q \cong q$. When (9) is used with this assumption, the frequency of the associated field is given by
3.4 Measurement of the cumulative OPD by white-light CCHM
The measurement of the CDTC was performed using the CCHM operating in a reflection mode [31]. The CCHM uses an incoherent light source, whose spatial and temporal coherence is controlled by adjustment of the source size in the aperture plane of the illumination path and the bandpass filtering. The experimental setup is shown in Fig. 2(A), where the light from a single point of the source plane is tracked through the system for simplicity. The light is collected and transformed by illumination lenses IL1-IL3. The illumination lenses create images of the light source in the back focal plane of microscope objectives MO1 and MO2 (both 10×, NA = 0.25) providing Köhler illumination conditions. When imaging the light source into the back focal plane of MO1 and MO2, the light passes through a binary phase grating G placed behind the illumination lenses and reflects from plane mirrors M1 and M2. The grating G is illuminated by slightly converging waves and deflects light into +1st and −1st diffraction orders (angular separation of 9.4°), creating the signal and reference paths of the CCHM. The used microscope objectives are finite corrected (fixed tube length) and the grating G is placed in their image plane (i.e. in the plane optically conjugated with the sample). The microscope planes optically conjugated with the sample are marked as the field planes FP in Fig. 2(A). In the used configuration, the grating G is projected on the signal and reference samples. This provides conditions of achromatic off-axis holography as initially proposed by Leith and Upatnieks [34]. In the measurement assessing the accuracy of the OPD restoration, the signal and reference samples are substituted by plane mirrors SM and RM perpendicular to the optical axis. Using beam splitters BS1 and BS2, the microscope objectives MO1 and MO2 create intermediate images of the signal and reference mirrors SM and RM in the object plane of a camera lens CL. The camera lens projects the intermediate images on a CCD (Ximea, MR4021MC-BH) providing the lateral magnification m = 4. When the mirrors M1 and M2 and the beam splitters BS1 and BS2 are adjusted in order to provide spatial overlapping of signal and reference images, the dispersion effects are canceled thanks to grating dispersion [34], and achromatic interference fringes are created at the CCD [31]. The hologram is spatially invariant with respect to the source point position and its structure is modulated by the achromatic carrier frequency. This property enables the spatial phase shifting and the single shot reconstruction of the CDTC. The hologram with an achromatic carrier frequency, corresponding to the sample evaluated in Fig. 7, is presented in Fig. 2(B). The CDTC accessible by Fourier transform of the hologram is shown in Fig. 2(C).
3.5 Accuracy assessment of the OPD reconstructed using broadband light
The measurement of the CDTC was realized with a halogen lamp (Euromex EK1) supplemented by two bandpass filters allowing operation in both quasi-monochromatic and spectrally broadband imaging modes. The power spectrum of light used in experiments was measured at the output plane of the microscope using a commercial spectrometer (Avantes ACS-SD2000). The measured spectral intensity was multiplied by the CCD efficiency and then normalized. The spectral intensity measured with a narrow bandpass filter (FB650-10, Thorlabs), providing the quasi-monochromatic light [full width at half maximum (FWHM) 10 nm], is shown in Fig. 3(A). The measured spectral intensity of white light provided by a halogen lamp is in Fig. 3(B). In the measurement, the spectrum of white light was shaped by a bandpass filter composed of two edgepass filters (longpass filter FELH0600, shortpass filter FESH0800, Thorlabs). The halogen light spectrum was modified to a right triangle shape (Fig. 4(A)) to demonstrate that the OPD restoration by a correctly designed associated field maintains the high accuracy even if a strongly asymmetric spectrum is used.
Fig. 3. Spectral intensity of light used in the measurement by CCHM: (A) Normalized spectral intensity of quasi-monochromatic light obtained with a narrow bandpass filter (FWHM 10 nm). (B) Measured normalized spectral intensity of white light from a halogen lamp.
