Image formation with spatially partially coherent light has been addressed in classic papers and monographs^[1–4] in depth but is nevertheless still the subject of acute scientific exploration^[5–10]. In the general case of a partially coherent illumination, the formed image cannot be simply expressed as a convolution of the object’s transparency and the system’s impulse response function, but rather involves four-dimensional (4D) correlation integrals. In order to simplify image computations, several numerical methods have been proposed including coherent-model representation^[11], outer-product expansion^[12], pupil shift matrix^[13], and the elementary-field approach^[14]. On the other hand, several somewhat empirical studies showed that the specially modeled complex degree of coherence (CDC) might have a significant impact on the formed image resolution, resulting in overcoming the Rayleigh diffraction limit^[15–17]. In particular, it was found in Ref. [15] that the twist phase can lead to an improvement of the two-point image resolution by an order of magnitude, but such a result only applies to axially symmetric points. Through manipulation of the CDC profile, the image resolution was shown to reach about 0.93dR (dR being the Rayleigh diffraction limit) for a partially correlated azimuthal vortex illumination^[16] and about 0.85dR for the Laguerre–Gaussian correlated illumination^[17]. Instead of using a certain type of illumination with a fixed CDC class, we propose to construct the optimal realizable CDC. Since the CDC cannot be devised arbitrarily, being a subject of several realizability conditions^[18], this makes the studies of the image resolution improvement by adjusting the CDC somewhat complex.

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^[20–26]. Owing to their particular source coherence properties, the radiated beams exhibit a variety of effects on propagation, including self-focusing^[20], self-splitting^[27], and self-steering^[28], and have already found applications in beam shaping, free-space optical communications, and optical trapping^[29–31]. Recently, the CDCs with spatially varying phase functions of the Schell type have also been successfully modeled^[32] using a simple sliding-function method, which substantially enriched available CDCs for asymmetric light manipulation^[33], including imaging applications. The sources with structured coherence have been synthesized in the laboratory with the help of spatial light modulators (e.g., Refs. [26,29]) and using the van Cittert–Zernike theorem (e.g., Ref. [17]).

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.

A schematic diagram of the telecentric imaging system is given in Fig. 1. Two lenses, L1 and L2 with focal lengths being f=f1=f2, constitute a typical 4f imaging system with unit image magnification. The object and its image are in the front focal plane of L1 and the rear focal plane of L2, respectively.

Let the illumination be radiated by a scalar, stationary source characterized by the CSD function W0(r1,r2,ω)=〈E*(r1,ω)·E(r2,ω)〉, where r1=(x1,y1) and r2=(x2,y2) are two position vectors in the object plane^[18]. Here, E denotes the electric field, where the asterisk and the angle brackets stand for complex conjugate and ensemble average, and ω is the angular frequency (its dependence will be omitted for brevity). With the (complex) object transmittance O(r), the CSD function behind the object becomes W0′(r1,r2)=O*(r1)O(r2)W0(r1,r2).(1)

If a coherent impulse response function between the object plane and the image plane is K(r,ρ), on the basis of coherence theory, the relation between the CSD functions in the image and the object planes is expressed via the integral Wim(ρ1,ρ2)=∫W0′(r1,r2)K*(r1,ρ1)K(r2,ρ2)d2r1d2r2,(2)where ρ1 and ρ2 are the position vectors in the image plane. For the telecentric system with a pupil, the impulse function K(r,ρ) is K(r,ρ)=−1λ2f2∫P(ξ)exp[−ikfξ·(r+ρ)]d2ξ,(3)where k=2π/λ with λ being the wavelength of light, and P(ξ) is the complex-valued transmission function of the pupil whose argument may contain the wave aberration of the system. For a genuine CSD of illumination, it suffices to be representable as^[19]W0(r1,r2)=τ*(r1)τ(r2)∫p(v)H0*(v,r1)H0(v,r2)d2v,(4)where τ(r) is a complex amplitude function, p(v) can be regarded as the power spectral density (hence, must be non-negative), and v is the 2D position vector in the spatial Fourier plane. For the Schell-model beam, the CDC is the Fourier transform (FT) of the p(v) function, which inspires us to construct the genuine partially coherent beams with any desired CDCs through adopting the suitable p(v) functions. H0 is an arbitrary kernel. Substitution of Eqs. (1) and (4) into Eq. (2), results in the image-plane spectral density Sim(ρ)=Wim(ρ,ρ): Sim(ρ)=∫d2vp(v)|∫τ(r)O(r)H0(v,r)K(r,ρ)d2r|2.(5)

