A NEW CONCEPT FOR SPECTROPHOTOMETRY OF EXOPLANETS WITH SPACE-BORNE TELESCOPES

This new concept for space transit spectroscopy, called densified pupil spectroscopy, achieves pupil spectrophotometry with a number of small telescopes aligned on an aperture plane such as the PLATO mission. This densified pupil spectroscopy allows us to perform highly stable spectrophotometry against telescope pointing jitter and deformation of the primary mirror instead of not having imaging capability. Figure 1 gives an overview of this concept. The densified pupil spectroscopy first divides the telescope aperture into a number of sub-pupils. Each sub-pupil is densified with two lens arrays. Here, focusing on the fact that the divided and densified sub-pupil can be treated as a point source, we discovered that a simplified spectrometer allows us to acquire the spectra of the densified sub-pupils on the detector plane, which is an optical conjugate with the primary mirror, by putting the divided and densified sub-pupils on the entrance slit of the spectrometer. The spectral resolution of this method is characterized by a diameter of the beam formed on the dispersive element, as discussed in Section 2.3. The beam diameter requires enlargement for higher spectral resolution. Thus, this concept realizes spectroscopy of the sub-pupils on the telescope aperture with simplified optics. This densified pupil spectroscopy has the following advantages compared to a conventional spectrometer in terms of the spectrophotometry. First, by producing an optical conjugation between a telescope aperture and a detector plane, this concept allows us to significantly reduce systematic noises due to telescope pointing jitter and distortion of the primary mirror, which prevent precise characterization of transiting exoplanets via the current space observatories such as the Spitzer Space Telescope and the HST. This concept also provides us with highly reliable scientific data by optically dividing the telescope aperture into a number of sub-apertures and simultaneously producing multiple equivalent spectra because the scientific data can be constructed without defective pixels and pixels hit by cosmic rays. In addition, the systematic noise can be further reduced through averaging of the multiple equivalent data in cases where the systematic noises are randomly added to the spectra. Instead of the averaging operation, various non-parametric data analysis techniques such as coherent analysis (Swain et al. 2010; Waldmann et al. 2012a) and independent component analysis (Waldmann 2012b, 2014) are also applied to this method. Finally, thanks to division the telescope aperture into multiple small sub-apertures, any optical requirement on the primary mirror is mitigated. Thus, this method reduces the systematic noise, which is a major issue in current space transit spectroscopy, and potentially achieves observation performance limited not by the instrumental systematic noise but by shot noise.

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In this section, we mathematically describe the wavefront propagation in the spectrometer of this densified pupil spectroscopy to understand its nature. From the standpoint of physical optics, this concept is mainly divided into two units: a pupil division/densifier unit and a conventional spectrometer. When the size of the densified sub-pupil formed by the former unit is close to the wavelength, the wavefront propagation in the spectrometer should not be written by geometric optics but by physical optics. In accordance with this consideration, the wavefront propagation in the spectrometer is mathematically described based on Fresnel propagation. The coordinate systems and parameters are set for this mathematical description as follows. The size of the entrance pupil is set to be D. The number of divisions along one dimension and the pupil densification are defined as n and η, respectively. The n² densified sub-pupils with size ηD/n are aligned in the re-imaged pupil plane; P2 in Figure 1. The P2 plane corresponds to the entrance slit of the spectrometer. The optics, composed of a collimator and camera optics in the spectrometer, produce an optical conjugate between the entrance slit and the detector plane. The coordinate systems in the entrance slit (P2), the plane just after the collimator optics (P3), the dispersive element plane (P4), the plane just before the camera optics (P5), and the detector plane (P6) are defined as (x₂, y₂), (x₃, y₃), (x₄, y₄), (x₅, y₅), and (x₆, y₆), respectively. The electric field in the coordinate system (x_i, y_i) is defined as E_i(x_i, y_i). The collimator and camera elements, whose focal lengths are f₁ and f₂ and aperture functions are A₁(x₃, y₃) and A₂(x₅, y₅), are, respectively, placed at distances of f₁ and 2f₁ + f₂ from the P2 plane along the vertical direction of the entrance plane, z. The dispersive element, which is mathematically described by M(x₄, y₄), is placed at a distance of f₁ from the collimator element along z. The detector is placed at a distance of f₂ from the camera optics.

