Abstract

Spectral cameras with integrated thin-film Fabry–Perot filters enable many different applications. Some applications require the detection of spectral features that are only visible at specific wavelengths, and some need to quantify small spectral differences that are undetectable with RGB color cameras. One factor that influences the central wavelength of thin-film filters is the angle of incidence. Therefore, when light is focused from an imaging lens onto the filter array, undesirable shifts in the measured spectra are observed. These shifts limit the use of the sensor in applications that require fast lenses or lenses with large chief ray angles. To increase flexibility and enable new applications, we derive an analytical model that explains and can correct the observed shifts in measured spectra. The model includes the size of the aperture and physical position of each filter on the sensor. We experimentally validate the model with two spectral cameras: one in the visible and near-infrared region and one in the short wave infrared region.

© 2018 Optical Society of America

1. INTRODUCTION

Spectral imaging is a technique that combines photography and spectroscopy. It allows one to capture an image of a scene and simultaneously sample the electromagnetic spectrum, providing a “fingerprint” for each point in the scene.

Some applications require the detection of spectral features that are only visible at specific wavelengths, and some require the quantification of small spectral differences that are undetectable with RGB color cameras [1]. Spectral cameras can therefore achieve higher selectivity in machine-learning applications and allow for physical interpretation of the data (e.g., material identification) [2,3].

In spectral camera designs with integrated thin-film Fabry–Perot filters, focused light from a finite aperture widens the transmittance peaks and shifts them toward shorter wavelengths [46]. The problem of assembling an accurate spectrum from the filter bank is complicated by the fact that the filters are patterned across the sensor. This makes the shift depend on both the physical position of each filter and the size of the aperture.

The shifts in wavelength limit the use of the sensor in applications that require lenses with large chief ray angles or require fast lenses (i.e., small f-number). The latter is often the case, since spectral filters are in general narrowband, which implies that more light or longer integration times are required. In this paper, we therefore study the effect of the f-number and the chief ray angle on the measured spectrum and derive a method to correct the spectral shift.

The main contribution of this paper is the use of the concept of the “effective refractive index” [7] and a convolution model of the effect of the aperture [8] in the context of spectral cameras with integrated thin-film filters that are patterned across an imaging sensor. We derive and use a compact equation to correct the spectral shifts observed in measurements. The presented method thus enables new applications and offers an increased flexibility for the sensor to be used in systems with different optical requirements.

The remainder of this paper is structured as follows. In Section 2, we discuss related work. In Section 3, we discuss the underlying theory and derive a method for wavelength correction. In Section 4, we validate the model experimentally, and in Section 5 we discuss the results and their limitations.

2. RELATED WORK

The presence of a finite aperture or, more generally, a varying angle of incidence, poses a challenge for many spectral imager designs [9,10]. Grating- and prism-based systems require careful alignment and good control of the angularity. This usually requires additional optical components (e.g., collimator). Spectral cameras with linear variable filters also need to take into account the angle of incidence. In Ref. [11], the authors analyze the effect of an aperture on their filter. The authors did not attempt to use their model to correct shifts in actual measurements. The authors of Ref. [12] briefly study the effect of an angle of incidence on interference filter-based designs. While they mention the concept of effective refractive index, they do not provide any practical results to describe the behavior for an actual aperture. The idea of using a convolution model for the effect of an aperture on the spectrum measured with a Fabry–Perot etalon is not new [8,13,14]. However, earlier results did not include thin-film Fabry–Perot filters placed behind an objective lens, but rather a Fabry–Perot spectrometer with an objective lens behind the etalon with the aperture in the imaging plane.

To the best of the authors’ knowledge, this is the first time that a practical method is proposed that corrects the shift in thin-film filters caused by an aperture, and that also takes into account the patterning on the spectral sensor such that the output of each filter is appropriately corrected and a more accurate spectrum can be assembled.

3. THEORY AND METHODS

In this section, we first introduce a model of a spectral imaging system with integrated thin-film filters (Section 3.A). We then discuss how the model of an ideal Fabry–Perot filter can be used to describe the effect of an angle of incidence on thin-film filters (Section 3.B). This is used to model the effect of a focused beam of light as a convolution of the transmittance with a kernel that depends on the f-number and the chief ray angle (Section 3.C). From this model a compact formula is derived that, for each filter on the sensor, can correct the shift in central wavelength (Section 3.D).

A. Spectral Imaging System Model

The imaging system used in this paper consists of an objective lens with a finite aperture (or more precisely, the exit pupil) that focuses the light onto pixels of an imaging sensor with integrated thin-film interference filters (Fig. 1). The output DN in digital numbers of a pixel with integrated filters can be modeled as

DN=λminλmaxT(λ)·L(λ)dλ.
Here L(λ) is the irradiance spectrum of the light incident at the pixel, and T(λ) is the transmittance of the filter measured under orthogonal collimated light conditions. The gain factors are assumed to be equal to 1 and are therefore omitted [15]. The limits of the integral describe the bandpass range of the spectral camera.

 

Fig. 1. Schematic representation of how the aperture focuses light on the spectral imaging sensor with integrated thin-film optical filters.

Download Full Size | PPT Slide | PDF

For a given f-number and chief ray angle θCRA, the effect of the aperture on the filter with central wavelength λcwl is modeled as a convolution of T(λ) with a kernel Kθcone,θCRA(λ), where the half-cone angle θcone characterizes the f-number:

DN=λminλmax(Kθcone,θCRA*T)(λ)·L(λ)dλ,
with * being the convolution operator, and the f-number being the ratio of the effective focal length to the diameter of the entrance pupil. The kernel Kθcone,θCRA is defined in Subsection 3.C.

Below in Table 1, the most important symbols are summarized.

Tables Icon

Table 1. List of the Main Symbols and Their Meaning

B. Collimated Light at Oblique Incidence

1. Ideal Fabry–Perot Model

The ideal Fabry–Perot etalon is a device that consists of two highly reflecting parallel mirrors such that reflected rays can interfere. By tuning the distance t between the plates, selection of narrow wavelength bands becomes possible (Fig. 2).

 

Fig. 2. Basic Fabry–Perot etalon. Two parallel near-perfect mirrors are separated by a material of refractive index ns and thickness t.

Download Full Size | PPT Slide | PDF

The Fabry–Perot has specific transmittance characteristics that are described by the Airy function [16].

For an angle of incidence ϕ, the transmittance peak centered at λcwl (at orthogonal incidence) shifts by

Δ(ϕ)=λcwl(1cosϕs)λcwl(1cosϕns),
where ns is the refractive index of the cavity material, and ϕs is the angle in the cavity material after refraction [16].

In practice, the Fabry–Perot design can be approximated using multilayer thin-film structures [16] with Bragg mirrors that are more complex, and which are discussed in the next section.

2. Ideal Fabry–Perot Model Applied to an All-Dielectric Single-Cavity Filter

A Fabry–Perot etalon approximated with a thin-film structure can be partially understood in terms of an ideal Fabry–Perot model. It can be proven that the shift in central wavelength for small angles ϕ is asymptotically equivalent to the shift of the transmittance of an ideal Fabry–Perot etalon with a cavity material with effective refractive index neff [7,16]. It has been shown that the small-angle approximation can remain valid up to 40° [7].

Therefore, for collimated light at an incident angle ϕ, the shift Δ (ϕ) in central wavelength of a transmittance peak of a thin-film Fabry–Perot filter is

Δ(ϕ)λcwl(1cosϕneff).
The shift of the transmittance spectrum T(λ) at a tilt angle ϕ can also be modeled as a convolution with a shifted Dirac distribution such that
Tϕ(λ)T(λ)*δ(λλcwl(1cosϕneff)).
We will now use this notation to model the effects of the aperture.

C. Focused Light from a Finite Circular Aperture

Focused light from a circular aperture can be interpreted as a distribution of incident angles. This distribution becomes more skewed for larger chief ray angles (Fig. 1). For each chief ray angle, the skewed cone of light can be decomposed in contributions that have identical angles of incidence (Fig. 3). This insight allows the problem to be treated as a weighted sum of contributions of the form Eq. (5).

 

Fig. 3. Decomposition in contributions with equal angles of incidence. The weight of a contribution is measured by the length of the blue arc within the aperture.

Download Full Size | PPT Slide | PDF

The decomposition can be formalized in the following way. Let x be the distance of the focusing plane to the aperture plane (or exit pupil), let R be the radius of the aperture for a given f-number, and let the chief ray angle be defined as θCRA=arctandx, with d being the off-center distance (Fig. 4).

 

Fig. 4. Decomposition of the light cone from the aperture in contributions of equal angle of incidence ϕ. The weight of each contribution is the infinitesimal area dA. Here d is the distance of the pixel from the optical axis. (a) Top view; (b) cut section view.

