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 [4–6]. 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

 

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

Download Full Size | PDF

For a given f-number and chief ray angle ${\theta }_{\mathrm{CRA}}$, the effect of the aperture on the filter with central wavelength ${\lambda }_{\mathrm{cwl}}$ is modeled as a convolution of $T(\lambda )$ with a kernel $K_{{\theta }_{\mathrm{cone}},{\theta }_{\mathrm{CRA}}}(\lambda )$, where the half-cone angle ${\theta }_{\mathrm{cone}}$ characterizes the f-number:

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

Table 1. List of the Main Symbols and Their Meaning

View Table | View all tables in this article

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 $n_s$ and thickness $t$.

Download Full Size | PDF

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

For an angle of incidence $\varphi$, the transmittance peak centered at ${\lambda }_{\mathrm{cwl}}$ (at orthogonal incidence) shifts by

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 $\varphi$ is asymptotically equivalent to the shift of the transmittance of an ideal Fabry–Perot etalon with a cavity material with effective refractive index $n_{\mathrm{eff}}$ [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 $\varphi$, the shift $\mathrm{\Delta }$ ($\varphi$) in central wavelength of a transmittance peak of a thin-film Fabry–Perot filter is

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 | 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 ${\theta }_{\mathrm{CRA}}=\arctan \frac{d}{x}$, 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 $\varphi$. The weight of each contribution is the infinitesimal area $\mathrm{d}A$. Here $d$ is the distance of the pixel from the optical axis. (a) Top view; (b) cut section view.

Download Full Size | PDF

The resulting transmittance ${\widehat{T}}_{{\theta }_{\mathrm{cone}},{\theta }_{\mathrm{CRA}}}(\lambda )$ is the sum of the transmittance functions $T_{\varphi }(\lambda )$ at different tilt angles [Eq. (5)]. The weight of each contribution is the infinitesimal area $\mathrm{d}A$ of the aperture that contributes to identical angles of incidence (Fig. 4). Therefore,

We proceed by rewriting the integral in terms of the angle of incidence $\varphi$ by substituting $r=x\tan \varphi$ such that

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

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

The shape of the convolution kernel $K_{{\theta }_{\mathrm{cone}},{\theta }_{\mathrm{CRA}}}$ models how the central wavelength and full width at half-maximum (FWHM) of the transmittance will change. For increasing ${\theta }_{\mathrm{cone}}$ and ${\theta }_{\mathrm{CRA}}$, the kernel $K_{{\theta }_{\mathrm{cone}},{\theta }_{\mathrm{CRA}}}$ 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 ${\theta }_{\mathrm{cone}}$ and ${\theta }_{\mathrm{CRA}}$. The mean value of each kernel according to Eq. (17) is marked with a circle.

Download Full Size | 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 $n_{\mathrm{eff}}=1.75$.

Download Full Size | PDF

 

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

Download Full Size | 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 $K_{{\theta }_{\mathrm{cone}},{\theta }_{\mathrm{CRA}}}$ 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:

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

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 ${\theta }_{\mathrm{cone}}=\arctan \frac{1}{2f_{\#,W}}\sim \frac{1}{2f_{\#,W}}$,

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 (${\theta }_{\mathrm{CRA}}=0$). 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.

Table 2. Properties of the Experimental Setup^a

View Table | View all tables in this article

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

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])

The effective refractive index $n_{\mathrm{eff}}$ 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]:

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 $n_{\mathrm{eff}}\approx 1.7$ for the VNIR sensor and $n_{\mathrm{eff}}\approx 1.75$ 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 | 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. 8–11. 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 | 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 | PDF

 

Fig. 11. Experiment 3 (SWIR). The reflectance of the sample is measured at different f-numbers at an off-axis position with ${\theta }_{\mathrm{CRA}}\approx 10.2°$. 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 | 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

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, ${\theta }_{\mathrm{cone}}=3.58°=0.0625\mathrm{rad}$. 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 ${\theta }_{\mathrm{cone}}=0$, then

We show the spatial image ($2048\text{pixels}\times 2048\text{pixels}$) 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 | 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:

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 $T(\lambda )$ 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

(A1)$$K_{{\theta }_{\mathrm{cone}},{\theta }_{\mathrm{CRA}}}(\lambda )=\frac{{\int }_{{\varphi }_{\min }}^{{\varphi }_{\max }}2x^2\gamma (\varphi )\frac{\tan \varphi }{{\cos }^2\varphi }\delta (\lambda -\mathrm{\Delta }(\varphi ))\mathrm{d}\varphi }{\pi R^2}$$

(A2)$$=\frac{{\int }_{{\varphi }_{\min }}^{{\varphi }_{\max }}2x^2\gamma (\varphi )\frac{\tan \varphi }{{\cos }^2\varphi }\delta (\lambda -\mathrm{\Delta }(\varphi ))\mathrm{d}\varphi }{\pi x^2{\tan }^2{\theta }_{\mathrm{cone}}}$$

