Coherence scanning and phase imaging optical interference microscopy at the lateral resolution limit

Peter Lehmann; Weichang Xie; Benedikt Allendorf; Stanislav Tereschenko

doi:10.1364/OE.26.007376

1. Introduction

Interference microscopy especially white-light or coherence scanning interferometry is a well-established and widely used technique for measuring 3D microstructures. In recent years there is a general trend to minimize structure dimensions, requiring to improve the lateral resolution capabilities of appropriate instruments. In context with coherence scanning interferometry (CSI), resolution enhancement, sometimes referred to as super-resolution, has been achieved by use of structured illumination techniques [1–3] or microsphere assistance [4–7], for example.

Nevertheless, these improvements were reached using more or less conventional interference microscopes and measurements were conducted based on either phase shifting or coherence scanning techniques. It should be noted, however, that this paper does not deal with super-resolution. We study the transfer characteristics of low coherence interference microscopy close to the Abbe limit of lateral resolution.

Although ISO 25178-604 [10] defines the term lateral resolution as “the smallest distance between two features which can be detected” there is no common use of this term in literature. Sometimes lateral resolution is simply assumed to equal the lateral resolution known from two-dimensional microscopic imaging [8, 9] and sometimes it is equated with the “lateral period limit” as defined also in ISO 25178-604 [10, 11].. Even if the same definition of lateral resolution as in conventional 2D microscopy is used, there are different approaches based on either spatially coherent (Abbe criterion [12, 13]) or incoherent illumination (Rayleigh resolution [8–11]). Furthermore, if the illumination is assumed to be spatially coherent, some authors assume that a plane wave travelling along the optical axis illuminates the object, i.e. a lateral resolution of λ/NA [6] instead of λ/(2 NA) [9] results. Sometimes the lateral resolution is even defined by λ/(4 NA) [8] based on the argument, that the resolved distance between two features equals half of the smallest grating period, which is resolved.

Despite of these definitions and even though coherence scanning interferometers are widely used, the transfer characteristics of these instruments are no longer clear, if height steps of the order of λ/4 appear or instruments operate close to their lateral resolution limit. Previous papers focus on batwings occurring in CSI at height steps and steep slopes even if the surface period is far away from the lateral resolution limit [25–28]. It should be noted that similar limiting effects were also observed in measurement results obtained by confocal optical microscopes at step height structure, surface gradient changes and strong local curvatures of the measured surface [37, 38].

In this paper we study the lateral resolution for reflecting rectangular grating structures first experimentally. We found that the profiles obtained from the phase of a CSI signal (called phase profiles throughout this paper) show an opposite sign compared to those profiles reconstructed from coherence scanning (called coherence profiles throughout this paper) using the same set of measurement data.

The wavelength spectrum of the measured signals is broadened due to the numerical aperture effect [14–17]. As a consequence, phase profiles can be determined analyzing CSI signals at different wavelengths. For smaller wavelength the measured phase profile changes its sign and corresponds to the coherence profile, but with much lower amplitude.

We were able to confirm these results by theoretical and numerical analysis applying either Kirchhoff theory or adapting the Richards-Wolf model [18] to interference microscopy.

2. Interference microscopy at high numerical aperture

For this study we used a Linnik interferometer built in our lab. The microscopic imaging system consists of two 100x microscope objective lenses of NA = 0.9 (Olympus MPLFLN100XBDP). The beam paths of the interferometer arms are coupled via a beam splitter cube in the afocal space. A tube lens generates an image on a scientific CMOS camera located in its focal plane. A LED light source illuminates a diffuser. Light is collected by a condenser lens and directed into the beam path through the beam splitter cube mentioned above, thus generating a defocused image of the diffuser in the object plane [19, 20]. Consequently, the pupil planes of the two microscope objectives are nearly fully illuminated and the illuminating aperture equals the imaging aperture, i.e. Abbe’s lateral resolution criterion $d = λ / (2 N A)$ should hold [13]. This means that a periodical grating structure can be resolved as long as for an incident beam with maximum angle of incidence $θ_{e, \max} = \arcsin (N A) / n_{0}$ the zero order reflected and diffracted rays of order + 1 or −1 are still captured by the aperture of the objective lens, i.e. $| θ_{s, \max} | = | θ_{e, \max} | = \arcsin (N A) / n_{0},$ with $n_{0}$ being the refractive index, which we set $n_{0} = 1$ assuming a non-immersion system and $θ_{s, \max}$ is the maximum angle under which diffracted light reaches the objective lens.

