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The intensity of scattered light is extremely sensitive to even small changes of illumination wavelength, incident angle, polarization states, or even the measurement position. To obtain light scattering distributions with varied parameters, time-consuming sequential measurement procedures are typically employed. Here, we propose a concept for the measurement of multiple properties at the same time. This is achieved by tailoring orthogonal frequency division multiplexing (OFDM) for light scattering measurement techniques to the required low inter-channel crosstalk performance. The concept is used for a highly-robust roughness and contamination characterization, to derive one-shot roughness information, as well as to characterize color and appearance.
© 2015 Optical Society of America
1. Introduction
Scattered light arises at surface and material imperfections which can be roughly categorized into (i) interface roughness, (ii) bulk index fluctuations and inclusions, (iii) defects, (iv) subsurface defects, and (v) contaminations [1,2]. Besides simply quantifying the amount and the distribution of scattered light, light scattering techniques have been effectively used to analyze these imperfections. On high-quality optical components, roughness-induced light scattering typically dominates. Hence, light scattering metrology has become an important tool for the characterization of surface roughness. The according fundamental theories (inverse scattering problem) have been proven to work very accurately and reliably [1,3–5].
Many practical applications, however, involve a superposition of different types of imperfections. For example, surfaces often exhibit not only roughness, but also contamination by particles. In order to ensure that the measured scattered light results just from a single type of imperfection, or to identify or even to separate different sources of scattered light, a variety of strategies have been developed. They usually involve multiple measurements of scattered light at different incident angles, illumination wavelengths, or polarization states. Multi measurement approaches have proven to be particularly useful to verify derived roughness information [6–8], to quantify roughness at buried interfaces [9], or to detect contamination [7] or subsurface damage [10,11]. Multiple measurements on optical components are moreover required to evaluate light scattering specifications at multiple nodes within the specified parameter range, e.g. inside a given bandwidth of illumination wavelengths or over a range of incident angles. Multiple measurements at different incident angles are usually needed to provide experimental input for accurate stray light (ray tracing) analysis [12], while realistic rendering applications require both measurements at different incident angles and different wavelengths [13].
However, state-of-the-art high-end goniometric instruments for light scattering measurements with high sensitivity and dynamic range [14–18] enable only a sequential characterization of different parameters, meaning that one measurement with one fixed set of parameters is performed at a time. This leads to a significant increase in measurement time and, moreover, increases the risk of non-stationary measurement conditions. As a consequence, interest has increased in developments that aim at parallelized measurement procedures. Complete or fractional light scattering distributions have been instantaneously acquired using either matrix sensor (e.g. CCD or CMOS) based concepts [19,20] or complex multi-detector assemblies [21,22]. The former are, however, usually limited in sensitivity and, moreover, either limited in the capability to detect near-specular scattered light (caused by stray light from additional optics to extend the covered angular range) or analogously exhibit a reduced covered angular range (no additional optics).
The sample position can be multiplexed by camera based instruments [23]. State-of-the-art approaches for wavelength-multiplexed measurements are mostly based on broad-band illumination and polychromator techniques [24–26], which are considerably limited in sensitivity or stray-light related cross-talk.
Goniometric instruments therefore still define the state-of-the-art. Existing approaches to multiplex wavelength or sample position are usually limited in sensitivity, near angular limit, or the covered angular range of scattering angles. Yet, there is a lack of concepts to multiplex incident angle, polarization, or coherence for light scattering measurements.
In this work, we introduce a previously unpublished, novel concept based on frequency division multiplexing (FDM) that allows measurements of scattered light with different parameters (for example, wavelength, polarization, coherence, angle of incidence, measurement position) to be performed simultaneously [27,28]. In contrast to other approaches, the proposed technique enables parallel measurements without restrictions in sensitivity, dynamic range, and spectral linewidth compared to high-end sequentially operating scatterometers. Furthermore, the mixing of different parameters in different parallel channels becomes possible (e.g. wavelength and/or polarization) enabling the development of one-shot capable sensors tailored to specific inverse scattering problems (e.g. to separate different light scattering mechanisms).
