Noise properties in a two-arm microscope imaging system with classical thermal light Yanfeng Bai,* Wenxing Yang, and Xiaoqiang Yu Department of Physics, Southeast University, Nanjing 210096, China *Corresponding author:
[email protected] Received 5 April 2010; revised 13 July 2010; accepted 17 July 2010; posted 22 July 2010 (Doc. ID 126533); published 13 August 2010
We present an analysis of the noise properties in a two-arm microscope imaging system. The aperture of the reference lens affects the imaging quality significantly. Using large apertures will enhance the resolution but also increase the noise. The effects from the distance the object is moved away from the original plane are also discussed, and we can obtain both good resolution and small noise by changing the distance. © 2010 Optical Society of America OCIS codes: 110.1650, 030.1640, 030.4280.
1. Introduction
Correlated imaging, usually called ghost imaging, is a method to nonlocally image an object through spatial intensity correlation measurement. Based on entangled photon pairs, the first coincidence imaging experiment was performed by Pittman et al. [1]. Their work led to many interesting studies, with results focused mainly on the field of quantum optics [2–4]. Soon Bennink et al. provided an experimental demonstration of correlated imaging by using a classical source [5]. Because correlated imaging with thermal light provided more potential applications by comparison with that under an entangled source, classical correlated imaging has been studied extensively in recent years, both experimentally and theoretically [6–10]. Our group theoretically investigated coincidence imaging with incoherent light by using classical statistical optics, based on which we gave a proposal to realize lensless Fourier-transform imaging and discussed its applicability in x-ray diffraction [11]. The experiment of lensless ghost imaging was performed by Scarcelli et al. for the first time [12]. Recently, the lensless ghost imaging experiments in which the reflected and scattered light from the object were detected by correlation measure0003-6935/10/244554-04$15.00/0 © 2010 Optical Society of America 4554
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ments present good promise for practical applications [13]. Very recently, Chan et al. theoretically studied two-color ghost imaging using either thermal or quantum light sources. They showed that a highquality image can be obtained even when the wavelength of light used in the object and reference arms are very difficult [14]. It is well known that resolution and noise are two important factors to characterize an imaging system. The properties of the resolution and the noise in correlated imaging systems have been discussed by many groups. Ferri et al. performed high-resolution ghost image and ghost diffraction experiments by using a single classical source of pseudothermal speckle light divided by a beam splitter [15]. Cheng et al. theoretically investigated the resolution and the noise in ghost imaging with a classical thermal light, and they discussed the effects from the spatial coherence of the source and the aperture in the imaging system [16]. Recently, our group reported a proofof-principle experimental demonstration of a twoarm microscope scheme and showed that, by measuring the second-order correlation of light fields, the two-arm imaging scheme is feasible for improving the resolution of a conventional lens-limited imaging system [17]. In this paper, intrigued by these studies, we consider a two-arm microscope system, which can be applied to some microscopes where the NAs of the