Download Full Size | PPT Slide | PDF

The resulting transmittance T^θcone,θCRA(λ) is the sum of the transmittance functions Tϕ(λ) at different tilt angles [Eq. (5)]. The weight of each contribution is the infinitesimal area dA of the aperture that contributes to identical angles of incidence (Fig. 4). Therefore,

T^θcone,θCRA(λ)=ApertureTϕ(λ)dAAperturedA,
where the normalization is required because of conservation of energy. Because of linearity, T^θcone,θCRA(λ) can be modeled as a convolution of the transmittance T(λ) at orthogonal collimated conditions with a kernel Kθcone,θCRA(λ) such that
T^θcone,θCRA(λ)=T(λ)*Apertureδ(λΔ(ϕ))dAAperturedA
=T(λ)*Kθcone,θCRA(λ).
To solve the integral in Kθcone,θCRA, we first rewrite it in terms of r such that dA is the area of an infinitesimal ring segment (Fig. 4):
dA=2γ(r)rdr,
where γ(r) is determined by the law of cosines in ΔXYZ as
γ(r)=Re(arccosd2R2+r22dr).
By taking the real part of the inverse cosine, the case in which there is a complete contributing circle within the aperture is also modeled (Fig. 3). This is because Rearccosz=π, for z1.

We proceed by rewriting the integral in terms of the angle of incidence ϕ by substituting r=xtanϕ such that

dA=2x2γ(ϕ)tanϕcos2ϕdϕ,
and γ is redefined as
γ(ϕ)=Re(arccostan2θCRAtan2θcone+tan2ϕ2tanθCRAtanϕ).
The factor 2x2γ(ϕ)tanϕcos2ϕ in Eq. (11) is the angular distribution of incident angles that was illustrated in Fig. 3.

Using Eqs. (11) and (12), the change of variables in the integral in Eq. (A4) gives

Kθcone,θCRA(λ)=ϕminϕmax2x2γ(ϕ)tanϕcos2ϕδ(λΔ(ϕ))dϕπR2,
where the integral in the denominator is equal to πR2, which is the area of the exit pupil. Here ϕmin and ϕmax are the smallest and largest incident angles coming from the aperture and are defined as
ϕmin={0ifθCRA<θconearctan(tanθCRAtanθcone)ifθCRAθcone,ϕmax=arctan(tanθCRA+tanθcone),
and illustrated in Fig. 3.

The complete analytical solution of Eq. (13) is given in Appendix A. An asymptotic approximation is

Kθcone,θCRA(λ)2neff2λcwlγ(neff2λλcwl)πtan2θcone,
which is valid for small values of λλcwl. This means that to a first approximation, the kernel is proportional to γ(ϕ), whose argument is an approximation of the inverse of Eq. (4) (See also Appendix A).

The shape of the convolution kernel Kθcone,θCRA models how the central wavelength and full width at half-maximum (FWHM) of the transmittance will change. For increasing θcone and θCRA, the kernel Kθcone,θCRA shifts and becomes wider (Fig. 5), and thus so does the transmittance (Fig. 6).

 

Fig. 5. Shape of the kernel [Eq. (15)] for different ratios of θcone and θCRA. The mean value of each kernel according to Eq. (17) is marked with a circle.

Download Full Size | PPT Slide | PDF

 

Fig. 6. Transmittance of a thin-film optical filter simulated with TFCalc for different apertures compared to applying the convolution kernel to the transmittance at collimated conditions. In this example neff=1.75.

Download Full Size | PPT Slide | PDF

 

Fig. 7. Experimental setups. (a) VNIR Snapscan with color filter; (b) SWIR Snapscan with reflectance target.

Download Full Size | PPT Slide | PDF

To verify the accuracy of the model, the effect of applying the kernel is compared to a reference simulation of a thin-film stack using TFCalc [17], which is based on the transfer-matrix method. The transfer-matrix method requires the thickness and refractive index for each layer of the thin-film stack. In contrast, the kernel requires only one parameter: the effective refractive index. The simulation with TFCalc shows a very good match with the kernel model (Fig. 6).

D. Method for Central Wavelength Correction

The kernel Kθcone,θCRA models the effect of the aperture on a filter. This allows system designers to predict the effective position and FWHM of the transmittance and simulate the impact on real-life measurements with a camera. For practical use a simplified equation is derived.

To quantify the shift in central wavelength, the mean of the kernel (when interpreted as a distribution) is used:

λ¯θcone,θCRA=λminλmaxλKθcone,θCRA(λ)dλλminλmaxKθcone,θCRA(λ)dλ=(normalized)λminλmaxλKθcone,θCRA(λ)dλ.
Calculating the exact mean of the kernel is not straightforward, and a closed-form analytical solution might not exist. Using asymptotic approximations (Appendix B), Eq. (16) becomes
λ¯θcone,θCRAλcwlneff2(θcone24+θCRA22)forθcone,θCRA0.
Thus for small angles, the shift in central wavelength is the sum of two independent contributions: the f-number and the chief ray angle. The validity of this approximation is further discussed in Appendix C.

To calculate the new central wavelength of a filter placed behind an aperture, the mean is subtracted from the original central wavelength such that

λcwlnew=λcwlλ¯θcone,θCRAλcwl(1θcone24neff2θCRA22neff2).
This correcting transformation can be efficiently applied for each individual filter on a sensor.

For practical use, we will reformulate Eq. (18) in terms of the working f-number f#,W and the off-center distance d of the filter. Assuming that θcone=arctan12f#,W12f#,W,

λcwlnewλcwl(1116f#,W2neff2d22x2neff2).

4. EXPERIMENTAL VALIDATION

In this section, Eq. (18) is used to correct the shifts in the spectrum measured with a representative experimental setup.

Two camera systems are used: imec’s Short Wave InfraRed (SWIR) and Visible Near InfraRed (VNIR) Snapscan cameras. We have selected commercial off-the-shelf lenses that allow for f-numbers lower than 2 and where the exit pupil is not farther than 40 mm from the focal plane so we can test for significant chief ray angles (up to 15°). We tested the lenses in regimes where there is no significant vignetting. For the VNIR lens used in the experiments, this assumption is only valid when it is used at a high f-number (i.e., f/8).

The effectiveness of the correction is demonstrated in three experiments. In the first experiment, the sample is centered on the optical axis (θCRA=0). The reflectance is measured at different f-numbers. In the second experiment, the transmittance of a color filter is measured at different chief ray angles at a high f-number. In the third experiment, the sample is positioned off-axis and scanned at different f-numbers. In all three cases, the central wavelength of the filters needs to be corrected.

A. Experimental Setup

For the VNIR region, imec’s VNIR Snapscan system (150 spectral bands) [18] is used with an Edmund Optics 16 mm C Series VIS-NIR fixed focal length lens. For the SWIR region, imec’s SWIR Snapscan system (128 spectral bands) [19] is used with an Optec 16 mm SWIR lens. The properties of the lenses in the setup are listed in Table 2.

Tables Icon

Table 2. Properties of the Experimental Setupa

Without loss of generality, the Snapscan system is chosen because it is a system that allows control of the chief ray angle. This is because the sensor is positioned behind the lens, and its position can be controlled using an integrated translation stage. The method remains applicable to other classes of spectral cameras, even with different patterning of the filters on the sensor.

The reflectance of the sample is measured using a flat-fielding approach. This involves scanning the same scene twice, i.e., once with the sample and once with a Spectralon white reference tile. For each pixel, an estimate of the reflectance is obtained by dividing the measurement of the sample by the measurement of the white tile after subtracting the dark image. In the case of a transmittance measurement, an image of a white tile with a transmission filter on the front side of the lens is used instead of a sample.

In terms of the spectral imaging model [Eq. (2)], the reflectance estimation can be formulated for each pixel as

reflectance=DNsampleDNdarkDNwhiteDNdark.
Different samples for each camera are used (Fig. 7). For the SWIR camera, the reflectance of a Spectralon calibrated multicomponent wavelength standard is measured. For the VNIR camera, the transmittance of a Thorlabs FGB67S colored glass filter is measured. A transmittance filter enables the convenient measurement of the same spectrum at different off-axis distances. It is important that the filter itself is not angle-dependent; therefore, colored glass is used instead of an interference filter.

B. Determining the Model Parameters

To apply the correction, first all model parameters need to be known. These parameters are: the effective refractive index, the working f-number, and the distance to the exit pupil.

Assuming that the distance to the exit pupil is known (Table 2), we briefly discuss the working f-number and different methods to estimate the effective refractive index.

The working f-number is calculated as ([20])

f#,W(1+mP)f#,
with m the linear magnification, P the pupil magnification, and f# the f-number when focused at infinity. The latter is usually equal to the f-number markings on the lens. In our experiments, the correcting factor (1+mP)=1.074 for the VNIR setup and 1.03 for the SWIR setup.

