Sub-Rayleigh-diffraction imaging via modulating classical light

Erfeng Zhang; Huizu Lin; Weitao Liu; Quan Li; Pingxing Chen

doi:10.1364/OE.23.033506

1. Introduction

In an imaging system, the spatial resolution is restricted by the Rayleigh diffraction limit [1]. Improving the spatial resolution of the imaging system is an interesting and important research topic in both classical and quantum optics. Nowadays, there have been many techniques to achieve sub-Rayleigh-diffraction imaging, such as confocal scanning microscopy [2], stimulated emission depletion microscopy [3], optical centroid measurement procedures [4–6], speckle illumination combing with correlation measurements [7–12], and so on. Meanwhile, quantum effects have also been applied to realize sub-Rayleigh-diffraction imaging [13–18]. Recently, Giovannetti et al. demonstrated theoretically that N-fold coherent and incoherent imaging schemes can be realized with N-photon strategies, which can exceed the Rayleigh diffraction limit [15]. Then, the experiments to realize the strategy with incoherent mixtures of coherent states were achieved by combining point-by-point illumination with N-photon detection [16, 17]. Later, two-photon sub-Rayleigh-diffraction quantum imaging schemes were realized experimentally using two-photon entangled sources [18]. However, in these experiments, either the scanning over the object with a point source is needed which is not simple to implement [16, 17], or the entangled source is employed which is difficult to generate, and also very weak nowadays [18].

In this paper, we show that the resolution enhancement beyond the Rayleigh diffraction bound can be achieved using classical light sources together with auto-correlation measurements, which rely on the idea of the N-photon strategies [15]. In these schemes, the classical light sources are obtained by modulating the amplitude distribution and wavefront of a laser beam randomly in time which are simple to implement. It is found that the peculiarities of the superposition of two-photon Fock states and the incoherent mixture of two-photon Fock states can be realized with the second-order auto-correlation measurements respectively, then the sub-Rayleigh-diffraction images can be obtained. Furthermore, these types of the light sources are easy to generate, and can be also very intense, which can bring the N-photon strategies to applications extensively in practice.

2. System setup and analytical results

The schematic of the sub-Rayleigh-diffraction imaging schemes based on a traditional imaging system is shown in Fig. 1. The light from the light source illuminates an object, then travels into an optical imaging system. In the optical imaging system, a thin lens with radius R and focal length f is placed at a distance D_o from the object, at a distance D_i from the image plane, and the distances D_o, D_i and f satisfy the Gaussian imaging equation: 1/D_o +1/D_i = 1/f. The image of the object can be obtained by disposing the intensities measured by the CCD camera at the image plane.

Fig. 1 The schematic for sub-Rayleigh imaging via modulating classical light.

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We start our analysis from a review of conventional imaging schemes. In conventional imaging schemes, the object is illuminated by a spatially coherent or incoherent source, and the field of the optical light at the image plane can be obtained [15]

E ({\vec{x}}_{i}) = \int d {\vec{x}}_{o} E ({\vec{x}}_{o}) T ({\vec{x}}_{o}) h ({\vec{x}}_{i}, {\vec{x}}_{o}),

where

E ({\vec{x}}_{o})

denotes the field at the object plane,

T ({\vec{x}}_{o})

is the object’s aperture function, and

h ({\vec{x}}_{i}, {\vec{x}}_{o})

is the point-spread function of the imaging system [19,20]:

h ({\vec{x}}_{i}, {\vec{x}}_{o}) \propto e^{i θ} somb (\frac{2 π R | {\vec{x}}_{o} + {\vec{x}}_{i} / m |}{λ D_{o}}),

in which

somb (\vec{x}) \equiv 2 J_{1} (\vec{x}) / \vec{x}

is the Airy function, λ is the wavelength of the light source, and m = D_i/D_o is the image magnification factor. In addition, the phase θ in Eq. (2) plays few role in the imaging process which can be ignored or compensated [19]. Then, the images can be obtained by measuring the intensities in the coherent and incoherent imaging schemes

I ({\vec{x}}_{i}) = E^{*} ({\vec{x}}_{i}) E ({\vec{x}}_{i})