This paper is organized as follows. After introducing the basic light scattering definitions, the concept for simultaneous scattering measurements is presented and analytically described. Different techniques to extend the relative dynamic range between the individual channels are discussed, the achieved performance is demonstrated, and applications are proposed.
2. Background
Distributions of scattered light are quantified by the Angle Resolved Scattering (ARS) function defined as the power ΔPs of the light scattered into the solid angle ΔΩs normalized to that solid angle and to the incident light power Pi [1,2]
Usually, ARS is simplified to a function of only the polar scattering angle. In Eq. (1) we explicitly extended the arguments to demonstrate the large range of variable parameters. Here, φs and θs are the azimuthal and polar scattering angles [Fig. 1], φi and θi accordingly describe the direction of the incident beam, λ is the illumination wavelength, X / Y are the sample coordinates, and p describes polarization states.
Fig. 1 Light scattering geometry and definitions: 1 - sample, 2 - incident beam, 3 - reflected beam, 4 - transmitted beam, θi - angle of incidence, φi - azimuthal angle of incidence, θs - polar scattering angle, φs - azimuthal scattering angle, ΔΩs - solid angle, X / Y - sample coordinates.
ARS corresponds to the radiant intensity scaled by the incident radiant flux, while the sometimes alternatively used bidirectional scattering distribution function (BSDF) corresponds to the surface radiance scaled by the irradiance [29,30]. It can be calculated from ARS by an additional cosine factor:
ARS is the more general form without singular points (at |θs| = 90°) and closer to measurement, while BSDF is more connected to the visual appearance and therefore preferably used in rendering applications – however, BSDF is strictly speaking only defined for surface scattering [31].Scattered light exhibits a direct proportionality to interface roughness properties. The power spectral density function (PSD) is the squared modulus of the Fourier transform of the topography and describes the power of different roughness components as a function of spatial frequencies fx and fy [4]. Integration of the PSD leads to the surface rms roughness and can be calculated from ARS measurements by [32]
where Q is the optical factor containing dielectric constants, polarization states, and scattering geometry. Spatial frequencies and scattering angles are connected by the grating equation [1]. To display ARS (normal incidence) or PSD in 2D-diagrams, an azimuthal averaging in polar coordinates can be performed.3. Multiplexing concept for ARS measurements
3.1 Working principle
The concept proposed in this work is based on frequency division multiplexing (FDM), which is particularly prevalent in telecommunication techniques [33–35]. Several modifications had to be introduced in order to tailor the approach for light scattering analysis, in particular to address the high dynamic ranges and crosstalk requirements.
Figure 2(a) displays a sketch of the proposed ARS concept for the special case of wavelength multiplexing. Light beams from different laser sources (1) are individually marked by optical modulators with different modulation frequencies (2). The beams are combined by dichroic beam splitters (4). The combined beam is guided through the beam preparation optics of the scatterometer (5), hits the sample (6), and the scattered light is detected by a single wavelength- and polarization-insensitive detector mounted on a goniometer (7). Demultiplexing is performed by signal processing (8) where the detector signal is separated into individual channels based on the reference signals (9) containing the modulation frequencies. Either bandpass filters or lock-in amplifiers (LIAs) could be used, however, only the latter offer the signal-to-noise ratio required for ARS measurements of high quality optical components. State-of-the-art sequentially operating light scattering instruments can be easily upgraded by additional optical components in the beam preparation system and additional LIAs, while most of the system can be left unmodified.
Fig. 2 (a) Proposed set-up for multiplexed light scattering measurements. (b) Selected ARS examples.
