1. Introduction
In this paper we consider a monochromatic plane wave incident normally on a metallic cylinder that has a thin dielectric coating. The TMz incident wave is linearly polarized along the z-axis of the coated cylinder, as shown in
Fig. 1
Fig. 1 Optical metrology system schematic: C = detector array, D = computer
, where the incident electric and magnetic fields are denoted by
Ezincz and
Hinc. The scattered electric field wave
Ezscattemanates at all angles
0≤|ϕ|≤π . For this scattering configuration the exact solution of Maxwell’s equations is well known [
11. C. A. Balanis, Advanced Engineering Electromagnetics (J. Wiley & Sons, 1989).
–
44. M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, 1969).
]. There is considerable literature for various practical situations, mainly in the microwave regime. However, our interest is motivated by the need for accurate and rapid metrology of various fluids used as lubricants and coating deposition on hypodermic needles and other medical devices. Typically, for these applications the product
2πa/λof incident laser radiation wavenumber and cylinder radius (
a) spans a range from
3×103to
1×106 radians.
Since many medical devices such as hypodermic needles are manufactured in a clean room environment, it is desirable to have a remote sensing means both for checking the presence or absence of the thin lubricant and also to measure its thickness,
T=b−a, typically
4μm or so. While not directly applicable to the clean room requirement, it is interesting to try using a laboratory microscope in inspecting for a thin micron-like film on a curved surface. Even using a modern microscope, one finds that imaging this thin film is quite difficult. Ellipsometry is the well-known measurement method for measuring the thickness of thin films on flat surfaces and sophisticated precise instruments using polarimetry are commercially available [
55. Handbook of Ellipsometry, H.G. Tompkins and E.A. Irene, Eds. (William Andrew, 2005).
]. To our knowledge none of these instruments will rapidly measure the thin film on the curved surface at optical wavelengths with the small radius (0.225 mm) that we are considering. The authors stress the applicability and compatibility of the novel
Fig. 1 apparatus to high-rate production of coated needles or wires, since neither polarization optics nor microscopic objectives are required in the optical train.
2. Electromagnetic theory for the scattered field
In order to determine an appropriate scattering angle for placement of the detector array in the metrology configuration shown in
Fig. 1, it is necessary to evaluate accurately the scattered field at very high frequencies over
πsteradians. The results presented in this paper show that it is feasible to compute the scattered field where needed by truncating its eigenfunction expansion infinite series to a finite sum, in contrast to previous authors using other methods [
66. G. James, Geometrical Theory of Diffraction for Electromagnetic Waves (Peregrinus, 2003).
,
77. V. Borovikov and B. Kinber, Geometrical Theory of Diffraction (Institute of Electrical Engineers, 1994).
]. The development of the eigenfunction expansion is summarized below.
The advantage to using a computationally feasible implementation of the eigenfunction expansion is that the electric field can then be computed simply and accurately at any point in the radiation field.
The tangential component of the electric field vanishes at the surface of the conducting cylinder at
ρ=a, and the tangential components of both the electric field and the magnetic field are continuous at the surface of the dielectric at
ρ=b. These conditions result, for each
m, in three independent linear equations in the coefficients
Am,
Bm, and
Cm. The requirement that the tangential electric field vanish at the surface of the conducting cylinder means that the coefficients
Cm can be eliminated setting
Eq. (4) to 0 at
ρ=a:
The linear equations for
Amand
Bmresulting from the requirement for continuity of the tangential component of the magnetic field
Hat the dielectric boundary
ρ=b is obtained from Faraday’s law
applied at
ρ=bto the electric fields
Escattand
Ediel defined in
Eqs. (3) and
(4). In this problem the electric fields involved are everywhere parallel to the z-axis, so the magnetic field
Hhas only radial and azimuthal components
Hρ and
Hϕ. The tangential component of the magnetic field at the surface of the dielectric at
ρ=b is the azimuthal component
Hϕ, which in this case is given by
The accuracy of values for
Amcomputed using
Eq. (11) depends on
Λmbeing well-conditioned [
88. G. Strang, Linear Algebra and its Applications (Thompson Brooks/Cole, 2006).
]. If
Λmis ill-conditioned, the relative accuracy of computed values for
Amwill be overly sensitive to roundoff errors, as discussed in more detail below.
3. Numerical analysis and computational accuracy
In order to generate satisfactory values for
Escatt using a truncated eigenfunction expansion,
two requirements must be met. First, the coefficients
Amused in
Eq. (12) must be numerically accurate. Second, the value of M chosen for
Eq. (12) must be large enough so that the terms omitted from the exact eigenfunction expansion in
Eq. (3) are negligible.