Fig. 4. Deviation $\mathrm{\delta } = \mathrm{\Delta } - {\mathrm{\Delta }_E}$ between the cumulative OPD $\mathrm{\Delta }$, numerically calculated using the measured spectrum of broadband light, and the comparative OPD ${\mathrm{\Delta }_E}$. (A) Measured power spectrum of broadband light with marked center of gravity frequency (green line), median frequency (red dashed line) and peak frequency (blue dashed line). (B) Deviation $\mathrm{\delta }$ obtained for $\mathrm{\Delta }$ restored from the CDTC by the center of gravity frequency ($4.637 \times {10^{14}}$ Hz, green curve), median frequency ($4.444 \times {10^{14}}$ Hz, red line) and peak frequency ($4.800 \times {10^{14}}$ Hz, blue line). (C)-(E) Deviation $\mathrm{\delta }$ evaluated in the linearity area (dashed-line area in B) for the center of gravity, peak and median frequency, respectively.
The measured spectra were used in numerical simulations of the OPD restoration performed to compare theoretical and experimental results. The phase $\mathrm{\Phi }$ was restored from the CDTC calculated using (7) for the triangle-like broadband spectrum shown in Fig. 4(A). The frequency of the associated field was determined as the center of gravity frequency of this spectrum, ${\nu _A} = 4.637 \times {10^{14}}$ Hz (${\lambda _A} = 647$ nm) (green line in Fig. 4(A)). The phase was mapped in the dependence on the comparative OPD ${\mathrm{\Delta }_E}$ varied in the range of ${\pm} 2.5$µm (${\mathrm{\Delta }_E}$ was set by changing the signal geometric path $\mathrm{\Delta }{g_S}$ while the reference path remained in focus). This range corresponds to the coherence length of the broadband light used. The cumulative OPD $\mathrm{\Delta }$ was calculated from the reconstructed phase $\mathrm{\Phi }$ considering the associated field with the constant frequency ${\nu _A}$. The deviation of the cumulative OPD from the comparative OPD given by $\mathrm{\delta } = \mathrm{\Delta } - {\mathrm{\Delta }_E}$ represents the accuracy of the measurement and is illustrated in Fig. 4(B) (green curve). The deviation $\mathrm{\delta }$ is acceptably small in the dashed-line area. Here, the phase dependence on the comparative OPD is linear and the frequency of the associated field remains constant (case related to Approach II). When the frequency of the associated field is deviated from the center of gravity frequency of the spectrum, the accuracy of the OPD restoration is lost. The red and blue curves in Fig. 4(B) show such erroneous OPD restoration performed with the peak and median frequency of the power spectrum (red and blue dashed lines in Fig. 4(A)). In this case, the cumulative OPD $\mathrm{\Delta }$ was restored from the phase $\mathrm{\Phi }$ using the associated field whose frequency was deviated from the center of gravity frequency by $\mathrm{\Delta }{\mathrm{\nu }_1} ={-} 0.193 \times {10^{14}}$ Hz ($\mathrm{\Delta }{\mathrm{\lambda }_1} = 27$ nm) and $\mathrm{\Delta }{\mathrm{\nu }_2} = 0.163 \times {10^{14}}\; $Hz ($\mathrm{\Delta }{\mathrm{\lambda }_2} ={-} 23$ nm), respectively. For the used broadband light with an asymmetric spectrum, the phase linearity allowing the use of the associated field with a constant frequency is maintained in the depth range of ${\pm} 1.25$ µm (approximately half the coherence length). In this range, $\mathrm{\delta }$ does not exceed 5 nm for a correct OPD restoration carried out using the center of gravity frequency (Fig. 4(C)). The OPD restoration performed with the frequencies deviating by $\mathrm{\Delta }{\mathrm{\nu }_1}$ and $\mathrm{\Delta }{\mathrm{\nu }_2}$ from the center of gravity frequency leads to the errors that change linearly with ${\mathrm{\Delta }_E}$ and have an opposite slope (Figs. 4(D)–4(E)). In the dashed line linearity area, an unacceptably large error of ${\pm} 50$ nm appears. To verify simulations shown in Figs. 4(C)–4(E) experimentally, the restoration of the CDTC carried out under gradual variations in the OPD is required.