In Eq. (3), the impulse response function is K(r,ρ)=K(−r−ρ). Therefore, the spectral density in Eq. (5) becomes $$\begin{gathered} Sim(ρ)=∫d2vp(v)|∫F(v,r)K(−r−ρ)d2r|2 \\ =∫d2vp(v)|IFT[F˜1(v,f)K˜(f)]{−ρ}|2, \end{gathered}$$(6)where F(v,r)=τ(r)O(r)H0(v,r), and the tilde and IFT stand for direct and inverse FT, respectively. It follows from Eq. (6) that the fast FT (FFT) algorithm can be applied to calculate the spectral density in the image plane. First, the integral in the absolute value symbol is evaluated for each v using the FFT algorithm; then, Eq. (6) is applied to integrate over all values of v. With the help of the pseudo-mode expansion^[34], the spectral density Sim(ρ) takes an approximate discrete form: Sim(ρ)=∑iN∑jNp(vxi,vyj)M(vxi,vyj,−ρ),(7)with M(vxi,vyj,−ρ)=|IFT[F˜1(v,f)K˜(f)]{−ρ}|2 and v=(vxi,vyj),(i,j=1,2,…,N). Equations (6) and (7) result in a fast calculation scheme for the image-plane spectral density. In particular, if the source of illumination has circular coherence^[35], Eq. (7) could further be reduced to a one-fold summation over v, greatly saving the calculating time. When the illumination is Schell-like, i.e., with the CDC depending on point separation, the spectral density Sim(ρ) becomes Sim(ρ)=∫d2vp(v)|IFT[F˜1(f−v)K˜(f)]{−ρ}|2,(8)where F1(r)=τ(r)O(r), and τ(r) is a complex amplitude function. For Schell-model sources, the kernel function in Eq. (4) has the form H0(v,r)=exp(2πiv·r). The result in Eq. (8) is the same as in Ref. [9].

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 x axis, symmetrical with respect to y=0, set at separation d. Then, the object transmission O(r)=δ(x−d/2,y)+δ(x+d/2,y)(9)is a pair of the Dirac-delta functions δ(·). On substituting Eq. (9) into Eq. (8) and setting τ(r)=exp(−r2/4σ02), where σ0 is the beam rms width, we obtain the expression for the image spectral density: Sim(ρ)=exp(−d2/8σ02)∫d2v×p(v){|S+|2+|S−|2+2Re[S+S−*μ(d,0)]},(10)where S±=P˜(ρx±d/2,ρy), P˜(ρ) is the FT of pupil P(u) with u=ξ/λf, and μ(Δx,Δy) is the CDC. With Δr=r2−r1=(Δx,Δy), relation μ(Δr)=W(r1,r2)W(r1,r1)W(r2,r2)=∫d2vp(v)exp(2πiv·Δr)∫d2vp(v)(11)was used in derivation of Eq. (10) (only valid for Schell-model sources).

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., P(ξ)=|P(ξ)|, being a hard circular aperture of radius R. Hence, S±=2πR2λ2f2J1[2πR(ρx±d/2)2+ρy2/λf]2πR(ρx±d/2)2+ρy2/λf,(12)where J1 is the Bessel function of the first kind and order one. Since in Eq. (10) S± are real functions, S+S−*=S+S−. This result is consistent with that reported in Ref. [17]. When illumination is incoherent, μ(d,0)=0, the minimum resolvable separation (MRS) of two pinholes is given by the well-known Rayleigh criterion: dR=0.61λf/R. The Rayleigh criterion states that the MRS is met if the position of the first zero of one pinhole image coincides with the position of the maximum point of the other pinhole image. However, the resolution will exceed the Rayleigh limit if the real part of the CDC at (d,0) has a negative value (the closer the CDC to −1, the higher the resolution). Nevertheless, owing to the non-negative definiteness of p(v), the CDC cannot be set arbitrarily. It is one of the primary factors limiting the performance of partially coherent illumination in imaging systems. In fact, the minimum value of Re[μ(d,0)] is about −0.3 for Laguerre–Gaussian correlated illumination of order six^[17]. In order to make Re[μ(d,0)] approach −1 (theoretical minimum value), we choose p(v) as p(v)=e−a2[(vx−b/2)2+vy2]+e−a2[(vx+b/2)2+vy2],(13)with real a and b. Substituting Eq. (13) into Eq. (11) yields μ(Δr)=e−π2Δr2/a2cos(πΔx/b),(14)i.e., being a cosine function with period 2b enveloped by a Gaussian function of width a/π.