objective lenses are limited. One can obtain higherresolution images by adding a reference arm, which usually has no limitation in size [17]. The properties of the noise in this two-arm microscope imaging system are discussed. The results show that a large aperture of the reference lens will lead to a good imaging resolution and large noise. We can obtain both good resolution and small noise when the object does not deviate from the original plane. 2. Model and Equations
A two-arm microscope imaging system is shown in Fig. 1. The light from a thermal source, which can be simulated by projecting a Nd:YAG laser onto a slowly rotating ground-glass disk [18], is divided by a beam splitter into two beams that travel on their respective paths to be detected at spatially separated detection systems. In the test arm, the beam propagates through an object and the objective lens of focal length f t , and then after propagation, it travels to the CCD Dt. In the reference arm, we suppose a pseudoplane (σ plane) at the symmetric position of the object with respect to the splitter. The beam travels through the reference lens of focal length f r and then reaches the CCD Dr. Obviously, the test arm is a conventional imaging setup, and the object distance z1 , the image distance z2 , and the focal length of the objective lens f t obey the Gaussian thin-lens equation: 1=z1 þ 1=z2 ¼ 1=f t . Based on the Rayleigh criterion, the resolution limit δx of the test arm is determined by the wavelength and the NA of the objective lens: δx ¼ 0:61λ= NA ≃ λz1 =at , where at is the aperture of the objective lens. For simplicity, we suppose that the distance z3 , z4 , and f r also satisfy the Gaussian thin-lens equation in the reference arm: 1=z3 þ 1=z4 ¼ 1=f r . Under the paraxial approximation, the impulse response functions of the two arms can be derived from the Huygens–Fresnel integral [19]:
eikz0 iπ 0 exp ðx − x1 Þ2 tðx0 Þ iλz0 λz0 at u1 0 x þ ; × sinc λz1 M1 Z
h1 ðx1 ; u1 Þ ¼
dx0
eikz0 iπ exp ðx″ − x2 Þ2 iλz0 λz0 ar u2 x″ þ ; × sinc λz3 M2
ð1Þ
Z
h2 ðx2 ; u2 Þ ¼
dx″
ð2Þ
where z0 is the distance from the source to the object, tðx0 Þ is the transmission function of the object imaged, and sincðξÞ sinðπξÞ=ðπξÞ represents the onedimensional amplitude point-spread function related to the object (reference) lens. M 1 and M 2 are the magnification of the test and the reference arms, respectively. Based on the previous works [6,20], the information of the object can be retrieved by measuring the correlation between the intensity fluctuations in the two detectors: Gðu1 ; u2 Þ ¼ hΔI 1 ðu1 ÞΔI 2 ðu2 Þi Z 2 ¼ Γðx1 ; x2 Þh1 ðx1 ; u1 Þh2 ðx2 ; u2 Þdx1 dx2 ; ð3Þ where ΔI i ¼ I i − hI i i, ði ¼ 1; 2Þ is the intensity fluctuations in the two detectors, and Γðx1 ; x2 Þ is the first-order correlation function of the thermal source. To study the noise in this imaging system, we first give the fluctuation in Gðu1 ; u2 Þ: ΔGðu1 ; u2 Þ ¼ fh½ΔI 1 ΔI 2 2 i − ½hΔI 1 ΔI 2 i2 g1=2 :
ð4Þ
Whereas the expression of h½ΔI 1 ΔI 2 2 i has been given in [16], we do not show it here. It should be noticed that thousands of images are measured and averaged to obtain a high-quality signal in the practical experiments [21]. Suppose that pthere ffiffiffiffiffi are N CCD frames, then ΔGðu1 ; u2 Þ will be N times smaller. We assume that the intensity distribution of the source is of the Gaussian type, then the first-order correlation function of a complete incoherent light can be written as 2 x þ x2 Γðx1 ; x2 Þ ¼ G0 exp − 1 2 2 δðx1 − x2 Þ; 4a
ð5Þ
where G0 is a normalized constant, and a is the transverse size of the source. 3. Analytical Results and Numerical Simulations
Fig. 1. Two-arm microscope imaging system.