Such verification cannot be easily done in transmission holographic imaging, because it requires a sample with adjustable thickness. This is why the reflection configuration was used. In the reflection mode, the OPD can be gradually changed by axial displacement of the mirror SM. To avoid random effects associated with the time-lapse detection of displacing mirror, we used the CCHM ability to measure spatial OPD variations at individual points of the field of view. In the performed experiments, the mirror SM was tilted; hence the CDTC measured at the points along the tilt direction was affected by the comparative OPD ${\mathrm{\Delta }_E} = 2\mathrm{\Delta }{\textrm{g}_\textrm{S}}$ changing linearly (${\textrm{n}_S} = 1$). In this measurement, the OPD was mapped over the full measurement range using a single CCD record. From the knowledge of the mirror tilt, the CCD pixel size and the magnification of the imaging system, the comparative OPD ${\mathrm{\Delta }_E}$ was determined for the individual lateral positions of the correlation pattern. The cumulative OPD $\mathrm{\Delta }$ was obtained from the CDTC phase measured at the related lateral positions. By comparing $\mathrm{\Delta }$ and ${\mathrm{\Delta }_E}$, the measurement accuracy $\mathrm{\delta }$ was assessed. Comparing the numerical and experimental results in Figs. 4 and 6 was performed for the same ${\mathrm{\Delta }_E}$ values but different optical paths in the transmission and reflection geometry. The geometrical path difference $\mathrm{\Delta }{\textrm{g}_\textrm{S}}$ in the experiment was shortened twice to compensate for the double light passage through the object area in the reflection imaging. The measurement was first performed with the quasi-monochromatic light (spectrum shown in Fig. 3(A)) and then repeated with the broadband light (spectrum shown in Fig. 4(A)). The CDTC documented in Fig. 5 was measured in a selected rectangular area whose width is 2 µm along the $x$-axis. The mirror SM was tilted in the $y - z$ plane; hence the $y$-coordinate can be converted to a depth z related to the comparative OPD ${\mathrm{\Delta }_E}$ introduced between the signal and reference fields. The results in Figs. 5(A)–5(C) were obtained by the measurement performed in a quasi-monochromatic light and demonstrate the argument evaluated in the interval $\langle - \pi ,\pi \rangle$, the modulus, and the real part of the CDTC, respectively. The CDTC was separated from the off-axis correlation pattern recorded in the CCHM applying the Fourier filtering. The profile of the real part of $\gamma $ in Fig. 5(D) corresponds to the dashed line in Fig. 5(C) and shows that for the quasi-monochromatic light the oscillation amplitude remains almost unchanged throughout the whole range of the OPD. The oscillation in Fig. 5(D) occurs with a constant frequency.
Fig. 5. CDTC measured in dependence on the comparative OPD ${\mathrm{\Delta }_E}$ (left column: quasi-monochromatic light with spectrum shown in Fig. 3(A), right column: broadband light with asymmetrical spectrum shown in Fig. 4(A)). (A) and (E) argument of CDTC, (B) and (F) modulus of CDTC, (C) and (G) real part of CDTC, (D) and (H) profile along the dashed line in (C) and (G).
Fig. 6. Deviation $\mathrm{\delta } = \mathrm{\Delta } - {\mathrm{\Delta }_E}$ between the cumulative OPD $\mathrm{\Delta }$ and the comparative OPD ${\mathrm{\Delta }_E}$ obtained experimentally using broadband light with asymmetrical power spectrum in Fig. 4(A). (A) Dependence of the unwrapped CDTC phase on ${\mathrm{\Delta }_E}$. (B) Deviation $\mathrm{\delta }$ evaluated for $\mathrm{\Delta }$ correctly restored using the center of gravity frequency (green line in Fig. 4(A)). (C) and (D) Deviation $\mathrm{\delta }$ evaluated for $\mathrm{\Delta }$ restored using the peak and median frequency (blue and red dashed lines in Fig. 4(A)).
