1. Introduction

Spectrometers, devices for analyzing the spectral content of light, is one of the most powerful characterization tools which play an important role in countless fields including scientific research, industry, and consumer electronics [1–5]. Conventional benchtop spectrometers are largely confined to laboratory use because they are generally bulky, complex, expensive, and delicate. To meet the requirements for portable and on-site applications as well as varying the functionality in consumer electronics, it becomes highly desired to develop flexible, low-cost, and easy-to-use spectrometers [6–10]. To date, several fabulous concepts have been conceived and realized. The majority of designs rely on the construction of dispersive elements to spit the incident light into different channels and an array of detectors to measure the intensity at each channel [11–13]. However, these devices are particularly demanding of strict incident angle and thus high-precision fabrication process. Furthermore, since the spectral resolution is determined by the number of spectral channels, the fabrication complexity will be drastically increased as the resolution improving. In addition to these carefully designed dispersive spectrometers, compressive sensing and machine learning based techniques have also been explored for computational spectrometers [14–19]. This class of spectrometers relies on measuring the transmission profile of waveguide or filter arrays. The target spectrum can be reconstructed from calibrated spectral response information encoded within detectors, waveguides, or filters. Although the size of this type of spectrometer can be reduced into subwavelength scale, it requires ultra-fine manufacturing equipment and ultra-precision fabrication process to integrate the functions of both wavelength selective and photodetection into an individual micro-/nano-structure which is almost impossible task in most laboratories. Designs based on Fourier Transform (FT) spectrometers have also been investigated using microelectromechanical systems (MEMS) and integrated photonic circuits [20–26]. These types of spectrometers generally use an interferometer to modulate the incident light into an interferogram, then convert the obtained interferogram into wavelength-dependent spectrum via FT method. Different from dispersive and computational spectrometers, FT-based spectrometers have no direct correspondence between wavelength and the spectral channel, which means that this type of spectrometers are immune to the variation of incident angle and fabrication errors. However, the nonlinear scanning of optical path difference (OPD) or fluctuation of interferometric intensity will lead to unreliable results.

To counterbalance the nonuniform scan of OPD, an important contribution has been reported in the literature by P. Malara et al. [27,28]. In their work, they demonstrated that a monochromatic reference interferogram can be used to calculate instantaneous OPD thanks to the fact that a full fringe corresponds to an OPD scan of one reference wavelength. According to this concept, the instantaneous OPD of the maximum points of reference interferogram can be calculated approximately by counting the number of full fringes. However, since the OPD of other points (except maximum points) are obtained by linear interpolation, it may cause some errors between the calculated value and actual one which may result in spectral deviation. Additionally, because the light beam propagating inside the liquid sphere is a divergent beam, wavelength-dependent Guoy phase shift may also influence the OPD calculated result. In terms of intensity fluctuation, the work proposed in Refs. [27,28] introduced a hypothesis that the modal coupling efficiency of the light backreflected in the fiber does not depends on incident wavelength and incident intensity so as to eliminate the influence of the variation of coupling efficiency. Indeed, this concept is very nice and the method is quite elegant. However, the fluctuation of interferometric intensity caused by the incident optical power variation or optical path disturbance still unsolved which will significant influence the spectral analysis result. In other words, eliminating the influence of interference intensity fluctuation on the result of spectral analysis result remains as an open issue.