Crosstalk requirements for light scattering techniques are especially challenging as the ARS of typical samples [Fig. 2(b)] extends over an extremely large dynamic range of more than 12 orders of magnitude (OOM) from the specular reflected power to considerably lower scatter levels (this corresponds to the 1012 fraction of the incident power). Grating-type samples are considered as being especially critical, as extremely low signals of scattered light at one wavelength can get easily superimposed by extremely high signals of a diffracted order of another wavelength from a different channel. Therefore, the elementary FDM principle had to be modified as described in the next section. Multiparallel ARS measurements with a relative dynamic range of at least 8 OOM between the individual channels are aspired, while the absolute dynamic range of the individual channels should yield 14 OOM to achieve a performance comparable to high-end sequential scatterometers. This would allow parallel ARS measurements on the vast majority of typical samples, including diffraction gratings, with different illumination wavelengths or polarizations. To this end, different possible techniques are discussed in the following section.
3.2 Reduction of crosstalk
In order to reduce the crosstalk between different channels, the following approaches are considered. Extraction of modulated signals from random background noise by means of LIAs is a widely used principle. The interested reader is referred to [36,37]. In this section, a model is derived for the special case of demodulating a rectangular modulated signal in the presence of other channels. For the sake of simplicity, signal processing is described for two channels only. The derived key results, however, also apply for multiple channels. A corresponding extension of the model is straightforward.
The rectangular modulated signal U1(t) with amplitude A1 and radial modulation frequency ω1 is superimposed by the rectangular modulated interfering signal U2(t) with amplitude A2 and radial modulation frequency ω2, which yields the signal U(t) at the detector:
The rectangular waveform is described by Fourier series of orders i and j. Demodulation in the LIA is performed by multiplying U(t) with a reference signal of same frequency as signal 1 and a phase shift [36]. As in most state-of-the-art LIAs the phase is extracted by evaluating two phase shifted reference signals [37], the phase is in the following assumed as zero to ease the mathematics. This is however not a principle limitation. The demodulated signal X1(t) is then calculated by:The time independent (DC voltage) output 2A1/π corresponds to the desired measurement signal of the LIA, while the residual unwanted summands of higher frequencies are attenuated by a low-pass filter with the transfer function H(ω) [36] where and represent the forward and inverse Fourier transform, respectively. This works very well to extract a signal buried in stochastic noise, as V1(t) consists apart from the DC output of attenuated broadband noise. For an interfering signal with high amplitude and a discrete frequency, however, the DC output is superimposed by a pronounced oscillation. As a result, measurement uncertainty increases and the readout of the signal amplitude can be systematically distorted. Evidently, crosstalk increases with a growing number of additional channels.Considering Eq. (6), it is now tempting to reduce crosstalk by simply increasing the order of the low-pass filter |H(ω)|. However, an increased filter order also results in longer settling times and thus unwanted longer measurement times, which would reduce the benefits from a parallelized measurement concept to a sequential approach. Alternative concepts to reduce crosstalk are preferable. Equation (5) shows that the higher harmonics of the rectangularly modulated signals contribute noise to the DC output. Simply changing the modulation form to sinusoidal consequently reduces crosstalk. Practically, a sinusoidal modulation can be realized by choosing a chopper slit width that has about the same diameter as the Gaussian laser beam profile or by simply employing power modulated sources. Crosstalk can also be reduced by configuring the modulation frequencies of all channels. As the attenuation by the low-pass filter increases with |ω2-ω1|, ω1, and ω2, it is often advantageous to choose high modulation frequencies that are separated as far as possible.
Digital low-pass filters exhibit zeros in the transfer function [Fig. 3] [38,39], which ideally allows a perfect attenuation of crosstalk from other channels. In its simplest form, the filter kernel of periodicity 2τ corresponds to a moving average filter of order N and can be described by the following transfer function in the frequency domain [38]:
The minima can be exploited to yield high attenuation of crosstalk from other channels with modulation frequencies that fulfill the orthogonality requirement [35] according towhere ζ are the corresponding time modulation frequencies with ω = 2πζ. This also holds for multiple channels.In order to optimize crosstalk beyond the limitations posed by the demodulation process, further techniques are required:
- •Electronic prefiltering is a simple yet effective method to decrease crosstalk or to reduce the influence on the dynamic reserve of the LIA . As the modulation frequencies of all channels are practically known, bandpass filters can be used to attenuate the signals of other channels before the signal is fed into the LIA. However, increased signal noise has to be considered resulting from the additional electronics.