The unwrapped phase restored from the CDTC measured in the quasi-monochromatic light shows that its dependence on ${\mathrm{\Delta }_E}$ is perfectly linear. The quasi-monochromatic light meets the assumptions used in Approach I; hence, the OPD can be reconstructed using the central frequency of the light spectrum, $\mathrm{\Delta } ={-} c\Phi /({2\pi {\nu_E}} )$, ${\nu _E} = 4.615 \times {10^{14}}$ Hz (${\lambda _E} = 650$ nm). The determined $\mathrm{\Delta }$ approached the comparative OPD ${\mathrm{\Delta }_E}$ with the high accuracy (dependence of $\mathrm{\Delta }$ on ${\mathrm{\Delta }_E}$ was closely fitted by a linear function with the slope $d\mathrm{\Delta }/d{\mathrm{\Delta }_E} = 1.004)$. The CDTC measured in the broadband light with asymmetric spectrum in Fig. 4(A) is documented in Figs. 5(E)–5(H). The unwrapped phase of the obtained CDTC maintains a good linearity in the depth range of ${\pm} 1.25$ µm (Fig. 6(A)), previously estimated by the numerical simulation. This verifies that Approach II is applicable in the used measurement range, and the oscillation of the CDTC can be mimicked by the associated field with a constant frequency. Using this strategy, the cumulative OPD was restored from the CDTC phase by the center of gravity frequency ${\nu _A}$, $\Delta ={-} c\Phi /({2\pi {\nu_A}} )$. For the power spectrum shown in Fig. 4(A), this frequency corresponds to ${\nu _A} = 4.637 \times {10^{14}}$ Hz (${\lambda _A} = 647$ nm). The deviation of the cumulative and comparative OPD, $\mathrm{\delta } = \mathrm{\Delta } - {\mathrm{\Delta }_E}$, is shown in Fig. 6(B). The standard deviation evaluated in the measurement range of ${\pm} 1.25$ µm is $\mathrm{\sigma } = 5$ nm. This proves that our method allows the high-accuracy determination of the OPD even when the broadband light with a strongly asymmetric spectrum is used. In Figs. 6(C) and 6(D), the deviation of the cumulative OPD is shown that was determined using the peak and median frequency$\; 4.8 \times {10^{14}}\; $Hz (625 nm) and$\; 4.444 \times {10^{14}}$ Hz (675 nm), respectively (blue and red dashed lines in Fig. 5(A)). In this case, the deviation $\mathrm{\delta }$ increases significantly and varies in the range of ${\pm} 50$ nm in the measured interval. The deviation $\mathrm{\delta }$ obtained experimentally for the center of gravity frequency and the shifted frequencies (Figs. 6(B)–6(D)) corresponds well to the numerical simulation performed for the same conditions (Figs. 4(C) – (E)).
3.6 Demonstration of the surface topography using broadband light
To demonstrate the practical applicability of the proposed approach, we performed measurement of the OPD introduced by reflective surface with a step change of height $\mathrm{\Delta }{\textrm{g}_\textrm{S}}$. The reflective surface was first measured in quasi-monochromatic light (FWHM 10 nm, spectrum in Fig. 3(A)) providing the comparative OPD ${\mathrm{\Delta }_E}$ shown in Fig. 7(A). The cross-section profile corresponding to the quasi-monochromatic light is marked by a black dashed line in Fig. 7(B). In the subsequent measurement, broadband light with an asymmetric spectrum shown in Fig. 4(A) was used and the off-axis correlation record was acquired. The cumulative OPD $\mathrm{\Delta }$ was reconstructed from the CDTC that was obtained from the correlation record by the spatial phase shifting. In the optimal reconstruction, the associated field with the frequency ${\nu _A} = 4.637 \times {10^{14}}$ Hz (${\lambda _A} = 647$ nm) determined by the first moment of the frequency spectrum was used. The cross-section of the reconstructed OPD profile is marked by a green line in Fig. 7(B). The cumulative OPD obtained using the center of gravity frequency was compared with ${\mathrm{\Delta }_E}$ provided by the quasi-monochromatic light. The average deviation of these two measurements was evaluated in the side elevated areas of the surface and gave a value of $\bar{\delta } = 13.3$ nm. To demonstrate the sensitivity of the OPD reconstruction to the associated field, the restoration was also performed with the peak and median frequency deviated from the center of gravity frequency to values $4.800 \times {10^{14}}\; $Hz (625 nm) and$\; 4.444 \times {10^{14}}\; $Hz (675 nm), respectively. The cross-sections of the reconstructed OPD profiles in Fig. 7(B) are marked by violet and red lines, respectively. The cumulative OPD reconstructed using the deviated frequencies was again compared with ${\mathrm{\Delta }_E}$ (Figures 7(D) and 7(E)) and in the side elevated areas the average deviation increased to $\bar{\delta } ={-} 93.4 $ nm and $\bar{\delta } = 103 $ nm for the peak and median restoration frequency of $4.800 \times {10^{14}}$ Hz and $4.444 \times {10^{14}}$ Hz, respectively.
