Exit pupil localization to correct spectral shift in thin-film Fabry-Perot spectral cameras

Thomas Goossens; Chris Van Hoof

doi:10.1364/OSAC.2.002217

1. Introduction

Spectral cameras combine photography and spectroscopy and thus measure a spectrum for each point in the scene. In recent years, integrated thin-film Fabry-Perot filters have enabled compact and lightweight spectral camera technology [1–3]. This makes the technology more accessible for use in a wide range of drone-based [4], medical [5], food science [6], industrial and agricultural applications.

Spectral camera technology is mostly used when physical interpretation of the spectrum is required or when regular (RGB) color cameras cannot discriminate between different samples [7]. When spectral cameras are used to discriminate between samples it is essential that the measured spectrum does not depend on the object’s position in the scene, which could also make physical interpretation impossible.

A key concern of thin-film based spectral cameras is therefore that thin-film filters are sensitive to the angle of incidence [8]. The larger the incident angle, the more the central wavelength of the filter shifts towards shorter wavelengths.

To form an image, the spectral camera requires a lens. A non-telecentric lens will focus the light onto each pixel at a different incident angle (Fig. 1a). Because thin-film filters are sensitive to this incident angle, this will cause undesired shifts in the measured spectra across the imaging plane (Fig. 1b). All Fabry-Perot based designs placed in the imaging plane of a focusing lens will experience this effect [1,9–12].

Fig. 1. The spectrum measured at an off-axis distance $d=7.7$ mm is shifted with respect to the on-axis measurement. For simplicity, without loss of generality, the entrance pupil and exit pupil coincide in Fig. 1a.

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In [13,14], we introduced methods that can be used to correct the shifts in measured spectra for a spectral camera where the lens focuses light onto each pixel as in Fig. 1a. The shift depends on the position $d$ of the filter, the size of the aperture and the distance $x$ to the exit pupil.

The distance to the exit pupil is often not known for arbitrary camera systems. This is because the position may depend on several factors including the focus and zoom settings, how the lens is mounted, and the presence of other optical components (e.g. filters) in the optical path. Sometimes the datasheet of the lens provides some information on the exit pupil position. But it is often not straightforward to apply this to an actual camera system.

In this article we introduce a practical method to estimate the distance to the exit pupil from the amount of spectral shift observed in the measurements. The method localizes the exit pupil using the same spectral camera setup used for imaging. Apart from a well chosen target spectrum, no additional equipment is required. In Section 5, we discuss how to select the optimal target spectrum.

2. Spectral shift

Each thin-film filter on the sensor has a so-called central wavelength $\lambda _{\textrm {cwl}}$. Only wavelengths in the neighborhood of $\lambda _{\textrm {cwl}}$ are transmitted. This wavelength is determined for collimated light conditions at normal incidence.

To visualize the spectrum, the output of each filter is plotted at its designed central wavelength (Fig. 1b). However, these central wavelengths $\vec {\lambda }_{\textrm {cwl}}$ may not be accurate when used in a camera and will depend on the position of the filter on the sensor. This is because thin-film interference filters are sensitive to the angle of incidence.

For increasing angle of incidence, the central wavelength of thin-film filters will shift towards shorter wavelengths [8]. The further the filter from the optical axis, the larger the incident angle will be.

In our setup we will compare the spectrum of identical targets positioned at the optical axis and at an off-axis position (Fig. 1a). Because the chief ray angle is nonzero for a pixel at a distance $d = 7.7$ mm from the optical axis, this spectrum appears to be shifted (Fig. 1b).

This shift can be corrected by updating the wavelengths at which the data points are plotted. When light is focused from an aperture at a chief ray angle $\theta _{\textrm {CRA}}$, the corrected central wavelengths are calculated as [13]

(1)$$\vec{\lambda}_{\textrm{cwl}}^{\textrm{new}} = \vec{\lambda}_{\textrm{cwl}} \left(1 - \dfrac{\theta_{\textrm{CRA}}^2}{2{n_{\textrm{eff}}^2}} - \dfrac{\theta_{\textrm{cone}}^2}{4{n_{\textrm{eff}}^2}} \right),\, \textrm{with}\,\, \theta_{\textrm{CRA}} = \arctan \dfrac{d}{x}.$$

Where the half-cone angle $\theta _{\textrm {cone}}$ is a parameter quantifying the working f-number [13]. The lower the f-number, the larger $\theta _{\textrm {cone}}$.

