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Abstract
Spectroscopy continues to provide possibilities for a deeper understanding of fundamental physical phenomena. Traditional spectral measurement method, dispersive Fourier transformation, is always limited by its realization condition (detection in the temporal far-field). Inspired by Fourier ghost imaging, we put forward an indirect spectrum measurement to overcome the limitation. The spectrum information is reconstructed via random phase modulation and near-field detection in the time domain. Since all operations are realized in the near-field region, the required length of dispersion fiber and optical loss are greatly reduced. Considering the application in spectroscopy, the length of required dispersion fiber, the spectrum resolution, the range of spectrum measurement and the requirement on bandwidth of photodetector are investigated.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Spectroscopy plays an important role in the development of science and technology [1–4]. In the early days of quantum mechanics, it provides further insight into atoms and molecules structure. With the resolution improvement of spectral measurement, the fine structure of atoms and the anomalous Zeeman effect are found, which provide strong evidence for the electron spin theory. Nowadays, spectral measurement has been extended to various fields, such as the identification of material structure by Raman spectroscopy, ultra-precision measurements using optical frequency combs, and even exploring the evolution of the universe via the spectrum of distant stars.
As a powerful tool in spectroscopy, dispersive Fourier transformation (DFT) maps the spectrum information into the temporal domain [5–11]. When the optical pulse experience sufficient dispersion to satisfy the temporal far-field condition, spectral measurement or spectrum-to-time conversion is achieved [12,13]. Specifically, the spectrum is obtained by direct measuring the temporal waveform. From a design perspective, the length of dispersion fiber $z$ is required to be much larger than ${ \tau }^{ 2 } / { \beta }_{ 2 }$, where ${\tau }$ is the pulse width and ${ \beta }_{ 2 }$ is the group-velocity dispersion coefficient. However, in the implementation, when ${ \tau }$ is large, the above condition is hard or even impossible to achieve. Besides, large dispersion is always accompanied with huge optical loss [10]. Even if the far-field condition is realized, the number of detectable photons will be tiny, the sensitivity of DFT being challenged.
Based on space-time duality, the idea of indirect imaging method, such as Fourier ghost imaging [14–21], in which the Fourier image is acquired in the near-field region, can be introduced to solve the above limitation. In FGI setup, the thermal light source is divided into two arms by the beam splitter. In the object arm, the light passes through the object and propagates freely to a point detector. In the reference arm, the light propagates freely to a CCD. When the distance between the source and the point detector equals to the distance between the source and the CCD, the Fourier image can be acquired by intensity fluctuation correlation between the results of the point detector and the CCD. Specially, the major function of the CCD is to record the light field variations. If the source is controllable and calculable, the reference arm can be omitted, which is known as computational FGI. Thus, the Fourier image reconstruction is based on the point detection results and the pre-computational reference light field.
Inspired by FGI, we put forward an indirect spectrum measurement scheme to overcome the limitation in DFT. Random phase modulation is introduced, and thus the spectrum acquisition (photon detection) is sufficient to be realized in near-field, instead of far-field. After many times of random modulations and photon detections, the spectrum is demodulated based on correlation algorithm. Since detection is performed in the near-field region, the required dispersion and optical loss are both greatly reduced. We analyze the key features of this spectrum measurement method, including the physical interpretation , the length of required dispersion fiber, the spectrum resolution, the range of spectral measurement, and the requirement on bandwidth of photodetector.
2. Indirect spectrum measurement method
Our spectrum measurement is realized by a phase modulator (PM), a dispersive fiber and a photodetector, as shown in Fig. 1. With the help of phase modulation, the spectrum information of the test pulses $A(t_{1})$ with pulse width ${ \tau }$ is obtain in the near-field region. Scheme contains a total of three parts: random phase modulation, indirect measurement in the near-field region, and spectrum reconstruction.
Fig. 1. Indirect spectrum measurement scheme. The repeatable input pulses are modulated by the phase modulator (PM) randomly. After passing through the dispersive fiber, the modulated pulses are detected by the photodetector (PD). The input pulse spectrum is retrieved through reconstruction algorithm.
