Experimental investigation of chirped amplitude modulation heterodyne ghost imaging

Long Pan; Long Pan; Chenjin Deng; Chenjin Deng; Zunwang Bo; Xin Yuan; Daming Zhu; Wenlin Gong; Wenlin Gong; Shensheng Han; Shensheng Han

doi:10.1364/OE.399006

1. Introduction

Ghost imaging (GI) can nonlocally image an unknown object by measuring the intensity correlation function between the reference light field and reflection signal of the target [1–7]. Because the photons reflected from the target converge on the same single-pixel detector, GI has the advantage of high detection sensitivity and in recent years has attracted increasing research interests especially in the fields of lidar imaging [8,9], remote sensing [10–12] and imaging in scattering media [13,14]. In 2012, Zhao et al. initially proposed a pulsed GI lidar scheme and obtained high-resolution image of the target at a detection range of approximately 1.0 km [8]. After this, a three-dimensional imagery of a scene was also demonstrated when the time-resolved technique was used to measure reflection signals of the target [11]. However, to obtain an image with a good signal-to-noise ratio (SNR), the intensity of the signal light should be considerably stronger than the intensity of the background light for pulsed GI lidar mechanism.

Typically, chirped modulation lidar can obtain the range information of a remote target with a long-pulse width and low peak power laser, although the intensity of the signal is much lower than that of the background noise [15,16]. Combining GI with the technique of chirped modulation, Yang et al. and Deng et al. have proposed heterodyne GI (HGI) and pulse-compression GI via coherent detection, respectively [17,18]. Futhermore, it is theoretically proven and by numerical simulation that GI via chirped modulation has the capability against strong background light environments [19], but so far there is no experimental demonstration to support this theory. Moreover, compared with pulsed GI lidar, except for spatial modulation, the source has temporal modulation for GI via chirped modulation [17,18]. Although some studies have been conducted on the influence of the characteristics of temporal modulation source (such as the modulation depth and the modulation duration) on the property of ranging and velocity measurement in chirped modulation lidar [20,21], there is no known relevant research report on GI via chirped modulation. In addition, because the detection SNR is related to the number of speckles received by the detector [22], HGI will have an obvious advantage in detection SNR and its deveices are cheaper and simpler in comparison with pulse-compression GI, which is more suitable in commercial application. In this paper, we propose an experimental system of chirped amplitude modulation heterodyne GI (CAM-HGI). The effects of background light, modulation depth and modulation duration of the signal source on CAM-HGI are experimentally investigated.

2. Theoretical model

The schematic of CAM-HGI is shown in Fig. 1. A 500 MHz bandwidth chirped waveform with the modulation time 500 $\mu$s is generated using the chirped waveform system and divided into two paths: one is directly connected to a mixer to serve as a local oscillator (LO), the other is linked to an electro-optical modulator (EOM 1) to produce CAM light as signal light. In the LO path, the chirped waveform is attenuated to a proper voltage using a constant attenuator before driving the mixer. In the signal path, a 1550 nm narrow-linewidth continuous fiber laser (CW laser 1, NKT: Koheras Basik) is modulated by the EOM 1 to produce the CAM light. The produced CAM light propagates successively through a delay fiber with 1 km length, a single-frequency fiber amplifier (SFAMP) and illuminates a digital micro-mirror device (DMD, Texas Instruments: DLP 7000) using the collimator 1. The delay fiber is used to generate an appropriate beat frequency between the detection signal and the LO signal, in our table-top experimental system. The DMD modulates the CAM light in spatial domain and produces a series of random patterns. The patterns reflected by DMD are imaged onto the target by the lens $f_1$ with focal length 150 mm. Next, the photons reflected from the target are collected by the lens $f_2$ with focal length 150 mm and injected into a multimode fiber with the core diameter 200 $\mu$m. The output light from the multimode fiber is detected using an 1.2 GHz photodiode (PD, ThorLabs: DET01CFC). The detection signal is amplified by a radio frequency amplifier (AMP 1, MITEQ: AM-3A-000110-1103) and de-chirped with the LO signal in the mixer. Finally, the de-chirped signal passes through a low-pass filter (LPF) and then is sampled by an analogy-to-digital (ADC) card in PC.