Here, we demonstrate a cheap and easy-to-fabricate FT spectrometer design that constructed by commonly used fiber optic components. The device employs a fiber-tip Fizeau interferometer to generate wavelength-dependent interferogram. The fiber-tip interferometer is composed of a gradient-index fiber (Grin fiber) with a cleaved end facet and a polished silicon wafer which is fixed at the end of a capillary (the diameter is around 127$\mathrm{\mu}$m). When a probe optical beam inputs into the fiber, two backward beams are obtained: one is the beam reflected partly from the fiber end; the other is the reflected beam from the polished silicon wafer and coupled back into the fiber. The two-beam interference gives signals dependent on the spacing between the fiber facet and the silicon wafer and the intensities of the two interference beams. When the fiber is pulled out from the capillary, the OPD and interference intensity changes, and generate a detectable interferogram. This behaves as a scanning interferometer used in conventional FT spectrometers. However, because the OPD scanning and the intensities of the two interference beams during pulling process is irregular and unpredictable, the detected interferogram cannot be directly transferred to the spectrum by FT method. In order to conquer this issue, an optical fiber Bragg grating (FBG) is used to extract the single-wavelength interferogram so as to obtain the exact instantaneous OPD and intensities of two interference beams at every moment. Consequently, all information needed for the reconstruction of incident spectrum can be achieved and then the incident spectrum can be retrieved by FT method. Comparing with other state-of-the-art works [21–23,29–32], especially the fabulous work present in Refs. [27,28], our device is independent of complicated fabrication procedures and expensive hardware, the OPD as well as interference intensity at every time moment is achieved in an accurate way rather than approximate method. Additionally, since the gradient-index fiber can convert the divergent light beam into collimated one, Guoy phase shift can be further suppressed. A resolution of 7.69 cm$^{-1}$ in the wavelength range of 1400 nm $\sim$ 1700 nm is achieved. Our concept can be easily reproduced in most laboratories. The results not only provide an alternative opportunity for developing cost-effective spectrometers, but also suggest a powerful pathway for the creation of new portable devices with spectral analysis function.

2. Operation mechanism

The configuration of the proposed design is illustrated in Fig. 1(a). The fiber-tip interferometer is composed of a fiber with a cleaved end facet and a polished silicon wafer which is fixed at the end of a capillary (the diameter is around 127$\mathrm{\mu}$m). A short section of gradient-index fiber (Thorlabs, GIF625) is spliced to the single-mode fiber to reduce the coupling loss of beam divergence and convert the divergent beam into collimated one so as to eliminate the Guoy phase shift. Since the reflectivity at the front/rear interfaces is very low (3% $\sim$ 5%), multiple reflections can be neglected, leading to a Fizeau interference model to be considered. When the incident light with spectral distribution I(k), k denotes the vacuum wave vector of incident light, is launched into the input port of the spectrometer and sent to the fiber-tip Fizeau interferometer via an optical fiber circulator, two backward beams are obtained: one is the beam reflected partly from the fiber end [front surface, as shown in Fig. 1(b)]; the other is the beam reflected from the silicon surface [rear surface, as shown in Fig. 1(b)] and couples back into the fiber. These two beams coupled back to SMF and interfere. Then, as shown in Fig. 1(a), an optical fiber coupler is utilized to split the interference beam into two: one goes directly into a photodetector (PD1), the other goes through a FBG via an optical fiber circulator and finally into another photodetector (PD2). PD1 and PD2 work synchronously. The phase difference between the two interference beams is a sum of OPD induced phase difference $k\delta$ ($\delta$ stands for OPD which equals to 2$nd$, where $n$ and $d$ denote the refractive index and the length of the interferometric cavity respectively) and a constant $\mathrm{\pi}$ due to half-wave loss.

 

Fig. 1. Spectrometer design. (a) Scheme of the proposed design. (b) Schematic illustration of the light beam propagation in the interferometric cavity. Incident light reflected at the two interfaces is indicated with arrows. I(k)$\varepsilon$(t), I(k)$\eta _1$(t), and I(k)$\eta _2$(t) are the instantaneous light intensities of the input, reflected by front surface, and reflected by rear surface, respectively. (c) Experimentally sampled time-domain interferograms of an incident light. I$_1$(t) and I$_2$(t) are the interferograms measured by PD1 and PD2, respectively.

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The operation of our spectrometer is very simple as illustrates in Fig. 1(a). As the optical fiber is gradually pulled out of the capillary, OPD increases monotonically over time $t$. Then the time-domain outputs recorded by PD1 and PD2 can be expressed as

As the length of interferometric cavity increases from 0 to $d$ throughout the pulling process, OPD increases monotonically from 0 to 2$nd$. From Eq. (1) we can find that each wavelength from incident spectrum produces its interferogram; these interferograms superimpose and inject into PD1. As a consequence, the output of PD1 is comprised of a slowly varying background signal [first two terms of Eq. (1)] on which is superimposed a time-varying cosine variation signal [third term of Eq. (1)]. The first two terms of Eq. (1) is determined by the instantaneous light intensities of the two interference beams which is independent of OPD. In other words, the first two terms of Eq. (1) representing the trend of the output of PD1 over time, have no influence on the spectral analysis, which should be canceled out (the strategy to remove these two terms will be explained in the following section).