Firstly, random phase modulation is introduced. Similar to the spatial FGI [16,17], many measurements of a sequence of repeated pulses $A(t_{1})$ with period $T$ are required. In each period, the test pulse is modulated by the PM with time-varying random phase $E(t_{1})$. The phases with time pixel ${ \sigma }_{ g }$ are randomly distributed within $[-\mathrm{\pi}, \mathrm{\pi} )$ and the length of phase modulation is ${ \sigma }_{ I }$. In different period, $E(t_{1})$ is different. Secondly, detection in the near-field region is sufficient. After passing through the z-long dispersion fiber, the light field $E_{o}(t_{o})$ is expressed as
Thirdly, spectrum is reconstructed via correlation algorithm. In our scheme, spectrum reconstruction relies on the pre-computational reference light field $E_{ r }(t_{r})$ [16,17], which is set as
Both the dispersion and the phase modulation in Eq. (1) and Eq. (3) are the same, satisfying reconstruction condition [16]. The spectrum is reconstructed via the intensity fluctuations correlation,
Usually, ${ \sigma }_{ g }$ is much smaller than ${ \sigma }_{ I }$ and thus $\left\langle {E}_{r}({ t }_{ i }){ {E}_{r} }^{ * }({ t }_{ r }) \right\rangle$ is approximately $\delta ({t_i} - {t_r} )$. By setting ${t}_{o}$ to zero, we get the square of spectrum $\widetilde { A } ({ t }_{ i } / { \beta }_{ 2 }z )$.
3. Key features analysis
3.1 Physics of indirect spectrum measurement in near-field region
In DFT, the separation of different spectrum components relies on sufficient dispersion. The waveform in the far-field region stands for the spectrum distribution and the test spectrum is directly obtained via photon detection. So we call DFT direct spectrum measurement. By contrast, our scheme is an indirect measurement, in which the mixture of different spectrum components is just needed. This spectrum mixture is acquired by random phase modulation and insufficient dispersion. Since the spectrum components are mixed, the reconstruction algorithm is needed to retrieve the spectrum. Note that, with the help of modulation and dispersion, the near-field region is enough to acquire the spectrum information, such that both the required dispersion fiber and the optical loss are greatly reduced, compared with DFT [5–7].
3.2 Required length of dispersion fiber
Since the required dispersion fiber and optical loss are greatly reduced, it raises the question of what is the shortest length of dispersion fiber or the requirement of dispersion. The required or shortest length of dispersion fiber is
as explained below.The spectrum and time can be connected by Fourier transform. Every spectrum component is related with the entire time domain, and and vice versa. For retrieving the spectrum, the point detection results are required to contain the information of all time pixels. As propagating in the dispersion fiber, the time pixels get broadened and overlapped with each other. If the length of dispersion fiber is short, only the neighbor pixels overlap and interference such that the point detection results contain only a part of time pixels’ information. With the length of dispersion fiber increasing, until the most separated two time pixels overlap and interference, the point detection results contain the information of all time pixels. Learning from the theory of speckles and spatial FGI [22], the length of required dispersion fiber in which the most separated time pixels overlap is $z_{c}$. It is clear that the smaller ${ \sigma }_{ g }$ or high bandwidth PM is helpful to reduce the required length of dispersion fiber and the optical loss.
The required dispersion fiber can be greatly reduced, the theoretical comparison is given below and the numerical comparison is given in the simulation part. In DFT, the length of dispersion fiber ${z_{{\rm {DFT}}}}$ is required to be much larger than ${ \tau }^{ 2 } / { \beta }_{ 2 }$. The radio is
The specific reduced length depends on $\tau$ and ${\sigma _g}$.
3.3 Spectral resolution
In Eq. (5), the image is in the form of a convolution, in which $\widetilde { A } (-{ t }_{ i } / { \beta }_{ 2 }z )$ is convoluting with the first-order correlation function. The width of the correlation function ${ \Delta t }$ and the fixed coefficient $1/{ \beta }_{ 2 }z$ determine the spectrum resolution together. That is
The width of the correlation function at arbitrary propagation distance $z$ is expressed as [23]
Especially, when $z={z_{c}}= { \sigma }_{ I }{ \sigma }_{ g } / 2\mathrm{\pi} { \beta }_{ 2 }$, ${ \Delta t }$ equals to ${ { \sigma }_{ g } }$. Thus the spectrum resolution of is
If the length of dispersion fiber is fixed, the spectrum resolution is determined by ${ { \sigma }_{ g } }$ or the bandwidth of PM.