The effective refractive index neff can be estimated in multiple ways. A first approach is to use the theoretical result for the first resonance mode of Fabry–Perot-like designs with a low-index cavity [16]:

neff=nL1nLnH+(nLnH)2,
where nL and nH are the low refractive index and high refractive index material parameters, respectively.

If one does not know the material parameters, there is another method. The alternative is that one can characterize the sensor under collimated light conditions at different wavelengths [21]. By repeating this procedure at different tilt angles, the effective index follows from the shifts in central wavelength. From this procedure, we then estimated neff1.7 for the VNIR sensor and neff1.75 for the SWIR sensor. These are the values that we will use for correction.

C. Data Analysis

In this section, we explain how the measurements from the experiments will be analyzed and how the graphs should be interpreted.

For each filter, the measured reflectance is plotted at its central wavelength in collimated light conditions at normal incidence. Therefore, without correction, a shift of the filters towards shorter wavelengths creates the illusion that the spectrum moves toward longer wavelengths (Fig. 8). The reflectance is normalized by its peak value.

 

Fig. 8. Experiment 1 (SWIR). The same sample is measured at different f-numbers. The shift becomes larger for smaller f-numbers, causing the spread of shifts in the graph. The shifts are corrected using Eq. (23). The working f-number f#,W [Eq. (21)] is used for correction; the legend shows the f-number f# as marked on the lens. (a) Uncorrected; (b) central wavelength, corrected.

Download Full Size | PPT Slide | PDF

To correct the shift in the data, the central wavelengths at which the reflectance is plotted are updated using Eq. (18). Note that the angles in Eq. (18) are in radians, while we will often refer to the angles in degrees.

To better explain the effect of the aperture on the measurements, we will also compare the real measurements to simulations.

For comparison with future work, two quantitative error measures are used: the correlation coefficient and the maximum error between the most shifted spectrum and the reference. The correlation and maximum error are referred to as “corr” and “maxerr,” respectively, in Figs. 811. The correlation coefficient measures the improvement in shift, and the maximum error will better capture the losses in detail (Fig. 10). To enable comparison, the uncorrected and corrected spectra were resampled on a common grid of wavelengths.

 

Fig. 9. Experiment 2 (VNIR). The transmittance of the color filter is measured at different off-axis distances at f/8, which is a high f-number. The shifts are corrected using Eq. (24). (a) Uncorrected; (b) central wavelength, corrected.

Download Full Size | PPT Slide | PDF

 

Fig. 10. Image of the scene from Fig. 7(a) at the 721 nm band before and after central wavelength correction (Fig. 9). The uniformity of the scene is significantly improved. The corners are outside the image circle of the lens. (a) Before correction; (b) after correction.

Download Full Size | PPT Slide | PDF

 

Fig. 11. Experiment 3 (SWIR). The reflectance of the sample is measured at different f-numbers at an off-axis position with θCRA10.2°. The shifts are corrected using Eq. (25). As a reference, the measurement at f/5.6 at zero chief ray angle is used. (a) Uncorrected; (b) central wavelength, corrected.

Download Full Size | PPT Slide | PDF

D. Experiment 1: Sample in Center (SWIR)

The sample is centered at the optical axis of the lens. In this experiment, the chief ray angle is equal to zero such that

λcwlnewλcwl(1116f#,W2neff2).
For example, for f/2, neff=1.75, and λcwl=1550nm, the shift is λcwl(1116f#,W2neff2)7.45nm. For calculating f#,W, Eq. (21) is used. In this example, f#,W=1.03·2=2.07.

The reflectance of the same sample is measured at different f-numbers, and Eq. (23) is used to correct the shifts in wavelength. After applying the correction, the spread between the measurements is significantly smaller (Fig. 8).

E. Experiment 2: Sample at Different Positions (VNIR)

Using the VNIR Snapscan, the transmittance spectrum is measured at different off-axis distances at f/8. At this high f-number, θcone=3.58°=0.0625rad. The selected distances d from the center are 0, 2.75, 4.4, and 5.5 mm, which correspond to chief ray angles of 0°, 7.5°, 12°, and 15°, respectively.

Assuming that θcone=0, then

λcwlnewλcwl(1θCRA22neff2)=λcwl(1d22x2neff2),
with x=21mm and neff=1.7. Because of the high f-number, as in the collimated case, we only expect to observe an offset with little deformation of the spectrum. And indeed, after correction there is a very good overlap of the spectra, showing almost no deformations (Fig. 9).

We show the spatial image (2048pixels×2048pixels) at a wavelength of 721 nm (Fig. 12). Because at 721 nm the spectrum has a strong edge, shifts cause a significant gradient in intensity. After correction, as desired, the image is spatially uniform.

 

Fig. 12. Experiment 3 simulated (SWIR). The offset and loss of detail are very similar to those observed in real measurements (Fig. 10). (a) Uncorrected; (b) central wavelength corrected.

Download Full Size | PPT Slide | PDF

F. Experiment 3: Sample Off-Center (SWIR)

The multicomponent sample is placed at an off-axis position such that the chief ray angle is about 10.2°. This adds an additional offset on top of the spread caused by the f-number. Therefore both effects need to be taken into account:

λcwlnewλcwl(1116f#,W2neff2d22x2neff2).
For example, if λcwl=1550nm, neff=1.75, x=35mm, and d=6.23mm, then θCRA=10.2°=0.178rad. Therefore, the offset λcwlθCRA22neff2=λcwld22x2neff28nm. For calculating f#,W, Eq. (21) is used.

We observe three important effects in the measurements (Fig. 10). First, there is still the effect of the cone angle causing a spread in the measurements. Second, there is an offset shift caused by the chief ray angle. Last, there is an increased loss of detail compared to Fig. 8. This loss is due to an increased bandwidth of the filters, which has a smoothing effect and cannot be corrected by simply shifting the central wavelengths. This loss in detail is modeled by the increased width of the kernel, which has a smoothing effect.

The observed effects can also be simulated using Eq. (2) (Fig. 11). For this purpose, the sensor used in the experiment was characterized under collimated light conditions at normal incidence [21]. In essence, for each filter T(λ) in Eq. (1) was measured. The effect of the aperture is then simulated using Eq. (2), combined with the calibrated reflectance spectrum of the Spectralon calibrated multicomponent tile.

5. DISCUSSION

The experimental results show that the analytical model can be successfully applied to correct the spectral shifts in the different experiments with minimal effort in terms of calibration.

The presented correction method takes into account the physical position of each individual filter such that, for each point in the scene, a more accurate spectrum can be assembled. There are many applications for which this contribution is very relevant. In material classification, for example, the method could help to reduce the within-class variation and improve the between-class separation. The method could also help to reduce errors in parameter quantification (e.g., agriculture indices [22]).

In our experiments, we realigned the spectra by plotting the data points at their corrected wavelengths. An alternative would be to interpolate the data points at their corrected wavelengths and resample at the original uncorrected wavelengths. From an end-user perspective, this makes more sense because the corrected output of each filter can always be interpreted at the same wavelength.

The method can correct the shifts in wavelength but not the loss in detail at larger apertures. For larger apertures, the filters have an increased bandwidth, which inherently implies a loss of detail. This only becomes an issue when the spectra contain high-frequency components that will be significantly smoothed.

We demonstrated that the derived approximation is sufficiently accurate to correct the shifts for the incident angles that were tested. To push the limits of the method, one could attempt to expand the asymptotic series to higher orders or use numerical approximations of the mean (See Appendix C).

The kernel can also be used as an alternative to investigate the effect of an aperture on filters instead of the transfer-matrix method. The advantage is that the kernel can be applied to the transmittance spectrum as it is measured, while the transfer-matrix method first requires a fitting of model parameters. Assuming that the user does not know what thin-film filters are deposited, the fitting problem becomes even harder to solve: the number of layers, the materials used, and the thickness of each layer need be fitted. In contrast, the effective refractive index is a single parameter that needs to be measured once and requires no further knowledge of the stack.

While assumed constant, the effective refractive index is in principle wavelength-dependent and could vary because of process tolerances. The effective refractive index is also different for harmonics of the cavity [16]. So while Eq. (22) might provide a good estimate, it might be even better to characterize the effective refractive index experimentally by tilting the sensor under collimated light with a monochromator. This characterization only needs to be done once.

The presented model does not take vignetting into account. While there are many machine vision applications (e.g., medical) that require high-quality lenses, this remains an important limitation, since vignetting implies that part of the light beam is cut off, which changes the distribution of incident angles. This will be investigated in future work.

6. CONCLUSION

Some applications require the detection of spectral features that are only visible at specific wavelengths and some require quantifying small spectral differences that are undetectable with RGB color cameras. The use of the spectral sensor in these applications is limited because with integrated thin-film optical filters an aperture causes undesired shifts in the measured spectra.

