Pulse-to-pulse alignment based on interference fringes and the second-order temporal coherence function of optical frequency combs for distance measurement

Jigui Zhu; Pengfei Cui; Yin Guo; Linghui Yang; Jiarui Lin

doi:10.1364/OE.23.013069

1. Introduction

Since Minoshima performed high-accuracy length measurement by utilizing an optical frequency comb in 2000, it has become a more and more important tool for accurate length measurement to meet the requirements of science and industry [1–3]. An optical frequency comb is a pulse laser source that could be referenced to a stable microwave clock whose frequency stability could be better than 10⁻¹⁵; such a prominent property makes the length measurement with a large non-ambiguity range, nanometer accuracy and direct traceability to the SI definition of the meter possible [4].

Several methods for absolute length measurement with optical frequency combs have been demonstrated [5–11]. These methods can be simply classified into two general methodologies by which characteristics of optical frequency combs they use for measurement. One focuses on the frequency domain characteristics and accomplishes measurement with the wavelength, as same as the traditional optical interferometry. The other one relies on the time domain characteristics and accomplishes measurement with the adjacent pulse repetition interval length. The latter methodology has an advantage for length measurement because only repetition frequency of the optical frequency comb is needed to be stabilized [12]. The target distance is determined in the form of a sum of a multiple of the adjacent pulse repetition interval length and a small length. So, it is important to find the position where the measurement pulse and the reference pulse overlap completely, which is also named pulse-to-pulse alignment. A few methods aimed to improve the accuracy of pulse-to-pulse alignment have been proposed. In 2008, M. Cui determined the displacement from the curve fitting of the envelope of the cross-correlation function, and the accuracy of the result could be better than one optical fringe [13]. In 2009, D. Wei first analyzed the temporal coherence function of optical frequency combs and proposed an analytical model with good agreement to experimental measurements [14]. In 2009, P. Balling developed a numerical model of pulse propagation in air and compared the results with experimental data for short distances. The relative agreement for distance measurement in known laboratory conditions was better than 10⁻⁷ [15]. In 2012, C. Narin proposed a method for high-accuracy gauge block measurement and determined the peak position by low-pass filter and differentiated processing. The combined standard uncertainty of the result was calculated to be 43 nm [16]. In 2014, G. Wu proposed a synthetic-wavelength interferometry of optical frequency combs for the pulse-to-pulse alignment, and nanometer level alignment accuracy was realized [17].

In this work, a method for pulse-to-pulse alignment based on interference fringes and the second-order temporal coherence function of optical frequency combs is proposed. The second-order interferogram of optical frequency combs is sharper than the first-order one and thus more appropriate for peak finding. The interference fringes are regard as the length reference for curve fitting rather than filtered out without use. Therefore, the non-ideal translation stage could be compensated theoretically with no additional complexity. The problem in distance measurement caused by the non-ideal translation stage could be solved by the proposed method. It is helpful to expand the application of the pulse-to-pulse alignment without the limitation of the translation stage. This paper is organized as follows. First, the second-order temporal coherence function of optical frequency combs is analyzed and a numerical model without consideration of the dispersion and absorption is given in Section 2. Next, the principle of the pulse-to-pulse alignment is demonstrated with the numerical model in Section 3. In Section 4, the absolute distance measurement with the pulse-to-pulse alignment method is proposed. Then, the experimental evaluation of the method performance is described in Section 5. Finally, the main conclusions and future work are summarized in Section 6.

2. Analysis of the second-order temporal coherence function

The features of an optical frequency comb have been summarized several times [18,19]. In frequency domain, the spectrum generated by a mode-locked femtosecond laser is a “comb” separated by the repetition frequency $f_{r e p}$ with the offset frequency $f_{C E O}$ from zero frequency. In time domain, the electric field packet of a pulse repeats every pulse repetition period $T_{R} = 1 / f_{r e p}$ with the carrier phase slipping by $Δ φ_{c e} = 2 π f_{C E O} / f_{r e p}$ , caused by the difference between the group and phase velocities in the cavity. A general pulse train emitted from a mode-locked femtosecond laser could be expressed as follow [14]:

E_{t r a i n} (t) = A (t) \exp (i ω_{c} t + i (φ_{0} + Δ φ_{c e} t)) \otimes \sum_{m = - \infty}^{+ \infty} δ (t - m T_{R})

where

A (t)

,

ω_{c}

and

φ_{0}

are the pulse envelope, the angular frequency of the center wavelength and the initial phase of carrier pulse, respectively.