- •Optical prefiltering can be performed by splitting the individual channels optically on different detectors. Yet, it is in this context conceptually only reasonable when a multiplexed polarization separating detection is demanded or the spectral sensitivity of the detection system needs to be extended.
- •Optical adaption: Crosstalk arises not necessarily as a result of too large ARS ratios of the individual channels, but rather because of too large signal ratios A2/A1 (or vice versa A1/A2). Hence, the relative dynamic range can be effectively increased if the signals of all channels are individually (optically) attenuated to yield signal ratios as close as possible to 1. If crosstalk effects can be tolerated up to an electrical dynamic range threshold of DRel = |lg(A2/A1)| = |lg(A1/A2)|, signal ratios between
However, this technique is especially interesting as high-end scatterometers also require an adaption technique to achieve the large dynamic range necessary to characterize high quality optical components. Sometimes this can be realized by changing detection gain levels, however, often an optical attenuation system in front of the detector or in the illumination system is necessary. Hence, the proposed optical adaption technique to reduce crosstalk can also be used to achieve the large overall dynamic range of the instrument.
3.3 Crosstalk performance
To demonstrate the achievable crosstalk performance for light scattering instruments and to confirm the model described in part 3.2, a set-up as displayed in Fig. 2 was arranged for two channels. A PMT was used as a detector; A 405 nm laser diode and a 2ω-Nd:YAG solid state laser were employed for illumination, both power modulated with a rectangular waveform.
In order to characterize crosstalk as a function of modulation frequency, laser 1 (405 nm, channel 1) was modulated at a fixed frequency of ζ1 = 999.8 Hz, while laser 2 (532 nm, channel 2) was modulated between ζ2 = 1 Hz to 1100 Hz. At first, only the light of laser 2 was guided to the detector and only the signal of channel 1 was recorded. The LIA was set to τ = 50 ms with a low pass filter of 6 dB/octave (equals N = 1 in Eq. (8). Calculations were accordingly performed based on Eq. (6). The results of these crosstalk measurements are shown in Fig. 4. Model and experiment are in good agreement. Only between ζ2 = 400 Hz and 700 Hz slight deviations in the crosstalk level are visible. Possible explanations are a non-uniform window function or the not ideally rectangular modulation waveform in the experiment. The pronounced peaks at odd divisors of ζ1 result from the higher orders of the rectangular modulated signal. The smaller peaks at even divisors indicate a slight asymmetry in the modulation waveform. The zero-crosstalk locations predicted by Eq. (9) are clearly represented in the measurement and constitute modulation frequencies where crosstalk shows optimal performance.
Fig. 4 Crosstalk from channel 2 into channel 1 as a function of the modulation frequency of channel 2.
At one of the optimal (ζ1 = 990.8 Hz, ζ2 = 945.8 Hz) and one of the non-optimal (ζ1 = 990.8 Hz, ζ2 = 940.8 Hz) modulation frequencies, crosstalk was examined in more detail. Figure 5 shows the influence of crosstalk regarding signal noise and signal distortion as a function of the signal ratio A2/A1 for different demodulation parameters. For each data point, 100 signal values were recorded. Calculations were performed accordingly. Signal distortion is characterized by deviation from the ideal signal V1,ideal, which corresponds to the signal V1 when no other channels are present (no crosstalk). Simulation and experiment show a good agreement, deviations can be explained by ordinary signal noise. As predicted, crosstalk performance considerably improves for adapted modulation frequencies. For signal noise and signal distortion thresholds of <10% and <2%, respectively, a relative electrical dynamic range of DRel = 2 OOM was achieved. The aspired 8 OOM relative dynamic range can be achieved by optical adaption by means of two optical density steps of 3 OOM, respectively.
Fig. 5 Signal noise and signal distortion as a function of the signal ratio A2/A1 and for different demodulation parameters.
