The scattering process can be thought of as probabilistic, with a statistical distribution of possible new directions of travel with respect to the original. In the case of scattering at a surface interface between two bulk materials, the new direction of travel is distributed according to a function called the Bidirectional Scattering Distribution Function (BSDF).
Introduction
Is there a practical and reliable way to calculate the optical surface BSDF from surface topography measurements? It is important to know and understand all the possible mechanisms by which scattered scattered light can originate from precision optical systems, as well as other factors that can affect performance.
Precision Imaging Optical Systems
Aberration Retrieval
The PSF is only measured indirectly by performing an autocorrelation of the pupillary function computationally reconstructed from the interferogram [27]. The pupillary function contains aberration information and the PSF is the Fourier transform of the autocorrelation of the pupillary function.
Denitions and Terminology
Bidirectional Scattering Distribution Function
The BSDF of a material is not directly measurable, since light sources and detectors actually have angular apertures or linear ranges. Assuming that the BSDF is not directly measurable, when referring to BSDF measurements, it should be understood to include any actual measurements that are weighted spatial or spectral averages of the true BSDF.
BRDF and BTDF
In this respect, BSDF is not fundamentally different from other physical measurements such as optical power spectra which are thought to have a true underlying value, which is imperfectly measured [34].
Denitions Specic to Optical Surfaces
- Total Integrated Scatter
- Surface Topography and Power Spectral Density
- Autocovariance Function
- Angular Resolved Scattering
- Surface PSD Relationship to Image Quality
TIS is the ratio of the surviving ux scattered out of the specular beam to the total surviving (non-absorbed) ux. The statistical nature of the surface topography is captured using the autocorrelation of the topographic function [11].
Optical Scatter
Rayleigh-Rice Scattering
The BSDF in this case is a function of the (first order grating equivalent) amount, β, by which the beam is spread from the direction of the mirror by a spatial surface frequency component of f cycles per distance unit. Note also that since the PSD for a zero mean height distribution typically levels o at a spatial frequency off = 0, the BSDF also levels o in the specular direction where β = 0.
Generalised Harvey-Shack Scattering
The PSD is expressed here as a function of spatial frequency in any direction along the surface which is only possible for isotropic surfaces that have no azimuthal dependence of the distribution. The general relationship between the surface topography function z(x, y), surface ACV, surface PSD, and surface BSDF is illustrated in Figure 2.3.
Scatter Models for Stray Light Analysis
The Harvey Models
For this and other reasons, the 2-parameter Harvey model presents a consistency problem for TIS determination. At the shoulder value β =l, the BSDF begins to tend toward the simple inverse power law behavior of the 2-parameter model, also exhibiting the log-log slope of s.
The K-Correlation Model
At the shoulder value of β =l, the BSDF begins to tend towards the simple inverse power law behavior of the 2-parameter model, which also exhibits the log-log slope of s. and polishing) manufacturing methods [61,66].
Scatter Model Selection
Surface Prolometry and BSDF
Improvements in measurement techniques and scattermeter design, combined with a very careful geometric definition of the source beam and sensor apertures, have made it possible to perform measurements increasingly closer to the mirror direction [11,21]. The instrument signature is usually measured before sample insertion and subtracted from a measurement taken while the sample is in place. Ultimately, there will be interest in the overall scattering performance of the complete optical system, which may include many curved surfaces on a number of different optical substrate materials and with different thin lm coatings.
If the scattering properties of the individual surfaces are known, it is possible to model the overall performance of the system, but as mentioned above, there is considerable room for uncertainty.
Segmented Aperture Interferometry
Focal Plane Irradiance
The Fresnel integral has within the integral, in addition to the linear terms in the Fraunhofer integral (equation 3.2), a complex exponential quadratic term as:. 3.7) However, in the paraxial approximation of a thin lens of focal length fl, the phase change occurring in the plane of the lens pupil can also be expressed as the square term of the pupil coordinates. When the phase retarder and diaphragm mask are placed at the front focal point of the lens (called the telecentric stop position), d = fl and the phase factor u2 = 1. In the cases we are interested in, the lens will have very little vignetting (spatial apodization of the wave amplitude fronts) eect.
Assuming that the transmission/vignette factor, V, of the lens can be neglected, and the incident amplitude, A, is set to the plane wave with segmented aperture phase delay function given in Equation 3.3, the resulting image plane amplitude distribution follows the pattern of Equation 3.6.
Source and Detector Aperture Eects
A cross-section of the normalized irradiance in the x-direction through the origin is shown in figure 3.6, where φl = 4πxλf0x2. As might be expected, turning the phase retarder will give rise to a moving interference fringe pattern. A slit source with the long axis in the y direction is therefore chosen for analysis, as this will have the least influence on the depth of the irradiance zero.