For a given value of
m, the reliability of the computed eigenfunction expansion coefficient
Amdepends on the sensitivity of its computed value to computer roundoff errors. The condition number
cm [
88. G. Strang, Linear Algebra and its Applications (Thompson Brooks/Cole, 2006).
] of the matrix
Λmbounds the effect of roundoff errors on computed values of the expansion coefficients
Amas follows. Let
where
xmand
ymrepresent numerically exact values as follows:
Roundoff errors present in the computed values for the Bessel function
Jm(k0b) and its derivative
J′m(k0b)in the right-hand side of
Eq. (9) can be viewed as introducing a perturbation
δymto the exact value of
ymdefined by
Eq. (13). This perturbation causes the computed value of
xmto contain an error term
δxm to which the perturbed version of
Eq. (13) applies:
The condition number cmbounds the relative error in the computed solution due to roundoff errors as follows:
A similar expression bounds the effect of roundoff errors in
Λmon errors on the computed solution in terms of the condition number
cm .
Equation (16) shows that if the condition number
cm is too large, a slight variation in the value of
ym caused by roundoff errors can cause unacceptably large errors in the computed value for
xm.
To provide an example, suppose that 16-digit double precision accuracy is employed and the computed values for the Bessel function
Jm(k0b) and its derivative
J′m(k0b)in
Eq. (11) are accurate to one part in
10−16. If the condition number of the matrix
Λm is
1010, then the resulting relative error in
Amis less than 0.0001%.
The Fourier coefficient magnitudes become physically negligible for values of m well below the threshold value of 3200 where the invertibility of Λmbecomes problematic.
It is apparent that the sequence of values Eestscatt(ϕl)is determined by the FFT of the sequence {AmHm(2)(k0ρ)} of length 2M.
The approach outlined above can be offered as a candidate for consideration for solving other scattering problems in which boundary value conditions give rise to a set of linear equations for the eigenfunction expansion coefficients. The procedure is to evaluate the condition number of the matrix associated with the boundary value matching linear equation set, and simultaneously monitor the value of the eigenfunction expansion coefficients. This is done successively for each eigenfunction coefficient until all remaining coefficients are negligible. If an excessive number of terms are required, or if the matrices to be inverted become ill-conditioned before the associated expansion coefficients are negligible, then another method must be considered. The advantage to the present method is that it gives a uniformly accurate expansion of the scattered field everywhere, without requiring additional error analyses.
4. Preliminary design for optical metrology system & discussions
From well-known ranging calculations, one can readily show that placing a CMOS detector array at distancesρranging from 30 mm to 200 mm will permit one to record the scatter pattern with adequate angular resolution. To illustrate these choices below, we study features in the scattered radiation at a fixed radius of 100 mm and consider a detector array of 10 mm to 14 mm with twelve million pixels. Also we assume a laser power on the order of one watt at wavelength λ equal to 488 nm in a spot size of 4 mm diameter. Summarizing, we normalize the following curves to an incident TMz electric field with amplitudeEzincof 25×103v/m that is consistent with the above-stated values. There is no trouble with signal level in this system relativeto typical CMOS detector noise characteristics.
4.1 Plane wave scattering by conducting circular cylinder with dielectric coating for 0 ≤ ϕ ≤ π
Fig. 5 Normalized scattered field intensity|Escatt|2, dielectric thickness = 4 microns
4.2 Backscattering for the TMz incident polarization
4.3 Forward scattering for the TMz incident polarization
There are two aspects to the measurement at hand. For a complete “fingerprint” of the thickness, recording scattered intensity over a large range of scattering angle at fine resolution is certainly the most conservative procedure. However, if the primary interest is in dielectric coating thickness determination, equivalent results can be obtained with less data storage and simpler software analysis if the data is appropriately sampled, as described in this section.
Moreover, the low-frequency modulation term apparent when the 4 μm coating is in place is a clear feature of the dielectric coating, and it provides an accurate way for measurement of the thickness, T.
To study this in further detail, it is helpful to spread out the plots in ϕ even further, as shown in
Figs. 10
Fig. 10 Scattered energy density, dielectric thickness = 0
through
12
Fig. 12 Scattered energy density, dielectric thickness = 4 microns
which are for dielectric thicknesses of 0, 4, and 8 microns respectively.
Fig. 11 Scattered energy density, dielectric thickness = 4 microns
Two classes of algorithms are immediately evident from these exact theoretical curves: in one the carrier frequency variation vs. thickness is read out and in the other the lower modulation frequency can be monitored. The details of these algorithms are beyond the scope of this paper