Silicon substrate significantly alters dipole-dipole resolution in coherent microscope

Influences of a substrate below samples in imaging performances are studied by reaching the solution to the dyadic Green's function, where the substrate is modeled as half space in the sample region. Then, theoretical and numerical analysis are performed in terms of magnification, depth of field, and resolution. Various settings including positions of dipoles, the distance of the substrate to the focal plane and dipole polarization are considered. Methods to measure the resolution of $z$-polarized dipoles are also presented since the modified Rayleigh limit cannot be applied directly. The silicon substrate and the glass substrate are studied with a water immersion objective lens. The high contrast between silicon and water leads to significant disturbances on imaging.


Introduction
Fluorescence imaging [16] suffers from photobleaching and blinking, and the imaging duration is limited by the photochemically toxic environment. Therefore, label-free imaging [13] is desired but requires improvements in resolution. With no or little modifications of the existing instruments, computational techniques [2; 8; 1] are used to enhance the resolution by making use of the point spread properties, which are commonly studied through the point spread function and aberration functions in optics.
A full electromagnetic wave approach, rather than ray optics (geometric optics), increases the applicability of the computational models of microscope to high NA systems. This was for example demonstrated before in the case of solid immersion lens [11; 5; 6], where computational modeling used for identification of the suitable pinhole dimension that allows better resolution by balancing the collection of light from longitudinal and lateral dipoles induced in the sample region. A more general version, without assuming solid immersion lens is available in [14]. While general, it lacks one inevitable aspect of microscopy, especially when used for biological imaging. This aspect is the presence of an interface due to the resting surface of sample, for example petri dish in Fig. 1(a) and the silicon substrate in Fig. 1 photonics-based microscopes [4; 3]. The importance of modeling the interface has been recognized before, and incoherent point spread functions of optical microscopes have been derived [9]. However, we are not aware of analogous 3D full-wave dyadic green function (DGF) for a coherent microscopy system. We do note that there have been works related to layered medium [10] in optical systems for optical memory and lithography. However, for a microscopy system involving the conventional microscope-objective and tube lens pair is currently not simulated. Here, we present the DGF of half space emulating an air or liquid-dipping objective based coherent microscopy system where the half space corresponds to the glass surface or silicon surface.

(b) in modern
Expanding the field emitted by a dipole into plane waves, the DGF of the concerned microscopy system is solved by analyzing the reflection and refraction behaviors of each plane-wave component and integrating all rays reaching the image region. Based on the closed-form solution of DGF, the lateral and longitudinal resolutions are studied with various settings. Quantifying the resolvability by saddle-to-peak ratio, the modified Rayleigh limit is used as the resolution criterion. Further, since z-polarized dipoles are shown as annuli [14] and the computation of saddle-to-peak ratio is not straightforward, efforts are made here to apply the modified Rayleigh limit to z-polarized dipoles. The influences of the substrate on the depth of field are also discussed. Taking the optical axis as the z axis, two local coordinate systems are built whose origins are located at the focal points of the objective region and the image region, respectively. These regions are respectively created by the objective lens (focused on the sample) and the tube lens (focused on the camera). The notation of quantities in the objective region employs the subscript "obj", while quantities in the image region, where camera sensors (scientific CMOS, CCD or emCCD) are positioned, are subscripted by "cam". Here, the general derivation is being made for the region close to the image plane and the center of camera pixel is used to represent the pixel assuming a pixel to be point-like detector. However, generalization to a large pixel or optical detector size (such as photodetector or multiphoton diode) can  be made by integrating the computed intensity over the area of the detection element.

Setup and notations
In the objective region, O obj denotes the focal point, n obj the refractive index of the medium in which sample is kept, k obj the wavenumber, f obj the focal length. r obj denotes the position of a dipole . This observation point in the objective region is defined to facilitate the derivation of the DGF and is not used in the final formulation of DGF. Remark that the vector quantities are noted by putting an arrow above and r obj stands for the radial distance to O obj . The notations in the image region are similarly defined.
The substrate is modeled as the lower half space and characterized by the position of the substrate interface z obj = z b obj (i.e., parallel with the x,y plane) and the refractive index n sub . The angular semiaperture of the objective lens is θ max obj . Both the objective lens and the tube lens are represented by Gaussian reference spheres (GRS) [14].

