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 , where the incident electric and magnetic fields are denoted by $E_z^{inc}z$ and $H^{inc}$. The scattered electric field wave $E_z^{scatt}$emanates at all angles $0\le \left|\varphi \right|\le \pi$ . For this scattering configuration the exact solution of Maxwell’s equations is well known [1–4]. 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 $2\pi a/\lambda$of incident laser radiation wavenumber and cylinder radius ($a$) spans a range from $3\times {10}^3$to $1\times {10}^6$ radians.

 

Fig. 1 Optical metrology system schematic: C = detector array, D = computer

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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 $4\mu m$ 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 [5]. 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.

In this research our major goal is to demonstrate from exact theory and careful computer simulation that the remote sensing approach shown in Fig. 1 is adequate and practical for determining the presence of a thin film coating on a cylinder, and for estimating its thickness. Section 2 presents a brief review of the theory. In Section 3 a new error analysis of the eigenfunction expansion is developed showing that a direct calculation of the eigenfunction series for $E_z^{scatt}$is feasible without recourse to the Watson transformation or the geometrical theory of optics [6,7]. Features of the simulated scattered electric field are discussed in Section 4 to show that it is practical, with a CMOS detector, to sense remotely enough of the scattered radiation to obtain the desired results on the presence of a thin film coating and its thickness in the$\mu m$range.

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 $\pi$steradians. 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 [6,7]. 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 incident plane wave electric field with wavenumber $k_0$shown in Fig. 1 takes the form

The scattered electric field has the form of outgoing waves [1–3], and therefore can be written as a superposition of Hankel functions of the second kind:

Inside the dielectric coating, the electric field can be written as a sum of Bessel functions of the first and second kind:

The tangential component of the electric field vanishes at the surface of the conducting cylinder at $\rho =a$, and the tangential components of both the electric field and the magnetic field are continuous at the surface of the dielectric at $\rho =b$. These conditions result, for each $m$, in three independent linear equations in the coefficients $A_m$,$B_m$, and $C_m$. The requirement that the tangential electric field vanish at the surface of the conducting cylinder means that the coefficients $C_m$ can be eliminated setting Eq. (4) to 0 at $\rho =a$:

The linear equations for $A_m$and $B_m$resulting from the requirement for continuity of the tangential component of the magnetic field $H$at the dielectric boundary $\rho =b$ is obtained from Faraday’s law

A second linear equation for $A_m$and $B_m$ is obtained by equating the tangential dielectric field in Eq. (3) at the surface of the dielectric boundary $\rho =b$to the total tangential electric field formed from the sum of Eqs. (2) and (3). The linear equations for $A_m$and $B_m$ take the form

A formal solution to Eq. (9) for $A_m$can be written in terms of the determinant of the 2 x 2 matrix ${\Lambda }_m$and its entries using Cramer’s rule, as follows:

The accuracy of values for $A_m$computed using Eq. (11) depends on ${\Lambda }_m$being well-conditioned [8]. If ${\Lambda }_m$is ill-conditioned, the relative accuracy of computed values for $A_m$will 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 $E^{scatt}$ using a truncated eigenfunction expansion,

For a given value of $m$, the reliability of the computed eigenfunction expansion coefficient $A_m$depends on the sensitivity of its computed value to computer roundoff errors. The condition number $c_m$ [8] of the matrix ${\Lambda }_m$bounds the effect of roundoff errors on computed values of the expansion coefficients $A_m$as follows. Let

Roundoff errors present in the computed values for the Bessel function $J_m(k_{0}b)$ and its derivative ${J^{\prime }}_m(k_{0}b)$in the right-hand side of Eq. (9) can be viewed as introducing a perturbation $\delta y_m$to the exact value of$y_m$defined by Eq. (13). This perturbation causes the computed value of $x_m$to contain an error term $\delta x_m$ to which the perturbed version of Eq. (13) applies:

The condition number $c_m$bounds the relative error in the computed solution due to roundoff errors as follows:

A similar expression bounds the effect of roundoff errors in ${\Lambda }_m$on errors on the computed solution in terms of the condition number$c_m$ . Equation (16) shows that if the condition number$c_m$ is too large, a slight variation in the value of$y_m$ caused by roundoff errors can cause unacceptably large errors in the computed value for $x_m$.

