Securing noise-adaptive selection of interference signal by nonlinear detection

Dong Wei; Masato Aketagawa

doi:10.1364/OE.26.019225

1. Introduction

Length measurement is crucial for both fundamental research and practical applications. Recently, femtosecond optical frequency combs (FOFCs) [1] have been developed for precise distance measurement [2–6]. In FOFC-based measuring systems, there are two parameters available as length rulers: the wavelength [3] and the adjacent pulse repetition interval length (APRIL), which is the physical length between two adjacent pulses [6–8], with both parameters being traceable to light speed in vacuum; the light speed in vacuum is the definition of the meter. The former and latter are inversely proportional to the frequency and repetition frequency, respectively, of the FOFC light source. In the case of an FOFC, its frequency, and of course, also its repetition frequency, can be easily stabilized. In other words, wavelength and APRIL can be easily stabilized and used as a physical measuring ruler.

Over the years, APRIL-based ranging systems have been studied extensively due to their simplicity and high energy efficiency [7–18]. The high energy efficiency of this methodology is derived from the fact that all frequency components contribute to the formation of interference fringes in a pulse-train interferometer. The basic principle of the pulse-train interferometer is to locate the maximum position of low-coherence interference-fringe envelopes obtained under linear displacement of the reference mirror in the interferometer. In all the schemes mentioned above, linear detectors have been used to measure interference fringes. The interference fringe signal is modulated to a specific frequency region. When this specific frequency area is superimposed by noise, there is no other option than to increase the number of measurements. As the number of measurement samples increases, the standard deviation of the measured values decreases; collection of as many samples as possible is needed for averaging.

Another solution to this problem is to shift the frequency region of the signal, for example, using a nonlinear detector that is sensitive to the squared intensity and yields the interference fringe signal of the fundamental wave and high-harmonic waves in different frequency regions. We study the possibility of distance detection using interference fringes of high-harmonic waves. As we will see in the Principles section, when the optical path length difference (OPLD) between two mirrors of the interferometer is zero, regardless of their modulation frequencies, the interference fringe signal of the fundamental wave and high-harmonic waves reach their maximum positions; the envelope peaks of the interference fringes due to the fundamental wave and high-harmonic waves match each other. Since the envelope peak of the interference fringes of the fundamental wave can be used to detect the zero OPLD position, the envelope peak of the interference fringes of the high-harmonic waves can also be used to find out the zero OPLD point. The use of the nonlinear detector increases the noise-adaptive selectability of the signal for signal processing. This noise-adaptive approach cannot be achieved with a linear detector.

To realize the shift of the frequency region of the signal, it is also possible to change the scanning frequency of the reference mirror. The variable range of scanning frequency of the reference mirror is limited to the operating frequency of actuator. For example, the operating frequency of a piezoelectric transducer (PZT) is in the region of several hundred Hz. Furthermore, in order to select the frequency, multiple measurements are required.

In this report, a scheme utilizing a nonlinear detector in a pulse-train interferometer is proposed. The paper is organized as follows. In Section 2, the principle of the formation of nonlinearly detected interference fringes is introduced first, then the signal processing scheme for securing noise-adaptive selection of the interference signal is presented. In Section 3, a proof-of-principle experiment is reported and the experimental results are analyzed. Some conclusions are drawn in Section 4.

2. Principles

We consider the light to be irradiated on a nonlinear detector. A Michelson-type interferometer with a nonlinear detector is shown in Fig. 1.

Fig. 1 Schematic diagram of a Michelson-type pulse-train interferometer with a nonlinear detector.