Using optical adaption during measurements is not necessary for light scattering distributions with a relative dynamic range of smaller than 2 OOM between the different channels. For wavelength multiplexing, this is probably true for the majority of typical samples (excluding e.g. gratings or diamond turned surfaces). Still, the necessary relative DRel refers to the signal ratio and not the ARS ratio. Hence, for different channels a difference in the detectivity of the sensor and the scattered intensity of the sample needs to be compensated by adapting the source powers to the experiment - or - by performing one optical adaption at the start of the measurements if the source powers are given.
4. Application
For the experimental part, the light scattering instrument AlbatrossTT [7] was adapted as described in part 3. The compact yet comprehensive table-top scatterometer [Fig. 6] was originally designed for flexible ARS measurements in the whole sphere around the sample; with sensitivities down to 10−9 1/sr and a dynamic range of 14 OOM; at illumination wavelengths of λ = 405 nm, 532 nm, and 640 nm; as well as for different detection and illumination polarization states. Instead of mechanical choppers, the sources (1) are power modulated by means of the reference signals (2), enabling higher modulation frequency, higher frequency stability, and the capability to perform sinusoidal modulations. The beam preparation system (4) is based on a spatial filter which is crucial to achieve a clean core beam; the scattered light of the sample (5) is recorded by a 3D-detection system (6) based on a photomultiplier tube. Three LIAs (7) are now used to perform parallel signal readout under optimized demodulation parameters. Additional attenuators (3) after each laser source can be used to adapt the illumination powers. A bigger laboratory type instrument [14] with comparable specifications was accordingly modified. However, this instrument is moreover capable for the characterization of very large samples (up to 700 mm in diameter) and at laser wavelengths ranging between 325 nm and 10.6 µm.
4.1 Wavelengths scaling analysis
As light scattering mechanisms show a characteristic function to the illumination wavelength, wavelength scaling analysis can be effectively used to discriminate between different light scattering sources and to derive reliable roughness information [7]. Because the concept provides almost identical illumination conditions at different wavelengths, errors as a result of deviations in the measurement position can be eliminated, ensuring a reliable post-measurement analysis.
Parallel multi-wavelength ARS measurements (λ = 405nm, 532 nm, and 640nm) were conducted on a polished nickel surface with all lasers illuminating the sample simultaneously and at the same position [Fig. 7(a)]. The experiments were performed at an angle of incidence of θi = 3°, s-polarized illumination, and solid angle sizes of ∆Ωs = 2.4·10−4 sr for ∆θs = 0.5° steps. All LIAs were set to τ = 50 ms with the low pass filter set to 12 dB/oct. Modulation frequencies where chosen to 7.005 kHz, 0.955 kHz, and 12.105 kHz with respect to Eq. (10), for channel 1 to 3, respectively. For a comparison to conventional measurements, the measurements were repeated sequentially for all three wavelengths. The parallel measurements show a good agreement to the sequentially performed ARS measurements. Rms deviations of 6.4%, 11.2%, and 5.0% (for λ = 405 nm, 532 nm, and 640 nm) are considered as insignificant compared to the combined measurement uncertainty of the instrument (𝛥ARS/ARS = 2% to 20% for k = 2 and as a function of θs [28]). However, the deviations were also affected by ordinary signal noise at low signal levels present in both sequential and parallel measurements.
Fig. 7 (a) Parallel ARS measurements at a polished aluminum surface with a nickel coating compared to conventional sequential ARS measurements. (b) The corresponding PSD functions indicate a non-topographic light scattering source, probably contamination.
For purely topographic scattering, the PSDs calculated from the ARS measurements by Eq. (3) should overlap within the limitations posed by measurement uncertainty [7], which is clearly not the case for this sample [Fig. 7(b)]. This indicates the presence of a non-topographic scattering source, most likely particulate contaminations. The derived PSDs are consequently not fully representing the real surface topography.