To provide an example, suppose that 16-digit double precision accuracy is employed and the computed values for the Bessel function $J_m(k_{0}b)$ and its derivative ${J^{\prime }}_m(k_{0}b)$in Eq. (11) are accurate to one part in ${10}^{-16}$. If the condition number of the matrix ${\Lambda }_m$ is ${10}^{10}$, then the resulting relative error in $A_m$is less than 0.0001%.

Figure 2 gives a plot of the condition number $c_m$ of the matrix ${\Lambda }_m$defined in Eq. (10) as a function of $m$for the case where the inner cylinder radius $a$ is 0.225 mm and the dielectric coating thickness $T$is 10 microns. For this geometry, $k_{0}b\approx 3025$. Figure 2 shows that ${\Lambda }_m$is well-conditioned when $m<k_{0}b$but becomes ill-conditioned when $m\ge 3200$. This gives an absolute limit on the number of terms $M$ that can be included in the finite sum approximation to the eigenfunction expansion in Eq. (12).

 

Fig. 2 Logarithm of condition number $c_m$of ${\Lambda }_m$

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Truncating the eigenfunction expansion infinite sum at a finite value causes the computed value $E_{est}^{scatt}$ to differ from the exact value $E^{scatt}$ by an amount that depends on the values of the Fourier coefficients $A_m{H}_m^{(2)}(k_0\rho )$ for $m>M$. Figure 3 shows a plot of the logarithm of $\left|A_m{H}_m^{(2)}(k_0\rho )\right|$ when the inner cylinder radius $a$ is 0.225 mm and the dielectric coating thickness $T$is 10 microns.

 

Fig. 3 logarithm of Fourier coefficient magnitude $\left|A_m{H}_m^{(2)}(k_0\rho )\right|$

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Figure 3 shows that the Fourier coefficient magnitudes are generally comparable in magnitude for $m<k_{0}b$but then decrease exponentially very rapidly. When the matrix ${\Lambda }_m$becomes ill-conditioned at approximately $m\approx 3200$, values of $\left|A_m{H}_m^{(2)}(k_0\rho )\right|$are seen to be physically negligible, and decrease monotonically at an approximately exponential rate.

The Fourier coefficient magnitudes become physically negligible for values of $m$ well below the threshold value of 3200 where the invertibility of ${\Lambda }_m$becomes problematic.

In the case considered here, use of the cylindrical coordinate frame enables use of the Fast Fourier Transform (FFT) to evaluate the computed scattered field. This extends application of the method to cases where a very large number of Fourier coefficients must be determined. The applicability of the FFT is apparent if Eq. (4) for the approximate scattered field $E_{est}^{scatt}$is evaluated at scattering angles ${\varphi }_l=l\frac{2\pi }{2M}$, for $-2M\le l\le 2M,$ so that

It is apparent that the sequence of values $E_{est}^{scatt}({\varphi }_l)$is determined by the FFT of the sequence $\left\{A_m{H}_m^{(2)}(k_0\rho )\right\}$ 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

We describe the optical metrology for measuring the thin lubricant coating on a polished metallic cylinder, as shown in Fig. 1. Since our theory and calculations are based on exact solutions of Maxwell’s equations, it will be clear and convincing that it is relatively simple and effective first to detect the presence or absence of said thin dielectric coating and secondly to measure the thickness in the range from 1 μm to 100 μm. The incident plane-polarized laser beam can be obtained from a low power He-Ne laser (10 mw) or from a medium power Argon laser (1 to 10 w). From prior studies of hypodermic needles, it is known that a 4 mm circular spot is effective to model the infinite laser beam (unless one wishes to study the variation of film thickness along the length of the needle). The scattered beam is measured in power by the CCD or CMOS detector array by recording a signal proportional to the energy density in the electric field, i.e., to ${\left|E^{scatt}\right|}^2$.