(A3)$$=\frac{{\int }_{{\varphi }_{\min }}^{{\varphi }_{\max }}2\gamma (\varphi )\frac{\tan \varphi }{{\cos }^2\varphi }\delta (\lambda -\mathrm{\Delta }(\varphi ))\mathrm{d}\varphi }{\pi {\tan }^2{\theta }_{\mathrm{cone}}}.$$

By introducing the indicator function $\mathrm{\Pi }=H(\varphi -{\varphi }_{\min })-H(\varphi -{\varphi }_{\max })$, the limits of the integral can be replaced such that

(A4)$$K_{{\theta }_{\mathrm{cone}},{\theta }_{\mathrm{CRA}}}(\lambda )=\frac{{\int }_{-\infty }^{\infty }2\gamma (\varphi )\frac{\tan \varphi }{{\cos }^2\varphi }\delta (\lambda -\mathrm{\Delta }(\varphi ))\mathrm{\Pi }(\varphi )\mathrm{d}\varphi }{\pi {\tan }^2{\theta }_{\mathrm{cone}}},$$

where $H(·)$ is the Heaviside step function.

Let

(A5)$$\mathrm{\Delta }(\varphi )={\lambda }_{\mathrm{cwl}}(1-\cos {\varphi }_{\mathrm{s}})$$

(A6)$$={\lambda }_{\mathrm{cwl}}(1-\cos (\arcsin \frac{\sin \varphi }{n_{\mathrm{eff}}}\left)\right)$$

(A7)$$={\lambda }_{\mathrm{cwl}}(1-\sqrt{1-\frac{{\sin }^2\varphi }{n_{\mathrm{eff}}^2}}),$$

and its inverse,

(A8)$${\mathrm{\Delta }}^{-1}(\lambda )=\arcsin \left(n_{\mathrm{eff}}\sqrt{\frac{\lambda }{{\lambda }_{\mathrm{cwl}}}(2-\frac{\lambda }{{\lambda }_{\mathrm{cwl}}})}\right)$$

(A9)$$=n_{\mathrm{eff}}\sqrt{\frac{2\lambda }{{\lambda }_{\mathrm{cwl}}}}+\mathcal{O}\left(\frac{{\lambda }^{3/2}}{{\lambda }_{\mathrm{cwl}}^{3/2}}\right),\frac{\lambda }{{\lambda }_{\mathrm{cwl}}}\to 0.$$

Here $\mathcal{O}(·)$ is the Big O notation and describes the limiting behavior of the error term of the approximation.

If we define $u=\lambda -\mathrm{\Delta }(\varphi )$, it follows that

(A10)$$\mathrm{d}u=\frac{{\lambda }_{\mathrm{cwl}}\cos (\varphi )\sin (\varphi )}{n_{\mathrm{eff}}^2\sqrt{1-\frac{\sin {(\varphi )}^2}{n_{\mathrm{eff}}^2}}}\mathrm{d}\varphi .$$

In Eq. (A4), we now replace $\mathrm{d}\varphi$ using Eq. (A10) and substitute $\varphi$ with ${\mathrm{\Delta }}^{-1}(\lambda -u)$. Then $\delta (\lambda -\mathrm{\Delta }(\varphi ))$ becomes $\delta (u)$. 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

(A11)$$\begin{gathered} K_{{\theta }_{\mathrm{cone}},{\theta }_{\mathrm{CRA}}}(\lambda )=\frac{2\gamma ({\mathrm{\Delta }}^{-1}(\lambda ))\tan ({\mathrm{\Delta }}^{-1}(\lambda ))}{{\cos }^2({\mathrm{\Delta }}^{-1}(\lambda ))\pi {\tan }^2{\theta }_{\mathrm{cone}}} \\ ·\frac{n_{\mathrm{eff}}^2\sqrt{1-\frac{{\sin }^2({\mathrm{\Delta }}^{-1}(\lambda ))}{n_{\mathrm{eff}}^2}}}{{\lambda }_{\mathrm{cwl}}\sin ({\mathrm{\Delta }}^{-1}(\lambda ))\cos ({\mathrm{\Delta }}^{-1}(\lambda ))}\mathrm{\Pi }({\mathrm{\Delta }}^{-1}(\lambda )). \end{gathered}$$

By grouping the terms, the closed-form analytical solution can be formulated as

(A12)$$K_{{\theta }_{\mathrm{cone}},{\theta }_{\mathrm{CRA}}}(\lambda )=\frac{g(\lambda )\gamma ({\mathrm{\Delta }}^{-1}(\lambda ))}{\pi {\tan }^2{\theta }_{\mathrm{cone}}}\mathrm{\Pi }({\mathrm{\Delta }}^{-1}(\lambda )),$$

with $\gamma$ as defined in Eq. (12) and the function $g(\lambda )$ containing all remaining terms. By applying trigonometric identities $g$ can be simplified such that