Comparing the third term of Eq. (1) with $\int I(k)\cos (k\delta +\pi )dk$ (the real part of the Fourier transforms of incident spectrum), the third term of Eq. (1) contains two extra time dependent factors: instantaneous interference intensity $2\sqrt {\eta _1(t)\eta _2(t)}$ and instantaneous OPD, $\delta (t)$. If these two time dependent factors can be eliminated, the real part of the Fourier transforms of incident spectrum can be obtained and the incident spectrum can be retrieved by FT method accordingly. For this task, the output signal of PD2 is used. Indeed, from Eq. (2) we can find that the amplitude of the time-varying cosine variation is the very $2\sqrt {\eta _1(t)\eta _2(t)}$. Since PD1 and PD2 are working synchronously, $2\sqrt {\eta _1(t)\eta _2(t)}$ in the third term of Eq. (1) can be removed subsequent to the normalized value of $2\sqrt {\eta _1(t)\eta _2(t)}$ be calculated from the output signal of PD2. It is important to note that all of the factors which may cause interference intensity fluctuations are contained in the parameters $\eta _1(t)$ and $\eta _2(t)$. Different from the work proposed in Refs. [27,28], in this work, the reflectivities of the two interfaces will no longer need to be calculated separately according to Sellmeier equations. Also, our method is more universal and precise. The only work need to do is to extract the normalized value of $2\sqrt {\eta _1(t)\eta _2(t)}$ from the output of PD2. Since the physical meaning of $2\sqrt {\eta _1(t)\eta _2(t)}$ is the instantaneous amplitude of time-domain interferogram recorded by PD2, this value can be worked out accuracy via calculating the difference between upper and lower envelopes of $I_2(t)$.

Now, the remaining issue is to get $\delta (t)$. Because the output of PD2 is the single-wavelength interferogram, the maximum points of the interferogram corresponds to constructive interference which means the OPD induced phase shift at the very moment $t$ equals to odd multiples of $\mathrm{\pi}$, viz., $\delta (t)=(2N+1)\times \lambda _{\rm {FBG}}/2, (N=0,1,2\cdots )$. Similarly, the minimum points correspond to destructive interference which means the OPD induced phase shift at the very moment $t$ equals to even multiples of $\pi$, viz., $\delta (t)=2N\times \lambda _{\rm {FBG}}/2, (N=0,1,2\cdots )$. The value of $\delta$ at other points can also be worked out by analogous method. Details of how to get instantaneous OPD is described in following section. According to this strategy, the instantaneous OPD, $\delta (t)$, can be achieved exactly rather than approximately.

Consequently, the real part of the Fourier transforms of incident spectrum $I(k)$ can be extracted. Here, we should note that the interferogram can have an arbitrary wavelength dependent phase without affecting the spectrum thanks to the fact that the power spectral density (spectrum) is real valued and one can eliminate the spurious phase term just by taking the magnitude of the inverse Fourier transform. It can be seen from the above analysis that although we do not have the prior information of $\eta _1(t)$, $\eta _2(t)$, and $\delta (t)$, $\int I(k)\cos (k\delta +\mathrm{\pi} )dk$ can still be extracted from the respective outputs of PD1 and PD2 by utilizing appropriate strategy in our design. This built-in self calibration mechanism can fix all variations caused by irregular scan of OPD, fluctuation of reflected beam intensity, and non-uniform sampling rate sampling rate.