, respectively:

\begin{array}{l} 〈 I_{c o h} ({\vec{x}}_{i}) 〉 \propto {| \int d {\vec{x}}_{o} T ({\vec{x}}_{o}) h ({\vec{x}}_{i}, {\vec{x}}_{o}) |}^{2}, \\ 〈 I_{i n c o h} ({\vec{x}}_{i}) 〉 \propto {\int d {\vec{x}}_{o} {| T ({\vec{x}}_{o}) |}^{2} | h ({\vec{x}}_{i}, {\vec{x}}_{o}) |}^{2}, \end{array}

where 〈·〉 denotes the time average. Here the image in the coherent imaging scheme is obtained by convolving the object’s aperture function

T ({\vec{x}}_{o})

with the point-spread function

h ({\vec{x}}_{i}, {\vec{x}}_{o})

. Meanwhile, the image in the incoherent imaging scheme is obtained by the convolution between the norm of the object’s aperture function

{| T ({\vec{x}}_{o}) |}^{2}

and the norm of the point-spread function

{| h ({\vec{x}}_{i}, {\vec{x}}_{o}) |}^{2}

.

In the following we will detail the formulation of the sub-Rayleigh-diffraction imaging schemes with classical field. In these schemes, the classical light sources are obtained by modulating a laser beam as shown in Fig. 1. In our analysis, each pixel in the modulator is temporally programmable both in amplitude and phase. Here the amplitudes $a ({\vec{x}}_{o})$ and the phases $φ ({\vec{x}}_{o})$ loaded on the pixels in the modulator are modulated randomly in time, and they satisfy the condition:

\begin{array}{l} a ({\vec{x}}_{o}) = a_{1} ({\vec{x}}_{o}) | \cos [φ_{1} ({\vec{x}}_{o})] |, \\ φ ({\vec{x}}_{o}) = π Π [φ_{1} ({\vec{x}}_{o})], \end{array}

where the random parameters

a_{1} ({\vec{x}}_{o})

are statistically independent and follow Rayleigh distribution, the random parameters

φ_{1} ({\vec{x}}_{o})

are statistically independent and uniformly distributed in the range [0,2π), and the function Π(x) is

Π (x) = {\begin{array}{l} 0, x \in [0, π / 2) or [3 π / 2, 2 π), \\ 1, x \in [π / 2, 3 π / 2) . \end{array}

In this case, the second-order correlation function of the complex-amplitudes in the source $E_{o} ({\vec{x}}_{o}) = a ({\vec{x}}_{o}) e^{i φ ({\vec{x}}_{o})}$ in the time average can be obtained [21–23]:

\begin{array}{l} 〈 E_{o}^{*} ({\vec{x}}_{o 1}) E_{o}^{*} ({\vec{x}}_{o 2}) E_{o} ({\vec{x}}_{o 3}) E_{o} ({\vec{x}}_{o 4}) 〉 \\ = 〈 a ({\vec{x}}_{o 1}) a ({\vec{x}}_{o 2}) a ({\vec{x}}_{o 3}) a ({\vec{x}}_{o 4}) e^{- i [φ ({\vec{x}}_{o 1}) + φ ({\vec{x}}_{o 2}) - φ ({\vec{x}}_{o 3}) - φ ({\vec{x}}_{o 4})]} 〉 \\ = \frac{{〈 a_{1}^{2} 〉}^{2}}{4} [δ ({\vec{x}}_{o 1} - {\vec{x}}_{o 2}) δ ({\vec{x}}_{o 3} - {\vec{x}}_{o 4}) + δ ({\vec{x}}_{o 1} - {\vec{x}}_{o 3}) δ ({\vec{x}}_{o 2} - {\vec{x}}_{o 4}) + δ ({\vec{x}}_{o 1} - {\vec{x}}_{o 4}) δ ({\vec{x}}_{o 2} - {\vec{x}}_{o 3})] . \end{array}

Meanwhile, the first-order correlation function can be easily obtained:

〈 E_{o}^{*} ({\vec{x}}_{o 1}) E_{o} ({\vec{x}}_{o 2}) 〉 = \frac{〈 a_{1}^{2} 〉}{2} δ ({\vec{x}}_{o 1} - {\vec{x}}_{o 2}) .