Here ${n_{\textrm {eff}}}$ is the effective refractive index of the filter. It is a dimensionless number that characterizes the sensitivity to the angle of incidence [8,15]. In this article we assume ${n_{\textrm {eff}}}$ to be known since it is a property of the filter and not of the lens.

Equation (1) is only valid if the lens experiences no vignetting. If vignetting is significant, an extended model is required [14]. For large f-numbers, $\theta _{\textrm {cone}}\rightarrow 0$ and vignetting becomes negligible such that the formula simplifies to

(2)$$\vec{\lambda}_{\textrm{cwl}}^{\textrm{new}} = \vec{\lambda}_{\textrm{cwl}} \left(1 - \dfrac{\theta_{\textrm{CRA}}^2}{2{n_{\textrm{eff}}^2}} \right),\, \textrm{with}\,\, \theta_{\textrm{CRA}} = \arctan \dfrac{d}{x}.$$

This makes Eq. (2) applicable for any lens with no appreciable pupil aberrations [16].

From this equation we will now derive a method to estimate the exit pupil position $x$.

3. Estimating the position of the exit pupil

Light is focused from the exit pupil onto each pixel. For Eq. (2) to be valid, the method requires that the size of the aperture is negligible (high f-number). This ensures that no significant smoothing of the spectra occurs and that vignetting can be neglected [13,14].

Let $\vec {\rho }_\textrm {on}$ be the spectrum measured on-axis and let $\vec {\rho }_\textrm {off}$ be the spectrum measured for the off-axis target. The method works by applying Eq. (2) to make $\vec {\rho }_\textrm {off}$ optimally match $\vec {\rho }_\textrm {on}$.

The correction, however, only updates the wavelengths at which the data is plotted. If the current guess of the exit pupil distance is $y$, then the updated wavelength vector becomes

(3)$$\vec{\lambda}_{\textrm{cwl}}(y) = \vec{\lambda}_{\textrm{cwl}} \left(1 - \arctan^2 \left(\dfrac{d}{y}\right)\cdot\dfrac{1}{2{n_{\textrm{eff}}^2}} \right).$$

These are then the wavelengths at which it is assumed that the spectrum $\vec {\rho }_\textrm {off}$ was actually sampled.

The spectrum $\vec {\rho }_\textrm {off}$ is then resampled at the wavelengths $\vec {\lambda }_{\textrm {cwl}}$ such that it can be compared to the on-axis spectrum. We define the resampled spectrum as

(4)$$\vec{\rho}_\textrm{corr}(y) = \textrm{interp}((\vec{\lambda}_{\textrm{cwl}}(y),\vec{\rho}_\textrm{off}),\vec{\lambda}_{\textrm{cwl}}).$$

Where one can use an interpolation method of choice.

The exit pupil estimation problem can then be formulated as the least square problem

(5)$$\hat{x} = \arg \min_{\substack{y}} \left \| {\vec{\rho}_\textrm{corr}}(y) - \vec{\rho}_\textrm{on} \right \|_2^2,$$

where $\hat {x}$ is the best estimate of the exit pupil.

This optimization problem can be easily solved with a sweep of possible exit pupil distances and by selecting the value that minimizes the merit function

(6)$$E_\textrm{lsq}(y;d) = \left \| {\vec{\rho}_\textrm{corr}}(y) - \vec{\rho}_\textrm{on} \right \|_2^2,$$

which is continuous with respect to $y$ and depends on the off-axis distance $d$.

4. Experiment and results

In this section we demonstrate the validity of our method. We proceed by using a lens for which the exit pupil is known and then change the position of the lens in a controlled way. This generates multiple cases for which the exit pupil position is then estimated.

We use imec’s visible, near-infrared Snapscan spectral camera (Fig. 2a) [2]. This camera allows accurate control of sensor in the image plane using a translation stage. The proposed method, however, can also be implemented for snapshot mosaic sensors [10] and for linescan sensors [1]. With the linescan, for both the on-axis and off-axis target, each filter will experience a different chief ray angle. Hence, in Eq. (2), $\theta _{\textrm {CRA}}$ will be different for each element of the vector $\vec {\lambda }_{\textrm {cwl}}$.

Fig. 2. The camera images a white diffuse surface. By placing a filter in front of the lens, the whole surface acts as one large uniform spectral target.

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The snapscan camera is used with a Schneider Cinegon 1.9/10 mm lens, for which a datasheet with the exit pupil position was available. The effective refractive index ${n_{\textrm {eff}}}$ of the filters is 1.7 [13].

