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We describe an interferometric method that enables to measure the optical path delay between two consecutive femtosecond laser pulses by way of dispersive interferometry. This method allows a femtosecond laser to be utilized as a source of performing absolute distance measurements to unprecedented precision over extensive ranges. Our test result demonstrates a non-ambiguity range of ~1.46 mm with a resolution of 7 nm over a maximum distance reaching ~0.89 m.
©2006 Optical Society of America
1. Introduction
The advent of femtosecond ultrashort pulse lasers has prompted various efforts to investigate new possibilities of advanced optical interferometry that were not possible with traditional sources such as CW lasers and white light. A notable example is the exploitation of a femtosecond laser for optical coherence tomography with the aim of performing high resolution biomedical imaging by way of either low-coherence interferometry [1] or complex spectral signal processing [2]. Another is the significant extension of the measurable distance of no periodic ambiguity by means of synthetic wavelength interferometry utilizing a femtosecond laser, which was carried out in the radio-frequency domain using a sequence of higher harmonics of the repetition rate [3] or in the optical frequency domain with flexible selection of two optical wavelengths within the optical comb [4]. The superior stability of the optical comb was also utilized to perform coherent interferometry for absolute distance measurement by overlapping two interfering pulses with adjustment of the pulse repetition rate [5]. The technique combines incoherent time-of-flight measurement with coherent optical interferometry to achieve optical fringe resolution over a range limited only by the coherence time of individual comb components, which can be of the order of 1 s [6, 7].
In this paper we describe a new way of dispersive interferometry devised to measure the optical path delay between two consecutive ultrashort pulses with high precision, which leads to an accurate means of absolute distance measurement using a femtosecond laser. The concept of dispersive interferometry was first introduced to measure an absolute distance using white light [8], of which measurable range was limited to a short range of a few micrometers. The exploitation of a femtosecond laser for the dispersive interferometry as attempted in this investigation permits producing an abundance of interference signals of monochromatic frequencies simultaneously. This advantage results in a significant extension of the measurable range far beyond the low-coherence limit of short pulses with no need of time-delay line of mechanical scanning, which may also finds applications in the field of Fourier interferometry for spectroscopy [9] and spectral interferometry in ultrafast technology [10].
2. Principles
The principle of dispersive interferometry proposed in this paper is applicable basically to most of two-arm interferometers and for convenience will be explained hereafter with regard to a Michelson type interferometer shown in Fig. 1. The light source is a Ti:Sapphire laser that emits a pulse train of ~10 fs pulse duration at a repetition rate of 75 MHz. The pulse train constitutes an optical comb spanning a spectral width of 80 THz about a central frequency of 375 THz in the frequency domain. Each mode of the optical comb is quasi-monochromatic with a line width of less than 1 MHz, bearing a temporal coherence length of ~150 m. The reference mirror MR is fixed stationary while the measurement mirror MM is movable along the optical axis of the measurement beam. The interference intensity between the reference and measurement beams is observed by use of a spectrometer that consists of a line grating and a line array of 3648 photodetectors. A FPE(Fabry-Perot Etalon) made from fused silica with 2.0 mm thickness is put before the spectrometer, of which the resonance filtering function trims down the mode density of the comb so that only ~3 filtered consecutive modes are selectively picked out to fall on each photodetector of the spectrometer.
The frequency-spread interference intensity provided by the spectrometer is described by use of the spectral power density g(ν), which is a function of the optical frequency ν in the form of
The mean intensity a(ν) and the modulation amplitude b(ν) are directly related to the spectral power density s(ν) of the source as follows:
where rr(ν) and rm(ν) denote the reflection coefficients of the reference and measurement mirrors, respectively. In general, the reflection coefficients tend to vary slowly with ν and for simplification, they may be assumed unity within the spectral range of s(ν). Consequently the spectral power density of Eq. (1) can be rewritten as
A typical distribution of the spectral power density g(ν) actually monitored in this investigation is shown in Fig. 2(a). Now the phase ϕ(ν) of Eq. (3) is generally expressed as
where α is the optical path delay that is explicitly given as α=2n(ν)L/c with n and c being the refractive index of air and the speed of light in vacuum, respectively. The distance L is the geometrical path length difference between the reference and measurement arms.
Fig. 1. Dispersive interferometer using femtosecond laser pulses. FS laser: femtosecond laser; OI: optical isolator; BS: beam splitter; CP: dispersion compensation plate; FPE: Fabry-Perot etalon; MR: reference mirror; MM: measurement mirror; CL: collimating lens. Inlets show the relative line density of the optical comb before and after filtering.
