Signal modeling in low coherence interference microscopy on example of rectangular grating

Weichang Xie; Peter Lehmann; Jan Niehues; Stanislav Tereschenko

doi:10.1364/OE.24.014283

1. Introduction

Besides confocal microscopy, low coherence interferometry has gained a wide range of applications in high-accuracy topography measurement. Recent developments in micro- and nano-technologies tend to require more challenging demands with respect to lateral resolution. Microscope objectives with high NA are usually applied to improve lateral resolution. It has been shown that the interference fringe spacing will be affected by an interference microscope objective, especially when the objective possesses a high NA [1–3]. As the NA increases, the distance between two fringes in the interference correlogram becomes greater than half of the central wavelength of the illuminating light. In this case, the wavelength applied in phase evaluation should be calibrated resulting in an effective wavelength. The ratio of the effective wavelength to the nominal wavelength depends on the incident angle and therefore on the NA. A useful parameter to study the measurement of step height using white light interferometry is the HWR value, which is the ratio of the step height to the effective wavelength. From the HWR the occurrence of the batwing effect at steep edges can be estimated. Many publications try to explain the relationship between effective wavelength and NA through analysis and measurement [1–6]. Most authors assume monochromatic illumination and a mirror-like measurement object [7–9]. This paper outlines a theoretical and experimental study on the signal formation through the example of rectangular gratings with respect to effective wavelength, HWR value and batwing effect, considering the influence of polychromatic illumination at different center wavelengths and high NA.

2. Signal modeling

Two methods of signal modeling are presented in the following subsections. The first model uses the Kirchhoff scalar theory to simulate the scattering at the rectangular grating and the reference mirror. The object and reference rays are superimposed and a lateral low pass filtering defined by the coherent amplitude transfer function takes place at the pupil plane of the objective lens. The second model utilizes a polarized vectorial calculation of the electromagnetic field near the focus of an aplanatic system, where an average over all polarization states considers the unpolarized case. This method is based on the Debye approximation and was first elegantly summarized and developed by Richards and Wolf [10]. After that, an interference signal is determined by adding the electromagnetic field of object and reference rays numerically. Both models integrate the interference intensity over several incident angles and illumination wavelengths in order to consider limited spatial and temporal coherence. A one-dimensional rectangular grating is assumed in the simulation. The signal modeling is therefore a two-dimensional modelling, namely the interference correlogram is obtained within the xz incident plane.

2.1. Kirchhoff-modeling

One of the most popular approaches to interpret the light scattering from rough surfaces is the Kirchhoff approximation [11, 12]. This theory assumes that the local curvatures of the surface are small enough to fulfill the tangent plane approximation, so that the surface can be treated as a flat plane locally and the Fresnel reflection coefficients can be applied. Therefore, the Kirchhoff approximation is highly appropriate for smooth surfaces without edges [13]. However, Beckmann and Spizzichino [12] pointed out that the presence of surface discontinuities such as rectangular corrugations is not itself a severe restriction of the Kirchhoff approximation. It is the ratio of the width of the corrugations and the separation between adjacent corrugations to the wavelength of illumination that are of primary importance. McCammon and McDaniel [14] concluded that Kirchhoff theory achieves good performance for surfaces with low surface heights, long correlation lengths and low frequencies and angles of incidence less than 60° (corresponding to a NA of 0.87). In case of a rectangular grating, whose period is much larger than the grating height and the illumination wavelength, the Kirchhoff-modelling should be appropriate to our concerns.

The object under investigation h(x₀, y₀) is a rectangular grating with given height h₀ and period Λ. The grating has a constant height value along the y direction. Thus,

h (x_{0}, y_{0}) = h (x_{0}) = \frac{h_{0}}{2} \cdot \frac{cos (2 π / Λ \cdot x_{0})}{| cos (2 π / Λ \cdot x_{0}) |} .

A schematic illustration of the scattering and imaging geometry is given in Fig. 1. The objective lens and tube lens represent the 2D Fourier operators and generate the Fourier transform of the electromagnetic field distribution into the object plane in the Fourier plane and an inverse Fourier transform into the image plane. Low pass filtering takes place in the Fourier plane, where the high spatial frequency components are filtered out by a rectangular amplitude transfer function. According to scalar Kirchhoff theory the scattered amplitude in the object arm of the interferometer can be formulated applying the tangent plane approximation and the far field approximation as described in Eq. 2 [15].