Using Kirchhoff diffraction theory (e.g [21], Eq. (1)) the surface height h(x,y) of a phase object is multiplied by the wavenumber and twice the cosine of the angle of incidence, i.e. $2 \cos θ_{e}$ . This can be interpreted as if the phase object would be illuminated perpendicularly by a wave of the effective wavelength $λ_{eff} = λ / \cos θ_{e}$ . However, the minimum angle of incidence is zero and thus the minimum effective wavelength equals the original wavelength λ. Considering a light source emitting a spectrum of minimum wavelength $λ_{1} = λ_{0} - Δ λ / 2$ and maximum wavelength $λ_{2} = λ_{0} + Δ λ / 2$ the wavelength spectrum of the measured interference signals will extend from $λ_{1}$ to $λ_{2} / \cos θ_{e, \max}$ . This is shown in Fig. 1 for an interference signal obtained with the Linnik interferometer described above using a blue LED. For comparison Fig. 1 shows also the spectral distribution of the used LED light source. Obviously, the measured spectrum obtained by Fourier transformation and rescaling of the interference signal resulting from a mirror-like object significantly differs from the spectrum of the light source. The signal spectra obtained from interference signals extend to much longer wavelengths (lower spatial frequency) compared to the light source spectra. It should be noted that, of course, this effect is closely related to the well-known numerical aperture effect, where the mean wavelength of a correlogram shifts to higher values as the NA increases [14–17]. The broadening of the spectrum is accompanied by an additional envelope of the interference signals shown in Fig. 1, which emerge from the limited depth of focus of the objective lens and can be interpreted as limited longitudinal spatial coherence [22]. In addition, Fig. 1 shows some discrepancies between simulation and measurement. These can be explained by dispersion effects, which are not considered in the simulation model so far. Furthermore, the illumination intensity distribution in the pupil plane of the objective lens influences the resulting spectral composition and thus may cause further deviations. For these reasons it is a difficult task to obtain the original spectrum of the light source from measured CSI signals.

Fig. 1 (a) Simulated CSI signal based on the Richards-Wolf model (see sect. 3.3) and (b) measured CSI signal using blue LED illumination; (c), (d) the corresponding spectra compared to the nominal spectrum of the LED (blue peaks).

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Due to the spectral broadening the measured interference signals can be phase analyzed even at longer wavelength, which are originally not present in the spectrum of the light source. Phase profiles for several spatial frequencies corresponding to different wavelengths were already obtained before [36].

Since we use narrowband illumination, we assume that longer wavelengths can be attributed to higher angles of incidence corresponding to better lateral resolution. This effect should be somehow similar to an appropriate annular aperture in the pupil plane of the objective lens. Figure 2 shows experimental results obtained from a rectangular grating structure of 0.3 µm period and 140 nm PV-amplitude (Simetrics RS-N). The phase of the signals was analyzed at different wavelengths (called evaluation wavelength λ_eval in the following sections) using lock-in detection [23] and the coherence peak was obtained via Hilbert transformation, which is very similar to the method introduced by Kino and Chim [39]. Obviously, the measured amplitude of the profile strongly increases as the evaluation wavelength increases. However, at a certain limit of the height-to-wavelength ratio HWR ≅ 0.25 [19] the amplitude is nearly zero [3, 24] and below this wavelength it increases again, but now the measured profiles are 180° phase shifted. Figure 3 shows profiles obtained from phase analysis assuming a maximum evaluation wavelength λ_eval of 900 nm, compared to the corresponding coherence profile again. It turns out that the amplitudes of the two corresponding profiles are comparable, but one profile is inverted, i.e. 180° phase shifted with respect to the other.