Using the datasheet of the lens and the Snapscan camera, it can be calculated that the exit pupil should be at a distance of 29.6 mm from the sensor. The distance to the exit pupil can now be controlled by adding spacer rings between the camera and the lens. We do this for additional distances of 0, 2, 3, 5, 7, 10, 12 and 15 mm.

For each case, at f-number $f/8$, we measure the transmittance of a Thorlabs FGB67S Colored Glass Bandpass filter (Fig. 2b). It is important that the filter is not an interference filter since it will also be sensitive to the incident angle. The chosen spectrum is measured on the optical axis and at multiple off-axis distances. The distance is then estimated using the described method.

The estimated distance $\hat {x}$ is compared to the calculation from the datasheet for each off-axis distance (Fig. 3). The estimates follow the ground truth very well. The estimates are less accurate for the off-axis target focused at $d=3.3$ mm. Two sets of merit functions that were used to obtain these estimates are displayed in Figs. 5a and b. The robustness of these estimates will be discussed in the next section.

Fig. 3. The method can predict the exit pupil position with varying accuracy. The estimates made with filters that are further from the optical axis (larger $d$) are more accurate.

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Once the distance to the exit pupil is obtained, the spectral shifts for the different positions on the sensor can all be corrected using Eq. 2 (Fig. 4). As desired, the spectra are now independent of the position of the target.

Fig. 4. When the exit pupil position is known ($x=29.6$ mm), undesired shifts for identical targets at different positions can be corrected using Eq. 2.

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During this experiment the focus setting may not be changed. The lens however does not need not to be focused onto the actual target. This is because we use a diffuse white surface and a transmittance filter in front of the lens. This ensures that all points in the scene will have the same spectrum. Each filter will still receive rays from each point of the exit pupil. Each ray originates from different points of the target surface, however since all rays have the same spectrum a complete incident light cone is still formed. Nevertheless, in practice one should focus the lens as it is going to be used in the actual application, because changing the focus can also change the position of the exit pupil.

5. Robustness

The method of finding the exit pupil is formulated as the minimization of a least square criterion (Eq. (5)). We discuss the relative robustness of the method for different system parameters and potential target spectra. The robustness is quantified using the curvature of the least square merit function in the neighborhood of the global minimum. The larger the curvature, the more robust the estimator will be for noise and artifacts.

5.1 Modeling the merit function

In our method, the merit function $E_\textrm {lsq}(y)$ needed to be minimized. This merit function was defined for discrete spectra that were measured using the filters on the sensor. To model the behavior of this merit function we will assume that the measured spectra are continuous. Thus, let $r(\lambda )\equiv r_\textrm {on}(\lambda )$ be the spectrum of the target that is measured on the optical axis. This function is the equivalent of vector $\vec {\rho }_\textrm {on}$ in Eq. (5).

The spectrum measured at an off-axis distance $d$ will be shifted. If the exit pupil position is $x$ then the shifted spectrum is

(7)$$r_\textrm{off}(\lambda) = r\left(\lambda \left( 1-\dfrac{d^2}{2{n_{\textrm{eff}}^2} x^2}\right)\right),$$

where we have used the small angle approximation $\arctan \left (\frac {d}{x}\right )\sim \frac {d}{x}$.

The shift of the measured spectrum is corrected by updating the wavelengths using Eq. (2). Therefore, if the exit pupil distance is assumed to be $y$, then the corrected spectrum is

(8)$$r_\textrm{corr}(\lambda) = r\left(\lambda \left( 1-\dfrac{d^2}{2{n_{\textrm{eff}}^2} x^2} +\dfrac{d^2}{2{n_{\textrm{eff}}^2} y^2}\right)\right).$$

Now the value of $y$ must be found such that $r_\textrm {corr}(\lambda )$ optimally fits $r(\lambda )$. The least square criterion becomes

(9)$$E(y) = \int_\Omega \left[r_\textrm{corr}(\lambda) - r(\lambda) \right]^2 \textrm{d}\lambda,$$

which is the continuous equivalent of $E_\textrm {lsq}(y)$. Here, $\Omega$ is the wavelength interval that is being used.