3.4 Range of spectral measurement
The product between the spectrum resolution and the number of spectrum resolutions is the range of spectral measurement. In the near-field region, the length of the per-computational light field in one period is ${ \sigma }_{ I }$ such that the number of spectrum resolutions is ${ \sigma }_{ I }/{ \sigma }_{ g}$. Thus the range of spectral measurement is
It is obvious that the higher the bandwidth of PM is, the larger the range of spectral measurement will be.
The Eq. (12) also has its physical meaning. There is one-to-one correspondence between the spectrum and the pixel in pre-computational reference light field. Thus, the range of spectral measurement is determined by the range of pre-computational reference light field, which is equals to the spectrum range of random phase modulation $1/{ \sigma }_{ g }$ [22].
3.5 Requirement of bandwidth of photodetector
As analyzed above, the function of photodetector is to obtain different superposition of input spectrum. To avoid the crosstalk from neighbor time points, the bandwidth of the photodetector should match the temporal size of time point or the width of the correlation function, which equals to ${ { \sigma }_{ g } }$ at the photodetector, according to the Eq. (10). That is to say, the bandwidth of the photodetector is required to be at least equal to the one of the PM. If the requirement on bandwidth is difficult to achieve, an optical shutter can be applied to select only one time pixel before the photodetector. The switching speed of optical shutter is also needed to match ${ { \sigma }_{ g } }$.
Finally, it is worth to note that there is a trade-off between the spectrum resolution and the required length of dispersion fiber. The spectrum resolution can not be improved by increasing ${ \sigma }_{ I }$ blindly, as the required length of dispersion fiber increases at the same time. CTFGI benefits from the high bandwidth PM, which can reduce the optical loss and increase the range of spectral measurement.
4. Simulation
We verify the above theoretical analysis in the following numerical simulations. The input pulse is in the Gaussian form, with the period 10 ${\rm {ns}}$. The full width at half maximum of inputs is 1 ${\rm {ns}}$. The bandwidth of the phase modulator is 100 ${\rm {GHz}}$ [24], which means ${ \sigma }_{ g }$ is 10 ${\rm {ps}}$. ${ \sigma }_{ I }$ is set as 4 ${\rm {ns}}$ and the number of periods is ${10^5}$. The required length of the dispersive fiber with dispersion −200 ${\rm {ps}}/({\rm {nm}} \cdot {\rm {km}})$ is 21 ${\rm {km}}$. The bandwidth of photodetector is also 100 ${\rm {GHz}}$ [25], matching the bandwidth of the PM. The simulation result is shown in Fig. 2. Both discrete spectrum and continuous spectrum are demonstrated. Take the fine structure of potassium atom as an example [26]. The simulation result and the standard spectrum distribution are shown in Fig. 2(a1) and (b1). Here, the Fig. 2(b1) and (b2) can also be understood as the measurement results via DFT, since only considering the dispersion. To quantitatively analyze the results of our method, mean-squared error (MSE) of the results compared to the standard spectrum is employed. The smaller the value of MSE is, the better quality of results will be. By statistics from simulations of 20 times, the MSE is $0.0163\pm 0.0003$. Take the stimulated Raman spectrum in a silicon waveguide as an example of continuous spectrum [7]. The simulation result via our method and the standard spectrum distribution are shown in Fig. 2(a2) and (b2). By statistics from simulations of 20 times, the MSE is $0.0162\pm 0.0004$. From the results of MSE, the results fit the standard spectrums well. The feature of our method, which is the spectrum can be obtained in the near field region, is verified in the above simulation. If our method is based on commercial high speed devices, such as phase modulator of 40 GHz (Thorlabs, Part number: LNP6119,) and photodetector of 50 GHz (Thorlabs, Part number: DXM50AF), the required length of the dispersive fiber will be 52 ${\rm {km}}$.
Fig. 2. The results of simulation in the upper row and the standard spectrum in the lower row. We give the examples of both continuous and discrete spectrum. The correlation value (c.v.) is normalized in simulations. The details are listed in the text.