Fig. 2 Measured surface profiles of a rectangular grating (Simetrics RS-N) of 300 nm period obtained with a high- resolution Linnik interferometer (NA = 0.9) using blue LED illumination. The profiles either result from the coherence peak or from phase analysis at different evaluation wavelengths λeval.

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Fig. 3 Surface profiles of the rectangular grating (Simetrics RS-N) of 300 nm period and an additional edge and plateau structure obtained with the high-resolution Linnik interferometer. The profiles either result from the coherence peak or from phase analysis at different evaluation wavelengths λeval.

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The height difference of the plateaus (right hand side of Fig. 3) is nearly independent of the chosen evaluation wavelength. At the edges batwings occur which systematically depend on the wavelength as we reported earlier [25–28].

In addition, Fig. 4 presents measured topographies obtained from a Blu-ray disc, where the track pitch is 0.32 µm and the groove depth is approximately 20 nm [40]. For illumination a blue LED was used resulting in an effective wavelength of 570 nm obtained from fringe spacing. The coherence topography (Fig. 4(a)) resolves the grooves of the Blu-ray Disc, whereas the groove structure can no longer be observed in the phase topography obtained for the center wavelength of the interference signals (Fig. 4(b)). However, for an evaluation wavelength λ_eval of 900 nm the groove structure can be observed again in Fig. 4(c), but the sign of the topography data differs from those of the coherence topography and the amplitude is reduced.

Fig. 4 Surface topographies obtained from a Blu-ray disc structure: (a) coherence topography, (b) phase topography using the center wavelength of 570 nm for phase analysis, (c) phase topography using a wavelength of 900 nm for phase analysis.

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From these experimental findings the following questions arise:

• Is it possible to confirm the lateral resolution enhancement by evaluating the phase of CSI signals at longer wavelength theoretically?
• How can we explain the inversion of the measured profiles obtained from phase analysis at longer wavelength?
• Which of the two measured profiles is correct and which one is inverted: the phase profile or the coherence profile?

3. Theoretical analysis of surface profiles obtained from either phase or coherence peak

In this section we start with an analytical formula for phase analysis, which considers only the zeroth and first order diffraction of a monochromatic electromagnetic wave scattered from a rectangular diffraction grating. A more appropriate result, which considers all diffraction orders captured by a given microscope objective lens and takes the wavelength spectrum of the light source into account, is based on the Kirchhoff-approximation and presented in subsection 3.2. Finally, subsection 3.3 demonstrates that similar results can be also achieved by an extended Richards-Wolf model, which calculates intensity distributions of interference patterns taken by focused and defocused microscopic interference patterns [18–20].

3.1 Analysis of phase imaging based on first order diffraction from rectangular profiles

In [21] we obtained the function r(x) describing the reflectance of a rectangular diffraction grating with respect to the phase:

r (x) = \exp {i ​ q_{z} h (x)} \approx \exp {i arc \tan (\frac{4}{π} \tan (q_{z} h_{0}) \cos (ω_{x} x))} .

In Eq. (1)

q_{z} = k (\cos θ_{e} + \cos θ_{s}) = \frac{2 π}{λ} (\cos θ_{e} + \cos θ_{s}) \approx \frac{4 π}{λ} \cos θ_{e} = \frac{4 π}{λ_{eff}},

where the last two expressions hold for

| θ_{e} | \approx | θ_{s} |,

i.e. the absolute value of the scattering angle

θ_{s}

equals the absolute value of the angle of incidence. The PV-amplitude

Δ h

of the rectangular grating equals

2 h_{0}

and

ω_{x} = 2 π / Λ

is the angular spatial frequency for the grating of period Λ. It should be noted that the variable

q_{z}

is always positive. Under this assumption, the reconstruction of the diffraction grating results in:

h (x) \approx \frac{arc \tan (\frac{4}{π} t a n (q_{z} h_{0}) \cos (ω_{x} x))}{q_{z}} .

