1. Introduction

2. Principles of the Data-Processing Algorithms

2.1 Proposed Algorithm 1: The Spectral Fringe Algorithm

The dispersive interferometry using OFC is typically implemented by a Michelson interferometer-type configuration, whereby the optical path difference (OPD) between the reference and measurement arms can be accurately determined by analyzing the interference spectrum with a proper data processing algorithm [45]. In this subsection, the conventional data processing algorithm is re-visited according to literature [47,50,51], based on which an improved data processing algorithm, namely the spectral fringe algorithm, is proposed.

When a laser beam emits from an OFC source, it will be separated by a beam splitter into two beams, which are reflected by mirrors in the reference arm and the measurement arm, respectively. The recombined beams interfere with each other to produce a spectral interference signal, which is subsequently detected by an optical spectrum analyzer (OSA). An OSA has a set of discrete spectral output data. For simplicity, the kth spectral output of OSA is assumed to correspond to the frequency f_k and the wavelength λ_k. Practically, it is difficult to ensure that the splitting ratio (α_R, α_M) of the beam splitter and the reflective index (r_R, r_M) of the mirrors in the two arms are exactly equal to each other. Therefore, the electric fields of the reference beam and the measurement beam, which correspond to the kth spectral output of the OSA, i.e., the kth mode of the optical frequency comb if the OSA has enough resolving power, can be expressed as:

Here, τ is the time delay caused by the optical path difference 2n(f)L between the reference beam and the measurement beam, which is explicitly given as τ = 2n(f)L/c where n and c are the refractive index of air and the light speed in a vacuum, respectively. The intensity of the corresponding interference signal can be written as:

For simplicity, S(f_k), which corresponds to the power spectrum of the OFC source, is referred to as the envelope component. The term in the brackets includes 1 and a cosine function, which is referred to as the interference fringe component with its unwrapped phase and amplitude defined by $\Phi$(f_k) and A, respectively. S(f_k), $\Phi$(f_k) and A can be written as follows:

$$S\left(f_k\right)=\frac{1}{2}·({\alpha }_R^2·r_R^2+{\alpha }_M^2·r_M^2)·E_0^2{·G}^{\text{'}2}\left(f_k\right)$$

(4)

$$\Phi \left(f_k\right)=2\pi f_k\tau =2\pi ·n+\phi (f_k)$$

(5)

$$A=\frac{2·{\alpha }_R·r_R{·\alpha }_M·r_M}{{\alpha }_R^2·r_R^2+{\alpha }_M^2·r_M^2}$$

(6)

where φ($f_k$) is the wrapped phase, and n is an integer showing the period number of the spectral interference fringe component. Consequently, the intensity of the spectral interference signal output from the OSA, which is a Fourier transform of Equation (3), can be rewritten as:

$$I\left(f_k\right)=S\left(f_k\right)·\left[1+A·\cos \left(2\pi f_k\tau \right)\right]$$

(7)

In the conventional data processing algorithm, the spectral interference signal shown in Equation (7) is directly inverse Fourier transformed into a time-domain function i(t) as follows:

$$i\left(t\right)=s\left(t\right)\text{⊗}\left[\delta \left(t\right)+\frac{1}{2}\delta \left(t-\tau \right)+\frac{1}{2}\delta \left(t+\tau \right)\right]=s\left(t\right)+\frac{A}{2}·S\left(t\tau \right)+\frac{A}{2}S(t+\tau )$$

(8)

where $\delta \left(t\right)$ is a unit impulse function, and s(t) is the inverse Fourier transform of S(f). Assuming the OFC source has a Gaussian-like power spectrum, both S(f) and s(t) will have Gaussian-like shapes. Three Gaussian-like pulses can then be observed in i(t), with their peaks located at -τ, 0, τ and amplitudes of $\frac{A}{2}$∙$\left|S\left(t-\tau \right)\right|$, A∙$\left|S\left(t\right)\right|$, $\frac{A}{2}$∙$\left|S(t+\tau )\right|$, respectively. The pulse at τ is subsequently picked out by utilizing a time-window centered at τ. Then, the picked-up pulse is Fourier transformed into the frequency domain as follows:

$$I^{\text{'}}\left(f\right)=\frac{A}{2}·S\left(f\right)·e^{-j2\pi f\tau }$$

(9)

The wrapped phase φ(f) can then be calculated by the arctangent function as:

$$\phi \left(f\right)={\tan }^{-1}\left\{-\frac{\mathit{Im}\left[I^{\text{'}}\left(f\right)\right]}{\mathit{Re}\left[I^{\text{'}}\left(f\right)\right]}\right\}$$

(10)

Since the wrapped phase φ(f) changes periodically within the range of [-π/2, +π/2] with the increase or decrease of the target distance L, L cannot be obtained directly from φ(f) due to the ambiguity in period number n in Equation (5). Therefore, it is necessary to calculate L by taking the first-order derivation of the wrapped or the unwrapped phase values as follows:

$$\frac{d\Phi \left(f\right)}{df}=\frac{d\phi \left(f\right)}{df}=\frac{4\pi n_{g}L}{c}$$