For analysis of the second-order temporal coherence function of optical frequency combs, a simple schematic based on a Michelson interferometer is used, as shown in Fig. 1. First, the pulse train emitted from an optical frequency comb is split by the beam splitter BS. Then two identical parts $E_{t r a i n 1} (t)$ and $E_{t r a i n 2} (t)$ are introduced into the reference arm and the measurement arm, respectively. Finally, they are recombined at the beam splitter BS and focused on a Barium Borate Crystal (BBO) to generate the second harmonic signal detected by the photodetector PD. When the two parts overlap in space, the interferogram can be got by scanning the reference mirror.

Fig. 1 Schematic of experimental setup.

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When $E_{t r a i n 1} (t)$ and $E_{t r a i n 2} (t)$ overlap in space, the total electric field at the BBO is

E_{t o t a l} (t) = E_{t r a i n 1} (t) + E_{t r a i n 2} (t)

The second harmonic intensity is proportional to the square of the fundamental intensity, so the intensity detected by the PD with a responding period

T_{d}

can be expressed as:

I = \sum_{m} 〈 {({| E_{t o t a l} (t) |}^{2})}^{2} 〉

where

m

is a positive integer that is the nearest one less than or equal to

T_{d} / T_{R}

. The time delay

τ

between

E_{t r a i n 1} (t)

and

E_{t r a i n 2} (t)

is defined as

τ = 0

when the two parts overlap completely; therefore, the second harmonic intensity can be expressed as:

\begin{matrix} I (τ) = \sum_{m} 〈 {({| E_{t r a i n 1} (t - τ) + E_{t r a i n 2} (t) |}^{2})}^{2} 〉 \\ = \sum_{m} \int_{T_{d}} {[E_{t r a i n 1} (t - τ) + E_{t r a i n 2} (t)] [E_{t r a i n 1}^{*} (t - τ) + E_{t r a i n 2}^{*} (t)]}^{2} d t \end{matrix}

The responding period of PD is much longer than the pulse repetition period, so the Eq. (4) can be expanded as:

\begin{matrix} I (τ) = \sum_{m} \int_{- \infty}^{\infty} [E_{t r a i n 1}^{2} (t - τ) E_{t r a i n 1}^{* 2} (t - τ) + E_{t r a i n 2}^{2} (t) E_{t r a i n 2}^{* 2} (t) + E_{t r a i n 1}^{* 2} (t - τ) E_{t r a i n 2}^{2} (t) \\ + E_{t r a i n 1}^{2} (t - τ) E_{t r a i n 2}^{* 2} (t) + 4 E_{t r a i n 1} (t - τ) E_{t r a i n 1}^{*} (t - τ) E_{t r a i n 2} (t) E_{t r a i n 2}^{*} (t) \\ + 2 E_{t r a i n 1} (t - τ) E_{t r a i n 1}^{* 2} (t - τ) E_{t r a i n 2} (t) + 2 E_{t r a i n 1}^{2} (t - τ) E_{t r a i n 1}^{*} (t - τ) E_{t r a i n 2}^{*} (t) \\ + 2 E_{t r a i n 1}^{*} (t - τ) E_{t r a i n 2}^{2} (t) E_{t r a i n 2}^{*} (t) + 2 E_{t r a i n 1} (t - τ) E_{t r a i n 2} (t) E_{t r a i n 2}^{* 2} (t)] d t \end{matrix}