U_{s; k, θ_{e}} (θ_{s}) = - \frac{j U_{0} e^{j k r}}{4 π r} \iint_{S} R_{s} (k, θ_{e}) e^{j ({\vec{k}}_{e} - {\vec{k}}_{s}) \cdot {\vec{r}}_{0}} ({\vec{k}}_{e} - {\vec{k}}_{s}) \cdot \hat{n} d S,

where R_s is the Fresnel reflection coefficient. k⃗_e and k⃗_s are the wave vectors of length k = 2π/λ of the incident and of the scattered wave, r⃗₀ is the coordinate of measurement object and n̂ is the normal vector to the object surface, with

{\vec{k}}_{e} = k (\begin{matrix} sin θ_{e} \\ 0 \\ - cos θ_{e} \end{matrix}), {\vec{k}}_{s} = k (\begin{matrix} sin θ_{s} \\ 0 \\ cos θ_{s} \end{matrix}),

{\vec{r}}_{0} = (\begin{matrix} x_{0} \\ 0 \\ h (x_{0}) \end{matrix}), \hat{n} = (\begin{matrix} 0 \\ 0 \\ - 1 \end{matrix}) .

θ_e and θ_s are the incident and scattering angle, respectively. For conciseness we define

\vec{q} = {\vec{k}}_{s} - {\vec{k}}_{e} = (\begin{matrix} q_{x} \\ q_{y} \\ q_{z} \end{matrix}) = k (\begin{matrix} sin θ_{s} - sin θ_{e} \\ 0 \\ cos θ_{s} + cos θ_{e} \end{matrix}) .

Fig. 1 Schematic illustration of imaging from the object plane to the image plane.

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Assuming P₁(θ_e) is the pupil function of the illuminating lens, which describes the effect of apodization and putting Eq. 4 and 5 into Eq. 2 results in:

\begin{array}{l} U_{s; k, θ_{e}} (θ_{s}) & = \frac{j U_{0} e^{j k r}}{\underset{A}{\underset{︸}{4 π r}}} P_{1} (θ_{e}) R_{s} (k, θ_{e}) q_{z} \\ \cdot \int exp {- j q_{z} h (x_{0})} \cdot exp (- j q_{x} x_{0}) d x_{0} . \end{array}

The coefficient A = jU₀e^jkr/4πr is dropped for further calculation, as we are finally interested in intensity for further envelope and phase evaluation. Analogously, the scattered amplitude of the reference arm can be written as

U_{r; k, θ_{e}} (θ_{s}) = P_{1} (θ_{e}) R_{r} (k, θ_{e}) q_{z} \cdot \int exp (- j q_{x} x_{0}) d x_{0} .

where the height value in the exponential function is zero, as its surface is plane and no depth scanning takes place.

Because of its periodicity, exp{− jq_zh (x₀)} in Eq. 6 can be expressed by a Fourier series:

exp {- j q_{z} h (x_{0})} = \sum_{n = - \infty}^{\infty} C_{n} exp (j \frac{2 n π}{Λ} x_{0}),

where

C_{n} = {\begin{array}{l} cos (q_{z} h_{0} / 2), & n = 0 \\ 0, & n is even \\ {(- 1)}^{0.5 (n + 1)} \cdot 2 j / n π \cdot sin (q_{z} h_{0} / 2) . & n is odd \end{array}

The diffraction order n for the object arm should be constrained by the aperture size of the microscope objective. The scattering angle of diffraction order n is

θ_{s, n} = arcsin (\frac{n λ}{Λ} + sin θ_{e}) .

θ_s,n should satisfy following equation:

- arcsin (NA) \leq θ_{s, n} \leq arcsin (NA) .

Thus,

n_{min} \leq n \leq n_{max}, and n \in ℤ,

with

n_{min} = ceil ((- NA - sin θ_{e}) \frac{Λ}{λ}), n_{max} = floor ((NA - sin θ_{e}) \frac{Λ}{λ}),

where ceil and floor are both operators. While ceil rounds the argument to the nearest integer greater than or equal to that argument, floor rounds the argument to the nearest less than or equal to it.

Substituting Eq. 8 and Eq. 12 in Eq. 6

\begin{array}{l} U_{s; k, θ_{e}} (θ_{s}) & = & P_{1} (θ_{e}) R_{s} (k, θ_{e}) \int \sum_{n_{min}}^{n_{max}} C_{n} q_{z} exp (j \frac{2 n π}{Λ} x_{0}) \cdot exp (- j q_{x} x_{0}) d x_{0} \\ = & P_{1} (θ_{e}) R_{s} (k, θ_{e}) \sum_{n_{min}}^{n_{max}} C_{n} q_{z} \int exp (j \frac{2 n π}{Λ} x_{0}) \cdot exp (- j q_{x} x_{0}) d x_{0} . \end{array}

The integration in Eq. 14 can be regarded as a Fourier transform with respect to the angular frequency q_x. Therefore,

U_{s; k, θ_{e}} (θ_{s}) = P_{1} (θ_{e}) R_{s} (k, θ_{e}) \sum_{n_{min}}^{n_{max}} C_{n} q_{z} δ (q_{x} - n \frac{2 π}{Λ}) .