3. Experimental results

To test our concept in practice, we performed an experimental analysis. An incident light with spectrum as shown in Fig. 6(c) is launched into the spectrometer and sent to the interferometer. By puling the optical fiber to make the gap between rear and front surfaces from 0 to $d$, $\delta$ scans from 0 to $2nd$ (in our case, $n$ approximately equal to 1 because the medium inside the interferometric cavity is air). During the pulling process, both output signals of PD1 and PD2 are recorded in real time. Fig. 1(c) gives the typical output signals recorded by PD1 and PD2. Initially, $d$ close to 0, the phase difference between two reflected beams approximately equal to $\mathrm{\pi}$ (destructive interference). As $d$ increases monotonically, $\delta$ increases as well, as a consequence the output of PD2 varies between constructive interference and destructive interference. For PD1, because all wavelengths of the spectral distribution arrive at the detector at the same time, varying $\delta$ gives a different superposition of interference intensities of all wavelengths. The incident spectrum can be reconstructed from these signals by suitable strategy. The analysis procedure of the recorded signals will be explained in detail in the following subsections. The brief description is as follows: To cancel the first two terms of Eq. (1), we first calculated the instantaneous mean values of the output of PD1 and PD2. Then, the first two terms of Eq. (1) can be removed by subtracting these instantaneous mean values from the original signals. Since $2\sqrt {\eta _1(t)\eta _2(t)}$ is the amplitude of the cosine variation, its normalized value at every moment $t$ can be acquired by subtracting the upper and lower envelops of the output of PD2. Subsequently, the normalized value of $\int I(k)\cos [k\delta (t)+\mathrm{\pi} ]dk$ and $I(k_{\rm FBG})\cos [k_{\rm FBG}\delta (t)+\mathrm{\pi} ]$ can be obtained by dividing normalized $2\sqrt {\eta _1(t)\eta _2(t)}$. The instantaneous value of $\delta$ at every moment $t$ can also be calculated exactly by proper strategy. Then the time domain interferograms can be transposed into spatial domain interferograms. Here we should note that the equally-spaced data points of the time-domain signal can not convert into equally-spaced data points in the spatial domain due to the nonlinear relationship between $\delta$ and $t$. In this case, we provide two strategies to retrieve the incident spectrum. As in practical application, the spectrometer should be calibrated to provide good accuracy, this procedure only need to be performed once for the spectrometer. The details of analysis procedure of the recorded signals is explained as follows:

3.1 Removing the slowly varying background of $I_1$(t) and $I_2$(t)

To remove the slowly varying background of $I_{\rm 1}(t)$ and $I_{\rm 2}(t)$, the instantaneous mean value of $I_{\rm 1}(t)$ and $I_{\rm 2}(t)$ must be obtained. For this task, we first got the maximum and minimum points of $I_{\rm 1}(t)$ and $I_{\rm 2}(t)$ via detecting its zero derivative points [Fig. 2(a)]. Let $m$ be the number of sampled points, then $t$ can be divided to $m$ parts, namely $t(1)$, $t(2)$, $\cdots$, $t({m})$. $I_1(t)$ can be divided to $m$ parts as well, namely $I_1[t(1)]$, $I_1[t(2)]$, $\cdots$, $I_1[t({m})]$. Define an array $p_1$ with all the time moment of the maximum points of $I_1(t)$. Then the recorded data can be illustrated as Table 1. The total number of sampled data is $m$, which means both arrays of the sampled $t$ and $I_1(t)$ contain $m$ elements. The exact time of the $i$th maximum point $t$ is $t=p_1(i)$ [as shown in Fig. 2(b)], the maximum value of $I_1(t)$ at this moment is $I_1[p_1(i)]$ accordingly.

 

Fig. 2. Extraction of the slowly varying background from the sampled $I_1(t)$ and $I_2(t)$. (a) The maximum and minimum points of the recorded signals, obtained by calculating zero derivative points of the sampled data. (b) Illustration of the strategy for the calculation of the upper envelope $f_{\rm up1}(t)$ of $I_1(t)$. (c) Black dash lines stand for the upper envelopes of both sampled signals. Black dotted lines represent the lower envelopes of both signals. Black solid line are the instantaneous mean values of $I_1(t)$ and $I_2(t)$. (d) Slowly varying background of $I_1(t)$. (e) Slowly varying background of $I_2(t)$. (f) Processed signal of $I_1(t)$ and $I_2(t)$, acquired by subtracting the slowly varying background from the original signals respectively.