For convenience of explanation, we assume the pulse envelope is Gaussian pulse model, expressed as $A \exp (- {(a t)}^{2})$ , where $A$ is the electric field amplitude and $a$ is the attenuation factors related to the pulse width. For a certain comb, the parameter $A$ and $a$ are determined. When the two parts overlap in space, there is an addition phase of $N Δ φ_{c e}$ between $E_{t r a i n 1} (t)$ and $E_{t r a i n 2} (t)$ . $N$ is a non-negative integer which is the nearest one less than or equal to $2 L / (C_{n} \times T_{R})$ . $L$ is the distance between the reference mirror and the measurement mirror; $C_{n}$ is the light velocity in air. So the Eq. (5) can be rewritten as:

I (τ) = \sum_{m} (I_{1} + I_{2} + I_{3} + I_{4})

where

\begin{matrix} I_{1} = \int_{- \infty}^{\infty} [E_{t r a i n 1}^{2} (t - τ) E_{t r a i n 1}^{* 2} (t - τ) + E_{t r a i n 2}^{2} (t) E_{t r a i n 2}^{* 2} (t)] d t \\ = \int_{- \infty}^{\infty} [I_{t r a i n 1}^{2} (t - τ) + I_{t r a i n 2}^{2} (t)] d t \\ = \frac{A^{4} \sqrt{π}}{16 a} \end{matrix}

\begin{matrix} I_{2} = 4 \int_{- \infty}^{\infty} [E_{t r a i n 1} (t - τ) E_{t r a i n 1}^{*} (t - τ) E_{t r a i n 2} (t) E_{t r a i n 2}^{*} (t)] d t \\ = 4 \int_{- \infty}^{\infty} I_{t r a i n 1} (t - τ) I_{t r a i n 2} (t) d t \\ = \frac{A^{4} \sqrt{π}}{8 a} \exp (- a^{2} τ^{2}) \end{matrix}

\begin{matrix} I_{3} = Re \int_{- \infty}^{\infty} [E_{t r a i n 1}^{* 2} (t - τ) E_{t r a i n 2}^{2} (t) + E_{t r a i n 1}^{2} (t - τ) E_{t r a i n 2}^{* 2} (t)] d t \\ = 2 Re \int_{- \infty}^{\infty} \frac{A {(t - τ)}^{2} A {(t)}^{2}}{16} \exp (2 (i ω_{c} τ + i N Δ φ_{c e})) d t \\ = \frac{A^{4} \sqrt{π}}{16 a} \exp (- a^{2} τ^{2}) \cos (2 ω_{c} τ + 2 N Δ φ_{c e}) \end{matrix}

\begin{matrix} I_{4} = 2 Re \int_{- \infty}^{\infty} [E_{t r a i n 1} (t - τ) E_{t r a i n 1}^{* 2} (t - τ) E_{t r a i n 2} (t) + E_{t r a i n 1}^{2} (t - τ) E_{t r a i n 1}^{*} (t - τ) E_{t r a i n 2}^{*} (t) \\ + E_{t r a i n 1}^{*} (t - τ) E_{t r a i n 2}^{2} (t) E_{t r a i n 2}^{*} (t) + E_{t r a i n 1} (t - τ) E_{t r a i n 2} (t) E_{t r a i n 2}^{* 2} (t)] d t \\ = 4 Re \int_{- \infty}^{\infty} (\frac{A {(t - τ)}^{3} A (t)}{16} + \frac{A (t - τ) A {(t)}^{3}}{16}) \exp (i ω_{c} τ + i N Δ φ_{c e}) d t \\ = \frac{A^{4} \sqrt{π}}{4 a} \exp (- \frac{3 a^{2} τ^{2}}{4}) \cos (ω_{c} τ + N Δ φ_{c e}) \end{matrix}

I_{1}

is a constant value caused by the linear superposition of the two parts own second harmonic intensity.

I_{2}

is an intensity correlation function of the two parts.