Fig. 1. The schematic diagram of CAM-HGI. The experimental system mainly consists of four parts: chirped waveform system, spatial-temporal modulation system of light field, background light simulating system, and post-processing system. EOM: electro-optical modulator, SFAMP: single-frequency fiber amplifier, DMD: digital micromirror device, BSC: beam splitter cube, PD: photodiode, AMP: amplifier, LPF: low-pass filter, WBRG: wideband random generator, EDFA: Erbium-doped optical fiber amplifier.

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In order to investigate the influence of the background light on CAM-HGI, a background light simulation system is constructed. The background light system is similar to the optical structure in the signal path as shown in Fig. 1. Another 1550 nm continuous laser light (EMcore 1782) is modulated by a 2 GHz wideband random waveform to produce the background light. Then the light passes through an Erbium-doped optical fiber amplifier (EDFA), a collimator, a rotating ground glass disk and a beam expander in sequence. The background light directly illuminates the object after the beam splitter (BS). We emphasize that the light is spatially modulated by the rotating ground glass disk to simulate the spatial coherence and the intensity fluctuation of background light. The background light from the rotating ground glass disk is expanded 5 $\times$ by the beam expander, which consists of the lens $f_3$ (with focal length 20 mm) and the lens $f_4$ (with focal length 100 mm) before the target is illuminated.

For the optical system shown in Fig. 1, the light field irradiated onto the target plane can be represented as $E(x,t) = {E_{bg}}(x,t) + {E_s}(x,t)$, where ${E_{bg}}(x,t)$ denotes the background light, ${E_s}(x,t)$ represents signal light. The background light can be expressed as ${E_{bg}}(x,t) = {s_{bg}}\left ( t \right ){E_{bg}}\left ( x \right )$, where ${s_{bg}}(t)$ stands for the temporal fluctuation induced by EOM 2, and ${E_{bg}}(x)$ is spatial modulation introduced by the rotating ground glass disk. The signal light field denotes as ${E_s}(x,t) = {s_s}(t){E_s}(x)$, where ${E_s}(x)$ is the spatial modulation, ${s_s}(t) = \left [ {1 + {s_{chirp}}\left ( t \right )} \right ]{\mathop {\ re}\nolimits } ct\left ( {{t \mathord {\left / {\vphantom {t T}} \right .} T}} \right )$ is the chirped-AM waveform and $T$ is the modulation duration. The chirped signal is ${s_{chirp}}(t) = m\cos \left ( {2\pi {f_0}t + \pi k{t^2}} \right )$, where $m$ is the modulation depth, $f_0$ is the starting frequency, and $k$ is the frequency modulation rate. The signal light and background light are converted into a photocurrrent $i(t)$. After the mixer and LPF, fast Fourier Transform is used to obtain an intensity spectrum which contains the beat frequency. Based on the results described in Ref. [19], the output spectrum $I\left ( f \right )$ can be denoted as

(1)$$\begin{aligned} I(f) & =FFT\{ [{s_{LO}}(t)i(t)] \otimes H(t)\}\\ & =\frac{{{m^2}}}{2}T{\textrm{sinc}} [T(f - {f_z})]\exp [j\phi ]\int {d{x_o}{{\left| {{E_s}({x_o},t)} \right|}^2}O({x_o})}\\ &+ {\textrm{FFT}} \left\{ {{s_{LO}}(t)\left[ {\left( {\int {d{x_o}{{\left| {{E_{bg}}({x_o},t)} \right|}^2}O({x_o})} } \right) \otimes H(t)} \right]} \right\}\\ & \buildrel \Delta \over = {I_s}(f) + {I_{bg}}(f), \end{aligned}$$

where ${s_{LO}}(t)$ is the LO signal, $H(t)$ is the impulse response function for receiving system, ${f_z}$ is the beat frequency, $\textrm{sinc}(x) = \sin (\pi x)/(\pi x)$, $\phi$ denotes constant delay phase, and $\otimes$ denotes convolution. Considering that the background light produces almost no interference with the signal light, the interference between them is neglected. In addition, the electronic noise of the chirped electrical signal is not considered in Eq. (1).