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Light emitted from an FOFC light source is divided into two pulse trains by a beam-splitter cube. The pulse trains reflected by the object and reference mirrors, propagating onto the nonlinear detector, are respectively written as $E_{train_o} (t)$ and $E_{train_r} (t + τ)$ . $τ$ is the time delay of the reference pulse train relative to the object pulse train and is determined by $τ = Z / c$ , where $Z$ is the OPLD between two pulse trains and $c$ is the speed of the light in vacuum. The resultant intensity detected by the nonlinear detector, which is sensitive to the squared intensity [19], can be expressed as

I (τ) = 〈 {{| E_{train_o} (t) + E_{train_r} (t + τ) |}^{2}}^{2} 〉,

where

〈 〉

represents a time-averaged operator and

| \dots |

denotes a modulus function. Equation (1) can be further written as

I (τ) = {[I_{ref} + I_{obj} + 2 Re 〈 E_{train_o} (t) E_{train_r}^{*} (t + τ) 〉]}^{2} .

Here the first two terms

I_{ref} = 〈 E_{train_r} (t + τ) E_{train_r}^{*} (t + τ) 〉

and

I_{obj} = 〈 E_{train_o} (t) E_{train_o}^{*} (t) 〉

are the intensities of reference and object pulse trains, and

A^{*}

represents the complex conjugate of A

The third term on the right side of Eq. (2) is the cross-correlation interference pattern that depends on the time delay between the two pulse trains. The phase difference between two beams arising from the OPLD is given by $φ = 2 π ν_{0} τ$ , where $ν_{0}$ is the central frequency of the FOFC laser. We can rewrite Eq. (2) as

I (τ) = {[I_{ref} + I_{obj} + 2 \sqrt{I_{ref} I_{obj}} | C_{t} (τ) | \cos (2 π ν_{0} τ)]}^{2} .

Here,

C_{t} (τ)

is the temporal coherence function of the light source. According to the Wiener–Khintchine theorem, we need to determine the power spectral density

S (ν)

of the light source.

C_{t} (τ)

is given by

C_{t} (τ) = F [S (ν)]

, and

F (\dots)

denotes the Fourier transform.

For an FOFC laser with a Gaussian spectrum, the power spectral density is normally given by [20]

S (ν) = \frac{2 \sqrt{\ln (\frac{2}{π})}}{Δ ν} \exp {- 4 (\ln 2) {(\frac{ν - ν_{0}}{Δ ν})}^{2}} \times {comb}_{ν_{rep}} (ν) .

Here,

{comb}_{ν_{rep}} (ν) \equiv \sum_{m = - \infty}^{\infty} δ (ν - m ν_{rep})

, and

Δ ν

is the half-power bandwidth of the single frequency of the light source. Substituting the Fourier transform of Eq. (4) into Eq. (3), and using

F [{comb}_{ν_{rep}} (ν)] = \frac{1}{ν_{rep}} {comb}_{\frac{1}{ν_{rep}}} (τ)

, we have

I (τ) = {[I_{ref} + I_{obj} + \frac{2 \sqrt{I_{ref} I_{obj}}}{ν_{rep}} \exp [- {(\frac{π Δ ν τ}{2 \sqrt{\ln 2}})}^{2}] \otimes {comb}_{\frac{1}{ν_{rep}}} (τ) \cos (2 π ν_{0} τ)]}^{2} .

The round-trip distance around the fiber ring of the FOFC laser is denoted

L = T_{R} \times c

. Using

ν_{0} = c / λ_{0}

and

1 / ν_{rep} = T_{R} / c = L

, the intensity function is rewritten as

I (Z) = {[I_{b g} + γ L \exp [- {(\frac{Z}{2 l_{c}})}^{2}] \otimes {comb}_{L} (Z) \times \cos (\frac{2 π}{λ_{0}} Z)]}^{2} .

Here,

I_{b g} = I_{ref} + I_{obj}

,

γ = 2 \sqrt{I_{ref} I_{obj}}

,

I_{b g}

represents the background intensity and

γ

is a constant.

l_{c} = c \sqrt{\ln 2} / π Δ ν

represents the coherence length of the single-pulse light source.