(A13)$$g(\lambda )=\frac{2n_{\mathrm{eff}}^2}{{\lambda }_{\mathrm{cwl}}}·\left(\frac{1-\frac{\lambda }{{\lambda }_{\mathrm{cwl}}}}{1-2n_{\mathrm{eff}}^2\frac{\lambda }{{\lambda }_{\mathrm{cwl}}}+n_{\mathrm{eff}}^2\frac{{\lambda }^2}{{\lambda }_{\mathrm{cwl}}^2}}\right)$$

(A14)$$=\frac{2n_{\mathrm{eff}}^2}{{\lambda }_{\mathrm{cwl}}}+\mathcal{O}\left(\frac{\lambda }{{\lambda }_{\mathrm{cwl}}}\right),\mathrm{for}\frac{\lambda }{{\lambda }_{\mathrm{cwl}}}\to 0,$$

which is approximately a constant function.

Now using Eqs. (A8) and (A13), the kernel can be approximated for $\lambda \ge {\lambda }_{\min }=\mathrm{\Delta }({\varphi }_{\min })\ge 0$ as

(A15)$$K_{{\theta }_{\mathrm{cone}},{\theta }_{\mathrm{CRA}}}(\lambda )\sim \frac{2n_{\mathrm{eff}}^2}{{\lambda }_{\mathrm{cwl}}}·\frac{\gamma \left(n_{\mathrm{eff}}\sqrt{\frac{2\lambda }{{\lambda }_{\mathrm{cwl}}}}\right)}{\pi {\tan }^2{\theta }_{\mathrm{cone}}}.$$

The indicator function can be omitted, since $\gamma ({\mathrm{\Delta }}^{-1}(\lambda ))$ is zero for $\lambda >{\lambda }_{\max }=\mathrm{\Delta }({\varphi }_{\max })$.

This approximation is plotted in Fig. 5. Notice that for ${\theta }_{\mathrm{CRA}}=0$, 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

(B1)$$\begin{gathered} E({\theta }_{\mathrm{cone}},{\theta }_{\mathrm{CRA}})={\overline{\lambda }}_{{\theta }_{\mathrm{cone}},{\theta }_{\mathrm{CRA}}} \\ ={\int }_{{\lambda }_{\min }}^{{\lambda }_{\max }}\lambda K_{{\theta }_{\mathrm{cone}},{\theta }_{\mathrm{CRA}}}(\lambda )\mathrm{d}\lambda . \end{gathered}$$

The Taylor expansion in ${\theta }_{\mathrm{CRA}}$ and ${\theta }_{\mathrm{cone}}$ around zero is

(B2)$$\begin{gathered} E({\theta }_{\mathrm{cone}},{\theta }_{\mathrm{CRA}})\sim E(0,0)+\frac{\partial E}{\partial {\theta }_{\mathrm{CRA}}}{|}_{(0,0)}{\theta }_{\mathrm{CRA}} \\ +\frac{\partial E}{\partial {\theta }_{\mathrm{cone}}}{|}_{(0,0)}{\theta }_{\mathrm{cone}}+\frac{1}{2}\frac{{\partial }^{2}E}{\partial {\theta }_{\mathrm{CRA}}^2}{|}_{(0,0)}{\theta }_{\mathrm{CRA}}^2 \\ +\frac{1}{2}\frac{{\partial }^{2}E}{\partial {\theta }_{\mathrm{cone}}^2}{|}_{(0,0)}{\theta }_{\mathrm{cone}}^2+\frac{{\partial }^{2}E}{\partial {\theta }_{\mathrm{cone}}{\theta }_{\mathrm{CRA}}}{|}_{(0,0)}{\theta }_{\mathrm{cone}}{\theta }_{\mathrm{CRA}}. \end{gathered}$$

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:

(B3)$${\int }_{{\lambda }_{\min }}^{{\lambda }_{\max }}{K}_{{\theta }_{\mathrm{cone}},{\theta }_{\mathrm{CRA}}}(\lambda )\mathrm{d}\lambda =1,\forall {\theta }_{\mathrm{cone}},{\theta }_{\mathrm{CRA}}>0.$$

Yet, for ${\theta }_{\mathrm{cone}},{\theta }_{\mathrm{CRA}}\to 0$, ${\lambda }_{\min },{\lambda }_{\max }\to 0$ 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 ${\theta }_{\mathrm{cone}}$, one can calculate that for ${\theta }_{\mathrm{CRA}}\to 0$, $\gamma (\lambda )\to \pi$ in Eq. (A12), and ${\lambda }_{\min }=0$. The integral then becomes