Analogous to Eq. (5), $\hat {x}$ will be the global minimizer and best estimate of $x$ such that

(10)$$\hat{x} = \arg \min_{\substack{y}} E(y).$$

5.2 Asymptotic approximation

The merit function $E(y)$ is asymptotically equivalent with $\hat {E}(y)$ such that

(11)$$\begin{aligned} E(y) & \quad \sim \quad \hat{E}(y),\quad\textrm{for}\, y\rightarrow x \nonumber\\ & \quad = \quad \dfrac{d^4}{4{n_{\textrm{eff}}^4}}\cdot \int_\Omega (\lambda r')^2 \textrm{d}\lambda \cdot \left(\dfrac{1}{x^2} - \dfrac{1}{y^2}\right)^2 \nonumber\\ & \quad = \quad \dfrac{d^4}{4{n_{\textrm{eff}}^4}}\cdot f(r) \cdot \left(\dfrac{1}{x^2} - \dfrac{1}{y^2}\right)^2. \end{aligned}$$

With $r'$ being the first derivative of $r(\lambda )$.

This follows from comparing the Taylor series expansions of Eq. (9) and Eq. (11) for $y\rightarrow x$. We have

(12)$$\hat{E}(y) = \dfrac{d^4}{x^6 {n_{\textrm{eff}}^4}}\cdot f(r) \cdot (x-y)^2 + \mathcal{O}((x-y)^3).$$

And by first expanding the integrand in Eq. (9), integration gives

(13)$$E(y) = \dfrac{d^4}{x^6 {n_{\textrm{eff}}^4}}\cdot f(r) \cdot (x-y)^2 + \mathcal{O}((x-y)^3).$$

Which confirms our ansatz in Eq. (11).

The merit functions of the real measurements are compared to the asymptotic approximation in Figs. 5a and b. The approximations fit the measurements up to a common scalar factor. This means that our approximation can correctly predict the relative curvature of the merit functions for different distances.

Fig. 5. Merit functions used to estimate exit pupil distance. The estimates are made at different off-axis distances. The global minimum for $x=39.6$ mm is less pronounced than for $x=29.6$ mm (Fig. 5a). The analytical approximations $\hat{E}(y)$ fit the real merit functions $E_{\textrm{lsq}}(y)$ well up to a common scalar factor.

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5.3 Effect of the system parameters

The coefficient of the quadratic term in Eq. (12) is

(14)$$\dfrac{d^4}{x^6 {n_{\textrm{eff}}^4}}.$$

With $x$ the exit pupil distance and $d$ the position on the sensor used to measure the shifted spectrum.

This factor shows that the curvature increases for larger $d$ (Fig. 5a) and smaller $x$. This is logical since the larger the distance $d$, the more the spectrum will be shifted and hence the more sensitive the method. In experiment, the estimates using the smallest distance $d$ gave less accurate results (Fig. 3).

The smaller the distance $x$, the larger the incident angles will be and hence the larger the shifts. The conclusion is that the method will be more robust for smaller exit pupil distances. This can be seen by comparing the merit functions for $x=29.6$ mm and $x=39.6$ mm in Figs. 5a and b respectively. This however comes as no real surprise, since for small $x$ the effect of spectral shift becomes significant, which was why we wanted to estimate $x$ in the first place.

5.4 Selecting the best target spectrum

In this section we introduce a criterion that can be used to identify the best spectrum from a given set of target spectra. It is important that the used target spectrum is not interference based, because then it will also be angle dependent.

The only factor in Eq. (11) that depends on the shape of the spectrum $r(\lambda )$ is

(15)$$f(r) = \int_\Omega (\lambda r')^2 \textrm{d}\lambda.$$

It can therefore be used as a criterion to select the most suitable target. The best target being the one with a spectrum that maximizes $f(r)$, and hence maximizes the curvature of the merit function.

The interpretation of Eq. (15) is that the square of the derivative should be as large as possible on the given wavelength interval. This will be true for spectra with many highly oscillatory features.

When selecting the best target spectrum, one needs to take into account that the spectral filters have a finite full width at half maximum (FWHM). Because of this, spectra with high frequency features could be significantly smoothed and might be less suited as a target spectrum. Indeed, when a highly oscillatory spectrum is significantly smoothed, the derivative $r'$ will tend to zero, and hence so will $f(r)$.

Therefore one should use a smoothed version of $r(\lambda )$. As this is mainly meant as a heuristic approach, a pragmatic way is to convolute $r(\lambda )$ with a rectangular window with identical full width half maximum (FWHM) as the filters. The smoothed spectrum is then used to evaluate Eq. (15).

In our experiments we made use of a Thorlabs FGB67S bandpass colored glass filter. It is a well-suited target spectrum since its transmittance is highly oscillatory in the 700 to 1000 nm region (Fig. 6). In the 1000 to 1300 nm region the transmittance is nearly flat ($r' \approx 0$ in Eq. (15)), making it unsuitable for exit pupil estimation. This is reflected in the values of $f(r)$ as numerical calculation shows that in the first region $f(r)$ is 527 times larger than in the bandpass region.