The sign of the arctan-function depends of the sign of its argument. Hence, it depends on the sign of the term $\tan (q_{z} h_{0})$ whether the surface profile resulting from phase analysis is reversed or not. Since the parameters in the argument of the tan-function are positive, the function takes a negative sign if the argument exceeds π/2, i.e. for

\frac{4 π}{λ_{eff}} h_{0} > \frac{π}{2} \to λ_{eff} / 4 < Δ h = 2 h_{0} .

If a quarter of the effective wavelength becomes smaller than the total height difference $Δ h$ of the rectangular grating, the sign of the phase profile will change. Equation (4) holds especially for small effective wavelengths, i.e. if the illuminating wavelength and the angle of incidence are both small. Although this result corresponds quite well to our experimental observation, neither the reduction of the measured surface amplitude is considered nor the dependence of the zeroth and first order diffracted components on the angle of incidence. Furthermore, the above formulae were derived for phase analysis only. Hence, it is not yet clear how the results of coherence peak evaluation will underpin these results.

3.2 Simulation of CSI signals from rectangular profiles using the Kirchhoff-Approximation

According to Abbe’s theory of microscopic imaging the field amplitude in the image plane of a microscope can be theoretically treated by an appropriate spatial frequency domain filtering of the electric field scattered by the phase object followed by a Fourier transform in order to consider the propagation from the pupil plane to the image plane [12, 13]. Figure 5 shows the basic idea of simulating CSI signals [29]. We assume that the measuring object behaves as a periodical phase grating. The phase grating is illuminated by a plane wave incident under an angle θ_e with respect to the optical axis. This results in the scattered far field [30]:

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

where

R_{s}

is the Fresnel reflection coefficient, θ_s is scattering angle, i.e. the polar angle of the wave vector of the diffracted wave,

{\vec{k}}_{e}

,

{\vec{k}}_{s}

are wave vectors of length

k = 2 π / λ

and

Fig. 5 Diffraction from a one-dimensional phase grating depending on the angle of incidence a) pupil plane coordinate k_x and diffraction pattern coordinate q_x assuming $θ_{s} = θ_{r}$ for zero order diffraction, b) diffraction pattern in the pupil plane at perpendicular incidence with zero order diffraction component at $k_{x} = k_{y} = 0$ , c) diffraction pattern in the pupil plane with zero order diffraction component at $θ_{e} = θ_{\max}, k_{x} = k \sin θ_{\max}, k_{y} = 0.$

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{\vec{k}}_{e} - {\vec{k}}_{s} = - \vec{q} = (\begin{matrix} k (\sin θ_{e} - \sin θ_{s}) \\ 0 \\ - k (\cos θ_{e} + \cos θ_{s}) \end{matrix}), \vec{r} = (\begin{matrix} x \\ 0 \\ h (x) \end{matrix}), \hat{n} = (\begin{matrix} 0 \\ 0 \\ - 1 \end{matrix}) .

For simplicity, we assume the two-dimensional case, i.e. incident and scattered rays are located in the xz-plane. The zeroth diffraction order of the resulting diffraction pattern corresponds to the position of the origin of the q_x coordinate, given by

q_{x} = k (\sin θ_{s} - \sin θ_{e}) = 0.

Under this assumption $θ_{s}$ equals the reflection angle $θ_{r}$ as it is plotted in Fig. 5 (a). In Fig. 5 (b) the objective’s pupil illumination is depicted for perpendicular incidence, where the coordinate q_x coincides with the coordinate $k_{x} = k \sin θ_{s} .$

According to Fig. 5 (c) for $θ_{e} = θ_{e, max}$ the whole diffraction pattern is shifted along the k_x axis. Due to the numerical aperture the maximum number of diffracted field components is limited by the circular aperture of the objective lens:

k_{x}^{2} \leq k^{2} \sin^{2} θ_{\max} = k^{2} {(N A)}^{2} .