Taking into account the physical position of each filter in the filter bank, the observed shifts of the measured spectra can be corrected using a compact formula. This was demonstrated experimentally with two realistic spectral imaging setups.

Our approach is of high practical relevance because it enables new applications and offers increased flexibility for using the sensor in different environments.

APPENDIX A: CLOSED-FORM SOLUTION OF THE KERNEL

In this section, we derive a closed-form solution of the kernel and derive the analytical approximation given in Eq. (15).

The kernel was defined as

Kθcone,θCRA(λ)=ϕminϕmax2x2γ(ϕ)tanϕcos2ϕδ(λΔ(ϕ))dϕπR2
=ϕminϕmax2x2γ(ϕ)tanϕcos2ϕδ(λΔ(ϕ))dϕπx2tan2θcone
=ϕminϕmax2γ(ϕ)tanϕcos2ϕδ(λΔ(ϕ))dϕπtan2θcone.
By introducing the indicator function Π=H(ϕϕmin)H(ϕϕmax), the limits of the integral can be replaced such that
Kθcone,θCRA(λ)=2γ(ϕ)tanϕcos2ϕδ(λΔ(ϕ))Π(ϕ)dϕπtan2θcone,
where H(·) is the Heaviside step function.

Let

Δ(ϕ)=λcwl(1cosϕs)
=λcwl(1cos(arcsinsinϕneff))
=λcwl(11sin2ϕneff2),
and its inverse,
Δ1(λ)=arcsin(neffλλcwl(2λλcwl))
=neff2λλcwl+O(λ3/2λcwl3/2),λλcwl0.
Here O(·) is the Big O notation and describes the limiting behavior of the error term of the approximation.

If we define u=λΔ(ϕ), it follows that

du=λcwlcos(ϕ)sin(ϕ)neff21sin(ϕ)2neff2dϕ.
In Eq. (A4), we now replace dϕ using Eq. (A10) and substitute ϕ with Δ1(λu). Then δ(λΔ(ϕ)) becomes δ(u). Therefore the result can be simplified by employing the sampling property of the Dirac distribution on the integrand. The integral is then equal to its integrand evaluated at u=0 such that
Kθcone,θCRA(λ)=2γ(Δ1(λ))tan(Δ1(λ))cos2(Δ1(λ))πtan2θcone·neff21sin2(Δ1(λ))neff2λcwlsin(Δ1(λ))cos(Δ1(λ))Π(Δ1(λ)).
By grouping the terms, the closed-form analytical solution can be formulated as
Kθcone,θCRA(λ)=g(λ)γ(Δ1(λ))πtan2θconeΠ(Δ1(λ)),
with γ as defined in Eq. (12) and the function g(λ) containing all remaining terms. By applying trigonometric identities g can be simplified such that
g(λ)=2neff2λcwl·(1λλcwl12neff2λλcwl+neff2λ2λcwl2)
=2neff2λcwl+O(λλcwl),forλλcwl0,
which is approximately a constant function.

Now using Eqs. (A8) and (A13), the kernel can be approximated for λλmin=Δ(ϕmin)0 as

Kθcone,θCRA(λ)2neff2λcwl·γ(neff2λλcwl)πtan2θcone.
The indicator function can be omitted, since γ(Δ1(λ)) is zero for λ>λmax=Δ(ϕmax).

This approximation is plotted in Fig. 5. Notice that for θCRA=0, the kernel is asymptotic to a rectangular window.

APPENDIX B: MEAN OF THE KERNEL

We use the mean value of the kernel, when interpreted as a distribution, as a measure of the shift in central wavelength. The mean is defined as

E(θcone,θCRA)=λ¯θcone,θCRA=λminλmaxλKθcone,θCRA(λ)dλ.
The Taylor expansion in θCRA and θcone around zero is
E(θcone,θCRA)E(0,0)+EθCRA|(0,0)θCRA+Eθcone|(0,0)θcone+122EθCRA2|(0,0)θCRA2+122Eθcone2|(0,0)θcone2+2EθconeθCRA|(0,0)θconeθCRA.
We will now determine the coefficients of this power expansion.

For collimated light at zero chief ray angle, there is no shift. Thus E(0,0) should be 0. By construction, the area of the kernel is constant:

λminλmaxKθcone,θCRA(λ)dλ=1,θcone,θCRA>0.
Yet, for θcone,θCRA0, λmin,λmax0 and thus converges, loosely speaking, to a Dirac distribution. Applying the sampling property in Eq. (B1) then implies that E(0,0)=0.

To calculate the coefficients of the monomials in θcone, one can calculate that for θCRA0, γ(λ)π in Eq. (A12), and λmin=0. The integral then becomes

0λmaxλg(λ)tan2θconedλ.
Since the upper limit of the integration, λmax, depends on θcone [Eq. (14)], we first substitute λ=vλmax such that
E(θcone,0)=01vλmax2g(vλmax)tan2θconedv.
Using a symbolic solver, the series expansion of the integrand can now be calculated. Each term of the expansion is then integrated such that
E(θcone,0)=λcwl(θcone24neff2+(14neff2)24neff4θcone4+O(θcone6)),forθcone0.
To calculate the coefficients of the monomials in θCRA, one can use the physical insight that for θcone0, the problem becomes equivalent to the pure tilt case, where the shift is equal to Δ(θCRA). Expanding this around zero gives
E(0,θCRA)=λcwl(θCRA22neff2+(34neff2)24neff4θCRA4+O(θCRA6)),forθCRA0.
Comparison with a numerical solution suggests that the contribution of the monomial θconeθCRA is negligible, and we will assume that it is zero. This assumption is supported by the experiments and is justified further in Appendix C.

Disregarding higher orders, the formula for the shift in central wavelength is approximated by

E(θcone,θCRA)λcwl(θcone24neff2+θCRA22neff2).

APPENDIX C: VALIDITY OF THE APPROXIMATION

To validate the quality of the approximation [Eq. (B8)], we compare it with the numerical solution of Eq. (B1) (Fig. 13).

 

Fig. 13. Approximation of the mean value compared to the numerical result (neff=1.7). For large chief ray angle, the shift is slightly overestimated.

Download Full Size | PPT Slide | PDF

For experiment 3, the error between the exact shift and approximation is small (1.6 nm at λcwl=1550nm) compared to the total correction of 18 nm.

The approximation error grows proportional to θcone2θCRA2. This suggests that in the power series expansion an important higher-order term was truncated.

Based on direct comparison with the numerical solution, we propose the following approximation:

E(θcone,θCRA)λcwl(θcone24neff2+θCRA22neff2θCRA2θcone2neff2).
This approximation agrees well with the numerical result (Fig. 14) and even holds up well for much larger values of θcone and θCRA (Fig. 15).

 

Fig. 14. Approximation of the mean value compared to the numerical result (neff=1.7). For large chief ray angle, the estimated shift becomes too large.

Download Full Size | PPT Slide | PDF

 

Fig. 15. For very large values of θcone and θCRA, the higher-order approximation [Eq. (C1)] fits the numerical solution very well, while the more simple approximation deteriorates (neff=1.7).

Download Full Size | PPT Slide | PDF

Acknowledgment

We thank Joren Vanherck for his valuable comments and discussions.

REFERENCES

1. P. Shippert, “Why use hyperspectral imagery?” Earth Sci. 70, 377–379 (2004).

2. G. Lu and B. Fei, “Medical hyperspectral imaging: a review,” J. Biomed. Opt. 19, 010901 (2014). [CrossRef]  

3. A. Gowen, C. O’Donnell, P. Cullen, G. Downey, and J. Frias, “Hyperspectral imaging—an emerging process analytical tool for food quality and safety control,” Trends Food Sci. Technol. 18, 590–598 (2007). [CrossRef]  

4. N. Tack, A. Lambrechts, P. Soussan, and L. Haspeslagh, “A compact, high-speed, and low-cost hyperspectral imager,” Proc. SPIE 8266, 82660Q (2012). [CrossRef]  

5. B. Geelen, N. Tack, and A. Lambrechts, “A compact snapshot multispectral imager with a monolithically integrated per-pixel filter mosaic,” Proc. SPIE 8974, 89740L (2014). [CrossRef]  

6. B. Geelen, C. Blanch, P. Gonzalez, N. Tack, and A. Lambrechts, “A tiny VIS-NIR snapshot multispectral camera,” Proc. SPIE 9374, 937414 (2015). [CrossRef]  

7. C. R. Pidgeon and S. D. Smith, “Resolving power of multilayer filters in nonparallel light,” J. Opt. Soc. Am. 54, 1459–1466 (1964). [CrossRef]  

8. P. A. Wilksch, “Instrument function of the Fabry-Perot spectrometer,” Appl. Opt. 24, 1502–1511 (1985). [CrossRef]  

9. N. Hagen and M. W. Kudenov, “Review of snapshot spectral imaging technologies,” Opt. Eng. 52, 090901 (2013). [CrossRef]  