Solution to dyadic Green's function
As sketched in Fig. 2, a unit dipole polarized along an arbitrary directionâ locates in the objective region.
After reflections by the substrate interface, the emitted waves propagate through the objective lens and the tube lens, and finally recorded at pixel locations on camera array. We denote the field solution as Following the derivation in [12], the solution to the DGF of the half space is a superposition of TE (s-polarized) and TM (p-polarized) parts, in which The first terms of Eq. (2a) and (2b) stand for the wave emitted by the dipole and the second terms for the wave reflected by the substrate interface. k ± obj = k xx + k yŷ ± k z objẑ denote the wave vector of the integrated plane-wave components. k z obj = k 2 obj − k 2 x − k 2 y has a non-negative imaginary part when k 2 x + k 2 y > k 2 obj .φ obj k ± obj andθ obj k ± obj are defined as the unit vector of the azimuthal and elevation axis for k ± obj . The reflection coefficients are obtained based on the continuity of tangential electric fields across the substrate interface and expressed as where k sub is the wavenumber of the substrate and k z sub is similarly defined with k z obj . Since we consider a far-field imaging microscope, i.e., the focal length f obj λ obj , only the far (propagating) fields need to be considered in the integration of (1), i.e., the integral region is restricted by k 2 x + k 2 y ≤ k 2 obj . With the method of stationary phase [14], the solution of fields (before the refraction by the objective lens) at any point on the objective lens GRS surface A is solved as Following the sine condition and the intensity law [14], after the refraction by the objective lens and the tube lens, the field at the tube lens GRS surface B is with expression where θ A obj and θ B cam are the elevation angle of A and B, respectively. The superposition of all rays from the surface of the tube lens yields the solution of DGF, i.e., The limit by the angular semiaperture is imposed by transforming the integral to the spherical coordinate system of the objective region, Solving the integral about φ obj analytically, the two-dimensional integral is transformed as a one-dimensional integral about θ obj only. Specifically, the expression of the DGF is then given as where α = k cam e i(k obj f obj +kcamfcam) 8iπ cos θ obj cos θ cam sin θ obj dθ obj , (9b) where The identity f obj sin θ obj = f cam sin θ cam has been used for the derivation. The above integrals can be simplified by using the assumption f cam f obj which yields the approximations sin θ cam ≈ 0 and cos θ cam ≈ 1 [11]. The lateral magnification and longitudinal magnification (assuming paraxial approximation and neglected reflections by the substrate interface) are derived in Supplement 1.

Investigations on resolution
Effects of the half space are studied by setting the refractive index of the lower half space as n sub Where not stated explicitly, the NA of the system by default is assumed to be 1.

Quantification of resolution
The (induced) charge of dipoles can have a high variance in the analysis of imaging performance [7].
To get rid of effects from values of charge, normalization is performed. Considering electric fields at points r (i) cam , i = 1, . . . , N , the normalized electric fields equal E( r . . , N } and the related quantities are denoted by putting a bar above, i.e.,Ē = E/E max . The normalized field due to two dipoles is computed asĒ( r cam , r s, obj ) denotes the normalized field due to the s-th dipole. Thus, the intensity of combined fields may have maximum not equal to 1. The silicon substrate is used in the example settings in Figs  Based on the image of two adjacent dipoles, as sketched by Fig. 3, the saddle-to-peak ratio I min /I max is computed, where I max is the maximum intensity of the combined field and I min the minimum between two peaks. According to the modified Rayleigh limit, the resolution is defined as the minimum distance between two dipoles when the ratio is ≤ 0.735.
Lateral resolution for z-polarized dipoles: The aforementioned criterion is based on the assumption that the image of a dipole is spot-like. This is true for x-polarized and y-polarized dipoles. However, as shown by Fig. 4(a), the image of a z-polarized dipole is an annulus. Then, observations along the x or y axis have two peaks, which make the above method for quantifying resolution inapplicable. We consider potential solution for defining resolution in the situation where at least one of the dipoles is z-polarized. Two cases are considered, two dipoles directed towardsẑ and one of them directed tox. We note that y-polarized dipoles are not considered due to the similar behaviors with x-polarized dipoles and straightforward generalizability.. Two dipoles are placed at (−∆x obj /2, 0, 0) and (∆x obj /2, 0, 0), respectively. In Fig. 4(b), we consider the situation when two z-polarized dipoles are separated by distance ∆x obj = 0.7λ (i.e., 350 nm). Even though the image of each dipole independently looks like annulus (see Fig. 4(a)), the image of two dipoles in vicinity does not look like a superimposition of two annulus (see Fig. 4 between the dipoles are due to destructive interferences. Along the x axis, using the DGF we determined that the intensity value is mainly contributed from the real part of the x-component of electric fields, denoted by E x , which is plotted in Fig. 4(d). One sees that destructive interference occurs due to the opposite signs of (E x ) between the dipoles.