I_{3}

and

I_{4}

are oscillation terms with the frequency of

2 ω_{c}

and

ω_{c}

respectively. From Eqs. (6)-(10), a usual simple second-order temporal coherence function can be calculated with specific pulse parameters of the optical frequency comb. Figure 2(a) shows an example of calculation for a Gaussian pulse (central wavelength 1582 nm) with a pulse width of 50 fs at environmental conditions of 20°C, 100 kPa and 50% rel. humidity. The refractive index is calculated by Modified Edlén equation [20]. Figure 2(b) shows an example of typical interference fringes actually measured experimentally. A good agreement between calculation results and detected interferogram is shown. Compared to the first-order temporal coherence function, the second-order one needs the second-harmonic generation and a higher beam power in the setup. To some extent, it could increase the complexity of the system. However, it is significant for the good property of the second-order temporal coherence function. The envelope peak of the second-order temporal coherence function is obviously sharper than the first order one. It could be calculated easily from Eqs. (6)-(10) that the contrast of the envelope peak to the background is 8:1. While, as shown in others’ work, the contrast of the first-order temporal coherence function is only 2:1 [10]. Therefore, the second-order temporal coherence function is more appropriate for the purpose of pulse-to-pulse alignment. On the other hand, the second-order temporal coherence function is sensitive to the chirp. However, the real pulse length could be determined from the second-order temporal coherence function, which makes it possible to estimate or even compensate the measurement pulse.

Fig. 2 Illustration of agreement between modeled shape and experimental interferogram. (a) Modeled shape. (b) Experimental interferogram.

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3. Pulse-to-pulse alignment principle

The envelope of the second-order temporal coherence function could be obtained from Eqs. (6)-(10) and expressed as:

S (τ) = \frac{A^{4} \sqrt{π}}{16 a} + \frac{3 A^{4} \sqrt{π}}{16 a} \exp (- a^{2} τ^{2}) + \frac{A^{4} \sqrt{π}}{4 a} \exp (- \frac{3 a^{2} τ^{2}}{4})

The maximum of the envelope is exactly corresponding to the position where the measurement pulse and the reference pulse overlap completely. In theory, the envelope could be got by Fourier transform. However, to obtain the relation between intensity and time delay in experiment with one optical frequency comb, a translation stage should be used to change the time delay; meanwhile, the intensity is recorded by an oscilloscope. So the nonlinear motion of the translation stage has an adverse effect on the data. It would distort the waveform of the relation between intensity and time delay. A specific quantitative evaluation of the effect in our experiment will be shown in Section 5. This effect could reduce the accuracy of the envelope got by Fourier transform. Although it could be controlled by calibrating the stage, the problem couldn’t be solved completely.

Therefore, to compensate the non-ideal translation stage, a curve fitting method combined with interference fringes is proposed for pulse-to-pulse alignment. The fringes are interference terms between the measurement pulses and the reference pulses; hence, the distance between two adjacent fringe peaks is approximately equal to the center wavelength of the comb. Using the adjacent interval length of the fringe peaks as the length reference, a high accuracy curve fitting for envelope peak finding could be accomplished. The steps of the method are as follow. First, the interferogram is got by scanning the reference mirror. Then, the intensities of the fringe peaks are extracted with their fringe orders as abscissa. Finally, the envelope peak is found by fitting the data from above step in the fringe orders-intensity coordinate system. The distance between the envelope peak and the brightest fringe could be got from the fitting result. Another key point is the choice of the fitting formula. From Eq. (11), the envelope is the superposition of a DC component and two Gaussian functions with different parameters. Therefore, a modified double Gaussian fitting which the two Gaussian functions have the same peak position is proposed for fitting. It could be expressed as:

I (x) = 3 k \exp (- {(\frac{x - b}{c_{1}})}^{2}) + 4 k \exp (- {(\frac{x - b}{c_{2}})}^{2}) + d

Where

k

,

b

,

c_{1}

,

c_{2}

and

d

are the parameters to be estimated. The parameter

b

is the position of the envelope peak on the fringe order abscissa. A simulation result illustrated in Fig. 3 demonstrates that this method has advantages over ordinary Gaussian fitting in an ideal environment.

