Optimizing illumination’s complex coherence state for overcoming Rayleigh’s resolution limit
1. Introduction
Image formation with spatially partially coherent light has been addressed in classic papers and monographs[1–
Application of the Bochner’s theorem has led to a simple strategy for devising genuine cross-spectral density (CSD) functions[19], hence the CDCs, and has resulted in a “zoo” of novel partially coherent sources[20
In this Letter, we analyze telecentric imaging systems with the most general partially coherent scalar illumination. We first use a pseudo-mode expansion to evaluate the image intensity as a sum of two-dimensional (2D) (not 4D) Fourier integrals. Our new result is not limited to commonly used Schell-like illumination: it is also suitable for non-uniformly correlated[20] or twisted[23] illumination. Then, using this result, we construct the CDC of illumination, in the Schell-model class, for two particular axially symmetric cases: (i) two points and (ii) three points located in the vertices of an equilateral triangle, and analyze possible resolution improvement.
2. Theoretical Analysis
A schematic diagram of the telecentric imaging system is given in Fig. 1. Two lenses, and with focal lengths being , constitute a typical imaging system with unit image magnification. The object and its image are in the front focal plane of and the rear focal plane of , respectively.
Let the illumination be radiated by a scalar, stationary source characterized by the CSD function , where and are two position vectors in the object plane[18]. Here, denotes the electric field, where the asterisk and the angle brackets stand for complex conjugate and ensemble average, and is the angular frequency (its dependence will be omitted for brevity). With the (complex) object transmittance , the CSD function behind the object becomes
If a coherent impulse response function between the object plane and the image plane is , on the basis of coherence theory, the relation between the CSD functions in the image and the object planes is expressed via the integral
In Eq. (3), the impulse response function is . Therefore, the spectral density in Eq. (5) becomes
3. Numerical Results
We will now use Eq. (8) to analyze the effect of the CDC on the image resolution under the Schell-model illumination whose CDC only depends on the difference between two position vectors. The Schell-model beams are readily experimentally generated and controlled[18]. Hence, the results derived latter will be more instructive. In the following examples, the illumination’s CDC could be optimized based on the coherent imaging theory, where the optimal imaging for the two-point object and the three-point object can be achieved by the phase difference between the adjacent points being π and 2π/3, respectively.
First, let the object be two pinholes located on the axis, symmetrical with respect to , set at separation . Then, the object transmission
Equation (10) is routinely used for the two-pinhole resolution analysis under Schell-model illumination. We assume that the system is aberration-free, i.e., , being a hard circular aperture of radius . Hence,
Figure 2 illustrates the CDC as a function of at the cross line for several values of . For the bigger value of , it implies that we can get a slower envelope function and faster modulation functions of the source CDC, namely the CDC will get a value closer to . For , the CDC minimum value is about , which is very close to the theoretical minimum value of . From Eq. (14), one may deduce the position of the minimum value by finding , which is , where . Hence, if ratio is sufficiently large, the position difference , where reaches the minimum value, i.e., the closest to zero solution, is about . Hence, the image of two pinholes reaches appreciable resolution if (distance between two points) for large enough .
Figures 3(a)–3(c) illustrate the density plots of the normalized spectral density illuminated by beams with the CDC in Eq. (14) for three values of . The distance between two pinholes is set as . In the calculation of the CDC function, we set . For comparison, the image of two pinholes illuminated by an incoherent source is illustrated in Fig. 3(d). As expected, the resolution of the two-pinhole image is gradually improved as the value increases. When , one can clearly distinguish the images of two points due to the negative correlation of the illumination at the pinholes. The corresponding cross lines of normalized spectral density () in Figs. 3(a)–3(d) are shown in Fig. 3(e). Under incoherent illumination, the ratio of the spectral density at to the spectral density maxima is about , whereas the ratio decreases to 0.0286 when illumination is cosine-Gaussian correlated with .
Fig. 3. Images of two pinholes under (a)–(c) partially coherent illumination (normalized Sim) for three values of ratio a/b; (d) incoherent illumination; (e) the cross lines (ρy = 0) of Sim in (a)–(d).