(11)

where $n_g$ = n(f) + (dn(f)/df)∙f, is the group refractivity of air determined by the central frequency of the source, and n(f) is the phase refractivity of air. L can then be obtained as:

$$L=\frac{c}{4\pi n_g}\cdot \frac{d\Phi \left(f\right)}{df}=\frac{c}{4\pi n_g}\cdot \frac{d\phi \left(f\right)}{df}$$

(12)

As can be seen above, in the conventional data processing algorithm for dispersive interferometry, the spectral interference signal from OSA shown in Equation (7) is directly inverse Fourier transformed into the time-domain function i(t) shown in Equation (8). In this conventional algorithm, the dead-zone, which is the unmeasurable distance range, is determined by the width of the pulse function s(t). Since $\tau$ is proportional to the target distance L, when L is smaller than a certain value, the three pulses of i(t) will overlap with each other. In this case, the time-window for selecting the pulse does not function well, which prevents the measurement of the distance.

On the other hand, in the proposed spectral fringe algorithm, the envelope component S(f) and the 1 in the brackets of Equation (7) are removed to leave only the cosine function, i.e., the interference fringe component as a modified spectral interference signal I_m(f) = $A·\cos \left(2\pi f\tau \right)$. Consequently, the inverse Fourier transform of the modified I_m(f) will generate a modified time function of i_m(t) with only two impulse-shaped pulses at -τ and τ, instead of the three Gaussian-like pulses at -τ, 0, τ in the conventional data processing algorithm. As a result, the distance between two pulses in i_m(t) by the spectral fringe algorithm is doubled due to the removal of the central pulse compared with that in i(t) by the conventional algorithm. In addition, the width of the impulse-shaped pulses i_m(t) is much narrower than that of the Gaussian-like pulses in i(t). These two effects can significantly shorten the dead-zone, which is the fundamental concept of the proposed spectral fringe algorithm shown below.

In the spectral fringe algorithm, the upper and lower envelopes of the detected spectral interference signal in Equation (7) are first evaluated based on Ref. [47] as follows:

The modified spectral interference signal I_m($f_k$) can then be obtained by:

$$I_m(f_k)=A·\cos \left(2\pi f_k\tau \right)=\frac{I\left(f_k\right)}{S\left(f_k\right)}-1$$

(15)

where

$$S\left(f_k\right)=\frac{1}{2}·\left[UE\right(f_k)+LE(f_k\left)\right]$$

(16)

$$A=\frac{\mathit{UE}\left(f_k\right)-\mathit{LE}\left(f_k\right)}{\mathit{UE}\left(f_k\right)+\mathit{LE}\left(f_k\right)}$$

(17)

Subsequently, an inverse Fourier transforming of Equation (15) gives rise to modified time function i_m(t) as follows:

$$i_m\left(t\right)={\mathit{FT}}^{-1}\left[Im\left(f_k\right)\right]=\frac{A}{2}·\delta \left(t+\tau \right)+\frac{A}{2}·\delta \left(t-\tau \right)$$

(18)

Differing from the conventional data processing algorithm, only two impulse-shaped pulses exist in the time domain, which are located at -τ and τ with an intensity of $\frac{A}{2}$∙$\delta \left(t+\tau \right)$ and $\frac{A}{2}$∙$\delta (t-\tau )$, respectively. The distance between the two pulses is twice of that in the conventional algorithm, which shortens the dead-zone to be half for the same pulse width. In addition, the width of the impulse-shaped pulses by the proposed spectral fringe algorithm is much narrower than that of the Gaussian-like pulses by the conventional algorithm, which further shortens the dead-zone. In other words, the impulse-shaped pulse at τ can be selected more easily and precisely by a time-window with a much narrower bandwidth for calculating the target distance based on Equations (9)-(12) with a significantly shortened dead-zone.

It should be noted that although the way of calculating the upper and lower envelopes of the spectral interference signal is similar, the proposed spectral fringe algorithm is different from that of Ref. [47] from the point of view of the phase calculation method. In Ref. [47], the phase of the spectral interference signal is directly obtained from taking arccosine of the modified spectral interference signal I_m(f) based on Equation (15) for the purpose of improving the performance of dispersive interferometry in long absolute distance measurement. On the other hand, in the proposed spectral fringe algorithm, the phase is obtained from Equations (9) and (10) by using the time-windowed impulse-shaped pulse of the modified time function i_m(t) in Equation (18) for the purpose of shortening the dead-zone close to the zero-position of measurement, which is a critical issue in millimeter-order short-range absolute distance measurement.

2.2 Proposed Algorithm 2: The Combined Algorithm

The spectral fringe algorithm is combined with the excess fraction method [52] as the combined algorithm. In a dispersive interferometer using OFC, the relationship between the kth OSA output of wavelength λ_k (λ_k = c/(2π$f_k$)) and the target distance L can be given by n_i∙L = (m_k + ε_k)∙λ_k, in which m_k is an integer, ε_k is the excess fraction part of the wavelength λ_k, and n_i is the refractive index. The excess fraction part ε_k can be calculated by the wrapped phase φ(f_k), which is ε_k = [φ(f_k) + 0.5π]/π. The uncertain integer number m_k can be obtained from the excess fraction part ε_k based on the excess fraction method.