Fig. 3 Comparison between fitting results of Gaussian fitting and modified double Gaussian fitting. Interference fringes (red dotted line), Gaussian fitting result (blue line), modified double Gaussian fitting result (pink line).

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In the pulse-to-pulse alignment mentioned above, the adjacent interval length of the fringe peaks is length reference for the curve fitting. So the difference between the adjacent interval length and the center wavelength is an important factor for the accuracy of the method. The adjacent interval length of the fringe peaks can be calculated by the derivation of Eq. (6), but the result is hardly expressed by an expression for that the derivation is a transcendental equation. Thus, based on the specific conditions used in Fig. 2, the adjacent interval lengths are calculated numerically with different pulse width and addition phase $N Δ φ_{c e}$ . The adjacent interval lengths are almost symmetrical about the brightest fringe. So taking the brightest fringe as coordinate origin and the fringe order as abscissa, the differences between the adjacent interval length and the center wavelength of right side of the brightest fringe with different parameters are shown in Figs. 4 and 5. With the same addition phase, it is demonstrated in Fig. 4 that the narrower pulse width corresponds to more difference between the adjacent interval length and the center wavelength. The appropriate pulse width should be chosen to ensure the accuracy of the method. In conditions mentioned above, the difference under 1 nm will be got with the pulse width wider than about 69 fs. On the other hand, with the same pulse width of 50 fs, the effect of the addition phase is illustrated in Fig. 5(b). A common obvious attenuation trend with different rate caused by the addition phase is found and the maximums of the difference are approximately equal, under 2 nm. From Fig. 5(a) and Fig. 5(b), the effect of the addition phase is mainly caused by the different relative position of the brightest fringe and the envelope peak. In application, the effect of the addition phase could be neglected contrasting with the effect of the pulse width. Therefore, using the adjacent interval length of fringe peaks as the length reference with appropriate pulse width, the peak finding could be achieved without the effect of the non-ideal translation stage. The adjacent interval length could also be calibrated beforehand for a better result.

Fig. 4 The difference between the adjacent interval length and the center wavelength with the pulse width of 20 fs (only 7 fringes), 30 fs, 40 fs, 50 fs and 100 fs.

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Fig. 5 The interferogram and the difference between the adjacent interval length and the center wavelength with the addition phase of 0, $π / 2$ , $π$ and $3 π / 2$ . (a) The interferogram corresponding to the different addition phase. (b) The difference between the adjacent interval length and the center wavelength corresponding to the different addition phase.

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4. Distance measurement principle

Figure 6 shows a schematic of the distance measurement system with the proposed pulse-to-pulse alignment method. The system is a combination of a balanced optical-path Michelson interferometer and two unbalanced optical-path Michelson interferometers. The balanced optical-path Michelson interferometer is made up of BS1, M_R1, M_T0 and the optical frequency comb. The balanced optical-path Michelson interferometer is used to determine the start point of the distance measurement. One unbalanced optical-path Michelson interferometer is composed of BS1, M_R1, M_T1 and the optical frequency comb. It is used to get the end point of the distance measurement. The other unbalanced optical-path Michelson interferometer is composed of BS2, M_R2, M_R3 and an auxiliary cw laser source. The purpose of the interferometer of the cw laser source is to measure the displacement of the translation stage during the measurement.

Fig. 6 Schematic of distance measurement system.

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When the mirror M_R1 is scanned by the translation stage, there will be two second-order temporal coherence functions recorded by the oscilloscope successively. Both of them would be processed with the pulse-to-pulse alignment method and got the distance between the envelope peak and the brightest fringe $d_{1}$ and $d_{2}$ . The distance between the two brightest fringes $d_{0}$ could be determined with the result of the interferometer of the cw laser. Figure 7 illustrates this data processing. Then the distance $L$ between the mirror M_T0 and the mirror M_T1 could be calculated as:

L = (N \times C_{n} \times T_{R} \pm (d_{0} - d_{1} + d_{2})) / 2

The integer parameter

N

could be determined by several methods, such as the ordinary TOF method or even a tape measure. The sign before the

d_{0}

could be determined by the calibrated piezoelectric transducer under the mirror M_T0. When the distance

L

is shortened by changing the position of the PZT, the sign before the

d_{0}

could be known by the change of the value of

(d_{0} - d_{1} + d_{2})

. On the other hand, when the distance

L

is closed to

N \times C_{n} \times T_{R}

, the two second-order temporal coherence functions would be overlapped. In this situation, the PZT is also used to change the position of the M_T0 to separate the two second-order temporal coherence functions.

Fig. 7 Interference fringes of the cw laser and the OFC.

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5. Experiments and results

The performance of the pulse-to-pulse alignment is evaluated by an ordinary Michelson interferometer at first. The experimental setup is shown in Fig. 6. The laser source is a mode-locked femtosecond fiber laser (C-Fiber, MenloSystems) whose repetition frequency is controlled by the repetition rate synchronization electronics (RRE-SYNCRO, MenloSystems) with the signal from a rubidium frequency standard (8040C, Symmetricom). The repetition frequency, pulse width, pulse shape, and output power of the fiber laser are 100 MHz, 50 fs, Gauss, and 33 mW, respectively. The center wavelength is 1582 nm and the spectral width is 101 nm. During the experiment, the environmental conditions are recorded for calculation of the refractive index of air.

The pulse train emitted from fiber laser is collimated by a collimator and introduced into the balanced optical-path Michelson interferometer. The second harmonic signal is detected by a Si gain detector (PDA36A, Thorlabs) and recorded by an oscilloscope (MSO4034, Tektronix) when the reference arm is scanned by a linear translation stage (M-521.DD1, PI). A typical interferogram measured experimentally is shown in Fig. 2(b). In the experiment, the measurement arm is equal to the reference arm. So the parameter N is zero and there is no addition phase between the measurement pulse and the reference pulse. In this situation, the distance between the envelope peak and the brightest fringe is zero which could be used for verification of the precision of pulse-to-pulse alignment.

The effect of the nonlinear motion of the translation stage is verified. Fifteen interferograms are got by scanning the reference mirror in the same controller parameters with the target mirror fixed. The fringe peaks are extracted and normalized by the intensity of the brightest fringe, respectively. Meanwhile, the adjacent interval lengths of these peaks are also extracted and normalized by the maximum of them in each interferogram. Figures 8(a) and 8(b) show the intensity normalization and the adjacent interval length normalization, respectively. The good agreement is shown in the intensity normalization, in contrast to the disagreement of the adjacent interval length normalization. Since the abscissa of Fig. 8(a) is the fringe order, the translation stage plays no role in the intensity normalization. Additionally, the good uniformity of data shown in Fig. 8(a) demonstrates the negligible effect of other factors like the stability of fiber laser during the experiment. However, for the adjacent interval length normalization shown in Fig. 8(b), the adjacent interval length result is recorded by the oscilloscope when scanning the translation stage. The non-ideal translation stage has direct influence on the adjacent interval length normalization. In the experiment, the variation of refractive index of air is 0.277 ppm corresponding to a variation under picometer level in the center wavelength. Meanwhile, the repetition frequency of the fiber laser is well locked to the rubidium frequency standard. The deviation of the adjacent interval length normalization result could be attributed to the nonlinear motion of the translation stage. Therefore, it could be demonstrated that the nonlinear motion of the translation stage has significant effect on the data which would impede subsequent data processing.

Fig. 8 Comparison between the intensity normalization and the adjacent interval length normalization of fifteen interferograms. The error bars indicate the standard deviation of each normalization result. (a) The intensity normalization results. (b) The adjacent interval length normalization results.