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Table 1. Data points sampled by PD1

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Here, since the range between two adjacent maxima is very small, the evolution of upper envelope in this interval can be regarded as linear. Consequently, every point of the upper envelope $f_{\rm up1}(t)$ can be calculated via the following strategy:

The lower envelope of $I_1(t)$ can be obtained by the same strategy. Consequently, the instantaneous mean value of $I_{\rm 1}(t)$ can be expressed as:

Similarly, the upper and lower envelopes of $I_2(t)$, namely $f_{\rm up2}(t)$ and $f_{\rm lo2}(t)$, can be got in the same way. Then the instantaneous mean value, $f_{\rm mid2}(t)$, of $I_2(t)$ can also be obtained [as shown in Fig. 2(c)]. The calculated $f_{\rm mid1}(t)$ and $f_{\rm mid2}(t)$ are illustrated in Fig. 2(d) and Fig. 2(e). By subtracting $f_{\rm mid1}(t)$ and $f_{\rm mid2}(t)$ from the sampled signals $I_1(t)$ and $I_2(t)$, the slowly varying background can be removed [as shown in Fig. 2(f)].

3.2 Making the interferogram independent of $\eta _1(t)$ and $\eta _2(t)$

Since the intensities of the two reflected beams changes constantly during the pulling process, the interference intensity (visibility) will vary with time accordingly. In order to extract the correct interferogram from the sampled signals, this influence of time-dependent interference intensity $2\sqrt {\eta _1(t)\eta _2(t)}$ must be eliminated. For this task, $I_2(t)$ is used. Theoretically, the signal detected by PD2 is a constant amplitude signal out of the characteristic of single-wavelength interferogram. But the practical amplitude of the sampled signal fluctuate because of the instability of interference intensity. From the third term of Eq. (2) we can find that $2\sqrt {\eta _1(t)\eta _2(t)}$ is determined by the envelope of sampled signal of $I_2(t)$, as shown in Fig. 3(a). Thus, $2\sqrt {\eta _1(t)\eta _2(t)}$ can be expressed by:

 

Fig. 3. (a) The normalized value of $2\sqrt {\eta _1(t)\eta _2(t)}$. The curve is obtained by subtracting the upper and lower envelopes of $I_2(t)$. The intensity range in this curve is normalized. (b) The time domain interferograms which is obtained by dividing the curve in Fig. 2(f) by normalized value of $2\sqrt {\eta _1(t)\eta _2(t)}$. The intensity range is normalized.

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Then, we have:

Consequently, the influence of instantaneous interference intensity fluctuation is counterbalanced. Fig. 3(b) depicts the interferograms independent of instantaneous interference intensity. Both interferograms plot in Fig. 3(b) are normalized with respect to their maximum.

3.3 Getting instantaneous OPD and spatial domain interferograms

Because the signal detected by PD2 is a single-wavelength interferogram, the maximum and minimum points of $I_2^{\rm 3rd}(t)$ correspond to the constructive and destructive interference. For constructive interference, the value of OPD plus $\lambda _{\rm FBG}/2$ (caused by half-wave loss) should be equal to integer multiples of $\lambda _{\rm FBG}$. For destructive interference, the value of OPD plus $\lambda _{\rm FBG}/2$ should be equal to odd multiples of half wavelength of $\lambda _{\rm FBG}/2$. Define two arrays $p_2$ and $d_2$ with all the time moment of the maximum and minimum points of $I_2^{\rm 3rd}(t)$, respectively. Thus the exact time of the $i$th maximum point of $I_2^{\rm 3rd}(t)$ is $p_2(i)$. The $i$th minimum point of $I_2^{\rm 3rd}(t)$ should be $d_2(i)$ as well. Because OPD increases from 0, the first maximum point must be in front of the first minimum point [as shown in Fig. 4(a)], viz., inequality $p_2(i)<d_2(i)$ must be true [as shown in Fig. 4(b)]. This can be illustrated with the assistant of Table 2 and Fig. 4. The total number of the data is $m$, which means both arrays of $t$ and $I_2^{\rm 3rd}(t)$ contain $m$ elements. The exact time of the $i$th maximum point is $t=p_2(i)$, and the exact time of the $i$th minimum point is $t=d_2(i)$. The value of $I_2^{\rm 3rd}(t)$ at maximum and minimum points are 1 and -1, respectively.