This corresponds to a low-pass filtering procedure and therefore limits the lateral resolution of an imaging system. It can be considered by modifying Eq. (5) by:

U_{s; k, θ_{e}} (θ_{s}) = P (θ_{e}, θ_{s}) R_{s} (k, θ_{e}) q_{z} \int e x p (- i q_{z} h (x)) e x p (- i q_{x} x) d x

neglecting the constant factor in front of the integral. In Eq. (8)

P (θ_{e}, θ_{s})

is the pupil function which considers the numerical aperture of the objective lens and, in addition, the effect of apodization if necessary [15].

The field reflected from the reference mirror can be treated in the same manner assuming $h (x) = 0 :$

U_{r; k, θ_{e}} (θ_{s}) = P (θ_{e}, θ_{s}) R_{s} (k, θ_{e}) q_{z} \int e x p (- i q_{x} x) d x,

In order to be able to simulate CSI signals we assume a periodical one-dimensional rectangular grating structure, i.e.:

h (x) = h_{0} rect (ω_{x} x) = h_{0} \cos (ω_{x} x) / | \cos (ω_{x} x) |

of amplitude h₀ and period

Λ = 2 π / ω_{x}

. As mentioned before, the measuring object can be seen as the corresponding phase grating

r (x)

according to Eq. (1). Now we replace the grating function by its Fourier series expansion, which results in:

\begin{array}{l} r (x) = \cos (q_{z} h_{0}) + \sum_{n = 1}^{\infty} {α_{n} \exp (i n ω_{x} x) + α_{- n} \exp (- i n ω_{x} x)}, \\ α_{n} = {\begin{matrix} 0 for even n \\ 2 i {(- 1)}^{(n + 1) / 2} \sin (q_{z} h_{0}) / (n π) for odd n \end{matrix}} \end{array}

where n represents the diffraction order. Using this expansion, our simulation procedure works as follows:

1. For a certain wavelength λ and angle of incidence θ_e we obtain the diffraction orders passing the aperture of the objective lens according to Fig. 5, multiply the corresponding electric fields by the phase factor $φ (m Δ z) = \exp (i k (\cos θ_{e} + \cos θ_{s}) m Δ z)$
related to the position m Δz of the depth scanner with the sampling interval Δz and add the terms belonging to different diffraction orders. Then we calculate the interference intensity after having added the constant amplitude of the reference wave. So far, this is a coherent superposition assuming plane wave illumination.
2. We repeat this procedure for a number of different angles of incidence θ_e corresponding to point sources at different locations in the pupil plane of the objective. We add the intensity values related to the field contributions calculated for the different angles of incidence. This corresponds to an incoherent superposition according to Köhler’s illumination [13]. The contributions belonging to different angles are weighted by the pupil function $P (θ_{e}, θ_{s}) = {(\cos θ_{e} \cos θ_{s})}^{1 / 2}$ , which considers an aberration-free system satisfying Abbe’s sine condition [15].
3. Next, we change the wavelength of light according to the given spectral distribution of the incident light and repeat steps 1 and 2. Finally, we add the resulting intensity values considering the spectral distribution. This results in a matrix of intensity values whose rows correspond to the x values of the sample and the columns correspond to the z = m Δz values of the depth scanner. Evaluating the columns using well-known CSI signal analysis techniques [23, 39] results in depth profiles obtained from our simulated interference signals.

In contrast to earlier CSI signal modeling approaches found in literature [31, 32] the procedure described above is not restricted to plane objects. Furthermore, one can easily extend our Kirchhoff model with respect to third dimension, i.e. y or q_y coordinate, and consider the angular and polarization dependence of the reflection coefficient [19, 20, 29]. In addition the Kirchhoff model described above should yield similar results compared to the foil model introduced in the literature, which also assumes that Kirchhoff’s assumptions apply [33, 34] but has not yet been proven for high NA values and rectangular grating structures. However, since the Kirchhoff model works only for rather low aspect ratios (height to groove width ratio) we consider PV-amplitudes of 20 nm and 100 nm for the following simulations.