For the scattered amplitude U_{r;k,θ_e} (θ_s) in the reference arm, no diffraction takes place, i.e., only zero order diffraction will be considered. Consequently, Eq. 7 becomes:

U_{r; k, θ_{e}} (θ_{s}) = P_{1} (θ_{e}) R_{r} (k, θ_{e}) q_{z, n = 0} δ (q_{x}),

where q_z,n=0 = 2k cosθ_e indicates the value of q_z in case of zero order diffraction.

Assuming P₂(θ_e) is the pupil function of the collecting lens and Δz is related to the axial position of the depth scanner, the intensity related to a single wavelength requires an incoherent integration over each incident angle θ_e. Therefore,

\begin{array}{l} I (x, Δ z; k) & = \int_{- θ_{e, max}}^{θ_{e, max}} (P_{1} (θ_{e}) P_{2} (θ_{e}) sin θ_{e} \\ {\cdot {\sum_{n_{min}}^{n_{max}} C_{n} q_{z} R_{s} (k, θ_{e}) exp (j \frac{2 n π}{Λ} x) \cdot exp (- 2 k cos θ_{e} Δ z) + 2 k cos θ_{e} R_{r} (k, θ_{e})})}^{2} d θ_{e} . \end{array}

where sinθ_e in the integrand is a weighting factor to each incident angle, since the area in the Fourier plane is proportional to sinθ_e (see Fig. 1). The term exp(−2k cosθ_eΔz) indicates an additional phase change in the object arm during the depth scan.

Since broadband illumination is used and the spectral sensitivity of the camera is not constant over the illumination spectrum, the intensity is finally calculated via integration over the wave number,

I (x, Δ z) = \int_{0}^{\infty} I (x, Δ z; k) F (k) S (k) d k,

where F(k) and S(k) are the spectral sensitivity of the camera and the illumination spectral density function, respectively.

2.2. Richards-Wolf-modeling

A vectorial study of a linearly polarized incident field introduced by Richards and Wolf [10] is based on the Debye approximation [16]. The electromagnetic field near the focus of an aberration-free imaging system can be calculated in a numerical way. This solution is not restricted to systems of low aperture. Due to conciseness a detailed derivation of the model is not given here. Figure 2 shows a schematic illustration of the optical setup according to Richards-Wolf-modeling. If the measurement object is located in the focus plane of the objective lens, the image will be formed in the detector plane, as illustrated by the dashed lines in Fig. 2. The interference intensity is recorded by the camera during a depth scan of a phase object (in our case the rectangular reflective grating). If, for example, the defocus of the grating has a value of Δz, as illustrated in Fig. 2, the image plane will have the same defocus under the assumption that the magnification is one. Since the defocus is much smaller than the focal length of tube lens, the maximum opening angle at the image side is the same as the objective angular aperture, namely arcsin(NA). For each point on the rectangular grating, there is a corresponding image point. Taking an image point o_c for example, a local coordinate system having o_c as its origin is built for modelling (see Fig. 3). The intensity at point P₀ and its vicinity in the detector plane will be calculated according to Richards-Wolf-modeling. P₀ equals the image point o_c in case of no defocus. For simplicity of modelling the magnification is assumed to be unity, yet the modeling remains valid even in case of other magnifications.

Fig. 2 Schematic illustration of the optical setup regarding Richards-Wolf-modeling.

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Fig. 3 Local coordinate system of defocused imaging regarding Richards-Wolf-modeling.

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The interference intensity of any point P on the detector plane according to Fig. 3 can be calculated. The local Cartesian coordinate system is transformed to a spherical coordinate system. s⃗ is a unit vector of an incident ray with an incident angle θ_e and an azimuth ϕ. For clearness the distance between point P₀ and P is intentionally sketched exceedingly large compared to the rectangular grating size. However, only distances in micrometers from P₀ are of interest due to the limiting size of the Airy disk, where diffraction will take effect in imaging.

The Richards-Wolf-modeling is simplified to be a two-dimensional consideration within the xz plane. Omitting the constant coefficient, the detected intensity can be described as the following sum [10]:

I (x, x_{c}; k, θ_{e}) = | E_{0}^{2} | + 2 | E_{1}^{2} | + | E_{2}^{2} |,

where

\begin{array}{l} E_{0} (x, x_{c}; k, θ_{e}) & = & P_{1} (θ_{e}) P_{2} (θ_{e}) \sqrt{cos θ_{e}} sin θ_{e} (1 + cos θ_{e}) J_{0} (k r_{P} sin θ_{e} sin θ_{P}) \\ \cdot [R_{s} (k, θ_{e}) exp (j 2 k Δ z cos θ_{e}) + R_{r} (k, θ_{e})], \\ E_{1} (x, x_{c}; k, θ_{e}) & = & P_{1} (θ_{e}) P_{2} (θ_{e}) \sqrt{cos θ_{e}} {sin}^{2} θ_{e} J_{1} (k r_{P} sin θ_{e} sin θ_{P}) \\ \cdot [R_{s} (k, θ_{e}) exp (j 2 k Δ z cos θ_{e}) + R_{r} (k, θ_{e})], \\ E_{2} (x, x_{c}; k, θ_{e}) & = & P_{1} (θ_{e}) P_{2} (θ_{e}) \sqrt{cos θ_{e}} sin θ_{e} (1 - cos θ_{e}) J_{2} (k r_{P} sin θ_{e} sin θ_{P}) \\ \cdot [R_{s} (k, θ_{e}) exp (j 2 k Δ z cos θ_{e}) + R_{r} (k, θ_{e})] . \end{array}