(b). Small intensities in the region
As the distance ∆x obj is increased, the signature of two dipoles becomes quite evident even though the image still does not look like a simple super-imposition of two annuli. Fig. 4(c) shows the image when ∆x obj = λ. Through observations in Fig. 4, we propose that the resolvability of two z-polarized dipoles separated along x-axis can be inferred using the intensity profile along the line defined by z cam = 0 and y cam = ±y peak cam , as shown by the dash-dotted lines in Fig. 4(b,c). y peak cam is the distance of peak value of intensities along y axis to the center of the annulus and can be numerically determined. Fig. 4(e) shows the variation of the sampled intensities when the two dipoles are moved further from each other along this line for the example configuration used in this paper. The decreased saddle-to-peak ratio with the increasing distance between the emitters reveals that the modified Rayleigh limit computed over the line defined by z cam = 0 and y cam = ±y peak cam can now be used for quantifying the resolution for the case of two z-polarized dipoles.
Lateral resolution for one x-and one z-polarized dipole: Images of the situation when a x- polarized dipole and a z-polarized are presented in the sample region are shown in Fig. 5(a) and (b). As seen, the only indicator of the presence of two dipoles is the asymmetry in the intensity profile in Fig. 5(a) whereas the presence of two dipoles is evident from the intensity profile in Fig. 5(b). Considering that the image of a z-polarized dipole is an annulus and not a spot, the observations for quantifying the resolution are taken along a line segment that starts at the center of the annulus and extends beyond the peak corresponding to the other dipole, such as shown using the dash-dotted lines in Fig. 5(a,b). This is contrary to considering a line segment that passes through both of the dipoles in the case of only x-polarized or y-polarized dipoles as well as considering a line that passes through the peak of annuli parallel to the locations of dipole in the case of two z-polarized dipoles. As shown in Fig. 5(c), as the distance ∆x obj increases, the initially merged peak splits into two peaks and the saddle-to-peak ratio decreases. The modified Rayleigh limit can therefore be applied. However, it is important to remark that the location of peaks of sampled intensities does not correspond to the accurate position of dipoles.

Depth of field (DOF)
Depth of field is investigated by moving the position of a dipole along the optical axis. The substrate interface is assumed at the plane of z b obj =−2 µm. As shown in Fig. 3 and ??, the image of a dipole at the focal point is a focused spot or an annulus with small side lobes. When the dipoles has the distance of 1.5λ to the focal plane, as presented in Fig. 6(a,c), the images become out of focus with high side lobes. We note also that the peak intensity is decreased as light spreads wider. Sampling intensities along the y axis (since higher slide lobes observed than along the x axis for the x-polarized dipole ), Fig.6(b,d) are obtained by varying the position of dipole with the silicon substrate, while (e,f) present the results with the glass substrate. Despite the polarization and the substrate material, the peak values, which are the intensities along the dashed lines, tend to decrease when the dipole moves further from the focal plane. Since small peak values indicate defocused images, FWHM (full width at half maximum) of the line graphs is used to qualitatively measure the depth of field. The above results are with NA= 1. Keeping the immersion medium as water, the effects of numerical aperture is studied by varying the angular semiaperture. As seen from Fig. 6(g), the DOF is smaller with the increasing NA. We also note that even though the DOF is usually reported as a single number, the sensitivity to the polarization of the dipole is not often reported such as done here. The variation of DOF when using glass or silicon substrate is not significant in Fig. 6(g). The x-polarized dipole generally have smaller focused sizes than the z-polarized dipole. Fig.   6(h) shows the sensitivity of the DOF to the location of the distances of the interface from the focus. We note a certain oscillatory behaviour of the value of DOF, which is more prominent in the case of silicon as compared to the glass substrate. The second case is due to the fact that it may have significantly higher resolution than the first case (see Supplement 1).

Comparison of lateral resolution for silicon and glass substrate.
Setting the positions of dipoles are (−∆x obj /2, 0, z obj ) and (∆x obj /2, 0, z obj ), the modified Rayleigh limit is obtained by increasing ∆x obj with the step 0.02λ and taking the minimal ∆x obj when the saddleto-peak ratio is ≤ 0.735. We assume that two dipoles have the distance to the focal plane less than or equal to λ and the substrate interface is below the focal plane up to 2λ, i.e. z b obj ∈ (0, −2λ]. The results without the presence of half space are also included for comparisons. Fig. 7 is quantified in λ. The region bounded by white dashed lines and filled with black represents the non-physical condition when the dipoles are below the substrate interface and therefore not applicable for study. The estimated resolutions with the glass substrate are presented in Fig. 7(e-h). Due the small refractive index difference with water, the variation of resolution follows the similar phenomena with the situation when no presence of the substrate (the associated results in Fig. 7(i)).