10. N. Gat, “Imaging spectroscopy using tunable filters: a review,” Proc. SPIE 4056, 50–64 (2000). [CrossRef]  

11. I. G. E. Renhorn, D. Bergström, J. Hedborg, D. Letalick, and S. Möller, “High spatial resolution hyperspectral camera based on a linear variable filter,” Opt. Eng. 55, 114105 (2016). [CrossRef]  

12. T. Skauli, H. E. Torkildsen, S. Nicolas, T. Opsahl, T. Haavardsholm, I. Kåsen, and A. Rognmo, “Compact camera for multispectral and conventional imaging based on patterned filters,” Appl. Opt. 53, C64–C74 (2014). [CrossRef]  

13. G. Hernandez, “Analytical description of a Fabry-Perot photoelectric spectrometer,” Appl. Opt. 5, 1745–1748 (1966). [CrossRef]  

14. G. Hernandez, “Analytical description of a Fabry-Perot spectrometer. 3: off-axis behavior and interference filters: erratum,” Appl. Opt. 18, 3364–3365 (1979). [CrossRef]  

15. J. R. Janesick, Photon Transfer (SPIE, 2007).

16. H. A. Macleod, Thin-Film Optical Filters (CRC Press, 2001).

17. Software Spectra, “Thin film design software for Windows, Version 3.5.15,” Software Spectra, 2009, http://sspectra.com/files/win_demo/manual.pdf.

18. J. Pichette, W. Charle, and A. Lambrechts, “Fast and compact internal scanning CMOS-based hyperspectral camera: the Snapscan,” Proc. SPIE 10110, 1011014 (2017). [CrossRef]  

19. P. Gonzalez, J. Pichette, B. Vereecke, B. Masschelein, L. Krasovitski, L. Bikov, and A. Lambrechts, “An extremely compact and high-speed line-scan hyperspectral imager covering the SWIR range,” Proc. SPIE 10656, 106560L (2018). [CrossRef]  

20. P. Agrawal, K. Tack, B. Geelen, B. Masschelein, P. Mateo, A. Moran, A. Lambrechts, and M. Jayapala, “Characterization of VNIR hyperspectral sensors with monolithically integrated optical filters,” in Image Sensors and Imaging Systems (Society for Imaging Science and Technology, 2016), pp. 1–7.

21. R. Kingslake, Optics in Photography, Vol. 6 of SPIE Press Monograph (SPIE, 1992).

22. P. Thenkabail, R. Smith, and E. De Pauw, “Hyperspectral vegetation indices and their relationships with agricultural crop characteristics,” Remote Sens. Environ. 71, 158–182 (2000). [CrossRef]  

References

  • View by:
  • |
  • |
  • |

  1. P. Shippert, “Why use hyperspectral imagery?” Earth Sci. 70, 377–379 (2004).
  2. G. Lu and B. Fei, “Medical hyperspectral imaging: a review,” J. Biomed. Opt. 19, 010901 (2014).
    [Crossref]
  3. A. Gowen, C. O’Donnell, P. Cullen, G. Downey, and J. Frias, “Hyperspectral imaging—an emerging process analytical tool for food quality and safety control,” Trends Food Sci. Technol. 18, 590–598 (2007).
    [Crossref]
  4. N. Tack, A. Lambrechts, P. Soussan, and L. Haspeslagh, “A compact, high-speed, and low-cost hyperspectral imager,” Proc. SPIE 8266, 82660Q (2012).
    [Crossref]
  5. B. Geelen, N. Tack, and A. Lambrechts, “A compact snapshot multispectral imager with a monolithically integrated per-pixel filter mosaic,” Proc. SPIE 8974, 89740L (2014).
    [Crossref]
  6. B. Geelen, C. Blanch, P. Gonzalez, N. Tack, and A. Lambrechts, “A tiny VIS-NIR snapshot multispectral camera,” Proc. SPIE 9374, 937414 (2015).
    [Crossref]
  7. C. R. Pidgeon and S. D. Smith, “Resolving power of multilayer filters in nonparallel light,” J. Opt. Soc. Am. 54, 1459–1466 (1964).
    [Crossref]
  8. P. A. Wilksch, “Instrument function of the Fabry-Perot spectrometer,” Appl. Opt. 24, 1502–1511 (1985).
    [Crossref]
  9. N. Hagen and M. W. Kudenov, “Review of snapshot spectral imaging technologies,” Opt. Eng. 52, 090901 (2013).
    [Crossref]
  10. N. Gat, “Imaging spectroscopy using tunable filters: a review,” Proc. SPIE 4056, 50–64 (2000).
    [Crossref]
  11. I. G. E. Renhorn, D. Bergström, J. Hedborg, D. Letalick, and S. Möller, “High spatial resolution hyperspectral camera based on a linear variable filter,” Opt. Eng. 55, 114105 (2016).
    [Crossref]
  12. T. Skauli, H. E. Torkildsen, S. Nicolas, T. Opsahl, T. Haavardsholm, I. Kåsen, and A. Rognmo, “Compact camera for multispectral and conventional imaging based on patterned filters,” Appl. Opt. 53, C64–C74 (2014).
    [Crossref]
  13. G. Hernandez, “Analytical description of a Fabry-Perot photoelectric spectrometer,” Appl. Opt. 5, 1745–1748 (1966).
    [Crossref]
  14. G. Hernandez, “Analytical description of a Fabry-Perot spectrometer. 3: off-axis behavior and interference filters: erratum,” Appl. Opt. 18, 3364–3365 (1979).
    [Crossref]
  15. J. R. Janesick, Photon Transfer (SPIE, 2007).
  16. H. A. Macleod, Thin-Film Optical Filters (CRC Press, 2001).
  17. Software Spectra, “Thin film design software for Windows, Version 3.5.15,” Software Spectra, 2009, http://sspectra.com/files/win_demo/manual.pdf .
  18. J. Pichette, W. Charle, and A. Lambrechts, “Fast and compact internal scanning CMOS-based hyperspectral camera: the Snapscan,” Proc. SPIE 10110, 1011014 (2017).
    [Crossref]
  19. P. Gonzalez, J. Pichette, B. Vereecke, B. Masschelein, L. Krasovitski, L. Bikov, and A. Lambrechts, “An extremely compact and high-speed line-scan hyperspectral imager covering the SWIR range,” Proc. SPIE 10656, 106560L (2018).
    [Crossref]
  20. P. Agrawal, K. Tack, B. Geelen, B. Masschelein, P. Mateo, A. Moran, A. Lambrechts, and M. Jayapala, “Characterization of VNIR hyperspectral sensors with monolithically integrated optical filters,” in Image Sensors and Imaging Systems (Society for Imaging Science and Technology, 2016), pp. 1–7.
  21. R. Kingslake, Optics in Photography, Vol. 6 of SPIE Press Monograph (SPIE, 1992).
  22. P. Thenkabail, R. Smith, and E. De Pauw, “Hyperspectral vegetation indices and their relationships with agricultural crop characteristics,” Remote Sens. Environ. 71, 158–182 (2000).
    [Crossref]

2018 (1)

P. Gonzalez, J. Pichette, B. Vereecke, B. Masschelein, L. Krasovitski, L. Bikov, and A. Lambrechts, “An extremely compact and high-speed line-scan hyperspectral imager covering the SWIR range,” Proc. SPIE 10656, 106560L (2018).
[Crossref]

2017 (1)

J. Pichette, W. Charle, and A. Lambrechts, “Fast and compact internal scanning CMOS-based hyperspectral camera: the Snapscan,” Proc. SPIE 10110, 1011014 (2017).
[Crossref]

2016 (1)

I. G. E. Renhorn, D. Bergström, J. Hedborg, D. Letalick, and S. Möller, “High spatial resolution hyperspectral camera based on a linear variable filter,” Opt. Eng. 55, 114105 (2016).
[Crossref]

2015 (1)

B. Geelen, C. Blanch, P. Gonzalez, N. Tack, and A. Lambrechts, “A tiny VIS-NIR snapshot multispectral camera,” Proc. SPIE 9374, 937414 (2015).
[Crossref]

2014 (3)

B. Geelen, N. Tack, and A. Lambrechts, “A compact snapshot multispectral imager with a monolithically integrated per-pixel filter mosaic,” Proc. SPIE 8974, 89740L (2014).
[Crossref]

G. Lu and B. Fei, “Medical hyperspectral imaging: a review,” J. Biomed. Opt. 19, 010901 (2014).
[Crossref]

T. Skauli, H. E. Torkildsen, S. Nicolas, T. Opsahl, T. Haavardsholm, I. Kåsen, and A. Rognmo, “Compact camera for multispectral and conventional imaging based on patterned filters,” Appl. Opt. 53, C64–C74 (2014).
[Crossref]

2013 (1)

N. Hagen and M. W. Kudenov, “Review of snapshot spectral imaging technologies,” Opt. Eng. 52, 090901 (2013).
[Crossref]

2012 (1)

N. Tack, A. Lambrechts, P. Soussan, and L. Haspeslagh, “A compact, high-speed, and low-cost hyperspectral imager,” Proc. SPIE 8266, 82660Q (2012).
[Crossref]

2007 (1)

A. Gowen, C. O’Donnell, P. Cullen, G. Downey, and J. Frias, “Hyperspectral imaging—an emerging process analytical tool for food quality and safety control,” Trends Food Sci. Technol. 18, 590–598 (2007).
[Crossref]

2004 (1)

P. Shippert, “Why use hyperspectral imagery?” Earth Sci. 70, 377–379 (2004).

2000 (2)

N. Gat, “Imaging spectroscopy using tunable filters: a review,” Proc. SPIE 4056, 50–64 (2000).
[Crossref]

P. Thenkabail, R. Smith, and E. De Pauw, “Hyperspectral vegetation indices and their relationships with agricultural crop characteristics,” Remote Sens. Environ. 71, 158–182 (2000).
[Crossref]

1985 (1)

1979 (1)

1966 (1)

1964 (1)

Agrawal, P.