θ_e is again the incident angle and P₁(θ_e), P₂(θ_e) are the pupil functions of the illuminating and collecting lenses, respectively. r_P, θ_P and ϕ_P are the radial distance, polar angle and azimuth angle in spherical polar coordinates. J_n is the Bessel function of the first kind and order n. In case of unpolarized illumination I(x, x_c; k, θ_e) in Eq. 19 is symmetrical with respect to the z_c-axis and therefore, independent of ϕ_P. Due to the two-dimensional scale of the model, ϕ_P = 0 applies. Therefore, r_P sinθ_P in the Bessel function can be replaced by |x_c|. Since the reference mirror is placed in the focus position, the Fresnel reflection coefficient of the reference mirror R_r(k, θ_e) is added to the exponential functions in Eq. 20. A factor of 2 is multiplied in the power of the exponential function, since a defocus of Δz in the object plane results a phase change proportional to 2Δz. In case of spatially incoherent illumination, the integration over the incident angle θ_e is calculated once the intensity I(x, x_c; k, θ_e) is determined according to Eq. 19. Finally an integration over all relevant wavenumbers is carried out:

I (x, Δ z) = \int_{0}^{\infty} \int_{0}^{θ_{e, max}} \sum_{x_{c} = - δ_{x}}^{x_{c} = - δ_{x}} {I (x, x_{c}; k, θ_{e})} d θ_{e} F (k) S (k) d k,

where F(k) and S(k) are again the spectral sensitivity of the camera and the spectral density function of the illumination. Equation 21 does not require a convolution or a low pass filtering, since diffraction is already considered by the summation of I(x, x_c; k, θ_e) at each single x position. δ_x is the half boundary within which diffraction is taken account. δ_x should be chosen to be at least the radius of the Airy disk at the corresponding wavelength. A large value of δ_x indicates high accuracy of modelling but high computation time.

3. Simulation results

Measurement results at rectangular gratings with height steps of the same order as the illumination wavelength show systematic errors caused by the batwing effect. This gives rise to extra nonlinearity of the ITF [17, 18]. Xie et al. [18] introduce a numerical model based on a theoretical approach, which takes diffraction into account by convolving the raw intensity data with the ideal incoherent point spread function (PSF). However, full longitudinal spatial coherence was considered in [18]. In our current signal modeling approach, also finite longitudinal spatial coherence caused by an extended illumination source and the high NA objective lens is considered. The lateral spatial incoherence results from the fact that the light source is extended rather than a point source. Throughout this paper we assume that the illumination is laterally spatially perfectly incoherent as Köhler illumination is, for example. Longitudinal coherence takes effect if a microscope objective is applied [19], in this case a finite depth of field depending on the NA value occurs. For high NA objectives, longitudinal coherence is dominant and therefore, the width of the inference correlogram is determined by the depth of field rather than by the temporal coherence length.

Temporal and spatial coherence can be considered independently from each other. Interference correlograms showing both, limited temporal and spatial coherence are introduced. Simulation results for a rectangular grating with a period of Λ = 6 μm and different height values h₀ are displayed in Fig. 4. Interferograms with different properties of temporal and spatial coherence at plateaus and edges are presented. At first, simulation results of the Kirchhoff-modeling are shown.

Fig. 4 Interference correlograms with different properties of temporal and spatial coherence, assuming P₁(θ_e) = P₂(θ_e) = 1 and R_s(k, θ_e) = 1. The black dashed line represents the correlogram from the upper plateau and the green dashed line the one from the lower plateau. The solid lines show the correlogram at the upper and lower edge position of the rectangular grating. (a) Correlogram assuming perfect temporal and spatial coherence: θ_e → 0, h₀ = λ₀/4. (b) Correlogram assuming partial temporal coherence (Gaussian envelope) and full spatial coherence: θ_e → 0, h₀ = λ₀/4, where λ₀ is the center wavelength of the Gaussian spectrum. (c) Correlograms assuming finite spatial coherence and full temporal coherence: NA = 0.9, h₀ = 0.2 μm and λ₀ = 0.6 μm. λ_eff is the effective wavelength resulting from the NA effect and λ_eff = 0.8 μm. (d) Correlograms assuming finite temporal and spatial coherence: θ ∈ [−arcsin(NA), arcsin(NA)], h₀ = 0.2 μm, λ₀ = 0.6 μm. λ_eff = 0.8 μm results.

