Rayleigh-Sommerfeld diffraction on a subwavelength scale: Theories and a resolution criterion

Classical resolution criteria such as Rayleigh's and Sparrow's are based on the fringe structure produced by Fresnel diffraction or Fraunhofer diffraction. However, these two diffraction theories are based on the assumption of large-aperture scale and are thus incapable of describing the diffraction of subwavelength structure. We find a singularity near the incident wave vector $k_0$ in a subwavelength aperture by considering Rayleigh-Sommerfeld diffraction. The diffraction fringe structure disappears once the aperture scale is smaller than a threshold value. A two-point resolution criterion unrelated to the fringe structure is proposed based on the second-order derivative of the field structure. Numerical results indicate that, for the illumination wavelength of $\lambda =500\mathrm{nm}$, resolution of two 100-nm rectangular holes under near-field Rayleigh-Sommerfeld diffraction is improved by about 300 and 20% in the $x$ and $z$ directions, respectively, compared with the Sparrow criterion, while the lateral resolution limit is reduced to 35 nm.

Figure 1

Plane-wave representation of one-dimensional RS diffraction.

Figure 2

Comparison of the normalized amplitude $\left|u/u\left(0\right)\right|$ between Rayleigh-Sommerfeld diffraction and Fresnel diffraction of one-dimensional rectangular apertures with different widths $d$, calculated by the first Rayleigh integral Eq. (2) and the Fresnel approximation Eq. (10), with boundary condition Eq. (11). $\lambda =500\mathrm{nm},z=10\mu \mathrm{m}$. The sampling interval on $x$ and $x^{\prime }$ is 5 nm.

Figure 3

Normalized amplitude $|u/u_0|$ of different structures below the wavelength level, calculated by the first Rayleigh integral Eq. (2) with boundary condition Eqs. (14). $\lambda =500\mathrm{nm}, z=1\lambda$. The three curves are shown as single-hole, double-hole, and three-hole diffraction results, respectively. The aperture $d$ of each hole is 0.2λ, and the central distance $r$ between the holes is 0.4λ. The sampling interval on $x$ and $x^{\prime }$ is 5 nm.

Figure 4

Comparison of second-order derivative resolution and classical resolution limits in general large-aperture incoherent Fraunhofer diffraction, calculated by Eq. (18). Panel (a) shows the normalized intensity $I\left(x\right)$, panel (b) shows its second derivative $I^{\prime \prime }\left(x\right)$ in the $x$ direction, $r$ is the lateral distance between the two point sources, and $a$ is the Rayleigh limit. Dashed, dot-continuous, continuous, and dot-dashed lines correspond to the Rayleigh limit, the Sparrow limit, the second-order derivative limit, and single-point imaging. Although the Sparrow limit appears when $r$ is shortened to around $0.83a$, we cannot see the two-peak structure from the image itself, but the second derivative retains the same extremum structure until $r$ is further reduced to $0.7a$.

Figure 5

Relative resolution limits of two-point coherent Fraunhofer diffraction under (a) different criteria (the second derivative limit over the Sparrow limit) and (b) resolution improvement vs phase difference $\sigma$, calculated by Eqs. (19) and (20), respectively.

Figure 6

Normalized intensity $I\left(x\right)$ (a) and its second derivative $I^{\prime \prime }\left(x\right)$ (b) of coherent RS diffraction results of two 100-nm-wide rectangular holes at different separate distance $r$ with no phase difference after axial propagation at $z=140\mathrm{nm}$. Dot-continuous, continuous, and dot-dashed lines correspond to the Sparrow limit, the second-derivative limit, and single-point imaging, respectively.

Figure 7

Comparison of the resolution limits at different axial propagation distances $z$ under RS diffraction. The dotted line corresponds to the Sparrow limit and the asterisked line corresponds to the second derivative limit, defined as the minimum two-hole distance $r$ in accordance with the second-order derivative resolution criterion and the Sparrow criterion. The on-axis sampling interval is 2.5 nm, and the lateral sampling interval is 5 nm.

Figure 8

Comparison of the normalized amplitude $|u/u_0|$ and the real part Re($u/u_0$) at a single moment of RS diffraction (a) and Fraunhofer diffraction (b), calculated by Eqs. (2) and (22), respectively, with boundary condition Eq. (11). The incident wavelength $\lambda =500\mathrm{nm}$. Aperture width $d=200\mathrm{nm},z=140\mathrm{nm}$ in (a) and $d=10\mathrm{cm}, z=5\mathrm{cm}$ in (b) (consider a focal lens with large NA), and the sampling interval on $x$ and $x^{\prime }$ is 5 nm. Although under such conditions the norms of the light field $\left|u\right|$ of RS diffraction and Fraunhofer diffraction share a similar size, the real part differs from the norm in panel (a) while it does not in panel (b), indicating that phase information is still stored in RS diffraction on such a scale while it is not in Fraunhofer diffraction.

Figure 9

The norm of light field $\left|u\right|$ in the half plane after the diffraction screen of a one-dimensional rectangular hole with different aperture sizes $d$ under the illumination of a monochromatic plane wave parallel to the optical axis, calculated by the first Rayleigh integral Eq. (2) and boundary condition Eq. (11) with dimensionless coordinates $x_1$ and $z_1$. Incident wavelength $\lambda =500\mathrm{nm}$, the axial sampling interval of $z_1$ is 0.1, and the lateral sampling interval on either $x^{\prime }$ or $x$ is $0.01d$. For clarification on axis values and the power transmission problem, please refer to Ref. [22].