If such a signal is measured via dispersive Fourier transformation, the required length of the dispersive fiber must be much larger than 3320 ${\rm {km}}$ to satisfy the temporal far field condition. Suppose the attenuation of the dispersive fiber is 1 ${\rm {dB}}/{\rm {km}}$, the optical loss caused by the fiber in our scheme is only 23.5 ${\rm {dB}}$, containing the loss of the dispersion fiber 21 ${\rm {dB}}$ and the PM 2.5 ${\rm {dB}}$. In contrary, the optical loss in DFT is at least 3320 ${\rm {dB}}$.
We also investigate the influence of ${ \sigma }_{ I }$ and ${ \sigma }_{ g }$ on the spectrum resolution, as shown in Fig. 3. In the upper row ${ \sigma }_{ I }$ is 4 ${\rm {ns}}$, 3 ${\rm {ns}}$ and 2 ${\rm {ns}}$ respectively, while the length of dispersive fiber is 21 ${\rm {km}}$, 15.8 ${\rm {km}}$ and 10.5 ${\rm {km}}$ respectively, according Eq. (6). From left to right, the spectrum resolution declines with decreasing of ${ \sigma }_{ I }$. In the lower row ${ \sigma }_{ g }$ is 10 ${\rm {ps}}$, 20 ${\rm {ps}}$ and 30 ${\rm {ps}}$, respectively. The length of dispersive fiber remains unchanged, $z=63$ ${\rm {km}}$. The spectrum resolution decreases from left to right as ${ \sigma }_{ g }$ increases.
Fig. 3. The influence of ${ \sigma }_{ I }$ and ${ \sigma }_{ g }$ on the spectrum resolution. ${ \sigma }_{ I }$ is 4 ${\rm {ns}}$, 3 ${\rm {ns}}$ and 2 ${\rm {ns}}$ respectively in the upper row. ${ \sigma }_{ g }$ is 10 ${\rm {ps}}$, 20 ${\rm {ps}}$ and 30 ${\rm {ps}}$ respectively in the lower row.
It is worth to note that the spectrum resolution is determined by the length of modulation $\sigma _I$. With $\sigma _I$ increasing, the spectrum resolution is improved. When $\sigma _I$ is shorter than the width of pulse, the improvement of spectrum resolution will be reflected in the measurement results, as shown in Fig. 3. However, when $\sigma _I$ is longer than the width of pulse, the improvement of spectrum resolution cannot be reflected in the measurement results. From the theory of Fourier transformation, spectrum and time are the conjugate pair of variables. The details of spectrum are related to the time duration. The longer the time duration is, the more details the spectrum may have. When $\sigma _I$ is longer than the width of pulse, the theoretical spectrum resolution is better than the spectrum details of the pulse. Thus, the measurement results will not change with $\sigma _I$ increasing.
5. Discussion and conclusion
Since the optical loss is greatly reduced, our scheme is possible to be applied into measuring the spectrum of single-photons. Taking the heralded single-photon source in atomic ensembles as an example [27–29], the pulse width of the single-photon is about several nanoseconds. It is possible to modulate the pulse under current technology. The photodetector is replaced by a single-photon detector and counting rate in each phase modulation is recorded. After modulated many times, the spectrum of the single-photon can be reconstructed via the correlation between the counting rates and the pre-computational reference light fields. The spectrum of single-photons can be so hard measured via DFT due to the huge optical loss. Beside, our scheme is possible to be applied in other wavelengths in which the dispersion fibers have high optical loss. Furthermore, the idea of our method can also be applied into other spectrum methods which are based on high dispersion and companied with huge optical loss.
Compared with DFT, we propose a low loss and easy-to-implement spectrum measurement scheme. By random phase modulation and detection in the time domain, the spectrum information is obtained in the near-field. The required length of dispersion fiber, the spectrum resolution, the range of spectrum measurement and the requirement of photodetector bandwidth are analyzed in theory. Our method is demonstrated by simulation, including both discrete spectrum and continuous spectrum. The phase modulation parameters which effect the spectrum resolution are also verified.
Funding
Research Program of National University of Defense Technology (ZK21-11); National Natural Science Foundation of China (62001484, 62105365).
Acknowledgments
The authors thank Rui-Jing He for useful discussions on dispersive Fourier transformation and Jun-Hao Gu for drawing the picture.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available but may be obtained from the corresponding author upon reasonable request.
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