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If, for example, only a single incident angle of illumination and a single wavelength are considered, full temporal and spatial coherence occurs as it is shown in Fig. 4(a). The incident angle θ is set to zero and the step height equals a quarter of the wavelength λ₀. The illumination of the measurement object is monochromatic with λ₀ = 0.6 μm. In this case, the effective wavelength λ_eff is the same as the nominal wavelength. The correlogram from the middle of upper plateau shows 180 degree phase difference compared to the one from the middle of the lower plateau. At the edge position destructive interference occurs so that the modulation depth of the correlograms at these positions drops dramatically. Figure 4(b) shows correlograms with limited temporal coherence (coherence length l_c is 2 μm) and full spatial coherence. The correlogram at the plateau having a fringe spacing of λ₀/2 shows a Gaussian envelope and equals an analytical result where finite temporal coherence and full spatial coherence is considered [18]. Furthermore, at the edge position, the maxima of the envelopes from the upper and lower plateau are much more apart from each other than the step height h₀. This leads to the well-known batwing effect [18]. In Fig. 4(c) full temporal coherence and finite spatial coherence is considered. In comparison to Fig. 4(a) and 4(b) an enlarged fringe spacing and a smaller depth of field due to the high NA can be observed. The simulation results depends on the pupil function.

Assuming

P_{1} (θ_{e}) = P_{2} (θ_{e}) = {cos}^{ς} θ_{e},

where ς is the apodization index, higher values of ς correspond to apodization that falls off faster with angle θ_e [4]. In Fig. 4 we assume ς = 0, namely P₁(θ_e) = P₂(θ_e) = 1, thus no apodization is considered. The effective wavelength on the plateau is obtained from the centroid wavelength of the interferogram spectrum. In case of the nominal wavelength λ₀ = 0.6 μm and NA = 0.9, an effective wavelength of 0.8 μm results. We define the NA factor as the ratio of the effective wavelength at plateau position to the nominal wavelength (centroid wavelength of the illumination). Therefore, a NA factor of 1.33 is calculated for Fig. 4(c), 4(d). This result is consistent with analytical results for the case, where the sine condition is satisfied and no apodization is present [4]. Additionally, no dependence of the Fresnel reflection coefficient on wavelength and incident angle is considered, namely R_s(k, θ_e) = R_r(k, θ_e) = 1. The step height is 0.2 μm, which is one fourth of the effective wavelength. Therefore, a phase difference of π between interferograms from the upper and lower plateau can be observed. Both correlograms at the upper and lower edge show destructive interference. In Fig. 4(d), both finite spatial and temporal coherence are considered. Obviously in case of NA of 0.9, the spatial coherence is dominant, as the depth of field due to high NA is smaller than the temporal coherence length.

Fig. 5 Simulated interference correlograms and their spectra. The blue curve in the spectrum represents the nominal spectrum of the LED, the red curve is the part of the spectrum applied to calculate the effective wavelength according to a centroid method. (a) Correlograms and their spectra using a royal blue LED. (b) Correlograms and their spectra using a red LED.

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Fig. 6 Upper figures: Profiles of a rectangular grating resulting from envelope evaluation (blue curve) and phase evaluation (red curve); Bottom figures: corresponding course of the effective wavelength. (a) Topography and effective wavelength using royal blue LED. (b) Topography and effective wavelength using red LED.

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Fig. 7 Graphical illustration of simulation results of different LED illumination, assuming NA = 0.9.

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Fig. 8 Fringe images of maximum contrast captured by the CMOS camera using red. (a) and royal blue. (b) LED illumination.

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Fig. 9 Measurement results using royal blue and red LED illumination. (a) Measured correlograms and their spectra using royal blue LED. The blue curve in upper figure represents the unfiltered correlogram, while the red curve shows the correlogram filtered by a Gaussian filter. The blue curve in the bottom figure represents the nominal spectrum of the LED, the red curve is the part of the spectrum applied to calculate the effective wavelength according to a centroid method. (b) Measured correlograms and their spectra using red LED. (c) Measured topography and effective wavelength using royal blue LED. Upper figure: Profiles of a rectangular grating resulting from envelope evaluation (blue curve) and phase evaluation (red curve); Bottom figure: corresponding course of the effective wavelength. (d) Measured topography and effective wavelength using red LED.

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In the simulation a rectangular grating with height h₀ = 0.191 μm and period Λ = 6 μm is considered as measuring object. For illumination low coherent LEDs with a Gaussian-like spectral density function S(λ) or S(k) are used. A graphic representation of the spectral density of the LEDs is shown in Fig. 10 in appendix A. The transfer characteristics of interference microscopes can be studied using different LED-illumination.