Then, at the image plane [21–23]:

\begin{array}{l} 〈 I^{2} ({\vec{x}}_{i}) 〉 = 〈 {[E_{i}^{*} ({\vec{x}}_{i}) E_{i} ({\vec{x}}_{i})]}^{2} 〉 \\ = \frac{{〈 a_{1}^{2} 〉}^{2}}{4} [{| \int d {\vec{x}}_{o} T^{2} ({\vec{x}}_{o}) h^{2} ({\vec{x}}_{i}, {\vec{x}}_{o}) |}^{2} + 2 {(\int d {\vec{x}}_{o} {| T ({\vec{x}}_{o}) |}^{2} {| h ({\vec{x}}_{i}, {\vec{x}}_{o}) |}^{2})}^{2}], \\ 〈 I ({\vec{x}}_{i}) 〉 = 〈 E_{i}^{*} ({\vec{x}}_{i}) E_{i} ({\vec{x}}_{i}) 〉 \\ = \frac{〈 a_{1}^{2} 〉}{2} \int d {\vec{x}}_{o} {| T ({\vec{x}}_{o}) |}^{2} {| h ({\vec{x}}_{i}, {\vec{x}}_{o}) |}^{2} . \end{array}

Using Eq. (8) we find that

Δ G_{'}^{1} ({\vec{x}}_{i}) \equiv 〈 I^{2} ({\vec{x}}_{i}) 〉 - 2 {〈 I ({\vec{x}}_{i}) 〉}^{2} \propto {| \int d {\vec{x}}_{o} T^{2} ({\vec{x}}_{o}) h^{2} ({\vec{x}}_{1}, {\vec{x}}_{o}) |}^{2},

which generalizes the conventional coherent imaging scheme in Eq. (3) to second-order intensity detection. The function that governs the spatial resolution in the imaging system is now

h^{2} ({\vec{x}}_{i}, {\vec{x}}_{o})

.

Meanwhile, another modulating mode is considered: the time-variable random amplitudes $a ({\vec{x}}_{o})$ and the time-variable random phases $φ ({\vec{x}}_{o})$ modulated by the modulator satisfy the condition:

\begin{array}{l} a ({\vec{x}}_{o}) = a_{1} ({\vec{x}}_{o}) a_{2} ({\vec{x}}_{o}) | \cos [φ_{1} ({\vec{x}}_{o})] |, \\ φ ({\vec{x}}_{o}) = \mod {π Π [φ_{1} ({\vec{x}}_{o})] + φ_{2} ({\vec{x}}_{o}), 2 π}, \end{array}

where the random parameters

a_{1} ({\vec{x}}_{o})

,

a_{2} ({\vec{x}}_{o})

are statistically independent and follow Rayleigh distribution, and the random parameters

φ_{1} ({\vec{x}}_{o})

,

φ_{2} ({\vec{x}}_{o})

are statistically independent and uniformly distributed in the range [0,2π). In this situation, the second-order and first-order correlation functions of the source can be expressed as [21–23]:

\begin{array}{l} 〈 E_{o}^{*} ({\vec{x}}_{o 1}) E_{o}^{*} ({\vec{x}}_{o 2}) E_{o} ({\vec{x}}_{o 3}) E_{o} ({\vec{x}}_{o 4}) 〉 \\ = \frac{{〈 a_{1}^{2} 〉}^{2} {〈 a_{2}^{2} 〉}^{2}}{4} [4 δ ({\vec{x}}_{o 1} - {\vec{x}}_{o 2}) δ ({\vec{x}}_{o 1} - {\vec{x}}_{o 3}) δ ({\vec{x}}_{o 1} - {\vec{x}}_{o 4}) + δ ({\vec{x}}_{o 1} - {\vec{x}}_{o 3}) δ ({\vec{x}}_{o 2} - {\vec{x}}_{o 4}) + δ ({\vec{x}}_{o 1} - {\vec{x}}_{o 4}) δ ({\vec{x}}_{o 2} - {\vec{x}}_{o 3})], \\ 〈 E_{o}^{*} ({\vec{x}}_{o 1}) E_{o} ({\vec{x}}_{o 2}) 〉 = \frac{〈 a_{1}^{2} 〉 〈 a_{2}^{2} 〉}{2} δ ({\vec{x}}_{o 1} - {\vec{x}}_{o 2}) . \end{array}