(B4)$${\int }_0^{{\lambda }_{\max }}\lambda \frac{g(\lambda )}{{\tan }^2{\theta }_{\mathrm{cone}}}\mathrm{d}\lambda .$$

Since the upper limit of the integration, ${\lambda }_{\max }$, depends on ${\theta }_{\mathrm{cone}}$ [Eq. (14)], we first substitute $\lambda =v{\lambda }_{\max }$ such that

(B5)$$E({\theta }_{\mathrm{cone}},0)={\int }_0^{1}v{\lambda }_{\max }^2\frac{g(v{\lambda }_{\max })}{{\tan }^2{\theta }_{\mathrm{cone}}}\mathrm{d}v.$$

Using a symbolic solver, the series expansion of the integrand can now be calculated. Each term of the expansion is then integrated such that

(B6)$$\begin{gathered} E({\theta }_{\mathrm{cone}},0)={\lambda }_{\mathrm{cwl}}(\frac{{\theta }_{\mathrm{cone}}^2}{4n_{\mathrm{eff}}^2}+\frac{(1-4n_{\mathrm{eff}}^2)}{24n_{\mathrm{eff}}^4}{\theta }_{\mathrm{cone}}^4+\mathcal{O}({\theta }_{\mathrm{cone}}^6)), \\ \mathrm{for}{\theta }_{\mathrm{cone}}\to 0. \end{gathered}$$

To calculate the coefficients of the monomials in ${\theta }_{\mathrm{CRA}}$, one can use the physical insight that for ${\theta }_{\mathrm{cone}}\to 0$, the problem becomes equivalent to the pure tilt case, where the shift is equal to $\mathrm{\Delta }({\theta }_{\mathrm{CRA}})$. Expanding this around zero gives

(B7)$$\begin{gathered} E(0,{\theta }_{\mathrm{CRA}})={\lambda }_{\mathrm{cwl}}(\frac{{\theta }_{\mathrm{CRA}}^2}{2n_{\mathrm{eff}}^2}+\frac{(3-4n_{\mathrm{eff}}^2)}{24n_{\mathrm{eff}}^4}{\theta }_{\mathrm{CRA}}^4+\mathcal{O}({\theta }_{\mathrm{CRA}}^6)), \\ \mathrm{for}{\theta }_{\mathrm{CRA}}\to 0. \end{gathered}$$

Comparison with a numerical solution suggests that the contribution of the monomial ${\theta }_{\mathrm{cone}}{\theta }_{\mathrm{CRA}}$ 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

(B8)$$E({\theta }_{\mathrm{cone}},{\theta }_{\mathrm{CRA}})\sim {\lambda }_{\mathrm{cwl}}(\frac{{\theta }_{\mathrm{cone}}^2}{4n_{\mathrm{eff}}^2}+\frac{{\theta }_{\mathrm{CRA}}^2}{2n_{\mathrm{eff}}^2}).$$

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 ($n_{\mathrm{eff}}=1.7$). For large chief ray angle, the shift is slightly overestimated.

Download Full Size | PDF

For experiment 3, the error between the exact shift and approximation is small (1.6 nm at ${\lambda }_{\mathrm{cwl}}=1550\mathrm{nm}$) compared to the total correction of 18 nm.

The approximation error grows proportional to ${\theta }_{\mathrm{cone}}^2{\theta }_{\mathrm{CRA}}^2$. 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:

(C1)$$E({\theta }_{\mathrm{cone}},{\theta }_{\mathrm{CRA}})\approx {\lambda }_{\mathrm{cwl}}(\frac{{\theta }_{\mathrm{cone}}^2}{4n_{\mathrm{eff}}^2}+\frac{{\theta }_{\mathrm{CRA}}^2}{2n_{\mathrm{eff}}^2}-\frac{{\theta }_{\mathrm{CRA}}^2{\theta }_{\mathrm{cone}}^2}{n_{\mathrm{eff}}^2}).$$

This approximation agrees well with the numerical result (Fig. 14) and even holds up well for much larger values of ${\theta }_{\mathrm{cone}}$ and ${\theta }_{\mathrm{CRA}}$ (Fig. 15).

 

Fig. 14. Approximation of the mean value compared to the numerical result ($n_{\mathrm{eff}}=1.7$). For large chief ray angle, the estimated shift becomes too large.

Download Full Size | PDF

 

Fig. 15. For very large values of ${\theta }_{\mathrm{cone}}$ and ${\theta }_{\mathrm{CRA}}$, the higher-order approximation [Eq. (C1)] fits the numerical solution very well, while the more simple approximation deteriorates ($n_{\mathrm{eff}}=1.7$).

Download Full Size | 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]