Fig. 10 Normalized spectral density of LEDs of different colors

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Figure 5 shows simulated correlograms and their Fourier spectra under different LED-illumination at the plateau and the edge position of the rectangular grating. To be able to compare the simulation results with measurement results, an apodization index ς = 0.5 is assumed, namely the pupil function of the objective and tube lens satisfies $P_{1} (θ_{e}) = P_{2} (θ_{e}) = \sqrt{cos θ_{e}}$ . Also the Fresnel reflection coefficients of the silicon grating R_s(k, θ_e) and of an aluminum mirror R_r(k, θ_e) are considered. The blue curve in the spectrum represents the nominal spectrum of the LED. The red curve is the spectrum of the corresponding correlogram applied to calculate the effective wavelength. The spectra of the correlograms differ a lot from the nominal spectrum of the LED illumination. This is due to the oblique incident rays leading to a larger fringe spacing in the correlogram and consequently a larger effective wavelength. While a broadening of the correlogram takes place at the edge position if the illumination is royal blue LED, a clear increase of the fringe spacing in the middle of the correlogram can be seen for red LED illumination. Both edge spectra, i.e. for royal blue and red LED show a constriction at 0.78 μm. This effect arises from destructive interference at the wavelength corresponding to fourfold the step height. The constriction occurs at 0.78 μm, but not exactly at 0.764 μm (four times of the step height) since the frequency resolution of the spectra is limited.

Figure 6 shows simulated grating profiles of envelope and phase evaluation for royal blue and red LED illumination. Also the effective wavelength course is presented. The HWR value of royal blue and red LED illumination is 0.191/0.557 ≈ 0.34 and 0.191/0.786 ≈ 0.24, respectively. Since the HWR value of red illumination is closer to 0.25, the envelope evaluation shows a much stronger batwing effect compared to the results of royal blue illumination. Yet the phase evaluation of red illumination shows no batwing effect but phase jumps, while a slight batwing effect can be seen even if the royal blue illumination is used. These results agree with a numerical simulation [18] and an analytical study [20] in our prior studies. A NA factor of 1.25 on the plateau position can be calculated, which is smaller than in case of no apodization. This is why we use an apodization index ς = 0.5, which may come from an inhomogeneous illumination in the pupil plane. It should be mentioned that the effective wavelength is not constant over the whole measurement object. The effective wavelength drops at edges when royal blue illumination is used, while it increases at edges in case of red illumination. This is due to the fact that a constriction occurs in the spectra at a wavelength of four times the step height. At royal blue illumination the spectrum at the plateau has its predominant part below 0.76 μm, while at red illumination it reverses.

The evaluation results for different LED light sources are graphically summarized in Fig. 7. The numerical data is shown in Table 1 in appendix B. The x axis represents the different LED center wavelengths. In addition to the color notation the center wavelength of the corresponding LED spectrum and the HWR value are given in the brackets. For clarity Fig. 7 has five vertical axes in different colors and scales. Results plotted in the figure correspond to the axis showing the same color. Independent of the LED color, we assume the pupil function P₁(θ_e) and P₂(θ_e) equal both $\sqrt{cos θ_{e}}$ . The calculated NA factor for simulations at a nominal NA 0.9 is about 1.25 (see blue curve), which is comparable with analytical results obtained in prior studies [4, 6]. The effective wavelength change at the edges shows a systematic characteristic. For illumination wavelengths with HWR higher than 0.25, like royal blue, blue, cyan and green LED illumination, the effective wavelength change is negative, which means that the effective wavelength at the edges is smaller than on the plateau. Additionally, for the other LED illumination wavelengths (except for amber) with HWR smaller than 0.25, the change is positive. The batwing height regarding the envelope evaluation reaches its maximum when an amber or red LED is used, both of which have a HWR value close to 0.25. For all HWR values different from 0.25, the batwing height drops gradually. The solid and dashed red curves represent the step height obtained from either the envelope or phase evaluation of the rectangular grating, respectively. Due to the batwing effect, the estimated step height resulting from the envelope evaluation is overestimated. The height value obtained by phase evaluation is almost constant between 190 and 191 nm.

Table 1. Overview of simulation results at different LED illumination

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The simulation results of Richards-Wolf-modeling is consistent to results of the Kirchhoff-Modeling and are presented in appendix C.

4. Experimental results

In this chapter we compare experimental results with simulation results obtained by use of the above mentioned simulation procedure. All measurements were performed using a Linnik interferometer with 100× magnification and a NA of 0.9. For the sake of conciseness the description of the experimental setup is given in appendix D.

A rectangular grating of a period of 6 μm and a height of 191 nm (Simetrics RS-N resolution standard) is measured. The NA factors for different LED illumination are between 1.20 and 1.24, which is approximately the same compared to results from Kirchhoff-modelling and Richards-Wolf-modelling under the consideration of apodization and angular dependence of the Fresnel reflection coefficient. The NA factor varies slightly with the wavelength instead of being constant.