We first derive the electric field formed on the dispersive element for each densified sub-pupil of size ηD/n on the entrance slit of the spectrometer. As explained in Section 2.1, n and η should be designed such that the following approximation formula is satisfied:

$$\begin{eqnarray}&&{f}_{1}\gg \displaystyle \frac{{(\eta D/n)}^{2}}{\lambda }.\end{eqnarray} \tag{ 1 }$$

According to Equation (1), the Fraunhofer diffraction pattern of the densified sub-pupil is formed on the plane (P3) through the collimator element:

$$\begin{eqnarray}{E}_{3}({x}_{3},{y}_{3}) & = & \displaystyle \frac{\mathrm{exp}\left(\tfrac{2\pi {{if}}_{1}}{\lambda }\right)}{i\lambda {f}_{1}}{A}_{1}({x}_{3},{y}_{3})\displaystyle \int {{dx}}_{2}\displaystyle \int {{dy}}_{2}\\ & & \times {E}_{2}\left(\displaystyle \frac{{x}_{2}}{{f}_{1}},\displaystyle \frac{{y}_{2}}{{f}_{1}}\right)\mathrm{exp}\left(\displaystyle \frac{-2\pi i({x}_{2}{x}_{3}+{y}_{2}{y}_{3})}{\lambda {f}_{1}}\right).\end{eqnarray} \tag{ 2 }$$

By using Fourier transform, the above equation can be written as

$$\begin{eqnarray}&&{E}_{3}({x}_{3},{y}_{3})=\displaystyle \frac{{f}_{1}\mathrm{exp}\left(\tfrac{2\pi {{if}}_{1}}{\lambda }\right)}{i\lambda }{A}_{1}({x}_{3},{y}_{3})\mathrm{FT}\left\{{E}_{2}\left(\displaystyle \frac{{x}_{2}}{{f}_{1}},\displaystyle \frac{{y}_{2}}{{f}_{1}}\right)\right\},\end{eqnarray} \tag{ 3 }$$

where FT{h} is the Fourier transform of a function h. The wavefront propagation from the plane P3 to the dispersive element can be described by the Fresnel diffraction equation. The electric field on the dispersive element is

$$\begin{eqnarray}{E}_{4}({x}_{4},{y}_{4}) & = & -\displaystyle \frac{\mathrm{exp}\left(\tfrac{4\pi {{if}}_{1}}{\lambda }\right)}{{\lambda }^{2}}\displaystyle \int {{dx}}_{3}\displaystyle \int {{dy}}_{3}{A}_{1}({x}_{3},{y}_{3})\\ & & \times \mathrm{exp}\left(\displaystyle \frac{\pi i({({x}_{3}-{x}_{4})}^{2}+{({y}_{3}-{y}_{4})}^{2})}{\lambda {f}_{1}}\right)\\ & & \times \mathrm{FT}\left\{{E}_{2}\left(\displaystyle \frac{{x}_{2}}{{f}_{1}},\displaystyle \frac{{y}_{2}}{{f}_{1}}\right)\right\}.\end{eqnarray} \tag{ 4 }$$

The above equation can be written by a convolution as follows:

$$\begin{eqnarray}{E}_{4}({x}_{4},{y}_{4}) & = & -\displaystyle \frac{\mathrm{exp}\left(\tfrac{4\pi {{if}}_{1}}{\lambda }\right)}{{\lambda }^{2}{f}_{1}^{2}}\left[\mathrm{exp}\left(\displaystyle \frac{\pi i({x}_{3}^{2}+{y}_{3}^{2})}{{f}_{1}\lambda }\right)\right.\\ & & \ast ({A}_{1}({x}_{3},{y}_{3})\mathrm{FT}\{{E}_{2}({x}_{2},{y}_{2})\})]({x}_{4},{y}_{4}),\end{eqnarray} \tag{ 5 }$$