Then, an generalization for incoherent imaging is obtained:

Δ G_{2}^{'} ({\vec{x}}_{i}) \equiv 〈 I^{2} ({\vec{x}}_{i}) 〉 - 2 {〈 I ({\vec{x}}_{i}) 〉}^{2} \propto {\int d {\vec{x}}_{o} {| T^{2} ({\vec{x}}_{o}) |}^{2} | h^{2} ({\vec{x}}_{i}, {\vec{x}}_{o}) |}^{2} .

It generalizes incoherent imaging to two-fold incoherent imaging. The attainable resolution is determined by ${| h^{2} ({\vec{x}}_{i}, {\vec{x}}_{o}) |}^{2}$ .

In our schemes, we employ the classical-states, thus these techniques in our schemes are immune to loss and noise, making them suitable for applications in practice. Furthermore, the second-order spatial auto-correlations $〈 I^{2} ({\vec{x}}_{i}) 〉$ contain the information of the sub-Rayleigh images and the conventional images. Then, by subtracting twice as much as the square of the first-order spatial correlations $2 {〈 I^{2} ({\vec{x}}_{i}) 〉}^{2}$ , the sub-Rayleigh images of the object can be obtained. Moreover, we generalize the conventional coherent and incoherent imaging to two-fold coherent and incoherent imaging, and the functions that determine the resolution in two-fold coherent and incoherent imaging, $h^{2} ({\vec{x}}_{i}, {\vec{x}}_{o})$ and ${| h^{2} ({\vec{x}}_{i}, {\vec{x}}_{o}) |}^{2}$ , have smaller spatial extent than that in conventional coherent and incoherent imaging, $h ({\vec{x}}_{i}, {\vec{x}}_{o})$ and ${| h ({\vec{x}}_{i}, {\vec{x}}_{o}) |}^{2}$ , respectively. Thus, the resolutions of two-fold coherent and incoherent imaging are better than that of conventional coherent and incoherent imaging respectively, which exceed the Rayleigh-diffraction limit. Meanwhile, it should be mentioned that there have been some similar schemes as the schemes we proposed, which employ the technology of speckle illumination combing with correlation measurements [8–10]. However, in the scheme employing pesudothermal light illumination [8], the image is obtained by auto-correlation measurement with background substraction

Δ G ({\vec{x}}_{i}) = 〈 {(I_{i} ({\vec{x}}_{i}) - 〈 I_{i} ({\vec{x}}_{i}) 〉)}^{2} 〉 \propto {(\int d {\vec{x}}_{o} {| T ({\vec{x}}_{o}) |}^{2} {| h ({\vec{x}}_{i}, {\vec{x}}_{o}) |}^{2})}^{2},

which is second-order power of the conventional incoherent intensity image, and the resolution is lower than that of the two-fold incoherent imaging system in our scheme. In the scheme employing random structured illumination [9, 10], there are two parts in the optical setup: the illumination imaging system and the collection imaging system. The resolution of the image obtained with the correlation measurements is limited by the illumination imaging system, and is unrelated to the collection imaging system. Meanwhile, in our scheme there is only one imaging scheme, and with the proposal in our scheme we can obtain higher resolution image than that with the conventional imaging scheme.

Then, we will discuss the experimental feasibility of these imaging schemes. In these schemes, the modulator which can perform amplitude and phase modulations is needed. This can be achieved by employing two spatial light modulators (SLMs), one for amplitude and another for phase modulation. This method has been used [24], and the proof-of-principle experimental setup of the modulator is shown in Fig. 2. The first SLM (ISLM) with two crossed polarizers P₁ and P₂ modulates the intensity of the beam and the second SLM (PSLM) performs phase modulation. The polarizer P₁ ensures that the polarization is appropriate for the intensity modulation of the ISLM. Then, the ISLM turns the polarization of the laser beam, leading to that the intensity can be spatially modulated with the polarizer P₂. With a 1:1 telescope consisting of lenses L₁ and L₂, the ISLM plane is imaged onto the PSLM plane. Moreover, the λ/2 wave plate turns the polarization to meet the requirement for phase modulation by the PSLM. The light reflected from the PSLM travels through the object, then travels into the imaging system, and is measured by the CCD camera. Moreover, there is perfect synchronization among the operations of the ISLM modulation, the PSLM modulation and the CCD camera measurement. Finally, the image of the object is reconstructed with the superposition of several frames of the measurement results by the CCD camera corresponding to different amplitude and phase modulations.