As illustrated in Fig. 1, the object mirror position varies with respect to the reference mirror; we label this variation $h$ , which occurs when the OPLD changes by $2 h$ . By shifting the reference surface $h$ , we can generate interferometric patterns as a function of OPLD, which may be expressed as

I (h) = {[I_{b g} + γ L \exp [- {(\frac{h}{l_{c}})}^{2}] \otimes {comb}_{L} (2 h) \times \cos (\frac{4 π}{λ_{0}} h)]}^{2} .

By expanding the factors in Eq. (7), $I (h)$ can be decomposed into three frequency components, $I_{f 0} (h)$ , $I_{f 1} (h)$ , and $I_{f 2} (h)$ at $f \approx 0$ , $f = \pm f_{0} = \pm 2 / λ_{0}$ , and $f = \pm 2 f_{0} = \pm 4 / λ_{0}$ , respectively, i.e., we obtain

I (h) = I_{f 0} (h) + I_{f 1} (h) + I_{f 2} (h) .

The individual terms in Eq. (8) are defined by

I_{f 0} (h) = I_{b g}^{2} + \frac{1}{2} γ^{2} L^{2} \exp [- 2 {(\frac{h}{l_{c}})}^{2}] \otimes {comb}_{L} (2 h),

I_{f 1} (h) = 2 I_{b g} \times γ L \exp [- {(\frac{h}{l_{c}})}^{2}] \otimes {comb}_{L} (2 h) \times \cos (\frac{4 π}{λ_{0}} h),

I_{f 2} (h) = \frac{1}{2} γ^{2} L^{2} \exp [- 2 {(\frac{h}{l_{c}})}^{2}] \otimes {comb}_{L} (2 h) \times \cos (\frac{8 π}{λ_{0}} h) .

Here, we used the relational expression

comb {(x)}^{2} = comb (x)

, and

\cos {(x)}^{2} = \frac{[1 + \cos (2 x)]}{2}

.

_{$I_{f 0} (h)$} corresponds to an intensity correlation with background. Since this term does not include distance information, it is not a quantity of interest. $I_{f 1} (h)$ is a sum of two mutually symmetric correlation functions of two pulse trains. The autocorrelation interference pattern appears when the pulse train interferes with itself in a balanced Michelson interferometer. The cross-correlation interference pattern appears when different pulse trains overlapped each other in an unbalanced Michelson interferometer. $I_{f 2} (h)$ represents an autocorrelation/cross-correlation of the second harmonic wave in a balanced/unbalanced Michelson interferometer.

At zero OPLD, using $\exp (0) = 1$ , ${comb}_{L} (0) = 1$ ( $\int_{- ε}^{ε} f (x) δ (x) d x = f (0), 0 < ε, ε \to 0$ ), and $\cos (0) = 1$ , we have

I_{f 1} (0) = 2 I_{b g} \times γ L

and

I_{f 2} (0) = \frac{1}{2} γ^{2} L^{2} .

We have the relation

\exp (- x) \leq 1

(

x \geq 0

) and

\cos (φ) \leq 1

. We note that

I_{f 1} (0)

and

I_{f 2} (0)

are at their maxima, since

\exp (- x)

and

\cos (φ)

reach their maxima, respectively, when

x = 0

and

φ = 0

.

The feature of the phenomenon at zero OPLD is twofold: first, the interference fringes of the fundamental wave from the FOFC laser is at its maximum, and second, the interference fringes of the second harmonic are at their maximum. The brightest point of the interference fringe is the peak of the envelope of the interference fringe. We can understand that the peak position of the envelope of the interference fringes of the second harmonic wave matches the peak position of the envelope of the interference fringes of the fundamental wave. We recall the fact that the peak position of the envelope of the interference fringes implies equal positions for the two mirrors, namely, the position of zero OPLD. These facts mean that we can determine the position of zero OPLD by finding the peak position of envelope of the interference fringes based on the second harmonic wave. The abovementioned matching also holds for interference fringes due to third- or higher-order harmonic waves, implying that all of the interference fringes based on the high-harmonic waves can be used to find the zero OPLD point.