P. Agrawal, K. Tack, B. Geelen, B. Masschelein, P. Mateo, A. Moran, A. Lambrechts, and M. Jayapala, “Characterization of VNIR hyperspectral sensors with monolithically integrated optical filters,” in Image Sensors and Imaging Systems (Society for Imaging Science and Technology, 2016), pp. 1–7.

Bergström, D.

I. G. E. Renhorn, D. Bergström, J. Hedborg, D. Letalick, and S. Möller, “High spatial resolution hyperspectral camera based on a linear variable filter,” Opt. Eng. 55, 114105 (2016).
[Crossref]

Bikov, L.

P. Gonzalez, J. Pichette, B. Vereecke, B. Masschelein, L. Krasovitski, L. Bikov, and A. Lambrechts, “An extremely compact and high-speed line-scan hyperspectral imager covering the SWIR range,” Proc. SPIE 10656, 106560L (2018).
[Crossref]

Blanch, C.

B. Geelen, C. Blanch, P. Gonzalez, N. Tack, and A. Lambrechts, “A tiny VIS-NIR snapshot multispectral camera,” Proc. SPIE 9374, 937414 (2015).
[Crossref]

Charle, W.

J. Pichette, W. Charle, and A. Lambrechts, “Fast and compact internal scanning CMOS-based hyperspectral camera: the Snapscan,” Proc. SPIE 10110, 1011014 (2017).
[Crossref]

Cullen, P.

A. Gowen, C. O’Donnell, P. Cullen, G. Downey, and J. Frias, “Hyperspectral imaging—an emerging process analytical tool for food quality and safety control,” Trends Food Sci. Technol. 18, 590–598 (2007).
[Crossref]

De Pauw, E.

P. Thenkabail, R. Smith, and E. De Pauw, “Hyperspectral vegetation indices and their relationships with agricultural crop characteristics,” Remote Sens. Environ. 71, 158–182 (2000).
[Crossref]

Downey, G.

A. Gowen, C. O’Donnell, P. Cullen, G. Downey, and J. Frias, “Hyperspectral imaging—an emerging process analytical tool for food quality and safety control,” Trends Food Sci. Technol. 18, 590–598 (2007).
[Crossref]

Fei, B.

G. Lu and B. Fei, “Medical hyperspectral imaging: a review,” J. Biomed. Opt. 19, 010901 (2014).
[Crossref]

Frias, J.

A. Gowen, C. O’Donnell, P. Cullen, G. Downey, and J. Frias, “Hyperspectral imaging—an emerging process analytical tool for food quality and safety control,” Trends Food Sci. Technol. 18, 590–598 (2007).
[Crossref]

Gat, N.

N. Gat, “Imaging spectroscopy using tunable filters: a review,” Proc. SPIE 4056, 50–64 (2000).
[Crossref]

Geelen, B.

B. Geelen, C. Blanch, P. Gonzalez, N. Tack, and A. Lambrechts, “A tiny VIS-NIR snapshot multispectral camera,” Proc. SPIE 9374, 937414 (2015).
[Crossref]

B. Geelen, N. Tack, and A. Lambrechts, “A compact snapshot multispectral imager with a monolithically integrated per-pixel filter mosaic,” Proc. SPIE 8974, 89740L (2014).
[Crossref]

P. Agrawal, K. Tack, B. Geelen, B. Masschelein, P. Mateo, A. Moran, A. Lambrechts, and M. Jayapala, “Characterization of VNIR hyperspectral sensors with monolithically integrated optical filters,” in Image Sensors and Imaging Systems (Society for Imaging Science and Technology, 2016), pp. 1–7.

Gonzalez, P.

P. Gonzalez, J. Pichette, B. Vereecke, B. Masschelein, L. Krasovitski, L. Bikov, and A. Lambrechts, “An extremely compact and high-speed line-scan hyperspectral imager covering the SWIR range,” Proc. SPIE 10656, 106560L (2018).
[Crossref]

B. Geelen, C. Blanch, P. Gonzalez, N. Tack, and A. Lambrechts, “A tiny VIS-NIR snapshot multispectral camera,” Proc. SPIE 9374, 937414 (2015).
[Crossref]

Gowen, A.

A. Gowen, C. O’Donnell, P. Cullen, G. Downey, and J. Frias, “Hyperspectral imaging—an emerging process analytical tool for food quality and safety control,” Trends Food Sci. Technol. 18, 590–598 (2007).
[Crossref]

Haavardsholm, T.

Hagen, N.

N. Hagen and M. W. Kudenov, “Review of snapshot spectral imaging technologies,” Opt. Eng. 52, 090901 (2013).
[Crossref]

Haspeslagh, L.

N. Tack, A. Lambrechts, P. Soussan, and L. Haspeslagh, “A compact, high-speed, and low-cost hyperspectral imager,” Proc. SPIE 8266, 82660Q (2012).
[Crossref]

Hedborg, J.

I. G. E. Renhorn, D. Bergström, J. Hedborg, D. Letalick, and S. Möller, “High spatial resolution hyperspectral camera based on a linear variable filter,” Opt. Eng. 55, 114105 (2016).
[Crossref]

Hernandez, G.

Janesick, J. R.

J. R. Janesick, Photon Transfer (SPIE, 2007).

Jayapala, M.

P. Agrawal, K. Tack, B. Geelen, B. Masschelein, P. Mateo, A. Moran, A. Lambrechts, and M. Jayapala, “Characterization of VNIR hyperspectral sensors with monolithically integrated optical filters,” in Image Sensors and Imaging Systems (Society for Imaging Science and Technology, 2016), pp. 1–7.

Kåsen, I.

Kingslake, R.

R. Kingslake, Optics in Photography, Vol. 6 of SPIE Press Monograph (SPIE, 1992).

Krasovitski, L.

P. Gonzalez, J. Pichette, B. Vereecke, B. Masschelein, L. Krasovitski, L. Bikov, and A. Lambrechts, “An extremely compact and high-speed line-scan hyperspectral imager covering the SWIR range,” Proc. SPIE 10656, 106560L (2018).
[Crossref]

Kudenov, M. W.

N. Hagen and M. W. Kudenov, “Review of snapshot spectral imaging technologies,” Opt. Eng. 52, 090901 (2013).
[Crossref]

Lambrechts, A.

P. Gonzalez, J. Pichette, B. Vereecke, B. Masschelein, L. Krasovitski, L. Bikov, and A. Lambrechts, “An extremely compact and high-speed line-scan hyperspectral imager covering the SWIR range,” Proc. SPIE 10656, 106560L (2018).
[Crossref]

J. Pichette, W. Charle, and A. Lambrechts, “Fast and compact internal scanning CMOS-based hyperspectral camera: the Snapscan,” Proc. SPIE 10110, 1011014 (2017).
[Crossref]

B. Geelen, C. Blanch, P. Gonzalez, N. Tack, and A. Lambrechts, “A tiny VIS-NIR snapshot multispectral camera,” Proc. SPIE 9374, 937414 (2015).
[Crossref]

B. Geelen, N. Tack, and A. Lambrechts, “A compact snapshot multispectral imager with a monolithically integrated per-pixel filter mosaic,” Proc. SPIE 8974, 89740L (2014).
[Crossref]

N. Tack, A. Lambrechts, P. Soussan, and L. Haspeslagh, “A compact, high-speed, and low-cost hyperspectral imager,” Proc. SPIE 8266, 82660Q (2012).
[Crossref]

P. Agrawal, K. Tack, B. Geelen, B. Masschelein, P. Mateo, A. Moran, A. Lambrechts, and M. Jayapala, “Characterization of VNIR hyperspectral sensors with monolithically integrated optical filters,” in Image Sensors and Imaging Systems (Society for Imaging Science and Technology, 2016), pp. 1–7.