Figure 8 shows fringe images captured by the CMOS camera for both, royal blue and red LED illumination, where the interference intensity modulation caused by the height difference of the rectangular grating can be seen. Figure 8(a) represents the result of using red LED illumination. Since the HWR value is approximately 0.25, a phase difference of the interference intensity is approximately π. Consequently, a high contrast of intensity maximum modulation results. In the case of royal blue LED illumination, as shown in Fig. 8(b), a lower contrast of intensity modulation occurs because the HWR value is about 0.34.

Measurement results using royal blue and red LED illumination are displayed in Fig. 9. Figure 9(a) and Fig. 9(b) show a similar shape of the correlogram and spectrum for both, the signal at plateau and edge compared to the simulation results according to Fig. 5(a) and Fig. 5(b). The correlograms at the edges in both, simulation and measurement show a broadening of the envelope for illumination with royal blue LED and a large fringe spacing compared to the correlograms at plateau in case of a red LED. The constriction of the spectrum is clearly seen at the wavelength corresponding to four times of the step height also in the measurement results. Both, envelope and phase evaluation in Fig. 9(c) show also a slight batwing effect, which agrees with simulation results in Fig. 6(a). Both, measurement and simulation results in case of using a red LED, as displayed in Fig. 9(d) and Fig. 6(b), show high batwings if the envelope is evaluated but no batwings if the phase is evaluated. Moreover, the measured effective wavelength is consistent with simulation. The grating heights obtained by envelope evaluation for royal blue and red illumination are 209 nm and 232 nm respectively, both showing an overestimation. The simulation results are consistent with the measurement ones.

An additional comparison of simulation and measurement results using amber led is shown in Fig. 14 in appendix E, where the greatest batwing effect appears at a HWR value of app. 0.25.

Fig. 11 Simulation results according to Richards-Wolf-modelling corresponding to Fig. 4 using the same parameters. The black dashed line represents the correlogram from the upper plateau and the green dashed line the one from the lower plateau. The solid lines show the correlogram at the upper and lower edge position of the rectangular grating. (a) Correlograms for temporally and spatially coherent illumination. (b) Correlograms for partial temporal coherence (Gaussian envelope) and full spatial coherence. (c) Correlogram for finite spatial coherence and full temporal coherence. (d) Correlogram for partial temporal coherence and finite spatial coherence.

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Fig. 12 Comparison of interferograms using Richards-Wolf-modelling compared to the analytic result from [19].The red and black curves are results of the analytic solution, which are the interferogram and its envelope respectively. The blue and green curves result from Richards-Wolf-modelling.

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Fig. 13 Schematic representation of the Linnik interferometer

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Fig. 14 Comparison of measurement and simulation results using amber LED illumination. The blue curve at the bottom diagrams of Fig. 14(a) and Fig. 14(b) represents the nominal spectrum of the LED, the red curve is the part of the spectrum applied to calculate the effective wavelength according to a centroid method. The upper figures in Fig. 14(c) and Fig. 14(d) show profiles of a rectangular grating resulting from envelope evaluation (blue curve) and phase evaluation (red curve).

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5. Conclusion

This paper deals with the impact of the HWR on measurement results of a rectangular grating obtained with low-coherence interference microscopy with respect to effective wavelength, NA factor and batwing effect if either the envelope or the phase of the occurring interference signal is evaluated. Two simulation models, a Kirchhoff- and a Richards-Wolf-model, assume a one-dimensional grating structure, where the intensity is calculated within the incidence plane (xz plane). A three dimensional extension of the modelling is intended in future. However, the simulated results show a good agreement with measurement results. Without apodization (P₁(θ_e) = P₂(θ₂) = 1), the NA factor of both Kirchhoff and Richards-Wolf-model is 1.33. If we assume $P_{1} (θ_{e}) = P_{2} (θ_{2}) = \sqrt{cos θ_{e}}$ , it drops to 1.25 and 1.23 respectively. The NA factor resulting from measurement covers the range from 1.20 to 1.24, which corresponds the latter case of apodization. The batwing effect in both, simulation and measurement results reaches its maximum if the HWR value is about 0.25. The effective wavelength along the rectangular grating is not constant. At the edges it tends to be lower than at plateaus if HWR is higher than 0.25, while at HWR values smaller than 0.25 it reverses.

The Kirchhoff-model confirmed its validity to the application to rectangular gratings with edges for a high NA value of 0.9. The Richards-Wolf-model is especially appropriate for high NA. No constraints with respect to the geometry of the object are known with regard to amplitude and correlation length. Concerning computational effort, the Kirchhoff-model is highly efficient, while Richards-Wolf-modeling is more time consuming. Furthermore, edge diffraction should also play a roll in experimental measurement, especially when polarization is considered [21]. So far, polarization is not taken into account neither by the Kirchhoff nor by the Richards-Wolf-model.