The estimated resolution in
The common observations consistent between the no-substrate and glass-substrate are presented here. In particular, we report the observations related to the resolution in the four cases when the z-location of the dipoles is changed. The resolution tends to be better when the two x-polarized dipoles and the x, zpolarized dipoles get closer to the focal plane. For the two z-polarized dipoles, the estimated resolutions range from 0.86λ to 0.9λ. The resolution for the x, y-polarized dipoles is a constant, except the cases which provide strong evidences for the dependency on the distance between the dipoles and the substrate interface. Moreover, for all considered settings of z obj and z b obj , the x, y-polarized two dipoles are imaged with the best resolution.
As seen from Fig. 7(a-d), the above-mentioned traits also appear with the silicon substrate. Examples are indicated by the dashed lines in Fig. 7(a,c,d). However, the high contrast of silicon (relative to water) leads to more complex trends. For instance, the resolution of the two x-polarized dipoles only has small variations when z obj ≤ 0.4λ, but is very sensitive to the position of substrate when the dipoles are moved further. Irrespective of the dipole polarization, the microscope with the silicon substrate shows a larger range of resolution than with the glass substrate. Interestingly, it can achieve higher resolutions for the best cases. Examples are given in Fig. 8(a-c), where the two x-polarized dipoles are not well separated in the images without the substrate or with the glass substrate but are clearly distinguishable using the silicon substrate. The estimations in Fig. 7 also reveal that samples on the focal plane may have lower-resolution images than those off the plane depending upon the position of the interface. Fig. 8(d,e) show that the nonresolvable x-polarized dipoles on the focal plane becomes resolvable when the dipoles are on the plane 0.6λ lower. Fig. 8(f,g) present the similar phenomenon for the x, z-polarized dipoles. However, it is should be noted that the shift of dipoles off the focal plane may lead to higher side lobes, as seen in Fig. 8(g). The longitudinal resolution is studied by putting two dipoles at the optical axis but separated by distance ∆z obj , which is increased from 0 by the step 0.02λ. When z b obj < −∆z obj , the two dipoles are assumed with z coordinates −∆z obj /2 and ∆z obj /2, respectively. Otherwise, the z coordinates equal z b obj and z b obj + ∆z obj to ensure two dipoles are above the substrate. Fig. 10(a-d) show the variation of saddleto-peak ratio with the increasing distance between dipoles and the varied position of the substrate. The ratio would be set as 1 when the two dipoles cannot be clearly distinguished (e.g., images of a single spot or with high side lobes). As seen, for the x, y-polarized dipoles, the ratio monotonically decreases as the distance between dipoles is larger with the glass substrate. However, for the x, x-polarized dipoles, the saddle-to-peak ratio shows oscillatory behaviors which can make the two dipole unresolvable even when the distance is larger than the modified Rayleigh limit, which is determined by the threshold of 0.735.
Setting z cam = 0, I xx is a function of ∆z obj and the normalized I (1) xx is plotted in Fig. 10(e). The local minima coincide with the minima of the line graphs of saddle-to-peak ratio for the three situations, i.e., without substrate and the presence of glass substrate and silicon substrate 500 nm below the focal plane. As seen, responses due to the two dipoles interfere destructively. In consequence, a minimum amplitude is observed at O cam , which leads to the resolvability of the two dipoles, but the positions of the peak values are shifted.