The excess fraction method is an effective approach to determine the absolute target length L by employing the measured excess fraction values of multiple wavelengths λ₁>λ₂>…>λ_i [53]. For the dispersive interferometer with OFC, multiple discrete spectral outputs of OSA with a narrow linewidth determined by the wavelength resolution of OSA, which can be as small as 0.02 nm [54], can be employed for the excess fraction method. The relationship between the target distance L and the used wavelengths can be expressed as:

where N_ireal is the unknown integer part of the wavelength order for each wavelength λ_i, ε_i is the measured fractional fringe value whose value is within the interval of [0, 1], and the subscript i is the wavelength number. Similarly, the nominal length $L\text{’}$ measured by another method with an error range of $\Delta L_{\mathrm{err}}$ can be expressed as:

Here, $N_i^{\text{'}}$ and ${\epsilon }_i^{\text{'}}$ represent the calculated integer part and the excess fraction part of the wavelength order, respectively. ${\epsilon }_i^{\text{'}}$ can be calculated by:

$${\epsilon }_i^{\text{'}}=\frac{2L^{\text{'}}}{{\lambda }_i}–\mathrm{INT}(\frac{2L^{\text{'}}}{{\lambda }_i})$$

(21)

in which function INT(x) means to obtain the integer part value of x. The difference c between the real length L and the nominal length $L^{\text{'}}$ is:

$$c=L-L^{\text{'}}=(m_i+{\delta }_i)·\frac{{\lambda }_i}{2}=m_i·\frac{{\lambda }_i}{2}+{\delta }_i·\frac{{\lambda }_i}{2}$$

(22)

where the first term m_i$·\frac{{\lambda }_i}{2}$ = (N_ireal - $N_i^{\text{'}}$)$·\frac{{\lambda }_i}{2}$ means the integer part of the difference c, and its value is varied by adjusting the desired value of $N_i^{\text{'}}$. The second term δ_i$·\frac{{\lambda }_i}{2}=$(ε_i - ${\epsilon }_i^{\text{'}}$)$·\frac{{\lambda }_i}{2}$ is stable and means the excess fractional part of c.

Now the issue of determining the integer part of the wavelength order N_ireal is transformed to determining the value of c. The maximum adjustment value ∆N_iT for the calculated integer part $N_i^{\text{'}}$ is:

$$∆N_{iT}=\mathit{INT}\left(\frac{2\Delta L_{\mathrm{err}}}{{\lambda }_i}\right)$$

(23)

where $\Delta L_{\mathrm{err}}$ means the error range of the nominal length. Hence, the acceptable adjustment integer value m_ij for the integer part of c is within the interval:

-∆N_iT ≤ m_ij = j ≤ ∆N_iT

(24)

As a result, the value for difference c can be adjusted by a variable m_ij,

$$c_j=(m_{\mathrm{ij}}+{\delta }_i)\text{·}\frac{{\lambda }_i}{2}$$

(25)

Considering the same adjusted difference value of c_j when using another wavelength λ_i+1, the fractional part of wavelength order caused by c_j can be calculated by:

$${\delta }_{(i+1)j}=\frac{2c_j}{{\lambda }_{i+1}}-\mathit{INT}\left(\frac{2c_j}{{\lambda }_{i+1}}\right)$$

(26)

Hence, the fractional part of $L^{\text{'}}$+c_j with using the wavelength λ_i+1 can be given by:

$${\epsilon }_{(i+1)j}={\delta }_{(i+1)j}+{\epsilon }_{(i+1)}^{\text{'}}$$

(27)

Finally, when the difference value between ε_(i+1)j and ε_(i+1) reach the minimum, the corresponding c_J is the optimum adjusted value, and the real length L of the target distance can be calculated by:

$$L=L^{\text{'}}+CJ$$

(28)

Figure 1 illustrates the detailed procedure of selecting the optimum difference c_J and the relationship between those above-mentioned parameters in the excess fraction method.

It is worth mentioning that similar to Ref. [46], the target distance L is given by n_i∙L = (m_k + ε_k)∙λ_k. However, the approach in the proposed combined algorithm for determining the integer and excess fraction part of wavelength λ_k is different from that of Ref. [46]. In Ref. [46], the phase φ(λ_k) for a specific wavelength λ_k, is obtained from a cosine fit of the interference signal, and the integer number of the wavelength m_k is determined by the result of dispersive interferometry. In addition, the aim of using the integer and excess fraction part of wavelength λ_k in Ref. [46] is not to measure the absolute distance L directly and make a comparison with the result of the dispersive interferometry, but to refine the final value of the distance L by averaging the results of different wavelengths.

3. Simulation and experiment Results

4. Conclusions

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