Figure 2 illustrates the CDC as a function of Δx/b at the cross line Δy=0 for several values of a/b. For the bigger value of a/b, 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 −1. For a/b=15, the CDC minimum value is about −0.957, which is very close to the theoretical minimum value of −1. From Eq. (14), one may deduce the position of the minimum value by finding dμ(Δx,0)/dΔx=0, which is sinθ+2(b/a)2θcosθ=0, where θ=πΔx/b. Hence, if ratio a/b is sufficiently large, the position difference Δx, where μ(Δx,0) reaches the minimum value, i.e., the closest to zero solution, is about Δx≈b(θ≈π). Hence, the image of two pinholes reaches appreciable resolution if b=d (distance between two points) for large enough a/b.

Figures 3(a)–3(c) illustrate the density plots of the normalized spectral density Sim(ρ)/[Sim(ρ)]max illuminated by beams with the CDC in Eq. (14) for three values of a/b. The distance between two pinholes is set as d=dR. In the calculation of the CDC function, we set b=dR. 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 a/b increases. When a/b=15, 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 (ρy=0) in Figs. 3(a)–3(d) are shown in Fig. 3(e). Under incoherent illumination, the ratio of the spectral density at ρ=0 to the spectral density maxima is about Sim(0)/[Sim(ρ)]max=0.735, whereas the ratio decreases to 0.0286 when illumination is cosine-Gaussian correlated with a/b=15.

To assess the MRS of two pinholes, we plot in Figs. 4(a)–4(c) their image for three values of d=b at a/b=15. The corresponding cross lines (ρy=0) are shown in Fig. 4(d). As separation distance d decreases, the image gradually blurs. When it is about 0.22dR, the ratio Sim(0)/[Sim(ρ)]max 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 a/b. As expected, the resolution monotonically decreases with the increase of a/b. When a/b=20, the MRS is about 0.18dR.

Three pinholes placed at the vertices of an equilateral triangle with side d can be characterized by transmission function $$\begin{gathered} O1(r)=δ(x,y−d/3)+δ(x−d/2,y+3d/6) \\ +δ(x+d/2,y+3d/6). \end{gathered}$$(15)

Using Eq. (15) in Eq. (8), we get for the image spectral density Sim(ρ)=exp(−d26σ2){|S1|2+|S2|2+|S3|2+2Re[S1S2*×μ(−d2,3d2)+S1*S2μ(−d2,−3d2)+S2S3*μ(d,0)]},(16)where S1=P˜(ρx,ρy+d3), S2=P˜(ρx+d2,ρy−d23), and S3=P˜(ρx−d2,ρy−d23). For the best resolution of such an object, the real part of the CDC at (−d/2,3d/2), (−d/2,−3d/2), and (d,0) must approach minimum possible (negative) values simultaneously. In order to obtain such a CDC, we set $$\begin{gathered} p(v)=e−a2[(vx−13b)2+vy2]+e−a2[(vx+123b)2+(vy+12b)2] \\ +e−a2[(vx+123b)2+(vy−12b)2]. \end{gathered}$$(17)

Substituting Eq. (17) into Eq. (11) results in the CDC in form μ(Δr)=13e−π2Δr2a2[2cos(πΔyb)e−iπΔx3b+ei2πΔx3b],(18)which is genuinely complex-valued.

Figure 5 shows variation of its real part with (Δx/b,Δy/b) for a/b=20 and the corresponding cross line at Δy/b=0. 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 Δy=0 by solving equation ∂Re[μ(Δr)]/∂Δx|Δy=0=0. When a/b is sufficiently large, the solution of this equation is Δx=±23b/3. Hence, if b=3d/2, the values of μ(−d/2,3d/2), μ(−d/2,−3d/2), and μ(d,0) are about −0.493, i.e., they approach the limiting value of −0.5 as a/b→∞.

Figures 6(a)–6(c) give the normalized spectral density of a three-pinhole image at three separation values for b=3d/2 and a/b=20. The corresponding images formed with incoherent light are shown in Figs. 6(d)–6(f). When d=dR, the three pinholes are clearly seen with illumination having CDC, as in Eq. (18), whereas they are barely distinguishable with incoherent light. As d decreases, the image gradually blurs. One can still barely distinguish three pinholes at d=0.4dR with partially coherent light; while for incoherent light, the image degenerates to a single bright spot [see Fig. 6(f)].

In summary, we analyzed imaging with partially coherent illumination by deriving the integral formula involving the shape p(v) function and correlation class H(r,v) 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 0.18dR. In this case, the minimum negative value of the designed CDC is ≈−0.97, being very close to the ideal minimum value of −1. In the three-pinhole scenario, the resolution of about 0.4dR 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 0.85dR)^[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 p(v) 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.