Conclusion
The dyadic Green's function (DGF) is solved as multiple Sommerfeld integrals. Our formulation includes the effect of reflections from the substrate over which the sample rests as well as realistically high numerical aperture for high NA water immersion systems. This allows us to emulate the 3D full-field effects and thereby more accurately study the resolution and the other effects, especially as a consequence of using high refractive index substrate such as silicon.
The lateral resolution is found dependent on the polarization of dipoles and also the position of the substrate interface in the case of silicon substrate. It is also demonstrated that such variation can be neglected when using glass substarte because of small refractive index contrast with water medium. It is also noted that simply using a high refractive index substrate can alter the range of resolution significantly.
However, the effect of high refractive index substrate on DOF is not quite significant.
The longitudinal resolution is studied with two cases, { p 1 , p 2 =x} and { p 1 =x, p 2 =ŷ}. In the former case, as the distance between dipoles increases, the saddle-to-peak ratio has oscillatory behaviors which can make the two dipole unresolvable even when the distance is larger than the conventional definition of resolution. In the latter case, with the glass substrate, the ratio monotonically decreases as the distance between dipoles is larger. However, it is not always true with the silicon substrate.
While this study indicate the possibility of achieving better resolution using silicon substrate, it also provides insight into large sensitivity of image behaviour on the relative position of interface and dipoles.
We expect that this analysis and the full-field derivation of DGF will open new avenues for exploring high refractive index materials as substrates for use in coherent microscopy. the amplitude of the integral. From the second summation term, the longitudinal magnification is derived as Remark that the derivation of M lat is with no assumptions or approximations, while the expression of M lon is with the assumptions that R TM , R TE ≈ 0 and θ max obj → 0. However, the assumption of paraxial approximation and neglected reflections by the interface may be invalid, especially when the substrate has a high contrast with the immersion medium.
Properties of dyadic Green's function: Only the resolution along the x direction is analyzed based on the following study of ellipticity. Putting a z-polarized dipole at the optical axis, the observed intensity would be the same if the x coordinate and the y coordinate of the observation point are exchanged, i.e., I( r ccd ) = I( r * ccd ), r ccd = [x ccd , y ccd , 0] and r * ccd = [y ccd , x ccd , 0]. This property can be derived from the solution of DGF in Eq. (9). The observed intensity is the summation of intensity of three field components, i.e., I = |I xz | 2 + |I yz | 2 + |I zz | 2 . The observation position impacts the field solution through ψ and γ. From the definitions in Eq. (10), we see cos ψ( r ccd ) = sin ψ( r * ccd ), sin ψ( r ccd ) = cos ψ( r * ccd ) and γ( r ccd ) = γ( r * ccd ). Consequently, I xz ( r ccd ) = I yz ( r * ccd ), I yz ( r ccd ) = I xz ( r * ccd ), and I zz ( r ccd ) = I zz ( r * ccd ). Thus, I( r ccd ) = I( r * ccd ). This identity reveals that no ellipticity for z-polarized dipoles. To check the ellipticity for x-polarized dipoles, values of FWHM associated with observations along the x axis and y axis are compared by varying the numerical aperture (NA) and the position of substrate, respectively. Despite the fact that the resolution would be improved with a higher NA lens, Fig. S1(a) shows that with the silicon substrate 1 µm below the focal plane (the dipole is placed at the focal point), the resolutions alongx andŷ only have small differences. Then, fixing NA= 1, Fig. S1(b) shows that the resolution has variations with different placements of the substrate. However, the maximum difference of FWHM w.r.t. the two observation directions is 0.048λ, which is quite small. Therefore, the inference regarding resolution along x-direction can be easily generalized to the resolution along y-direction and only the lateral resolution along the x direction is analyzed in the paper.
More properties about the dyadic Green's function, first, the resolution would be the same if the polarization of two dipoles is exchanged. Second, the x-polarized dipole and the y-polarized dipole behaves similarly and have approximately the same resolution. This conclusion is made by realizing that when observing field at r ccd for the x-polarized dipole and at r * ccd for the y-polarized dipole, since I xx ( r ccd ) = −I (2) xx ( r * ccd ), I xy ( r ccd ) = I xy ( r * ccd ) and I zx ( r ccd ) = I zy ( r * ccd ), we have the identity I( r ccd ) = I( r * ccd ), i.e., the image of the y-polarized dipole would the same with the x-polarized dipole after exchanging the x and y coordinate of observation points. Together with the small ellipticity, we conclude that the lateral resolutions w.r.t. these two polarizations are with little difference.
The second case is due to the fact that it may have significantly higher resolution than the first case, as shown by Fig. 7. Sinusoid-like observations along the optical axis: The fields in Fig. 10(g) behave as a modulated sinusoid with a high (spatial) frequency. z cam impacts the field solution through the exponential term in Eq. (11). With the knowledge that cos θ cam ≈ 1 (due to the condition f cam f obj ) and e ikcamzcam is a sinusoid about z cam , it is straightforward to do the investigation with the rewritten form of Eq. (11), Denote the integral in (S.4) asĨ. Fig. S2 shows the value ofĨ, which also behaves as a sinusoid but with small amplitude and low frequency. Therefore, the combined field is a sinusoid whose amplitude is modulated byĨ.