Fig. 2 The proof-of-principle experimental setup of the modulator.

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3. Numerical simulated results

Now, let us consider the setup of sub-Rayleigh-diffraction imaging schemes displayed in Fig. 1, and the numerical simulated results are shown in Fig. 3. In the numerical simulation, a laser beam with wavelength λ = 532 nm illuminates a modulator which can modulate the amplitude distribution and wavefront of the light field. Right behind the modulator, there is a three-point object with the distances d₁ = 33.8 µm and d₂ = 41.6 µm. Then the light travels into an optical imaging system. The parameters of the imaging system are R = 4 mm and D_o = D_i = 2 f = 50 cm. Finally, the intensities of the light at the image plane are measured by a CCD camera. The images of the two-fold coherent and incoherent imaging are reconstructed with Eq. (9) and Eq. (12), respectively. For comparison, the images of the conventional coherent and incoherent imaging are obtained with Eq. (3). The simulated results reconstructed by the four different imaging schemes with 4000 realizations of amplitude and phase modulations are shown in Fig. 3. As shown in Fig. 3, the three-point can be resolved using the scheme of two-fold incoherent imaging, however, in the conventional incoherent imaging scheme only the farther two-point can be resolved, and the nearer two-point can not be resolved. The image of two-fold coherent imaging is almost the same as that of incoherent imaging, and its resolution is better than that of the image obtained by the conventional coherent imaging, in which the three-point cannot be resolved. These simulated results validate the analytical results that an enhancement of the spatial resolution in the two-fold coherent and incoherent imaging is obtained compared with the conventional coherent and incoherent imaging, respectively.

Fig. 3 The simulated results reconstructed by the four different imaging schemes.

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4. The physical picture

From the analytical results, it can be found that the imaging formulae in Eqs. (9) and (12) obtained by our schemes are the same as that in the schemes proposed by Giovannetti et al. [15]. Then, an interesting question arises: what causes the sub-Rayleigh imaging effect in our schemes? In the following, we will consider the physical picture of these imaging schemes in the photon-counting regime. When a coherent state: $| ψ 〉 = \int d {\vec{x}}_{o} | 1_{{\vec{x}}_{o}} 〉$ is modulated in both amplitudes and phases which satisfy the condition expressed in Eq. (4), the single-photon and two-photon states in the time average can be expressed as:

\begin{array}{l} ρ^{(1)} = \frac{〈 a_{1}^{2} 〉}{2} \int d {\vec{x}}_{o} | 1_{{\vec{x}}_{o}} 〉 〈 1_{{\vec{x}}_{o}} |, \\ ρ^{(2)} = \frac{〈 a_{1}^{2} 〉}{4} (2 \int d {\vec{x}}_{o 1} | 1_{{\vec{x}}_{o 1}} 〉 〈 1_{{\vec{x}}_{o 1}} | \cdot \int d {\vec{x}}_{o 2} | 1_{{\vec{x}}_{o 2}} 〉 〈 1_{{\vec{x}}_{o 2}} | + \int d {\vec{x}}_{o 1} | 2_{{\vec{x}}_{o 1}} 〉 \cdot \int d {\vec{x}}_{o 2} 〈 2_{{\vec{x}}_{o 2}} |) . \end{array}

Note that the quantity $\int d {\vec{x}}_{o 1} | 2_{{\vec{x}}_{o 1}} 〉 \cdot \int d {\vec{x}}_{o 1} 〈 2_{{\vec{x}}_{o 2}} |$ inside the round brackets on the right hand of ρ⁽²⁾ in Eq. (14) is a superposition of two-photon Fock states. By use of Eq. (9), we can obtain the quantum correlation function [25]