From well-known ranging calculations, one can readily show that placing a CMOS detector array at distances$\rho$ranging 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 $\lambda$ 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 amplitude$E_z^{inc}$of $25\times {10}^3$v/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

Using Eq. (12) we plot curves in Figs. 4 -6 for scattering from a metallic cylinder with radius $a$ at 0.225 mm with dielectric coating thicknesses T of 0$\mu m$, 4$\mu m$ and 10$\mu m$respectively, over a range of scattering angles $\phi$ from 0 to $\pi$. The normally incident TMz polarized plane wave travels in the ( + x) direction. Figures 4-6 show values for ${\left|E^{scatt}\right|}^2$, with all curves normalized to the incident electric field amplitude of 25 x 10³ V/m that corresponds approximately to a laser power level of 1 watt in a uniform beam diameter of 4 mm. There is much signature material at various angles and these data clearly show an ease of classification and measurement for the determination of thickness. The curves also show much hidden detail so that finer resolution is needed in the curves as is given below.

 

Fig. 4 Normalized scattered field intensity${\left|E^{scatt}\right|}^2$, dielectric thickness = 0

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Fig. 6 Normalized scattered field intensity ${\left|E^{scatt}\right|}^2$, dielectric thickness = 10 microns

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Fig. 5 Normalized scattered field intensity${\left|E^{scatt}\right|}^2$, dielectric thickness = 4 microns

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4.2 Backscattering for the TMz incident polarization

While not central to our optical metrology system, the backscattering of radiation shows an interesting resonance for a thickness in the neighborhood of one micron with a dielectric coating index of refraction $n_0=1.5$. Figure 7 shows backscattered energy density for a 20 degree range in the backscatter region. We plot ${\left|E^{scatt}\right|}^2$normalized to 25 x 10³ V/m incident electric field for a thin one-micron dielectric film. It has a multiple frequency-modulated (FM) spectrum. Detailed consideration is beyond the scope of this paper.

 

Fig. 7 Backscattered energy intensity${\left|E^{scatt}\right|}^2$, dielectric thickness = 1 micron

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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.

First, we plot the energy density curves for a limited twenty-degree range, i.e., $0\le \phi \le \pi /9$. Figures 8 and 9 show scattered field energy density curves for dielectric coating thicknesses T at 0 $\mu m$thickness and 4 $\mu m$, respectively. We plot these on a linear y-axis scale in order that one can see the small amplitude differences. Again, the complex frequency of the oscillations is apparent. Clearly, the high frequency term is similar in both scatterers and a quick calculation shows one that the spatial frequency is determined by the angular scattering width of$\lambda /2b$radians. This radian frequency changes by a small percentage as the film thickness T increases, as one can verify by close study of the figures.

 

Fig. 8 Scattered energy density, dielectric thickness = 0 microns

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Fig. 9 Scattered energy density, dielectric thickness = 4 microns

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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 through 12 which are for dielectric thicknesses of 0, 4, and 8 microns respectively.

 

Fig. 10 Scattered energy density, dielectric thickness = 0

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Fig. 12 Scattered energy density, dielectric thickness = 4 microns

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Fig. 11 Scattered energy density, dielectric thickness = 4 microns

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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

5. Summary

A laser optical metrology system is shown to be effective for the rapid and remote determination of the presence or absence as well as the thickness of a thin dielectric coating on a hypodermic cylinder. In the approach presented, a TMz polarized plane wave is incident on the dielectric coated cylinder oriented along the z-axis, and analysis of the scattering pattern is used to determine the thickness of a thin dielectric coating. Exact solutions for the scattered wave are given with the eigenvalue expansion coefficients written explicitly for the scattered field $E^{scatt}$. Computational accuracy of the resulting values is analyzed using a new application of the condition numbers of the matrices representing the boundary value matching equations. Clear signatures in the scattered electric field are seen for various thicknesses of the thin dielectric film over the range of scattering angles from 0 to 180 degrees. Neither direct forward scattering nor back-scattering are ideal choices for the problem at hand. The scattered field is used to show that placement of a CMOS detector in the range of scattering angles from 5 to 10 degrees is well-suited for the metrology of the thin dielectric film. Scattered field characteristics suitable for use for pattern recognition determination of thickness include amplitude, carrier frequency, and low-frequency modulation. It is interesting to see that the modern desktop computer provides the means for applications of the calculation of exact solutions of Maxwell's equations to the optical scientist who need not be a specialist in electromagnetic waves.