With this understanding of the interference-fringe formation, we now turn to the fringe analysis procedure for the determination of the position of zero OPLD. A Fourier transform method is used to filter out the modulated different-signal-frequency components to obtain the envelope of the interference fringes. The procedure is described as follows. First, a Fourier transform is applied to a calculated fringe pattern, such as that shown in Fig. 2. The spectrum of the Fourier-transformed fringe pattern is then obtained, as shown in Fig. 3. Note that to see the details of the spectrum, we have subtracted the background intensity from the fringe pattern. The inset in Fig. 3 shows a magnification of the non-zero spectrum portion. Next, sections of the spectra of the fundamental and second harmonic waves are selected using appropriate filtering windows. The selected spectrum of the fundamental and second harmonic waves are then inverse Fourier-transformed; the absolute values of the inverse Fourier-transformed signals give the envelope functions of the original waves, in both cases, as indicated by the dotted red lines in Figs. 4(a) and (b), which respectively show the reconstructed envelopes of the fringe pattern of the fundamental and second harmonic waves for the example calculated interference fringes shown in Fig. 2. In Fig. 4(c), we show the difference between two normalized envelopes. It is apparent from this figure that the two peaks of the envelopes are matched to each other.

Fig. 2 Calculated interference fringes.

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Fig. 3 Calculated spectrum of Fourier-transformed fringe pattern, with a magnified portion of the nonzero spectrum (inset).

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Fig. 4 Reconstructed envelopes of the fringe pattern of (a) the fundamental wave and (b) the second harmonic wave, and (c) the difference of the two normalized envelopes.

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With this understanding of the proposed method, we next discuss the validation experiments.

3. Optical experiments

To gain the advantage of its ease of transformation between an APRIL-based interferometer [21] and a white-light interferometer, we used a conventional Michelson interferometer. The details of the Michelson interferometer are the same as those described in the Principles section (Section 2). Since the details of the FOFC laser source are the same as in [22], further description is omitted. We use this method to realize the shift of OPLD by periodically moving the reference mirror in the reference arm of the interferometer. To control the OPLD of the reference arm, a PZT was introduced into the reference arm of the interferometer. A schematic sketch (not to scale) of the experimental setup is shown in Fig. 5. By shifting the reference mirror, we can generate interferometric patterns as a function of OPLD. The interference fringe is recorded by a nonlinear high-speed fiber-coupled detector (DET01CFC/M, Thorlabs, Inc.). This detector is an Indium Gallium Arsenide (InGaAs) photodiode detector. Its detectable wavelength range, bandwidth, max peak power, rise time and fall time are 800–1700 nm, 1.2 GHz, 70 mW, <1 nanosecond and <1 nanosecond, respectively.

Fig. 5 Optical scheme of proof-of-principle experiment.

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The data are then processed by the procedure outlined in Section 2. In this way, envelope functions of the fundamental and second harmonic waves, which are a function of the change of the object mirror position, are generated as outputs. The interference signal and envelope reproduced are shown in Fig. 6.

Fig. 6 Reconstructed interference signal (dotted line) and envelope (solid line) of the (a) fundamental and (b) second harmonic waves.

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Figure 7 shows the differences between two calculated envelope peaks. The average value of the difference is −10.18 ± 163.15 nm. It is apparent that two envelope peaks were matched in time. Note that the purpose of this experimental study was to prove the matching between two envelope peaks. As a result, we proved that we can determine the zero position of the object mirror using the interference fringes of not only the fundamental but also of the second harmonic wave for the first time. The effectiveness of the proposed method has been demonstrated in this work. In this experiment, we only detected the secondary harmonics. With a proper detector, using the same procedure, we can also determine the zero position of the object mirror using the interference fringes of other higher-order harmonics.