The dimensions of the slit source projected onto the image plane will be denoted by sx and sy.
Bilaterally Symmetric Segmented Pupils
These convolutions, although difficult to solve analytically1, can be computed numerically using dense x2andy2 sampling and without using Fourier methods. The complex, incoherent OTF is then calculated as the normalized complex autocorrelation (denoted ?) of the generalized pupil function. The incoherent PSF is proportional to the Fourier transform of the incoherent OTF and the OTF is the autocorrelation of the pupil function (see Figure 2.2).
As a first check on the result in Equation 3.15, the rectangular diaphragm arrangement described by Equation 3.3 and illustrated in Figure 3.3 will be used and the result compared with Equation 3.12.
Circular Pupils
The Fourier transform of the pupil apodization function in Equation 3.16 is the convolution of the Fourier transform of the circular and step function. This follows from the convolution theorem, which states that the Fourier transform of the functional product is the convolution of the Fourier transforms of the factors [76]. Carrying out the Fourier transform of this complex pupil function is analytically demanding and was therefore not pursued further.
Therefore, we searched for an alternative approach to the solution for the fractional Hilbert phase mask, which was found in the form of the Nijboer-Zernike approach [28].
Imperfect Nulling
- Polychromaticity
- Non-Uniform Illumination or Vignetting
- Geometry Errors
- Aberrations
In principle, it is possible to achromatize the phase delay or at least to reduce the magnitude of the dependence of delay on wavelength. Non-uniform illumination of the pupil or lens vignetting effectively introduces a spatial amplitude apodization effect. This will have to be taken into account when analyzing the eectiveness and depth of the interference null.
For example, if the edges of the apertures were not perfectly straight or orthogonal, this would have an effect on the PSF.
Summary
Some of the most demanding imaging applications are those in microscopy, astronomy, space optics, and lithography. The design and fabrication of lens assemblies for ultraviolet lithography is a key technology in the fabrication of state-of-the-art electronic integrated circuits (ICs). Figure 4.2 shows the completed opto-mechanical assembly of the Zeiss® Starlith 1000 lithographic lens operating at a wavelength of λ=248 nm [80].
A variety of phenomena can degrade the contrast or quality of the image produced by a precision optical system.
Optical Design and Opto-mechanical Design
The method of support affects things like the self-weight distortion of the optical component under the influence of gravity. Components can move due to temperature fluctuations as mechanical components expand and contract. Ghost images are controlled by manipulating the geometric layout of the optical components, as well as by using thin optical lms (anti-reflection coatings).
Considering the image quality required of a system, as well as the operating environment in which the system must maintain performance, combined optical and opto-mechanical designs must fit within manufacturing tolerance bands to meet these requirements [87] .
Optical Manufacture
System integration involves the construction of the optical sub-assemblies and the system from the components. The methods, controls, and processes used for system assembly can affect the quality of the resulting system [87]. These manufacturing defects and deficiencies, both in optical and opto-mechanical components and assemblies, will generally result in degraded image quality.
Operational Environment
Strong radiation (eg protons, gamma rays) encountered in space can cause damage to glass types that are not designed to withstand such exposure [104,105]. While the above effects refer to optical scattering due to features occurring at different physical size scales, it is important to note that there is also background scattering caused by the fundamental physical grain (atoms, molecules) of the materials that make up the optical system. In particular, gases between optical components and glass materials will exhibit both elastic and inelastic molecular scattering within the bulk of the material.
In the case of EUV lithographic systems, evacuation of the system is mandatory because air strongly absorbs EM radiation at the operating wavelength of λ=13.5 nm.
Summary
Normalization of the Zernike functions is not always included by authors in the definition (Equation 1) (excluded by Born and Wolf [1], included by Noll [7] and Thiboset al. [6] for example). The inner product of a function Φ(ρ, θ) with the Zernike functions is defined as. 5) The inner products of the Zernike function with them-. This code was used for calculating the field in the focal plane for the circular pupil fractional Hilbert mask.
The normalization of Zernike functions is not always included in the definition (Equation (1)) by authors (except by Born and Wolf [1], included by Noll[7] and Thibos et al.[6] for example). For the defocus parameterf¼0, one can use the result of the classical Nijboer-Zernike (CNZ) theory in which the sum over l in equation(43) reduces to a single Bessel term as[4]. The convergence of the CNZ result for the circular pupil, the partial Hilbert mask (Equations (40), (41) and (45) with the complex-valued coefficients Bmn calculated with equation (48) or 38) was tested with increasing N ( value maximum of n).