Δ G_{1}^{'} ({\vec{x}}_{i}) \equiv Tr {ρ^{(2)} {[E^{(-)} ({\vec{x}}_{i})]}^{2} {[E^{(+)} ({\vec{x}}_{i})]}^{2}} - 2 {Tr [ρ^{(1)} E^{(-)} ({\vec{x}}_{i}) E^{(+)} ({\vec{x}}_{i})]}^{2} \propto Tr {\int d {\vec{x}}_{o 1} | 2_{{\vec{x}}_{o 1}} 〉 \cdot \int d {\vec{x}}_{o 2} 〈 2_{{\vec{x}}_{o 2}} | {[E^{(-)} ({\vec{x}}_{i})]}^{2} {[E^{(+)} ({\vec{x}}_{i})]}^{2}},

where

E^{(-)} ({\vec{x}}_{i})

and

E^{(+)} ({\vec{x}}_{i})

denote the negative-frequency and positive-frequency field operators, respectively. Moreover, the positive-frequency field operator

E^{(+)} ({\vec{x}}_{i})

can be expressed as [25]

E^{(+)} ({\vec{x}}_{i}) = \int d {\vec{x}}_{o} a ({\vec{x}}_{o}) T ({\vec{x}}_{o}) h ({\vec{x}}_{i}, {\vec{x}}_{o}) .

It has been proved that the two-fold coherent imaging scheme can be realized with the superposition of two-photon Fock states $\int d {\vec{x}}_{o 1} | 2_{{\vec{x}}_{o 1}} 〉 \cdot \int d {\vec{x}}_{o 2} 〈 2_{{\vec{x}}_{o 2}} |$ [15]. From Eq. (15), it can be seen that the two-fold coherent imaging scheme can be achieved by modulating the amplitudes and phases of the laser beam. Similarly, when the amplitudes and phases which are modulated by the modulator satisfy the condition in Eq. (10), it can be easily obtained that there are the incoherent mixture of two-photon Fock states $\int d {\vec{x}}_{o 1} | 2_{{\vec{x}}_{o 1}} 〉 \cdot \int d {\vec{x}}_{o 2} 〈 2_{{\vec{x}}_{o 2}} | δ ({\vec{x}}_{o 1} - {\vec{x}}_{o 2})$ in the two-photon density matrix ρ⁽²⁾. Furthermore, with the help of Eq. (12), the two-fold incoherent imaging is obtained with the incoherent mixture of two-photon Fock states $\int d {\vec{x}}_{o 1} | 2_{{\vec{x}}_{o 1}} 〉 \cdot \int d {\vec{x}}_{o 2} 〈 2_{{\vec{x}}_{o 2}} | δ ({\vec{x}}_{o 1} - {\vec{x}}_{o 2})$ [15]. These analytical results show that by modulating the amplitude distribution and wavefront of the coherent light beam, the classical light sources exhibiting the features of the superposition of two-photon Fock states and the incoherent mixture of two-photon Fock states can be generated, leading to two-fold coherent and incoherent imaging schemes, respectively.

5. Conclusion

In summary, we propose that sub-Rayleigh-diffraction images can be obtained with a specially designed classical light. By modulating the amplitude distribution and wavefront of a laser beam, the light sources can exhibit the peculiarities of the superposition of two-photon Fock states and the incoherent mixture of two-photon Fock states, respectively. Combing auto-correlation measurements, the schemes we proposed generalize coherent and incoherent imaging to two-fold coherent and incoherent imaging, and the images beyond the Rayleigh diffraction limit can be obtained. The results show the possibility to improve the resolution of imaging system through controlling the amplitude distribution and wavefront of classical light combined with auto-correlation measurements. Furthermore, the modulating methods to realize higher-fold coherent and incoherent imaging can be generalized, which can further improve the imaging quality.

Acknowledgments

This work is supported by the National Natural Science Foundation of China (NSFC) (Project Nos. 11004248, 11374368 and 61405251).

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Sub-Rayleigh-diffraction imaging via modulating classical light

Abstract

1. Introduction

2. System setup and analytical results

3. Numerical simulated results

4. The physical picture

5. Conclusion

Acknowledgments

References and links

Cited By

Figures (3)

Equations (16)

Optics Express