Letalick, D.

I. G. E. Renhorn, D. Bergström, J. Hedborg, D. Letalick, and S. Möller, “High spatial resolution hyperspectral camera based on a linear variable filter,” Opt. Eng. 55, 114105 (2016).
[Crossref]

Lu, G.

G. Lu and B. Fei, “Medical hyperspectral imaging: a review,” J. Biomed. Opt. 19, 010901 (2014).
[Crossref]

Macleod, H. A.

H. A. Macleod, Thin-Film Optical Filters (CRC Press, 2001).

Masschelein, B.

P. Gonzalez, J. Pichette, B. Vereecke, B. Masschelein, L. Krasovitski, L. Bikov, and A. Lambrechts, “An extremely compact and high-speed line-scan hyperspectral imager covering the SWIR range,” Proc. SPIE 10656, 106560L (2018).
[Crossref]

P. Agrawal, K. Tack, B. Geelen, B. Masschelein, P. Mateo, A. Moran, A. Lambrechts, and M. Jayapala, “Characterization of VNIR hyperspectral sensors with monolithically integrated optical filters,” in Image Sensors and Imaging Systems (Society for Imaging Science and Technology, 2016), pp. 1–7.

Mateo, P.

P. Agrawal, K. Tack, B. Geelen, B. Masschelein, P. Mateo, A. Moran, A. Lambrechts, and M. Jayapala, “Characterization of VNIR hyperspectral sensors with monolithically integrated optical filters,” in Image Sensors and Imaging Systems (Society for Imaging Science and Technology, 2016), pp. 1–7.

Möller, S.

I. G. E. Renhorn, D. Bergström, J. Hedborg, D. Letalick, and S. Möller, “High spatial resolution hyperspectral camera based on a linear variable filter,” Opt. Eng. 55, 114105 (2016).
[Crossref]

Moran, A.

P. Agrawal, K. Tack, B. Geelen, B. Masschelein, P. Mateo, A. Moran, A. Lambrechts, and M. Jayapala, “Characterization of VNIR hyperspectral sensors with monolithically integrated optical filters,” in Image Sensors and Imaging Systems (Society for Imaging Science and Technology, 2016), pp. 1–7.

Nicolas, S.

O’Donnell, C.

A. Gowen, C. O’Donnell, P. Cullen, G. Downey, and J. Frias, “Hyperspectral imaging—an emerging process analytical tool for food quality and safety control,” Trends Food Sci. Technol. 18, 590–598 (2007).
[Crossref]

Opsahl, T.

Pichette, J.

P. Gonzalez, J. Pichette, B. Vereecke, B. Masschelein, L. Krasovitski, L. Bikov, and A. Lambrechts, “An extremely compact and high-speed line-scan hyperspectral imager covering the SWIR range,” Proc. SPIE 10656, 106560L (2018).
[Crossref]

J. Pichette, W. Charle, and A. Lambrechts, “Fast and compact internal scanning CMOS-based hyperspectral camera: the Snapscan,” Proc. SPIE 10110, 1011014 (2017).
[Crossref]

Pidgeon, C. R.

Renhorn, I. G. E.

I. G. E. Renhorn, D. Bergström, J. Hedborg, D. Letalick, and S. Möller, “High spatial resolution hyperspectral camera based on a linear variable filter,” Opt. Eng. 55, 114105 (2016).
[Crossref]

Rognmo, A.

Shippert, P.

P. Shippert, “Why use hyperspectral imagery?” Earth Sci. 70, 377–379 (2004).

Skauli, T.

Smith, R.

P. Thenkabail, R. Smith, and E. De Pauw, “Hyperspectral vegetation indices and their relationships with agricultural crop characteristics,” Remote Sens. Environ. 71, 158–182 (2000).
[Crossref]

Smith, S. D.

Soussan, P.

N. Tack, A. Lambrechts, P. Soussan, and L. Haspeslagh, “A compact, high-speed, and low-cost hyperspectral imager,” Proc. SPIE 8266, 82660Q (2012).
[Crossref]

Tack, K.

P. Agrawal, K. Tack, B. Geelen, B. Masschelein, P. Mateo, A. Moran, A. Lambrechts, and M. Jayapala, “Characterization of VNIR hyperspectral sensors with monolithically integrated optical filters,” in Image Sensors and Imaging Systems (Society for Imaging Science and Technology, 2016), pp. 1–7.

Tack, N.

B. Geelen, C. Blanch, P. Gonzalez, N. Tack, and A. Lambrechts, “A tiny VIS-NIR snapshot multispectral camera,” Proc. SPIE 9374, 937414 (2015).
[Crossref]

B. Geelen, N. Tack, and A. Lambrechts, “A compact snapshot multispectral imager with a monolithically integrated per-pixel filter mosaic,” Proc. SPIE 8974, 89740L (2014).
[Crossref]

N. Tack, A. Lambrechts, P. Soussan, and L. Haspeslagh, “A compact, high-speed, and low-cost hyperspectral imager,” Proc. SPIE 8266, 82660Q (2012).
[Crossref]

Thenkabail, P.

P. Thenkabail, R. Smith, and E. De Pauw, “Hyperspectral vegetation indices and their relationships with agricultural crop characteristics,” Remote Sens. Environ. 71, 158–182 (2000).
[Crossref]

Torkildsen, H. E.

Vereecke, B.

P. Gonzalez, J. Pichette, B. Vereecke, B. Masschelein, L. Krasovitski, L. Bikov, and A. Lambrechts, “An extremely compact and high-speed line-scan hyperspectral imager covering the SWIR range,” Proc. SPIE 10656, 106560L (2018).
[Crossref]

Wilksch, P. A.

Appl. Opt. (4)

Earth Sci. (1)

P. Shippert, “Why use hyperspectral imagery?” Earth Sci. 70, 377–379 (2004).

J. Biomed. Opt. (1)

G. Lu and B. Fei, “Medical hyperspectral imaging: a review,” J. Biomed. Opt. 19, 010901 (2014).
[Crossref]

J. Opt. Soc. Am. (1)

Opt. Eng. (2)

I. G. E. Renhorn, D. Bergström, J. Hedborg, D. Letalick, and S. Möller, “High spatial resolution hyperspectral camera based on a linear variable filter,” Opt. Eng. 55, 114105 (2016).
[Crossref]

N. Hagen and M. W. Kudenov, “Review of snapshot spectral imaging technologies,” Opt. Eng. 52, 090901 (2013).
[Crossref]

Proc. SPIE (6)

N. Gat, “Imaging spectroscopy using tunable filters: a review,” Proc. SPIE 4056, 50–64 (2000).
[Crossref]

N. Tack, A. Lambrechts, P. Soussan, and L. Haspeslagh, “A compact, high-speed, and low-cost hyperspectral imager,” Proc. SPIE 8266, 82660Q (2012).
[Crossref]

B. Geelen, N. Tack, and A. Lambrechts, “A compact snapshot multispectral imager with a monolithically integrated per-pixel filter mosaic,” Proc. SPIE 8974, 89740L (2014).
[Crossref]

B. Geelen, C. Blanch, P. Gonzalez, N. Tack, and A. Lambrechts, “A tiny VIS-NIR snapshot multispectral camera,” Proc. SPIE 9374, 937414 (2015).
[Crossref]

J. Pichette, W. Charle, and A. Lambrechts, “Fast and compact internal scanning CMOS-based hyperspectral camera: the Snapscan,” Proc. SPIE 10110, 1011014 (2017).
[Crossref]

P. Gonzalez, J. Pichette, B. Vereecke, B. Masschelein, L. Krasovitski, L. Bikov, and A. Lambrechts, “An extremely compact and high-speed line-scan hyperspectral imager covering the SWIR range,” Proc. SPIE 10656, 106560L (2018).
[Crossref]

Remote Sens. Environ. (1)

P. Thenkabail, R. Smith, and E. De Pauw, “Hyperspectral vegetation indices and their relationships with agricultural crop characteristics,” Remote Sens. Environ. 71, 158–182 (2000).
[Crossref]

Trends Food Sci. Technol. (1)

A. Gowen, C. O’Donnell, P. Cullen, G. Downey, and J. Frias, “Hyperspectral imaging—an emerging process analytical tool for food quality and safety control,” Trends Food Sci. Technol. 18, 590–598 (2007).
[Crossref]

Other (5)

P. Agrawal, K. Tack, B. Geelen, B. Masschelein, P. Mateo, A. Moran, A. Lambrechts, and M. Jayapala, “Characterization of VNIR hyperspectral sensors with monolithically integrated optical filters,” in Image Sensors and Imaging Systems (Society for Imaging Science and Technology, 2016), pp. 1–7.