According to the HGI theory [17,19], the image of the object $\left \langle {G\left ( {{x_r}} \right )} \right \rangle$ can be reconstructed by computing the correlation between the intensities ${I_r}\left ( {{x_r}} \right )$ preloaded on DMD and the intensities $I\left ( f \right )$ recorded by the detection system,

(2)$$\left\langle {G\left( {{x_r}} \right)} \right\rangle = \left\langle {\delta I\left( f \right)\delta {I_r}\left( {{x_r}} \right)} \right\rangle,$$

where $< \cdot >$ denotes ensemble average over independent speckle patterns, $\delta I(f) = I(f) - \left \langle {I(f)} \right \rangle$, and $\delta {I_r}({x_r}) = {I_r}({x_r}) - \left \langle {{I_r}({x_r})} \right \rangle$. The presence of background light and detection noise introduces a spurious offset [23], which prevents retrieving the original image in absolute units. The noise associated to the stochastic variable $G\left ( {{x_r}} \right )$ can be expressed as [24,25]

(3)$$\left\langle {\Delta {G^2}({x_r})} \right\rangle = \left\langle {\delta {I^2}\left( f \right)\delta I_r^2\left( {{x_r}} \right)} \right\rangle - {\left\langle {\delta I\left( f \right)\delta {I_r}\left( {{x_r}} \right)} \right\rangle ^2} \approx \left\langle {\delta {I^2}(f)} \right\rangle \left\langle {\delta I_r^2({x_r}} \right\rangle.$$

Following [23,26], the minimum variation of $\Delta \left \langle {G({x_r})} \right \rangle$ is defined as the signal that needs to be detected, thus the imaging SNR of CAM-HGI is

(4)$${{\textrm{SNR}} _{\textrm{CAM - HGI}}} = \frac{{\left[ {\Delta \left\langle G \right\rangle } \right]_{\min }^2}}{{\left\langle {\Delta {G^2}} \right\rangle }}.$$

According to Ref. [19], by substituting Eq. (1) into Eq. (2) and Eq. (3), we can obtain the reconstructed image

(5)$$\left\langle {G({x_r})} \right\rangle = \frac{{{m^2}}}{2}T{A_{coh,s}}\left\langle {{I_r}({x_r})} \right\rangle \left\langle {{I_s}(f)} \right\rangle O({x_r}),$$

and the associated noise

(6)$$\left\langle {\Delta {G^2}} \right\rangle = \frac{{{m^4}}}{4}{T^2}{A_{coh,s}}{A_{beam}}{\left\langle {{I_r}({x_r})} \right\rangle ^2}{\left\langle {{I_s}(f)} \right\rangle ^2}\overline {{O^2}} \left( {1 + \frac{2}{{{m^4}}}\frac{{{\tau _{bg}}}}{T}\frac{{{A_{coh,bg}}}}{{{A_{coh,s}}}}\frac{1}{{{\sigma ^2}}}} \right),$$

where ${A_{coh,s}}$ is the coherent area of signal light, ${A_{beam}}$ denotes the area of signal light, ${\tau _{bg}}$ is the coherent time of background light, and ${A_{coh,bg}}$ is the coherent area of background light. In addition, $\sigma = \frac {{\left \langle {{I_s}(f)} \right \rangle }}{{\left \langle {{I_{bg}}(f)} \right \rangle }}$ denotes the irradiation SNR, which is the signal power to background power ratio on the target plane. By substituting Eq. (5) and Eq. (6) into Eq. (4), and after averaging over $N$ independent measurements, we can obtain the imaging SNR of CAM-HGI