Simulated profiles assuming Λ = 300 nm, NA = 0.9, and a central wavelength of 460 nm (blue LED) are shown in Fig. 6. Obviously, the coherence profiles are inverted in both cases, whereas the amplitude and in Fig. 6(b) also the sign of the phase profiles depend on the evaluation wavelength. In summary, there is a satisfying agreement between these simulation results and the measurement results displayed in Fig. 2 and Fig. 4.

Fig. 6 Profiles obtained from Kirchhoff simulations assuming a rectangular input profile of 140 nm PV amplitude (a) and 20 nm PV amplitude (b) and either coherence peak or phase analysis at different evaluation wavelengths.

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3.3 Simulation of CSI signals from rectangular profiles using the Richards-Wolf model

The Richards-Wolf model is based on the Debye diffraction integral of the distribution of the electromagnetic field near the focus of an aberration-free system which images a point source [18, 35]. We adapted and extended this model to an interferometric configuration with spatially incoherent illumination. For the sake of conciseness we will briefly introduce this modeling. A more detailed introduction is given in previous work [19, 20].

Figure 7 shows a schematic illustration of the setup assumed for the Richards-Wolf modeling of an interferometric system. $θ_{e}$ and ${θ^{'}}_{e}$ are the angles of incidence related to the object and image plane, respectively. During the depth scan different height positions of the measuring object lead to different interference intensity distributions on the detector, which can be calculated using the extended Richards-Wolf model. These intensity distributions are finally evaluated using coherence peak and phase analysis algorithms to reconstruct the topography of the measuring object.

Fig. 7 Schematic illustration of the Richards-Wolf model applied to an interferometric system.

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The integration over the incident angle ${θ^{'}}_{e}$ on the detector is transferred to an integration over the incident angle $θ_{e}$ on the measuring object by applying the sine condition. In addition, an integration over the wavenumber k weighted by S(k) taking the spectral distribution of the light source and the spectral sensitivity of the camera into account needs to be performed. This results in the formula

I (x, Δ z; x_{c}) = \int_{k_{\min}}^{k_{\max}} \int_{0}^{\arcsin N A} P (θ_{e}) \cos θ_{e} \sin^{2} θ_{e} J_{0}^{2} (k \sin θ_{e} | x_{c} |) \cdot | E_{int} |^{2} d θ_{e} S (k) d k

where

\begin{array}{l} P (θ_{e}) = P_{1}^{2} (θ_{e}) P_{2}^{2} (θ_{e}) and \\ E_{int} = R_{s} (k, θ_{e}) Γ (x, x_{c}; k, θ_{e}) \exp {i 2 k [m Δ z + h (x, x_{c})] \cos θ_{e}} + R_{r} (k, θ_{e}) . \end{array}

_{$P (θ_{e})$}describes the weighting factor related to the pupil function. $P_{1} (θ_{e})$ and $P_{2} (θ_{e})$ are the pupil functions of the illumination and collecting lenses. Furthermore, in the simulation we assume: $P_{1} (θ_{e}) = P_{2} (θ_{e}) = \sqrt{\cos θ_{e}}$ corresponding to [15]. The function $Γ (x, x_{c}; k, θ_{e})$ we introduced in our model represents a heuristical modeling of the shadow factor [19, 20], which describes the intensity distribution assuming a plane wave illumination with wavenumber $k$ and incident angle $θ_{e}$ . $R_{s} (k, θ_{e})$ and $R_{r} (k, θ_{e})$ are the Fresnel reflection coefficients of the measuring object and the reference mirror. J₀ is the zeroth-order Bessel function of the first kind. $h (x, x_{c})$ is the corresponding object height within a local coordinate system, as shown in Fig. 8, where the vector $\vec{s}$ represents the incident ray. $2 δ_{x}$ is the interval where diffraction is considered and thus related to the Airy disc diameter. Therefore, integration over $x_{c}$ takes diffraction into account [18–20]:

Fig. 8 Local and global coordinate systems used for the Richards-Wolf model.