To assess the MRS of two pinholes, we plot in Figs. 4(a)–4(c) their image for three values of at . The corresponding cross lines () are shown in Fig. 4(d). As separation distance decreases, the image gradually blurs. When it is about , the ratio is just 0.735, reaching the MRS of two pinholes. Figure 4(e) shows the dependence of the MRS of two pinholes on the value of . As expected, the resolution monotonically decreases with the increase of . When , the MRS is about .
Fig. 4. (a)–(c) Images (Sim) of two pinholes with three values of d under partially coherent illumination with a/b = 15; (d) cross lines (ρy = 0) of Sim in (a)–(d); (e) dependence of resolution on ratio a/b.
Three pinholes placed at the vertices of an equilateral triangle with side can be characterized by transmission function
Using Eq. (15) in Eq. (8), we get for the image spectral density
Substituting Eq. (17) into Eq. (11) results in the CDC in form
Figure 5 shows variation of its real part with for and the corresponding cross line at . In Fig. 5(a), there are six minimum regions located on the vertices of a regular hexagon. Three of them (denoted by white circles) are the sought minimum points. Figure 5(b) shows that the position of the minimum point in the right white circle is (1.15, 0). In fact, it is possible to obtain the positions of minimum points on axis by solving equation . When is sufficiently large, the solution of this equation is . Hence, if , the values of , , and are about , i.e., they approach the limiting value of as .
Figures 6(a)–6(c) give the normalized spectral density of a three-pinhole image at three separation values for and . The corresponding images formed with incoherent light are shown in Figs. 6(d)–6(f). When , the three pinholes are clearly seen with illumination having CDC, as in Eq. (18), whereas they are barely distinguishable with incoherent light. As decreases, the image gradually blurs. One can still barely distinguish three pinholes at with partially coherent light; while for incoherent light, the image degenerates to a single bright spot [see Fig. 6(f)].
Fig. 6. (a)–(c) Images (Sim) of three pinholes with different separations under the illumination of partially coherent beams with the CDC in Eq. (18 ). (d)–(f) The corresponding images of three pinholes with incoherent illumination.
4. Conclusion
In summary, we analyzed imaging with partially coherent illumination by deriving the integral formula involving the shape function and correlation class on the basis of the pesudo-mode expansion and FFT algorithm. By applying this formula to Schell-like light with predesigned CDC, we found that the image resolution of two pinholes can reach a value as low as . In this case, the minimum negative value of the designed CDC is , being very close to the ideal minimum value of . In the three-pinhole scenario, the resolution of about is achieved for each two-point pair. As compared with the previous work, in which we had improved the image resolution using the Laguerre–Gaussian correlated illumination (the image resolution reached only )[17], the current work has demonstrated that one can substantially improve the image resolution of a given object through the optimization design of the illumination’s coherence state. Here, we provide our suggestion for experimental realization of a specific Schell-model illumination. As suggested by Ref. [36], generating a specific Schell-model illumination in our work is actually to generate a function, where the intensity distribution is denoted on the round ground glass disk in the lab. The beam with any desired intensity distribution could be generated by a hologram loaded on a spatial light modulator. We can flexibly adjust the intensity distribution if we choose the different holograms. More details could be found in Ref. [36]. We anticipate that the idea of the active illumination that we introduced can be applied in a variety of the currently used conventional imaging systems, including microscopy and diffraction tomography, and it may generally stimulate the understanding and advancement of optical image formation mechanisms.
[1]
[4] B. J. Thompson. Image formation with partially coherent light. Prog. Opt., 1969, 7: 169.
[8] K. Yamazoe. Two models for partially coherent imaging. Opt. Lett., 2012, 29: 2591.
[18]
[22] L. Ma, S. A. Ponomarenko. Optical coherence gratings and lattices. Opt. Lett., 2014, 39: 6656.
[25] Z. Mei. Hyperbolic sine-correlated beams. Opt. Express, 2019, 27: 7491.
[30] Y. Zhou, H. Xu, Y. Yuan, J. Peng, J. Qu, W. Huang. Trapping two types of particles using a Laguerre–Gaussian correlated Schell-model beam. IEEE Photon. J., 2016, 8: 6600710.
Chunhao Liang, Yashar E. Monfared, Xin Liu, Baoxin Qi, Fei Wang, Olga Korotkova, Yangjian Cai. Optimizing illumination’s complex coherence state for overcoming Rayleigh’s resolution limit[J]. Chinese Optics Letters, 2021, 19(5): 052601.