(7)$$\begin{aligned} {{\textrm{SNR}} _{\textrm{CAM - HGI}}} &= \frac{N}{{{N_{sp}}}}\frac{{\Delta O_{\min }^2}}{{\left[ {1 + \frac{2}{{{m^4}}}\frac{{{\tau _{bg}}}}{T}\frac{{{A_{coh,bg}}}}{{{A_{coh,s}}}}\frac{1}{{{\sigma ^2}}}} \right]\overline {{O^2}} }}\\ &\approx \left\{ \begin{array}{ll} \frac{N}{{{N_{sp}}}}\frac{{\Delta O_{\min }^2}}{{\overline {{O^2}} }}\frac{{{m^4}{\sigma ^2}T{A_{coh,s}}}}{{2{\tau _{bg}}{A_{coh,bg}}}}, &\frac{2}{{{m^4}}}\frac{{{\tau _{bg}}}}{T}\frac{{{A_{coh,bg}}}}{{{A_{coh,s}}}}\frac{1}{{{\sigma ^2}}}\,\,\, \gg\,\, 1\\ \frac{N}{{N_{sp}}}\frac{\Delta O_{\min }^2}{{\overline {{O^2}}}}, &\frac{2}{{{m^4}}}\frac{{{\tau_{bg}}}}{T}\frac{{{A_{coh,bg}}}}{{{A_{coh,s}}}}\frac{1}{{{\sigma ^2}}}\,\,\, \ll 1\,\, \end{array} ,\right.\end{aligned}$$

where ${N_{sp}} = {{{A_{beam}}} \mathord {\left / {\vphantom {{{A_{beam}}} {{A_{coh,s}}}}} \right.} {{A_{coh,s}}}}$ is the number of speckles on the target plane. From Eq. (7), we can find that the SNR of CAM-HGI is not only related with the measurement number $N$, the number of speckles ${N_{sp}}$, and the coherent area of signal light ${A_{coh,s}}$, which is the same as the results of GI with only spatial modulation in the noiseless case [23], but also depends on the irradiation SNR $\sigma$, the modulation depth $m$ and the modulation duration $T$. In addition, it is obviously observed that the quality of CAM-HGI increases with $\sigma$, $m$ and $T$. In the following experiments, we will mainly give the later verifications.

In order to quantitatively measure the reconstruction quality of CAM-HGI, the reconstruction fidelity is estimated by calculating the peak signal-to-noise ratio (PSNR)

(8)$${\textrm{PSNR}} = 10{\log _{10}}\left[ {\frac{{{{({2^n} - 1)}^2}}}{{{\textrm{MSE}} }}} \right].$$

Here the bigger the PSNR value is, the better the quality of the recovered image is. For a 0$\sim$255 gray-scale image, $n=8$ and MSE represents the mean square error of the reconstruction ${O_{\textrm{CAM - HGI}}}(x_i)$ with respect to the original object $O(x_i)$,

(9)$${\textrm{MSE}} = \frac{1}{{{N_{\textrm{pix}}}}}{\sum_i {\left[ {{O_{\textrm{CAM - HGI}}}({x_i}) - O({x_i})} \right]} ^2},$$

where ${N_{\textrm{pix}}}$ is the pixel number of the image ${O_{\textrm{CAM - HGI}}}(x_i)$.

3. Experimental results

Here we will experimentally verify the analytical results mentioned in section 2. As shown in Fig. 1, the specific parameters of the experimental schematic are set as follow: the linewidth of fiber CW laser 1 is less than 1 KHz, the starting frequency is ${f_0} = 100$ MHz, and the pulse repetition frequency is set at 1 KHz by controlling the on-off keying of AD9914. The transverse size of the pattern at the DMD plane is set as 41 $\mu m$ $\times$ 41 $\mu m$ and we choose a reflection area with 64 $\times$ 64 pixels (one pixel is equal to the transverse size of the patterns) for projection. The probability distribution of these patterns obeys Bernolli distribution with a $50\%$ duty ratio and the total measurement number is $N = 10000$. The distance between the target and the lens ${f_1}/{f_2}$ is approximately 2 m. The target consists of two letters $GI$, whose width and height are about 9.3 mm and 8.2 mm, respectively.

3.1 Relationship between the irradiation SNR and CAM-HGI

Before verifying the effect of the irradiation SNR on CAM-HGI, we have tested some important parameters based on Eq. (7), such as the coherence areas of both the signal light ${A_{coh,s}}$ and the background light ${A_{coh,bg}}$ at the target plane, and the coherent time of the background light ${\tau _{bg}}$. When the target is replaced with a near infrared charge-coupled device (NIR CCD, Xenics: Bobcat 640) at the target plane, the coherence areas of the signal light and the background light are ${A_{coh,s}} = {(0.48)^2}$ $m{m^2}$ and ${A_{coh,bg}} = {(0.76)^2}$ $m{m^2}$ by measuring the distribution of normalized second-order correlation function [27]. In addition, the coherent time of the signal light ${\tau _c}$ and the background light ${\tau _{bg}}$ are about 1 ms and 1000 ns, respectively.