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I (x, Δ z) = \int_{- δ_{x}}^{δ_{x}} I (x, Δ z; x_{c}) d x_{c} .

Results of the Richards-Wolf modeling are shown in Figs. 9 and 10. Figure 9 explains the inversion of the coherence profile for small surface periods: The batwings which can be seen in Fig. 9(d) overlap in Fig. 9(c) leading to an overestimated profile height. Due to the lateral resolution limit the coherence profile of Fig. 9(b) no longer shows height steps but only a sinusoidal shape remaining from the batwings. The profiles plotted in Fig. 10 agree quite well with the measurement results presented in Fig. 2. In addition, Fig. 11 compares two profiles taken from Fig. 10 with simulation results using the Kirchhoff model according to subsection 3.2. Strong deviations of the corresponding profiles calculated under the same assumptions appear. Since the aspect ratio is 0.93 here, the discrepancies between the two models may be explained by the fact that the Kirchhoff model fails for higher aspect ratios.

Fig. 9 Simulation results based on Richards-Wolf modeling for profiles measured by CSI assuming different grating periods Λ and illumination with blue LED and NA = 0.9, black: original rectangular profiles with PV-amplitude of 140 nm, blue: coherence profiles, red: phase profiles for λ_eval = 600 nm (with fringe order determination based on coherence peak position).

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Fig. 10 Simulation based on Richards-Wolf modeling for a rectangular grating of period Λ = 0.3 µm (bottom profile), black: coherence profile obtained from the coherence peak position, color: phase profiles obtained for different evaluation wavelengths ( $λ_{eval}$ = 400 nm, 500 nm, 550nm, 650 nm and 900 nm).

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Fig. 11 Comparison of Kirchhoff and Richards-Wolf modeling for a rectangular grating of period Λ = 0.3 µm and 140 nm PV-amplitude, showing coherence profiles and phase profiles obtained for an evaluation wavelength of 900 nm.

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4. Conclusion

This paper discusses several effects which occur if reflecting rectangular structures of high spatial frequency are to be measured with coherence scanning interferometry using visible light and objective lenses of high numerical aperture.

With respect to the questions formulated at the end of section 2 we draw the following conclusions:

Due to the numerical aperture effect, lower spatial frequencies (longer wavelengths) occur in the measured interference signals, which are related to oblique angles of incidence on the measuring object. Therefore, the phase of the measured signals can be analyzed even at higher wavelengths, leading to an improved lateral resolution and a measured profile amplitude, which is closer to the real amplitude.

In addition, it turns out that the surface topography obtained from the position of the coherence peak (coherence profiles) seems to show a good lateral resolution. However, measured grating structures of high spatial frequency will be inverted as a consequence of the nonlinear transfer characteristics of the instruments for the given amplitude and period.

Starting from evaluation wavelength longer than four times the step height with decreasing evaluation wavelength the amplitudes of the phase profiles tend to zero. For evaluation wavelength below four times the step height the amplitude increases again but with an opposite sign of the profile height.

These effects are analyzed and confirmed theoretically by different models, starting with a simple calculation for a rectangular phase grating, continuing with an extended Kirchhoff model and finally applying the Richards-Wolf model, which seems to be most appropriate in this context. The main difference between Kirchhoff- and Richards-Wolf model is that the latter is not restricted with respect to small aspect ratios as it introduces a shadow factor to consider the illumination close to the edges.

Funding

Deutsche Forschungsgemeinschaft (DFG): Grant no. LE 992/6-2.

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Coherence scanning and phase imaging optical interference microscopy at the lateral resolution limit

Abstract

1. Introduction

2. Interference microscopy at high numerical aperture

3. Theoretical analysis of surface profiles obtained from either phase or coherence peak

3.1 Analysis of phase imaging based on first order diffraction from rectangular profiles

3.2 Simulation of CSI signals from rectangular profiles using the Kirchhoff-Approximation

3.3 Simulation of CSI signals from rectangular profiles using the Richards-Wolf model

4. Conclusion

Funding

References and links

Cited By

Figures (11)

Equations (16)

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