Abstract
Ghost imaging, also called correlation imaging, offers a novel imaging method, being able to image without any spatially resolved detector or even with only a single-pixel detector. A typical correlation imaging system consists of two arms of light. One arm, the reference arm, is equipped with a spatially resolved detector (such as a CCD camera) to record the intensity distribution of the light field. While the other arm, the object arm, is equipped with a bucket detector, which collects the transmitted or reflected light from the object. The light beams in two arms are usually created by separating the beam from the light source into two, such that the intensity distributions on the two arms are strongly correlated. After a variety of speckle patterns are sequentially illuminated onto the object, an image of the object can be obtained by calculating the correlation function (over all those frames of patterns) between the signals from the CCD and the bucket detector[
Figure 1.Schematic diagram of a typical setup for computational ghost imaging with an SLM.
In most occasions, the series of speckle fields are regulated by an SLM loading different random phases
From Eq. (
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On the other hand, because of the random fluctuations, many frames of different patterns are required to retrieve the image. Typically, the more complex the object is, the more varieties of speckle patterns are required. Therefore the shortest time to obtain a ghost image will be limited by the refreshing rate of the speckle patterns. Or, increasing the refreshing rate of the speckle patterns will be an effective way to improve the speed of correlated imaging. A high imaging speed can be vital if the object to be imaged is in motion. In addition, considering practical utility, environment factors such as temperature, humidity, atmospheric turbulence, and atmospheric scattering will affect the quality of ghost images, which are usually changing over time. A possible way to reduce such influence is to accomplish the imaging process as soon as possible, which also requires a high refreshing rate of the speckle fields. Given that speckle fields can be calculated by light fields from the light source, it is important to explore better and faster schemes for the regulation of light sources.
Considering computational ghost imaging with an SLM, the refreshing rate of the speckle patterns is determined by the refreshing rate of the random phase mask loaded on the SLM, which is typically refreshed frame by frame. That means only after all the pixels of the SLM are refreshed, the next phase mask can be loaded. This turns out to be the main limitation of the refreshing rate of speckle patterns. However, the response speed of each pixel on the SLM could be much faster if we control those pixels independently. At the same time, when controlling the phase of every pixel, it is reasonable that the response of the pixel will be more stable if the input is a smooth signal, such as sinusoidal signal, rather than a random modulation signal. What is more, a sinusoidal modulation signal can be much easier and more precisely generated compared to random modulation signals.
Here, we propose to control every pixel of the SLM with a sinusoidal signal independently to increase the refreshing rate of the speckle patterns. The sinusoidal signals for different pixels are sets of different frequencies so that the phase differences between the pixels will vary with time, therefore also the generated speckle patterns at the object plane. Without a loss of generality, the frequencies of all the sinusoidal signals are considered to be integers, for convenience. Additionally, it is noticed that the speckle fields will show temporal periodicity due to the periodicity of the sinusoidal signals. The frequency of the periodic speckle field is the largest common divisor of those of the sinusoidal signals. Meanwhile, the entire imaging process should be finished within one period. Otherwise, speckle fields will be repeated, and the corresponding results of the bucket detector will also be repeated, thus giving no more information on the object. So, frequencies should be set to make the fluctuation of the speckle patterns as random as possible, or to make the period of the speckle fields long enough such that there are enough different speckle patterns within one period. To achieve this, the frequency of all those different sinusoidal signals are set to be coprime. In this case, the temporal period of the speckle fields will be 1 s.
Since the phase loaded on each pixel of the SLM is varying with time, a pulsed laser will be better than a continuous wave in practice to achieve high-quality images. By setting the pulse duration short enough, the phase loaded on each pixel can be taken as constant during the pulse. At the same time, the time interval between two pulses should be long enough such that the modulated phases for these two pulses are not similar to each other. Under this situation, one pulse provides one frame of speckle pattern, thus the refreshing rate of the speckle fields is identical to the repetitive frequency of the laser pulse. Therefore, the refreshing rate of the speckle patterns is no longer directly limited by the refreshing rate of the SLM.
Specifically, the sinusoidal signal loaded on each pixel of the SLM can be written as
Figure 2.Histogram to show the phase distribution. For the left one, the amplitude of the sinusoidal function is
Then the light field of the
As an example, the effective size of the SLM is set to be
Figure 3.Ghost imaging results of a binary object, with the phase modulation loaded on the SLM set to be (a) coprime-frequencied sinusoidal, (b) random modulation signal, and (c) sinusoidal signals of the non-coprime different frequencies.
To estimate the performance of our method, we also measure the image quality using the mean square error (MSE) and investigate its behavior with an increasing number of frames. By changing the repetitive rate
A more detailed comparison between our method and random modulation is shown in Fig.
Figure 4.Comparison between coprime-frequencied modulation and random modulation by the MSE varying with the number of frames, which does not show much difference.
At the same time, the performance of our method is related to the combination of those sinusoidal frequencies and the sampling rate. We tried four different combinations with the modulation frequencies chosen to be 100 prime numbers in four different intervals, namely,
Figure 5.Image quality measured with the MSE varying with the number of frames, with the modulation frequencies chosen to be 100 prime numbers within the interval of (a) [103, 701], (b) [2, 701], (c) [4003, 5003], and (d) [2, 5003].
However, it can still be more than ten times higher than the highest modulation frequency of pixels on the SLM. This is concluded by observing the beginning spot of the flat part of the MSE curve. For these two cases of low frequency, the number at the inflection point is very close to 10000, while the inflection point does not show up until 50000 frames for high modulation frequency (up to 5 kHz) cases. At the same time, increasing the separation between neighboring modulation frequencies can also help to enhance the quality of the image. From Fig.
In practice, our proposal can be demonstrated with current techniques. The only thing that might be challenging is to control every pixel of the SLM individually, which contains no difficulty in principle. The upper bound of the effective refreshing rate of the speckle patterns is tightly related to the highest frequency that can be loaded on each pixel of the SLM. In addition, the signal for every pixel is a fixed sinusoidal signal, thus the controlling part can be simplified accordingly.
In conclusion, we present and discuss a coprime-frequencied sinusoidal modulation method to increase the refreshing rate of speckle patterns, thus to increase the speed of ghost imaging based on an SLM. Every pixel of an SLM is proposed to be modulated individually with sinusoidal signals instead of random ones. The imaging quality of our method can be very close to that of random modulation. While the refreshing rate of the speckle patterns can be more than ten times higher than the highest frequency that can be responded to by each pixel of the SLM. The parameters of each modulation signal are also discussed to enhance the imaging quality.
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