4.2 One-shot roughness measurements
Figure 8(a) displays the results of parallel ARS measurements performed with the scatterometer described in [14] at λ = 405 nm and 808 nm on a silicon substrate. The corresponding PSD functions [Fig. 8(b)] show a good overlap and a fractal slope, which indicates that the PSD represent the surface roughness. In these graphs also the spatial frequencies which correspond to the scattering angles θs = −5° and −40° are marked. These are equally spaced over the entire covered spatial frequency range and can already be used to determine the slope of the fractal PSD and thus the rms-roughness [40]. Hence, by using two (single point) detectors at these scattering angles and the two characterization wavelengths, one-shot roughness measurements can be performed. The measurement time can be reduced by a factor of 4 compared to sequential measurements with just one detector. If the sample position is additionally scanned, roughness maps covering the complete sample surface can be generated [40]. Similarly, wavelength scaling could be analyzed by one-shot measurements if wavelengths and scattering angles were adapted to yield overlapping points in coordinates of spatial frequencies.
4.3 Characterization of color and appearance
Multiwavelength light scattering measurements are especially interesting on samples where the wavelength scaling characteristic is hard to predict. Figure 9 displays BRDF measurements performed on a butterfly wing of a male doxocopa cherubina in the reflection hemisphere, at θi = 60°, and parallel at three illumination wavelengths. Obscurations caused by the detector moving in the incident beam were interpolated. BRDF is used here instead of ARS to achieve proportionality to the visually perceived luminance of the sample. The white cross marks the direction of the incident beam, while the white circle marks the direction of a specular reflected beam (not present for this surface).
Fig. 9 Parallel BRDF measurements performed on a butterfly wing at three wavelengths in the reflection hemisphere (top). The measurements were combined to a single RGB color plot (bottom). Bottom left show a focus variation microscope measurement of the topography.
It is striking how the butterfly wing shows a completely different characteristic at different illumination wavelengths. I.e., at 640 nm, the wing seems to be an almost perfect Lambertian scattering surface. Yet, at 405 nm and 532 nm illumination, a glossy almost specular backscattering behavior can be observed, while other directions exhibit scatter levels comparably low to those of black coatings used to reduce stray light in optical systems. This rather strange behavior is caused by a complex connection of optical effects with the photonic crystal structure on the wing [41], causing the butterfly to beautifully appear in different colors when being viewed from different directions.
The three illumination wavelengths were combined in a single RGB plot to illustrate the perceived color. The accuracy of the color calculation can be further increased by using more sampling points in the VIS spectrum.
5. Summary and conclusions
A novel concept was proposed that allows performing wavelength, polarization, coherence, and / or scattering geometry multiplexed light scattering measurements. This is especially helpful to simply decrease set-up and operation time, and to design one-shot-capable light scattering instruments tailored to tackle specific inverse problems (e.g. roughness measurement or particle detection). Moreover, time-critical observations can be made simultaneously at different parameters and non-stationary effects can be avoided. The concept is based on frequency division multiplexing (FDM), a technique mostly prevalent in telecommunications, which was tailored in the frame of this publication to meet the extreme requirements of light scattering metrology. In contrast to state-of-the art developments using polychromator techniques and broad band sources, the concept enables ARS measurement at the same sensitivity and dynamic range as high-end sequentially performing scatterometers.
Experiments were performed to verify the performance. The concept enables ARS measurements to be performed with over 8 OOM relative dynamic range between the individual channels, while the absolute dynamic range of the individual channels yield 14 OOM. This was achieved by a combination of orthogonal modulation frequencies and optical adaption. Parallel multiwavelength ARS measurements compared to sequentially performed measurements showed average rms deviations of <8% over a dynamic range of 3 OOM. These deviations include contribution of ordinary signal noise especially at low signal levels comparably present in classical sequential measurements. However, these uncertainties are comparable to the instrument measurement uncertainties and can consequently be considered small.
Applications covering wavelength scaling, as well as color, appearance, and roughness characterization were presented.
Acknowledgments
The authors would like to thank T. Herffurth and M. Hauptvogel (all Fraunhofer IOF) for interesting discussions and contribution to measurements.
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