 

Fig. 4. Process of extracting the instantaneous OPD and spatial domain interferograms from the sampled data. (a) Single-wavelength time domain interferogram sampled by PD2. We first find the maximum and minimum points of the recorded signals via calculating zero derivative points of sampled data. The time moment corresponds to maximum points is $t=p_2(1)$, $p_2(2)$, $\cdots$, $p_2({\rm max})$. Similarly, the time moment corresponds to minimum points is $t=d_2(1)$, $d_2(2)$, $\cdots$, $d_2({\rm max})$. Because the length of interferometric cavity increases from 0, the first maximum is always in front of the first minimum point, viz., $p_2(1)<d_2(1)$. Consequently, $p_2(i)$ must in front of $d_2(i)$. (B) Illustration of the strategy for calculating the instantaneous $\delta (t)$. (C) Comparison of the time domain interferogram (top curve) and spatial domain interferogram (bottom curve) obtained from the raw data detected by PD2. We can find that the irregular time domain interferogram is converted to regular spatial interferogram by using the strategy proposed in this work. (E) Comparison of the time domain interferogram (top curve) and spatial domain interferogram (bottom curve) obtained from the raw data detected by PD1. The sampled time domain interferogram of input spectrum successfully transposed into spatial interferogram by utilizing our method.

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Table 2. Data points of ${I_2^{\rm 3rd}(t)}$

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The instantaneous OPD, $\delta (t)$, can be obtained by the following method:

Then at every time point $t(i)$, there is a corresponding $\delta (i)$ [as shown in Table 3 and Fig. 4(c)]. Consequently, time domain $I_2^{\rm 3rd}(t)$ can be exactly converted to $I_2^{\rm 3rd}(\delta )$ [spatial domain, Fig. 4(d)]. Because PD1 and PD2 are working synchronously, the $i$th point of the $I_1^{\rm 3rd}(t)$ has the same OPD of the $i$th point of the $I_2^{\rm 3rd}(t)$, thus the time domain of $I_1^{\rm 3rd}(t)$ can be converted into spatial domain $I_1^{\rm 3rd}(\delta )$ as well [as shown in Fig. 4(e)]. Different from the method proposed in Refs. [27,28], the OPD achieved via our strategy is an accurate value rather than an approximate one.

Table 3. Transposing the time domain interferograms into spatial domain interferograms

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3.4 Retrieving the incident spectrum

Since the hand pulling is generally non-uniform process, the equally-spaced data points of the time domain interferogram $I_1^{\rm 3rd}(t)$ cannot be converted into equally-spaced data points of spatial domain interferogram $I_1^{\rm 3rd}(\delta )$, as illustrated in Fig. 5(a). Therefore, FT method cannot be applied directly to $I_1^{\rm 3rd}(\delta )$. Here, we introduce two different strategies which can be followed to retrieve the incident spectrum. In the first approach, the nonuniform data can be converted into equal interval data by means of resampling or interpolation. Since the spacing between adjacent points is very small, the evolution process can be approximately regarded as linear. Then a new oversampled $I_1^{\rm 3rd}(\delta )$ can be achieved [as shown in Fig. 5(b)]. The data spacing of the new $I_1^{\rm 3rd}(\delta )$ is regular, thus the input spectrum can be retrieved by FT method input spectrum. The second approach relies on the Lomb–Scargle algorithm [33–35]. In this case, the input spectrum can be directly worked out from the unevenly sampled data. Other sort of similar methods can also be utilized to retrieve the input spectrum from the nonuniform data [36–39].

 

Fig. 5. Making the data equally spaced. (a) Spatial interferogram of incident spectrum obtained by the method proposed in this work. The upper insert shows the highlighted region in more detail. The lower insert shows the sampling comb in the highlighted region, from which we can find that the spacing between two adjacent points is nonuniform. (b) New spatial interferogram of incident spectrum after interpolation. The upper inset again shows the highlighted region in more detail. The lower insert shows the new sampling comb, from which we can see that the irregular signal turns to be regular.