When the modulation depth $m$ is fixed at 1.0 and the modulation duration is set as $T=500$ $\mu s$ the irradiation SNR $\sigma$ is shifted by changing the background light power and keeping the signal light power at 11.8 mW. Figures 2(a)–(g) illustrate the reconstruction results of CAM-HGI when the irradiation SNR are −36.7 dB, −35.2 dB, −29.7 dB, −26.7 dB, −21.5 dB, −16.7 dB and 0.2 dB, respectively. It is clearly seen that CAM-HGI can obtain an image with high SNR although the irradiation SNR is approximately −30 dB. In addition, the curves of both $\rm {SNR}_{\textrm{CAM - HGI}}$-$\sigma$ and PSNR-$\sigma$ are shown in Fig. 2(h) and Fig. 2(i). From Fig. 2(h), $\rm {SNR}_{\textrm{CAM - HGI}}$ is proportional to ${\sigma ^2}$ when the irradiation SNR is very low ( $10\lg (\sigma ) < - 20$ dB, which is corresponding to the case of $\frac {2}{{{m^4}}}\frac {{{\tau _{bg}}}}{T}\frac {{{A_{coh,bg}}}}{{{A_{coh,s}}}}\frac {1}{{{\sigma ^2}}}\,\,\, \gg \,\,1$ described by Eq. (7)). Both the imaging results of CAM-HGI and PSNR-$\sigma$ are consistent with the curve of $\rm {SNR}_{\textrm{CAM - HGI}}$-$\sigma$.

Fig. 2. Experimental results of the influence of irradiation SNR on CAM-HGI. (a) -(g) are the reconstructed images in different irradiation SNR. (a) $10\lg (\sigma ) = - 36.7$ dB; (b) $10\lg (\sigma ) = - 35.2$ dB; (c) $10\lg (\sigma ) = - 29.7$ dB; (d) $10\lg (\sigma ) = - 26.7$ dB; (e) $10\lg (\sigma ) = - 21.5$ dB; (f) $10\lg (\sigma ) = - 16.7$ dB; (g) $10\lg (\sigma ) = 0.2$ dB; (h) the curve of $\rm {SNR}_{\textrm{CAM - HGI}}$-$\sigma$; (i) the curve of PSNR versus $\sigma$.

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3.2 Influence of the modulation depth on CAM-HGI

In this case, the modulation duration is $T=500$ $\mu s$, the signal light power is kept at 11.8 mW and the irradiation SNR is set as −26.7 dB. When the amplitude of the chirped waveform of EOM 1 is controlled, Figs. 3(a)–(g) give the imaging results in the case of the modulation depth $m$ = 0.2, 0.25, 0.3, 0.4, 0.5, 0.7 and 1.0, respectively. Similar to Fig. 2, the curves of $\rm {SNR}_{\textrm{CAM - HGI}}$-$m$ and PSNR-$m$ are shown in Fig. 3(h) and Fig. 3(i). It is clearly seen that when the modulation depth $m$ is lower than 0.5, the quality of CAM-HGI degrades significantly, which is also coincident with the curves of $\rm {SNR}_{\textrm{CAM - HGI}}$-$m$ and PSNR-$m$. In addition, we emphasize that the relationship of $\rm {SNR}_{\textrm{CAM - HGI}}$-$m$ for the analytical result described by Eq. (7) is satisfied only when the detection SNR is sufficient high [28]. However, because the detection SNR in the experiments is lower than 10 dB, the curve of $\rm {SNR}_{\textrm{CAM - HGI}}$-$m$ is deviated in comparison with the theoretical result in Eq. (7).

Fig. 3. Experimental results of the effect of modulation depth $m$ on CAM-HGI. (a)-(g) are the imaging results when $m$ = 0.2, 0.25, 0.3, 0.4, 0.5, 0.7 and 1, respectively; (h) the curve of $\rm {SNR}_{\textrm{CAM - HGI}}$-$m$; (i) the curve of PSNR-$m$.