Fig. 7 Variations in difference between two envelope peak points.

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We believe that the interferometer generates the system noise (e.g., the PZT vibration). We now discuss briefly the influence of noise. A comparative quantitative investigation of models in the presence of different noise is beyond the scope of this paper, but it is possible to make a few qualitative remarks based on our experiences. If we assume the noise spectrum band overlaps with the fundamental frequency band of the signal, the proposed scheme is effective even in the presence of noise because the higher harmonics of the signal have their spectral band centered at a higher frequency, far away from the noise spectrum, as indicated above. If additive complex field noise is superimposed on the complex signal fields as amplitude noise or phase noise, the nonlinearly detected noise will also generate a higher harmonic spectrum at a frequency that overlaps with the spectrum of the harmonics of the signal, and in this case the proposed scheme may not work well. Nonetheless, there is no doubt that the proposed method provides multiple choices compared to conventional methods. We will report further verification of this in an upcoming study.

As the final point of this discussion, since the above experiment was conducted with a conventional Michelson interferometer, we consider whether the proposed method can be applied to an APRIL-based interferometer [21]. An APRIL-based interferometer is composed of a balanced Michelson interferometer and an unbalanced Michelson interferometer. Potential noise reduction via the method we demonstrated means that it is a candidate for providing a way forward for the application of an unbalanced Michelson interferometer in practical length measurement. The results of investigations towards this end will be presented in our next report.

4. Conclusion

We have presented a novel scheme for securing noise-adaptive selection of an interference signal using nonlinear detection. The proposed principle utilizes the matching between the peaks of the interference-fringe envelopes of different frequency components (namely, the fundamental wave and one of its harmonics). The experimental results demonstrated the feasibility of the proposed method, confirming the temporal matching of the envelope peaks of the interference fringes of the fundamental wave and its second harmonic. To the best of our knowledge, this is the first report regarding the securing of noise-adaptive selection of interference signals by nonlinear detection. Because the experimental setup consists of a general-purpose arrangement, the method is adequate for checking the operational process at each pixel of not only a white-light interferometer but also a pulse-train interferometer. Finally, it is anticipated that the present technique will be a powerful metrological tool for surface profiling and length measurements.

Funding

Japan Society for the Promotion of Science (JSPS) KAKENHI Grant-in-Aid for Young Scientists (B) (Grant Number 17K17743).

References and links

1. J. Ye and S. T. Cundiff, Femtosecond optical frequency comb: principle, operation, and applications (Springer, 2005).

2. I. Coddington, W. C. Swann, L. Nenadovic, and N. R. Newbury, “Rapid and precise absolute distance measurements at long range,” Nat. Photonics 3(6), 351–356 (2009). [CrossRef]

3. S. A. van den Berg, S. van Eldik, and N. Bhattacharya, “Mode-resolved frequency comb interferometry for high-accuracy long distance measurement,” Sci. Rep. 5, 14661 (2015). [CrossRef] [PubMed]

4. Y. L. Chen, Y. Shimizu, J. Tamada, Y. Kudo, S. Madokoro, K. Nakamura, and W. Gao, “Optical frequency domain angle measurement in a femtosecond laser autocollimator,” Opt. Express 25(14), 16725–16738 (2017). [CrossRef] [PubMed]

5. G. Wu, M. Takahashi, K. Arai, H. Inaba, and K. Minoshima, “Extremely high-accuracy correction of air refractive index using two-colour optical frequency combs,” Sci. Rep. 3(1), 1894 (2013). [CrossRef] [PubMed]

6. T. Kato, M. Uchida, and K. Minoshima, “No-scanning 3D measurement method using ultrafast dimensional conversion with a chirped optical frequency comb,” Sci. Rep. 7(1), 3670 (2017). [CrossRef] [PubMed]

7. D. Wei and M. Aketagawa, “Comparison of length measurements provided by a femtosecond optical frequency comb,” Opt. Express 22(6), 7040–7045 (2014). [CrossRef] [PubMed]