R. Kingslake, Optics in Photography, Vol. 6 of SPIE Press Monograph (SPIE, 1992).

J. R. Janesick, Photon Transfer (SPIE, 2007).

H. A. Macleod, Thin-Film Optical Filters (CRC Press, 2001).

Software Spectra, “Thin film design software for Windows, Version 3.5.15,” Software Spectra, 2009, http://sspectra.com/files/win_demo/manual.pdf .

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (15)

Fig. 1.
Fig. 1. Schematic representation of how the aperture focuses light on the spectral imaging sensor with integrated thin-film optical filters.
Fig. 2.
Fig. 2. Basic Fabry–Perot etalon. Two parallel near-perfect mirrors are separated by a material of refractive index ns and thickness t.
Fig. 3.
Fig. 3. Decomposition in contributions with equal angles of incidence. The weight of a contribution is measured by the length of the blue arc within the aperture.
Fig. 4.
Fig. 4. Decomposition of the light cone from the aperture in contributions of equal angle of incidence ϕ. The weight of each contribution is the infinitesimal area dA. Here d is the distance of the pixel from the optical axis. (a) Top view; (b) cut section view.
Fig. 5.
Fig. 5. Shape of the kernel [Eq. (15)] for different ratios of θcone and θCRA. The mean value of each kernel according to Eq. (17) is marked with a circle.
Fig. 6.
Fig. 6. Transmittance of a thin-film optical filter simulated with TFCalc for different apertures compared to applying the convolution kernel to the transmittance at collimated conditions. In this example neff=1.75.
Fig. 7.
Fig. 7. Experimental setups. (a) VNIR Snapscan with color filter; (b) SWIR Snapscan with reflectance target.
Fig. 8.
Fig. 8. Experiment 1 (SWIR). The same sample is measured at different f-numbers. The shift becomes larger for smaller f-numbers, causing the spread of shifts in the graph. The shifts are corrected using Eq. (23). The working f-number f#,W [Eq. (21)] is used for correction; the legend shows the f-number f# as marked on the lens. (a) Uncorrected; (b) central wavelength, corrected.
Fig. 9.
Fig. 9. Experiment 2 (VNIR). The transmittance of the color filter is measured at different off-axis distances at f/8, which is a high f-number. The shifts are corrected using Eq. (24). (a) Uncorrected; (b) central wavelength, corrected.
Fig. 10.
Fig. 10. Image of the scene from Fig. 7(a) at the 721 nm band before and after central wavelength correction (Fig. 9). The uniformity of the scene is significantly improved. The corners are outside the image circle of the lens. (a) Before correction; (b) after correction.
Fig. 11.
Fig. 11. Experiment 3 (SWIR). The reflectance of the sample is measured at different f-numbers at an off-axis position with θCRA10.2°. The shifts are corrected using Eq. (25). As a reference, the measurement at f/5.6 at zero chief ray angle is used. (a) Uncorrected; (b) central wavelength, corrected.
Fig. 12.
Fig. 12. Experiment 3 simulated (SWIR). The offset and loss of detail are very similar to those observed in real measurements (Fig. 10). (a) Uncorrected; (b) central wavelength corrected.
Fig. 13.
Fig. 13. Approximation of the mean value compared to the numerical result (neff=1.7). For large chief ray angle, the shift is slightly overestimated.
Fig. 14.
Fig. 14. Approximation of the mean value compared to the numerical result (neff=1.7). For large chief ray angle, the estimated shift becomes too large.
Fig. 15.
Fig. 15. For very large values of θcone and θCRA, the higher-order approximation [Eq. (C1)] fits the numerical solution very well, while the more simple approximation deteriorates (neff=1.7).

Tables (2)

Tables Icon

Table 1. List of the Main Symbols and Their Meaning

Tables Icon

Table 2. Properties of the Experimental Setupa

Equations (49)

Equations on this page are rendered with MathJax. Learn more.

DN=λminλmaxT(λ)·L(λ)dλ.
DN=λminλmax(Kθcone,θCRA*T)(λ)·L(λ)dλ,
Δ(ϕ)=λcwl(1cosϕs)λcwl(1cosϕns),
Δ(ϕ)λcwl(1cosϕneff).
Tϕ(λ)T(λ)*δ(λλcwl(1cosϕneff)).
T^θcone,θCRA(λ)=ApertureTϕ(λ)dAAperturedA,
T^θcone,θCRA(λ)=T(λ)*Apertureδ(λΔ(ϕ))dAAperturedA
=T(λ)*Kθcone,θCRA(λ).
dA=2γ(r)rdr,
γ(r)=Re(arccosd2R2+r22dr).
dA=2x2γ(ϕ)tanϕcos2ϕdϕ,
γ(ϕ)=Re(arccostan2θCRAtan2θcone+tan2ϕ2tanθCRAtanϕ).
Kθcone,θCRA(λ)=ϕminϕmax2x2γ(ϕ)tanϕcos2ϕδ(λΔ(ϕ))dϕπR2,
ϕmin={0ifθCRA<θconearctan(tanθCRAtanθcone)ifθCRAθcone,ϕmax=arctan(tanθCRA+tanθcone),
Kθcone,θCRA(λ)2neff2λcwlγ(neff2λλcwl)πtan2θcone,
λ¯θcone,θCRA=λminλmaxλKθcone,θCRA(λ)dλλminλmaxKθcone,θCRA(λ)dλ=(normalized)λminλmaxλKθcone,θCRA(λ)dλ.
λ¯θcone,θCRAλcwlneff2(θcone24+θCRA22)forθcone,θCRA0.
λcwlnew=λcwlλ¯θcone,θCRAλcwl(1θcone24neff2θCRA22neff2).
λcwlnewλcwl(1116f#,W2neff2d22x2neff2).
reflectance=DNsampleDNdarkDNwhiteDNdark.
f#,W(1+mP)f#,
neff=nL1nLnH+(nLnH)2,
λcwlnewλcwl(1116f#,W2neff2).
λcwlnewλcwl(1θCRA22neff2)=λcwl(1d22x2neff2),
λcwlnewλcwl(1116f#,W2neff2d22x2neff2).
Kθcone,θCRA(λ)=ϕminϕmax2x2γ(ϕ)tanϕcos2ϕδ(λΔ(ϕ))dϕπR2
=ϕminϕmax2x2γ(ϕ)tanϕcos2ϕδ(λΔ(ϕ))dϕπx2tan2θcone
=ϕminϕmax2γ(ϕ)tanϕcos2ϕδ(λΔ(ϕ))dϕπtan2θcone.
Kθcone,θCRA(λ)=2γ(ϕ)tanϕcos2ϕδ(λΔ(ϕ))Π(ϕ)dϕπtan2θcone,
Δ(ϕ)=λcwl(1cosϕs)
=λcwl(1cos(arcsinsinϕneff))
=λcwl(11sin2ϕneff2),
Δ1(λ)=arcsin(neffλλcwl(2λλcwl))
=neff2λλcwl+O(λ3/2λcwl3/2),λλcwl0.
du=λcwlcos(ϕ)sin(ϕ)neff21sin(ϕ)2neff2dϕ.
Kθcone,θCRA(λ)=2γ(Δ1(λ))tan(Δ1(λ))cos2(Δ1(λ))πtan2θcone·neff21sin2(Δ1(λ))neff2λcwlsin(Δ1(λ))cos(Δ1(λ))Π(Δ1(λ)).
Kθcone,θCRA(λ)=g(λ)γ(Δ1(λ))πtan2θconeΠ(Δ1(λ)),
g(λ)=2neff2λcwl·(1λλcwl12neff2λλcwl+neff2λ2λcwl2)
=2neff2λcwl+O(λλcwl),forλλcwl0,
Kθcone,θCRA(λ)2neff2λcwl·γ(neff2λλcwl)πtan2θcone.
E(θcone,θCRA)=λ¯θcone,θCRA=λminλmaxλKθcone,θCRA(λ)dλ.
E(θcone,θCRA)E(0,0)+EθCRA|(0,0)θCRA+Eθcone|(0,0)θcone+122EθCRA2|(0,0)θCRA2+122Eθcone2|(0,0)θcone2+2EθconeθCRA|(0,0)θconeθCRA.
λminλmaxKθcone,θCRA(λ)dλ=1,θcone,θCRA>0.
0λmaxλg(λ)tan2θconedλ.
E(θcone,0)=01vλmax2g(vλmax)tan2θconedv.
E(θcone,0)=λcwl(θcone24neff2+(14neff2)24neff4θcone4+O(θcone6)),forθcone0.
E(0,θCRA)=λcwl(θCRA22neff2+(34neff2)24neff4θCRA4+O(θCRA6)),forθCRA0.
E(θcone,θCRA)λcwl(θcone24neff2+θCRA22neff2).
E(θcone,θCRA)λcwl(θcone24neff2+θCRA22neff2θCRA2θcone2neff2).

Metrics