Fig. 6. The transmittance of the Thorlabs FGB67S filter is highly oscillatory in the 400 to 1000 nm range.

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5.5 Effect of a tilted image sensor

In the model it is assumed that the image sensor is parallel to the exit pupil plane. However, in a real camera, the sensor might be (non-intentionally) tilted by a small zenith angle $\alpha$. As will be shown, this causes the global minimum of the merit function to shift.

We simplify the analysis by assuming that the tilt is such that the angles $\theta _{\textrm {CRA}}$ and $\alpha$ are approximately additive. The effective chief ray angle will then be $\theta _\textrm {CRA}' = \theta _{\textrm {CRA}} + \alpha \approx \frac {d}{x} + \alpha$. A full analysis on the effect of a tilt with an arbitrary azimuth angle is out of the scope of this work.

A tilt of the image sensor affects both the on-axis and off-axis measurement. The on-axis measurement will be shifted in accordance with Eq. (2) such that

(16)$$\lambda_\textrm{cwl,on}^\textrm{new}=\lambda_{\textrm{cwl}}\left(1-\dfrac{\alpha^2}{2{n_{\textrm{eff}}^2}}\right).$$

The off-axis spectrum will be shifted such that

(17)$$\lambda_\textrm{cwl,off}^\textrm{new}=\lambda_{\textrm{cwl}}\left(1-\dfrac{\theta_{\textrm{CRA}}'^2}{2{n_{\textrm{eff}}^2}}\right) = \lambda_{\textrm{cwl}}\left(1-\dfrac{\left(\theta_{\textrm{CRA}} +\alpha\right)^2}{2{n_{\textrm{eff}}^2}}\right).$$

The relative shift between the off-axis and on-axis measurement is then

(18)$$\lambda_\textrm{cwl,on}^\textrm{new} - \lambda_\textrm{cwl,off}^\textrm{new} = \lambda_{\textrm{cwl}} \dfrac{\theta_{\textrm{CRA}}\left(2\alpha+\theta_{\textrm{CRA}}\right)}{2{n_{\textrm{eff}}^2}}.$$

The method will correct this difference using a factor of the form $\lambda _{\textrm {cwl}}\frac {d^2}{2{n_{\textrm {eff}}^2} y^2}$. Equating this factor to Eq. (18) and solving for the estimated exit pupil distance $y$ gives

(19)$$y = \dfrac{d}{\theta_{\textrm{CRA}}\sqrt{1+\dfrac{2\alpha}{\theta_{\textrm{CRA}}}}} \approx \dfrac{x}{\sqrt{1+\dfrac{2\alpha}{\theta_{\textrm{CRA}}}}} \sim x \cdot\left( 1 - \dfrac{\alpha}{\theta_{\textrm{CRA}}}\right),\quad \dfrac{\alpha}{\theta_{\textrm{CRA}}} \rightarrow 0,$$

which corresponds to a relative error of

(20)$$\left|\dfrac{\Delta x}{x}\right| \sim \left|\dfrac{\alpha}{\theta_{\textrm{CRA}}}\right|.$$

It shows that the larger $\theta _{\textrm {CRA}}$, the lower the impact of a small tilt.

In the case of our experiment the largest chief ray angle is about 15 degrees. Thus, if there were to be an $\alpha =1$ degree tilt angle, this could give a 6 percent error on the exit pupil position estimation. This corresponds to an error of about 2 mm.

6. Conclusion

We introduced the idea of using the angular sensitivity of thin-film filters to estimate the distance to the exit pupil. In this paper we focused on spectral cameras but in principle the method need not remain restricted to this application as it might also be used as a generic method to localize the exit pupil.

We demonstrated a practical method to estimate the distance to the exit pupil using a thin-film based spectral camera. This distance is a key system parameter that is required to correct the undesired shifts in measured spectra.

Only a standard spectral camera setup and a well chosen target are required. We have therefore enabled easier calibration of thin-film based spectral cameras for both camera designers and end-users.

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Exit pupil localization to correct spectral shift in thin-film Fabry-Perot spectral cameras

Abstract

1. Introduction

2. Spectral shift

3. Estimating the position of the exit pupil

4. Experiment and results

5. Robustness

5.1 Modeling the merit function

5.2 Asymptotic approximation

5.3 Effect of the system parameters

5.4 Selecting the best target spectrum

5.5 Effect of a tilted image sensor

6. Conclusion

References

Cited By

Figures (6)

Equations (20)

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