The numerical results about the imaging signal Gðu2 Þ and its fluctuation ΔGðu2 Þ can be obtained by substituting Eqs. (1), (2), and (5) into Eqs. (3) 20 August 2010 / Vol. 49, No. 24 / APPLIED OPTICS
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and (4). During this process, a further point that should be noted is that an imaging detector instead of a pointlike or bucket detector is used in the test arm. Here a double slit with slit width of 0:02 mm and distance between two slits of 0:04 mm is chosen as the object imaged, and it is placed at z0 ¼ 100 mm from the light source. z1 ¼ z3 ¼ 375 mm and z2 ¼ z4 ¼ 750 mm are chosen to obtain the same magnification M 1 ¼ M 2 ¼ 2; furthermore, we assume that the objective and the reference lenses have the same focal length f t ¼ f r ¼ 250 mm, and the aperture of the objective lens is fixed at at ¼ 6 mm. Figure 2 shows the effects from the aperture of the reference lens on the resolution and the noise. By considering a reference arm with ar ¼ 6 mm, a relatively clear image with good resolution (the signal-to-noise ratio is about 3) can be reconstructed via the correlation measurement between the two detectors, and the imaging resolution increases by enhancing the aperture of the reference lens. The two slits are separated
Fig. 2. Normalized imaging signal Gðu2 Þ (dot–dash curve) and its fluctuation ΔGðu2 Þ (solid curve) versus u2 for (a) ar ¼ 6 mm, (b) ar ¼ 10 mm, and (c) ar ¼ 15 mm. Other parameters are chosen as λ ¼ 532 nm, a ¼ 0:5 mm. 4556
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clearly when ar ¼ 15 mm, as shown in Figs. 2(b) and 2(c). Meanwhile, we also note that the noise increases with an increasing aperture of the reference lens. When the object moves a short distance ΔzðΔz < z1 Þ away from the original plane along the axial direction, the impulse response function of the test arm is rewritten as eikðz0 þΔzÞ iπðx0 − x1 Þ2 exp iλðz0 þ ΔzÞ λðz0 þ ΔzÞ 0 a x u þ 1 : ð6Þ × tðx0 Þsinc t λ z1 − Δz z2 Z
h1 ðx1 ; u1 Þ ¼
dx0
The impulse response function of the reference arm keeps unchanged. By replacing Eq. (1) with Eq. (6), we can discuss the effects from Δz on the resolution and the noise. To show the effect from Δz clearly, a double slit with slit width of 0:04 mm and distance between two slits of 0:08 mm is chosen. Other parameters are chosen as z1 ¼ z3 ¼ 150 mm, z2 ¼ z4 ¼ 300 mm, at ¼ 2 mm, ar ¼ 1 mm, and f t ¼ f r ¼ 100 mm. At the same time, the defocus effect in the test arm is ignored. The resolution limit of the test arm δx ≃ 1:22λz1 =at ¼ 0:049 mm is smaller than the two-slit separation of 0:08 mm. Thus, we cannot distinguish the two slits only by the imaging setup of the test arm. Meanwhile, a relatively clear image can be obtained when the object is at the original plane by considering the additional reference arm, as shown in Fig. 3(a). It is shown that the resolution degrades when the object is moved away from the
Fig. 3. Normalized intensity correlation Gðu2 Þ (dot–dash curve) and its fluctuation ΔGðu2 Þ (solid curve) as a function of u2 for (a) Δz ¼ 0 and (b) Δz ¼ 100 mm.
This work is partially supported by the National Natural Science Foundation of China (NSFC) (grants 10904015, 10947129, and 10704017). References
Fig. 4. Amplitude of the noise versus Δz. Other parameters are the same as those in Fig. 3.
original plane [see Fig. 3(b)]. We also note that the amplitude of the noise increases with an increase of Δz. So the object should be placed at the original plane to improve both the resolution and the signalto-noise ratio at the same time. To get a deeper insight into the modulation effect of Δz on the noise, we depict the amplitude of the noise as a function of Δz in Fig. 4. It is shown that, while the amplitude of the noise slightly enhances for a small Δz, the change gradually becomes obvious for a large distance Δz. The results can be explained based on the expression of the signal-to-noise ratio, which can be obtained from Eqs. (3) and (4): 2 1=2 hI 1 ðu1 ÞihI 2 ðu2 Þi þ2 −1 Gðu1 ; u2 Þ 2 1=2 1 þ2 −1 ¼ ; ð7Þ V
ΔGðu1 ; u2 Þ ¼ Gðu1 ; u2 Þ
where hI 1 ihI 2 i is the background term, which cannot be used to obtain the imaging information [20]. Obviously, the noise is in inverse proportion to the imaging visibility V. From the results given in [17], we know that the visibility of the images degrades when Δz increases, which will result in an increase of the noise amplitude, as shown in Fig. 4. 4. Conclusions
In conclusion, we have investigated the noise properties in a two-arm microscope imaging system with classical thermal light. It has been shown that the noise enhances when the aperture of the reference lens is increased, though the corresponding resolution increases. In addition, we can obtain both good resolution and small noise by changing the distance the object is moved away from the original plane.
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