Fourier-transforming Eq. (3) in consideration of Eq. (4) leads to the result of
where δ(τ) is Dirac delta function and τ is the variable representing the optical path delay. S(ν) is the Fourier transform of s(ν). The spectral power density g(ν) is a real function, so its Fourier transform becomes symmetrical about τ=0. And, three peaks appear at locations -α, 0, and α, as illustrated in a typical experimental result shown Fig. 2(b). The peaks are convoluted by the function S(τ) and tend to be not sharp enough so that the exact value of α can be determined directly from the τ-domain. For the reason, only the peak at α is isolated by use of a band-pass filter of finite width and then inverse Fourier-transformed into the ν-domain, whose result is derived as
where i=√-1. Subsequently, the phase ϕ(τ) can readily be obtained through the arctangent operation of [11]
The arctangent operation gives the phase ϕ(ν) in its wrapped value confined within the range of -π to π, as illustrated in Fig. 2(c). Consequently, the unwrapped value of ϕ(ν) bears an offset from the true absolute phase value of ϕ(ν) as shown in Fig. 2(d). Thus the first-order slope of the unwrapped phase is taken, which can be related to the distance L with no effect of the offset as
where N=n+(dn/dν)ν. Note N represents the group index of refraction of air, which is a function of the center frequency of the light source. Finally the distance L is determined from the relation of
Fig. 2. Data processing procedure for measurement of L; (a) dispersed interference intensity captured by the spectrometer, (b) band-pass filtering of the peak at α, (c) wrapped phase, and (d) unwrapped phase.
The described Fourier-transform method of determining L demands that the peak at α in the τ-domain be securely separable from its neighboring peak appearing at the origin. This is, however, not always the case when measuring a small distance. Eq. (5) indicates that all the peaks are convoluted by S(τ), so they go overlapped when α reduces below a certain threshold (w) that is related to the temporal width of S(τ). This situation limits the minimum measurable distance following the relation of LMIN=c/(2Nw), which is found ~5 µm in our configuration.
The maximum extent of α is also restricted, in this case by the Nyquist sampling limit that is described as α≤1/(2p). The frequency resolution p represents the spacing of the two neighboring modes actually sampled by the spectrometer. This upper limit of α is converted to the distance as LNAR=c/(4Np), which is referred to as the non-ambiguity range (NAR). If all the modes of the optical comb are possibly be sampled, the frequency resolution p would equal the pulse repetition rate, resulting in a LNAR of 2.0 m that is identical to the cavity length of the femtosecond laser. However, due to the practical limitation in the total number of photodetectors available for the spectrometer in use, the modes need to be filtered using a FPE (Fig. 1). In this circumstance, p is decided by the free spectral range (F.S.R.) of the FPE, accompanying a reduction of LNAR, which is measured 1.458 mm in our current configuration.
An important advantage of using a femtosecond pulse laser instead of traditional white light is that the maximum measurable distance is not simply constrained by LNAR. When the distance to be measured is larger than LNAR, its interference signal is sampled in a folded state as illustrated in Fig. 3(a). This aliasing phenomenon results in the repeated triangle-shaped variation of the phase value as the distance increases beyond LNAR, as illustrated in Fig. 3(b). The maximum measurable distance LMAX is therefore extendable up to the temporal coherence length defined by the line width (Δ) of the optical comb modes, which is worked out to be more than ~150 m. In practice, as several FPE-filtered modes are sampled by each photodetector of the spectrometer as in our current configuration, the temporal coherence length is determined by the mode spacing of the filtered optical comb. Table 1 summarized the performance of our current configuration with reference to the case of ideal sampling in which FPE filtering is excluded.
Table 1. Performance of the dispersive interferometer with FPE filtering in comparison to the case of ideal sampling.
3. Experimental results
The experimental data shown in Fig. 3(b) was obtained while moving the measurement mirror (Fig. 1) over a distance range of 5.8 mm repeatedly with a step motion of 10 µm. The non-ambiguity range LNAR was measured 1458.5 µm with a measuring resolution of 7 nm. The measuring resolution is mainly affected by two factors; the hardware resolution of analog-to-digital conversion of the spectrometer and the computational resolution of the Fourier-transform software adopted for determining L. To improve the measuring resolution, the spectrometer resolution needs to be as fine as possible with a high level of S/N ratio. In addition, a large size of zero padding needs to be incorporated in the Fourier-transform of measured g(ν) of Eq. (5) together with an appropriate Hanning window to enhance the computational resolution in the ν-domain.