Fig. 7. Measurement of the OPD introduced by reflective surface with a step change of height in CCHM using quasi-monochromatic light (Fig. 3(A)) and broadband light with asymmetric spectrum (Fig. 4(A)). (A) The measured comparative OPD ${\mathrm{\Delta }_E}$ reconstructed from the correlation record acquired in quasi-monochromatic light. (B) Cross-sections of the OPD reconstructed in quasi-monochromatic light (black dashed line) and broadband light using different restoration frequencies (green line: center of gravity frequency $4.637 \times {10^{14}}$ Hz; violet line: peak frequency $4.800 \times {10^{14}}$ Hz; red line: median frequency $4.444 \times {10^{14}}$ Hz). (C) Deviation of the cumulative OPD restored using the center of gravity frequency from the comparative OPD provided by the quasi-monochromatic light. (D), (E) The same as in (C) but for the peak and median frequency, respectively.
4. Discussion and conclusions
In conclusion, we contributed to the phase concept of low coherence light by revealing the compatibility between the phase of white light and the OPD. The white-light phase was introduced using the CDTC obtained from the correlation records provided by the phase sensitive experiments. In the proposed approach, the phase of CDTC was directly connected with the cumulative OPD integrating optical path variations imposed by a dispersion phase object on the individual spectral constituents of broadband light. The relationship between the white-light phase and the cumulative OPD was found through an associated field that mimics the intensity oscillations of the correlation record. In the general case, the phase of the CDTC changes nonlinearly when the measured OPD varies. This means that the frequency of the associated field is not kept constant over the entire range of the measured OPD. Here, the approaches were examined showing that even the monochromatic associated field is well applicable, when its frequency is given by the first moment of the power spectrum. The developed concept of the OPD reconstruction from the phase of broadband light was used in CCHM measurements verifying good agreement with the theory. In the calibration measurement providing variations in the measured OPD by tilting the mirror, the accuracy of 5 nm was achieved in broadband light with an asymmetric frequency spectrum (bandwidth of 150 nm, shape of a right triangle). In the surface topography measurement performed with the same illumination, an average deviation of $\bar{\delta } = 13.3$ nm was achieved provided the OPD reconstruction was performed correctly through the frequency of the associated field determined by the first moment of the power spectrum. Measurements also showed that the average deviation increased significantly to $\bar{\delta } = 100$ nm when the peak or median frequency was used in the OPD restoration (frequencies deviated about ${\pm} 4$% from the center of gravity frequency).
Our study supports new trends favoring the use of white light in the QPI experiments [24,25]. The results show that the high accuracy and precision of the measurement can be maintained even when using spectrally broadband light but processing of the correlation records beyond the quasi-monochromatic approach is needed. The developed method does not require modifications in the implementation of experiments but needs knowledge of the frequency spectrum of the light source and the spectral filters used.
Funding
Grantová Agentura České Republiky (21-01953S); Vysoké Učení Technické v Brně (No. FSI-S-20-6353).
Disclosures
The authors declare no conflicts of interest.
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