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The envelope peaks of the fifteen interferograms are got by the method described above and Gaussian curve fitting without interference fringes, respectively. For our method, the fringe peaks are extracted and the envelope peak is obtained by curve fitting of the fringe peaks in the fringe orders-intensity coordinate system; while, for the Gaussian curve fitting, the envelope peak is obtained by curve fitting of the data recorded by the oscilloscope directly. Figure 9 shows the comparison of the distance between the envelope peak and the brightest fringe got by different method. Our method is verified by the improvement of standard deviation from 27.3 nm to 8.5 nm, as shown in Fig. 9. An average distance of 133 nm between the envelope peak and the brightest fringe is also found which is different to the theoretical model in the condition of $N = 0$ . It is caused by the dispersion in the BBO and the optical misalignment of the experiment setup. In length measurement, the difference could be removed by compensation, regarded as an inherent error.

Fig. 9 Comparison of peak-finding method.

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Then, a distance measurement experiment is carried out by the method described in Section 4. The adjacent pulse repetition interval length is about 1.5 m corresponding to the pulse repetition frequency. For the lack of a translation stage could cover a range of 1.5 m, we set the distance between the mirror M_T0 and the mirror M_T1 about 1.5 m with a tape measure. Hence, the parameter N is 1. Then the mirror M_T1 is moved by four steps at an increment of 5 μm. The small step is limited by the range of the PZT we used to change the position of the mirror M_T1. The movement of the mirror M_T1 is determined by a commercial Michelson interferometer (XL-80, Renishaw) which is not shown in Fig. 6. The absolute distance between the mirror M_T0 and the mirror M_T1 is measured from the experimental interference fringes, as shown in Fig. 10. For each position, the distance is measured for 5 times and the average result is taken. The increment of the absolute distance results is calculated and compared with the results of the commercial interferometer, as shown in Fig. 11. The first absolute distance result is regarded as the start point, so the first deviation is zero.

Fig. 10 The experiment interference fringes.

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Fig. 11 Comparison of the experimental results.

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The deviations range from −156 to 168 nm. It should be noted that the uncertainty of Edlén equation is 10⁻⁸. For the distance of about 1.5 m, an error of tens of nanometers will be induced by the change of the refractive index of air. Moreover, air turbulence, thermal expansion and vibration of the optical platform, instability of the pulses and measurement errors of the environmental conditions also introduce deviations into the experiment result. For the temperature of the platform is not monitored during experiment, the mean reason of the deviations may be the thermal expansion of the platform. A better result could be obtained by installing reflection on a plate of low thermal expansion such as the Zerodur plate. The performance of our method could be further improved by better knowledge of environmental conditions and more stabilized optical components.

6. Conclusion and future work

We proposed a pulse-to-pulse alignment method based on interference fringes and the second-order temporal coherence function of optical frequency combs for absolute distance measurement. We studied the second-order temporal coherence function of optical frequency combs and developed a numerical model of the function with an assumption of Guassion pulse. With the numerical model, the difference between the adjacent interval length of fringe peaks and the center wavelength is analyzed without consideration of the dispersion and absorption. The feasibility of using adjacent interval length of fringe peaks as length reference is demonstrated. The absolute distance measurement with the pulse-to-pulse alignment method is also proposed. The method is potential to expand the application of time of flight based on one optical frequency comb without the limitation of the non-ideal translation stage. The property of the method is verified by a proof-of-the-principle experiment and a distance measurement experiment in laboratory conditions.

In future work, we plan to improve the numerical model with consideration of the dispersion and absorption. With the improved model, we will measure a long distance in air using time of flight based on one optical frequency comb.

Acknowledgments

We thank Youjian Song and Haosen Shi for the informative discussions. This work was supported by the National Natural Science Funds for Distinguished Young Scholar (51225505) and the National Natural Science Foundation of China (51405338, 51305297).

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Pulse-to-pulse alignment based on interference fringes and the second-order temporal coherence function of optical frequency combs for distance measurement

Abstract

1. Introduction

2. Analysis of the second-order temporal coherence function

3. Pulse-to-pulse alignment principle

4. Distance measurement principle

5. Experiments and results

6. Conclusion and future work

Acknowledgments

References and links

Cited By

Figures (11)

Equations (13)

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