where ∗ is the convolution operator. By using the convolution theorem and the following equation,

$$\begin{eqnarray}&&{\mathrm{FT}}^{-1}\left\{\mathrm{exp}\left(\displaystyle \frac{\pi i({x}_{3}^{2}+{y}_{3}^{2})}{\lambda {f}_{1}}\right)\right\}\\ &&\qquad =\displaystyle \frac{\lambda {f}_{1}}{2}\mathrm{exp}\left(-\displaystyle \frac{\pi i({x}_{2}^{2}+{y}_{2}^{2})}{\lambda {f}_{1}}+\displaystyle \frac{\pi i}{2}\right),\end{eqnarray} \tag{ 6 }$$

Equation (5) is rewritten as

$$\begin{eqnarray}&&{E}_{4}({x}_{4},{y}_{4})\\ &&=-\lambda {f}_{1}\mathrm{exp}\left(\displaystyle \frac{4\pi {{if}}_{1}}{\lambda }+\displaystyle \frac{\pi i}{2}\right)\mathrm{FT}\left\{\mathrm{exp}\left(-\displaystyle \frac{\pi i({x}_{2}^{2}+{y}_{2}^{2})}{\lambda {f}_{1}}\right)\right.\\ &&\quad \times ({\mathrm{FT}}^{-1}\{{A}_{1}({x}_{3},{y}_{3})\}\ast \;\left.\left.{E}_{2}\left(\displaystyle \frac{{x}_{2}}{{f}_{1}},\displaystyle \frac{{y}_{2}}{{f}_{1}}\right)\right)\right\}.\end{eqnarray} \tag{ 7 }$$

By applying a large size collimator element to this system, ${\mathrm{FT}}^{-1}\{{A}_{1}({x}_{3},{y}_{3})\}\simeq \delta (\tfrac{{x}_{2}}{{f}_{1}})\delta (\tfrac{{y}_{2}}{{f}_{1}})$. Since the diameter of each densified sub-pupil on the entrance slit, ηD/n, is fully smaller than the length of f₁, the electric field on the dispersive element is rewritten as

$$\begin{eqnarray}&&{E}_{4}({x}_{4},{y}_{4})=-\lambda {f}_{1}\mathrm{exp}\left(\displaystyle \frac{4\pi {{if}}_{1}}{\lambda }+\displaystyle \frac{\pi i}{2}\right)\mathrm{FT}\left\{{E}_{2}\left(\displaystyle \frac{{x}_{2}}{{f}_{1}},\displaystyle \frac{{y}_{2}}{{f}_{1}}\right)\right\}.\end{eqnarray} \tag{ 8 }$$

When Equation (1) is satisfied, the Fraunhofer diffraction pattern of the densified sub-pupil on the entrance slit is formed on the dispersive element. Comparing Equations (3) with (8), the Fraunhofer diffraction pattern is kept from the plane P3 just after the collimator optics to the dispersive element. As a result, a tilt of the phase in the densified sub-pupil, corresponding to the telescope pointing error, does not change the incident angle to the dispersive element but its irradiated area by the diffraction image. Thus, the spectrum formed on the detector plane is very stable against the telescope pointing jitter.

We next derive the electric field formed on the detector plane. The electric field just after the dispersive element is

$$\begin{eqnarray}&&{E}_{4}^{\prime }({x}_{4},{y}_{4})={E}_{4}({x}_{4},{y}_{4})M({x}_{4},{y}_{4}).\end{eqnarray} \tag{ 9 }$$