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3.3 Impact of the modulation duration on CAM-HGI

Finally, the experimental demonstration of the dependence of the modulation duration $T$ on CAM-HGI is shown in Fig. 4. Here, the signal light power is kept at 11.8 mW, the irradiation SNR is −26.7 dB, and $m$ is fixed at 1.0. When the modulation duration $T$ is successively selected as 50 $\mu s$, 100 $\mu s$, 166.7 $\mu s$, 250 $\mu s$ and 500 $\mu s$, the corresponding CAM-HGI reconstruction results and the curves of both $\rm {SNR}_{\textrm{CAM - HGI}}$-$T$ and PSNR-$T$ are shown in Figs. 4(a)–(j). It is observed that the quality of CAM-HGI increases with modulation duration $T$ but is not proportional to $T$, which is caused by a low detection SNR.

Fig. 4. Experimental results of the dependence of the modulation duration $T$ on CAM-HGI. (a)-(e) are the reconstruction results when $T$ = 50 $\mu s$, 100 $\mu s$, 166.7 $\mu s$, 250 $\mu s$ and 500 $\mu s$, respectively. (f) the curve of $\rm {SNR}_{\textrm{CAM - HGI}}$-$T$. (g) the curve of PSNR-$T$.

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4. Discussion

Generally, CAM-HGI has both the advantages over narrow pulsed GI and pulse-compression GI via coherent detection. Similar to narrow pulsed GI, both the detection SNR and the imaging range of CAM-HGI can be improved as the optical aperture of receiving system is increased [29]. However, when the transmitting energy is very large especially in remote sensing, the modulation speed of light field for narrow pulsed GI is usually very low because of the restriction of the repetition frequency of the laser, which will lead to a low imaging frame frequency and does not satisfy the requirement for imaging moving target [30,31]. Different from narrow pulsed GI, high transmitting energy can be achieved by a long duration accumulation and high modulation speed of light field can be also guaranteed for CAM-HGI [32–34]. Similar to pulse-compression GI via coherent detection, the enhancement of detection SNR of CAM-HGI is realized by the interference between the signal light and the LO light and photons accumulation [18]. Futhermore, because the LO light does not correlate with the background light, high detection SNR can be held although the intensity of background light is largely stronger than that of the signal light [19]. However, for pulse-compression GI via coherent detection, in order to keep the coherence of detection, the optical aperture of receiving system is usually very small and the detector can only statistically receive one speckle, which means that the detection SNR is very low [22]. Although the quality of pulse-compression GI via coherent detection can be enhanced by multi-input detection structure [18], the detection system is of high cost and complicated. Different from pulse-compression GI via coherent detection which can be realized by amplitude or phase modulation, the technique of HGI can only exploit the type of amplitude modulation and the detection process of CAM-HGI is incoherent so that the detection SNR can be increased with the optical aperture of receiving system. In addition, the detection system is simple and only one single-pixel detector is used, thus a low-cost and miniaturized device can be developed by the technique of CAM-HGI, which has a great potential application in lidar imaging, autonomous vehicle and imaging through scattering media.

5. Conclusion

In conclusion, we have experimentally investigated the influence of the background light, the modulation depth and modulation duration of the signal light on CAM-HGI. The experimental results demonstrate that the imaging quality of CAM-HGI increases with the modulation depth and modulation duration of the signal light. In addition, we also show that CAM-HGI has a capability against strong background light. This technique of CAM-HGI will be useful to laser imaging against environmental disturbance, autonomous vehicle and imaging through scattering media.

Funding

Youth Innovation Promotion Association of the Chinese Academy of Sciences (2013162-2017); Defense Industrial Technology Development Program of China (D040301).

Disclosures

The authors declare no conflicts of interest.

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Experimental investigation of chirped amplitude modulation heterodyne ghost imaging

Abstract

1. Introduction

2. Theoretical model

3. Experimental results

3.1 Relationship between the irradiation SNR and CAM-HGI

3.2 Influence of the modulation depth on CAM-HGI

3.3 Impact of the modulation duration on CAM-HGI

4. Discussion

5. Conclusion

Funding

Disclosures

References

Cited By

Figures (4)

Equations (9)

Optics Express