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Fig. 6. Retrieving the incident spectrum. (a)Getting spectral calibration function. Obtained spectral calibration function $f(w)$ by the means of $f_{\rm ft}(w)/f_{\rm in}(w)$. $f_{\rm ft}(w)$ is calculated by FT method as illustrated in MM4, $f_{\rm in}(w)$ is obtained from commercial OSA ( YOKOGAWA AQ6370B). Inserts shows the spectra $f_{\rm ft}(w)$ and $f_{\rm in}(w)$. (b) The calculated input spectrum of our device without calibration step. The spatial interferogram of this spectrum is shown in Fig. 5(b). The black dot line shows the same spectrum measured by commercial OSA. (c) The calculated input spectrum of our device with calibration step. The input spectrum is the same as depicted in (b). The black line is again the spectrum measured by commercial OSA.

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From the spatial interferogram as plotted in Fig. 5(b), the optical spectrum can be retrieved by means of FT method. In the ideal case, the retrieved spectrum should be in accordance with the incident spectrum. In practice, however, because the sensitivity of photodetectors, transmission performance of circulator and coupler, and reflectivity of two interfaces vary with the wavelength more or less, the retrieved spectrum may have non-negligible difference with input spectrum if appropriate calibration method is not adopted [Fig. 6(b)]. To fix this issue, the relationship between the calculated spectrum and the actual spectrum at every wavenumber, namely spectral calibration function $f(w)$, should be predetermined. Here, $f(w)$ is defined as

In order to get $f(w)$, the same input spectrum is measured by commercial OSA and our spectrometer simultaneously. The spectrum measured by OSA is regarded as actual spectrum $f_{\rm in}(w)$, and the spectrum by our spectrometer is the very $f_{\rm ft}(w)$. Then, we can get the spectral calibration function $f(w)$, as plotted in Fig. 6(a). Consequently, more accurate spectrum $f_{\rm re}(w)$ can be achieved on the basis of obtained $f_{\rm ft}(w)$ [Fig. 6(c)]:

It is worthy to note that, for each spectrometer the calibration step is required before spectra measurement. Theoretically, any incident spectrum can be utilized to obtain the spectral calibration function $f(w)$. However, in practice, considering the measurement range of the spectrometer, the incident spectrum with broadband wavelength will be preferred.

3.5 Self-calibration ability

 

Fig. 7. Experimental measurement of the incident spectrum. (a) Outputs of PD1 (red line) and PD2 (blue line) in different measurements (incident spectrum in these measurement is identical). (b) $2\sqrt {(\eta _1(t))\eta _2(t)}$ as a function of $t$ in different experiments. This instantaneous value is calculated from the sampled data shown in (a) by the proposed method as describe in Fig. 3. (c) $\delta (t)$ as a function of $t$. This instantaneous value is calculated from the sampled data shown in (a) by the proposed method as describe in Fig. 4. (d) Extracted spatial interferograms from sampled data shown in (a). (e) Comparison of the measured spectra under different experiments. The measured spectra almost coincide and are consistent with OSA (blue dash line).

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Fig. 8. (a) Outputs of PD1 (red line) and PD2 (blue line) for different incident spectra. The raw data is provided in Ref. [40]. (b) Spatial interferograms of the incident spectra obtained by our method and theoretical analytical method. (c) Several different incident spectra measured by our spectrometer and compared to measurements by commercial OSA (black dash line).

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In order to test the self-calibration ability of our spectrometer, we first applied the spectrometer to experimentally measure the same incident spectrum under different hand pulling process. Fig. 7(a) gives the recorded outputs of PD1 and PD2 in four experiments. Since hand-pulling is an irregular and non-reproducible process, the outputs of PD1 and PD2 seems quite different. $2\sqrt {\eta _1(t)\eta _2(t)}$ and $\delta (t)$ are calculated by the method described above and depicted in Fig. 7(b) and Fig. 7(c), respectively. Then all variations during the hand pulling process can be fixed to achieve regular spatial interferograms [as depicted in Fig. 7(d)]. By utilizing FT method, incident spectra under different experiments can be retrieved. Fig. 7(e) presents the comparison of the measured spectra by our spectrometer and commercial OSA, from which we can find that the spectra measured under different hand-pulling processes almost coincide with each other as well as the spectrum measured by commercial OSA.