Fig. 3. (a) Aliasing effect (envelop folding), (b) triangle-shaped variation of measured phase and (c) Test result with a repeated step motion of 500 µm over a distance range of 100 mm.
When measuring distances larger than LNAR, an important issue is to determine the integer multiple of LNAR to obtain absolute distances, i.e., L=mLNAR±f where m is the integer multiple and f is the fraction being measured directly from the dispersive interferometer. To determine the integer multiple m, the scheme of synthetic wavelength interferometry presented in Ref.[3] has been added to our configuration. Using the pulse repetition rate along with its higher-order harmonics, the synthetic wavelength interferometer provides a large absolute measuring range up to 4.0 m with a resolution far less than LNAR. This allows the integer m to be accurately determined so that large distances more than LNAR can be measured. Fig. 3(c) shows a test result in which a step motion of 500 µm was repeatedly induced over a distance range of 100 mm.
Figure 4 shows an application of the dispersive interferometer in which the thickness of a glass plate was measured. Interference takes place between two pulses; one reflected from the front surface and the other from the rear surface of the glass sample. The optical thickness was measured for various samples made from BK7 and fused silica, along with subsequent determination of the geometrical thickness in consideration of the group refractive index of each material taken at 375 THz. Test results revealed that the measurement repeatability lies within a standard deviation of ~30 nm for glass samples of varying thickness in the range of 1 to 2 mm. Table 2 summarizes two representative test results. For a sample with known thickness, this dispersive method can also be used to determine the precise value of its group refractive index.
Fig. 4. Optical layout for measuring the thickness of a glass plate. FS laser: femtosecond laser; OI: optical isolator; BS: beam splitter; FPE: Fabry-Perot etalon; S: glass sample.
Table 2. Two representative results of glass thickness measurement.
Dominating sources of errors affecting the measurement accuracy are mainly attributed to three factors; the refractive index of air, the frequency stability of the optical comb, and the sampling frequency linearity of the spectrometer. Firstly, the refractive index of air is influenced by various environmental parameters such as temperature, pressure, and humidity, of which variation is directly related to the measured distance L with the relation of δL/L=-δN/N as implied in Eq. (7). Without control, especially for temperature, the measurement error tends to rise over one part in 105, which can be reduced below one part in 106, possibly down to 108, with temperature control within ±1.0°C together with compensation of the refractive index by use of the Edlen’s formula. Secondly, the frequency stability of the femtosecond laser yields a contribution of δL/L=δν/ν. All the modes of the optical comb can be stabilized collectively by way of controlling the carrier offset and the repetition rate concurrently. Without frequency control, the contribution usually exceeds one part in 106 in the laboratory condition mainly due to the cavity elongation of the femtosecond laser. However, the state-of-the-art control permits the frequency stabilization to reach an extreme level of one part in 1015 [12]. Thirdly, the frequency sampling linearity of the spectrometer is attributable to not only the fabrication imperfection of the dispersive grating and also the geometrical misalignment between the grating and the photodetector arrays within the spectrometer. The linearity error results in wrong reading of the dispersed frequencies, which causes the same level of error as that of the frequency instability of the optical comb. Currently, commercially available high quality spectrometers are subject to 1×10-5 linearity error, which is considered the most dominating error source for the dispersive interferometer built in this investigation.
4. Conclusion
We have investigated the new scheme of dispersive interferometry using a femtosecond pulse laser instead of white light. Our test results showed that the use of femtosecond laser pulses allows the non-ambiguity range to be extended to ~1.46 mm with a measuring resolution of 7 nm. The minimum measurable distance was constrained to a range of 5 µm, and the maximum measurable distance was extended to 0.89 m in association with a Fabry-Perot Etalon for absolute distance measurements. The method was applied to the thickness measurement of glass plates, of which overall measurement uncertainty was estimated to be in the level of one part in 105. The exploitation of a femtosecond laser for the dispersive interferometry as attempted in this investigation permits producing an abundance of interference signals of monochromatic frequencies simultaneously. This advantage results in a significant extension of the measurable range far beyond the low-coherence limit of short pulses with no need of time-delay line of mechanical scanning, which may also finds applications in the field of Fourier interferometry for spectroscopy and spectral interferometry in ultrafast technology.
References and links
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