8. D. Wei, K. Takamasu, and H. Matsumoto, “A study of the possibility of using an adjacent pulse repetition interval length as a scale using a Helium–Neon interferometer,” Precis. Eng. 37(3), 694–698 (2013). [CrossRef]

9. Y. Yamaoka, K. Minoshima, and H. Matsumoto, “Direct Measurement of the Group Refractive Index of Air with Interferometry between Adjacent Femtosecond Pulses,” Appl. Opt. 41(21), 4318–4324 (2002). [CrossRef] [PubMed]

10. J. Ye, “Absolute measurement of a long, arbitrary distance to less than an optical fringe,” Opt. Lett. 29(10), 1153–1155 (2004). [CrossRef] [PubMed]

11. M. Cui, R. N. Schouten, N. Bhattacharya, and S. A. Berg, “Experimental demonstration of distance measurement with a femtosecond frequency comb laser,” J. Eur. Opt. Soc. Rapid. Publ. 3, 08003 (2008). [CrossRef]

12. H. Matsumoto, X. Wang, K. Takamasu, and T. Aoto, “Absolute Measurement of Baselines up to 403 m Using Heterodyne Temporal Coherence Interferometer with Optical Frequency Comb,” Appl. Phys. Express 5(4), 046601 (2012). [CrossRef]

13. C. Narin, T. Satoru, T. Kiyoshi, and M. Hirokazu, “A new method for high-accuracy gauge block measurement using 2 GHz repetition mode of a mode-locked fiber laser,” Meas. Sci. Technol. 23(5), 054003 (2012). [CrossRef]

14. X. Wang, S. Takahashi, K. Takamasu, and H. Matsumoto, “Space position measurement using long-path heterodyne interferometer with optical frequency comb,” Opt. Express 20(3), 2725–2732 (2012). [CrossRef] [PubMed]

15. X. Wang, S. Takahashi, K. Takamasu, and H. Matsumoto, “Spatial positioning measurements up to 150m using temporal coherence of optical frequency comb,” Precis. Eng. 37(3), 635–639 (2013). [CrossRef]

16. W. Sudatham, H. Matsumoto, S. Takahashi, and K. Takamasu, “Verification of the positioning accuracy of industrial coordinate measuring machine using optical-comb pulsed interferometer with a rough metal ball target,” Precis. Eng. 41, 63–67 (2015). [CrossRef]

17. W. Sudatham, H. Matsumoto, S. Takahashi, and K. Takamasu, “Non-contact measurement technique for dimensional metrology using optical comb,” Measurement 78, 381–387 (2015). [CrossRef]

18. B. Xue, Z. Wang, K. Zhang, J. Li, and H. Wu, “Absolute distance measurement using optical sampling by sweeping the repetition frequency,” Opt. Lasers Eng. 109, 1–6 (2018). [CrossRef]

19. F. Träger, Springer handbook of lasers and optics (Springer, 2007).

20. D. Wei, S. Takahashi, K. Takamasu, and H. Matsumoto, “Analysis of the temporal coherence function of a femtosecond optical frequency comb,” Opt. Express 17(9), 7011–7018 (2009). [CrossRef] [PubMed]

21. D. Wei, S. Takahashi, K. Takamasu, and H. Matsumoto, “Time-of-flight method using multiple pulse train interference as a time recorder,” Opt. Express 19(6), 4881–4889 (2011). [CrossRef] [PubMed]

22. D. Wei and M. Aketagawa, “Analysis of the second harmonic generation of a femtosecond optical frequency comb,” Opt. Eng. 53(12), 122604 (2014). [CrossRef]

Securing noise-adaptive selection of interference signal by nonlinear detection

Abstract

1. Introduction

2. Principles

3. Optical experiments

4. Conclusion

Funding

References and links

Cited By

Figures (7)

Equations (13)

Optics Express