To further demonstrate the robustness, several different incident spectra also been measured by our spectrometer and briefly summarized in Fig. 8. The test results once again show that the retrieved spectra match nicely with the spectra measured by commercial OSA. Raw data of the sampled signals as we show in Dataset 1 (Ref. [40]).

4. Discussions

The spectral resolution of our device depends on the maximum OPD achieved in the hand pulling process which can be quantified as: $\delta _{\rm max}^{-1}$, where $\delta _{\rm max}$ stands for the maximum OPD recorded by photodetectors. Although the resolution is inversely proportional to OPD, it cannot be increased infinitively. It is because the light propagating in the interferometric cavity is generally not perfectly collimated beam, the optical power coupled back into SMF decays with the increases of $d$, larger $d$ means that the signal detected by photodetectors is very weak, resulting in a degraded signal. Besides, due to the variability of hand pulling dynamics, the maximum OPD recorded in each experiment is slightly different. As experimental results shown in Fig. 7(c), the maximum OPD measured in our experiment is around 1500 $\mathrm{\mu}$m, corresponds to a resolution up to 7.69 cm$^{-1}$. There is no deny the fact that other ways to enlarge the OPD scanning range also help improve resolution.

In addition to spectral resolution, bandwidth is another crucial metric of spectrometer performance. In our case, the spectral analysis range is determined by the bandwidth of optical components and sample spacing of the spatial interferogram. The bandwidths of optical fiber circulator and coupler are in 1400 $\sim$ 1700 nm range (5882 $\sim$ 7143 cm$^{-1}$). The sample spacing relies on pulling velocity and sampling rate. For example, when the pulling velocity is 100 $\mathrm{\mu}$m/s and the acquisition rate is 1kHz (1000 points per second), the mean spatial sample spacing is around 0.1 $\mathrm{\mu}$m, the maximum wavenumber can be retrieved by this spectrometer is 50000 cm$^{-1}$. The spectral range of the spectrometer can be designed flexibly according to practical requirements by choosing the bandwidth of optical components, pulling velocity and sampling rate. It is worth to point out that the spatial sample spacing of conventional FT spectrometer usually equals to half wavelength of the reference laser (the acquisition of the signal is triggered by the zero-crossing of the interferogram generated by this reference laser), which can only analysis spectrum with longer wavelength (comparing with reference laser). While in our spectrometer, as show in experimental results [Fig. 6(c), Fig. 7(e), and Fig. 8(c)], spectrum with wavelength components much shorter than $\lambda _{\rm FBG}$ (1540nm, $k_{\rm FBG}$ = 6493 cm$^{-1}$) can also be correctly retrieved. The fundamental concept introduced here gives a new strategy to construct compact and portable spectrometer with customized performance. Although the device was not optimized for any specific figures of merit, the performance is already comparable to existing benchtop devices. The performance of the proposed spectrometer compared with other state-of-the-art spectrometers is listed in Table 4. Although the performance of our device is not the best, but our device is independent of the complicated fabrication procedures, easy to use, and highly flexible instrument with many different applications. We envision that our work also can inspire a new generation of compact, robust, and cheap spectrometers for portable applications.

Table 4. The performance of spectrometers in recent literature

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It is important to note that although we take hand pulling as the driving way to carry out the experimental verification, any other forms of force can also be used to drive the spectrometer. Taking advantage of the self calibration mechanism, all instabilities can be fixed so that the exact spatial interferogram of incident spectrum can be extracted from recorded signals regardless of the specific OPD scanning process.

5. Conclusion

In summary, a miniature Fourier transform spectrometer based on fiber-tip Fizeau interferometer is proposed and demonstrated. The unique design of our device provides permanent calibration and stability to guarantee accurate and reliable measurements. This provides flexibility in building various practical devices to meet different requirements as well as improve the stability of the spectrometers. Because our device is based on a general principle, almost zero cost, and can be easily replicated, it may offer an opportunity for non-professionals (especially in undeveloped areas) to build their own spectrometers to meet the requirements of practical applications. Also, based on this concept, a vast number of devices can be developed to fit specific applications, ranging from compact spectrometers for portable sensing to wearable electronics with spectral analysis function for health diagnosis. This work may pave a new way to construct miniature spectrometers for practical applications.