Appendix

A Normalized spectral density of LEDs

The normalized spectral density of Luxeon Rebel Color LEDs applied in the simulation is plotted in Fig. 10.

B. Simulation results at different LED illumination

An overview of simulation results of the Kirchhoff-modeling using different LED illumination is given in Table 1, which corresponds to Fig. 7.

C. Simulation results of Richards-Wolf-modeling

The Richards-Wolf-modeling is based on the numerical calculation of the point spread function in the near of focus region. Figure 11 shows the interference correlograms at plateau and edge positions for different temporal and spatial coherence. Again, the pupil functions of objective and tube lens P₁(θ_e) and P₂(θ_e) are assumed to be unity. The Fresnel reflection coefficient is independent of wavelength and incident angle, namely F_s(k, θ_e) = 1.

In summary, Fig. 11 is similar to results of the Kirchhoff-Modeling in Fig. 4. In case of finite spatial coherence and full temporal coherence, where the longitudinal spatial coherence is taken into account, the envelope of the interferogram is a sinc-function [19]. Figure 12 gives a comparison of interferogram using Richards-Wolf-modelling to the analytic result from [19]. The red and black curves are obtained from the analytic solution, which are the interferogram and its envelope, respectively. The blue and green curves resulting from the Richards-Wolf-modelling differ only a little from the analytical curves. The NA factor of 1.33 is the same compared to the result from Kirchhoff-modelling. If the Fresnel reflection coefficient is considered depending on wavelength and incident angle, apodization is taken in to account additionally, where $P_{1} (θ_{e}) = P_{2} (θ_{e}) = \sqrt{(cos θ_{e})}$ , the NA factor drops to 1.23, which slightly differs compared to the value of 1.25 resulting from the Kirchhoff-modelling.

D. Experimental setup of Linnik interferometer

The Linnik interferometer is equipped with a replaceable illumination fixture such that each LED introduced in Fig. 10 can be used. A schematic representation of the Linnik interferometer is shown in Fig. 13.

The basic components of our self-assembled Linnik interferometer are listed in the following table:

Table 2. Basic components

View Table | View all tables in this article

E. Additional comparison of simulation and measurement results

Figure 14 represents both, simulation and experimental results for amber LED illumination. Both, correlograms corresponding to the edge obtained from simulation and measurement show a decrease of modulation depth. Again, a constriction of the spectrum at the wavelength corresponding to four times of the step height can be observed. Results of envelope evaluation show the batwing effect, whereas the profiles obtained from phase evaluation reproduce the rectangular grating very well except for some phase jumps. The effective wavelength slightly increases at edges in both, simulation and measurement.

Acknowledgment

The authors would like to thank the Deutsche Forschungsgemeinschaft (DFG, LE 992/6-2) for the support of this project.

References and links

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LED color	Royal blue	Blue	Cyan	Green	Amber	Red	Deep-red
Nominal (centroid) wavelength (nm)	444	463	499	523	590	628	654
Effective wavelength (nm)	557	581	624	653	740	787	820
NA factor (−)	1.25	1.25	1.25	1.25	1.25	1.25	1.25
HWR value (−)	0.34	0.33	0.31	0.29	0.26	0.24	0.23
Δλ_eff at edge (nm)	−35	−37	−37	−33	12	64	92
Batwing height (nm)	13	28	69	131	231	232	199
h₀ according to envelope evaluation (nm)	201	203	206	209	219	223	225
h₀ according to phase evaluation (nm)	190	190	190	190	190	190	191

LED color	Royal blue	Blue	Cyan	Green	Amber	Red	Deep-red
Nominal (centroid) wavelength (nm)	444	463	499	523	590	628	654
Effective wavelength (nm)	557	581	624	653	740	787	820
NA factor (−)	1.25	1.25	1.25	1.25	1.25	1.25	1.25
HWR value (−)	0.34	0.33	0.31	0.29	0.26	0.24	0.23
Δλ_eff at edge (nm)	−35	−37	−37	−33	12	64	92
Batwing height (nm)	13	28	69	131	231	232	199
h₀ according to envelope evaluation (nm)	201	203	206	209	219	223	225
h₀ according to phase evaluation (nm)	190	190	190	190	190	190	191

Signal modeling in low coherence interference microscopy on example of rectangular grating

Abstract

1. Introduction

2. Signal modeling

2.1. Kirchhoff-modeling

2.2. Richards-Wolf-modeling

3. Simulation results

4. Experimental results

5. Conclusion

Appendix

A Normalized spectral density of LEDs

B. Simulation results at different LED illumination

C. Simulation results of Richards-Wolf-modeling

D. Experimental setup of Linnik interferometer

E. Additional comparison of simulation and measurement results

Acknowledgment

References and links

Cited By

Figures (14)

Tables (2)

Equations (22)

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