The electric field formed just before the camera optics is described through the Fresnel diffraction equation as follows:

$$\begin{eqnarray}{E}_{5}({x}_{5},{y}_{5}) & = & \displaystyle \frac{\mathrm{exp}\left(\tfrac{2\pi {{if}}_{2}}{\lambda }\right)}{i\lambda {f}_{2}}\displaystyle \int {{dx}}_{4}\displaystyle \int {{dy}}_{4}{E}_{4}^{\prime }({x}_{4},{y}_{4})\\ & & \times \mathrm{exp}\left(\displaystyle \frac{\pi i{({x}_{4}-{x}_{5})}^{2}+{({y}_{4}-{y}_{5})}^{2}}{\lambda {f}_{2}}\right).\end{eqnarray} \tag{ 10 }$$

Since the electric field formed on the detector plane becomes the Fraunhofer diffraction pattern of the electric field just before the camera optics,

$$\begin{eqnarray}{E}_{6}\left(\displaystyle \frac{{x}_{6}}{{f}_{2}},\displaystyle \frac{{y}_{6}}{{f}_{2}}\right) & = & -\displaystyle \frac{\mathrm{exp}\left(\tfrac{4\pi {{if}}_{2}}{\lambda }\right)}{{\lambda }^{2}{f}_{2}^{2}}\displaystyle \int {{dx}}_{5}\displaystyle \int {{dy}}_{5}\\ & & \times {A}_{2}({x}_{5},{y}_{5})\mathrm{exp}\left(\displaystyle \frac{2\pi i({x}_{5}{x}_{6}+{y}_{5}{y}_{6})}{\lambda {f}_{2}}\right)\\ & & \times \displaystyle \int {{dx}}_{4}\displaystyle \int {{dy}}_{4}{E}_{4}^{\prime }({x}_{4},{y}_{4})\\ & & \times \mathrm{exp}\left(\displaystyle \frac{\pi i({({x}_{4}-{x}_{5})}^{2}+{({y}_{4}-{y}_{5})}^{2})}{\lambda {f}_{2}}\right).\end{eqnarray} \tag{ 11 }$$

By describing it with a convolution, the above equation can be written as follows:

$$\begin{eqnarray}&&{E}_{6}\left(\displaystyle \frac{{x}_{6}}{{f}_{2}},\displaystyle \frac{{y}_{6}}{{f}_{2}}\right)\\ &&=-\displaystyle \frac{\mathrm{exp}\left(\tfrac{4\pi {{if}}_{2}}{\lambda }+\displaystyle \frac{\pi i}{2}\right)}{{\lambda }^{2}{f}_{2}^{2}}\left[{\mathrm{FT}}^{-1}\left\{\mathrm{exp}\left(-\displaystyle \frac{\pi i({x}_{4}^{2}+{y}_{4}^{2})}{\lambda {f}_{2}}\right)\right\}\right.\\ &&\quad \times {\mathrm{FT}}^{-1}\{{E}_{4}^{\prime }({x}_{4},{y}_{4})\}]\ast {\mathrm{FT}}^{-1}\{A({x}_{5},{y}_{5})\}.\end{eqnarray} \tag{ 12 }$$

Given that the size of the camera element is fully large as well as the collimator, ${\mathrm{FT}}^{-1}\{A({x}_{5},{y}_{5})\}\simeq \delta \left(\tfrac{{x}_{6}}{{f}_{2}}\right)\delta \left(\tfrac{{y}_{6}}{{f}_{2}}\right)$. By inserting Equations (8) and (9) into the above equation, the electric field on the detector plane becomes

$$\begin{eqnarray}{E}_{6}\left(\displaystyle \frac{{x}_{6}}{{f}_{2}},\displaystyle \frac{{y}_{6}}{{f}_{2}}\right) & = & -\displaystyle \frac{{f}_{1}}{{f}_{2}}\mathrm{exp}\left(\displaystyle \frac{4\pi i({f}_{1}+{f}_{2})}{\lambda }\right)\\ & & \times \left[{\mathrm{FT}}^{-1}\{M({x}_{4},{y}_{4})\}\ast {E}_{2}\left(\displaystyle \frac{{x}_{2}}{{f}_{1}},\displaystyle \frac{{y}_{2}}{{f}_{1}}\right)\right].\end{eqnarray} \tag{ 13 }$$

The intensity distribution on the detector plane is simply expressed as a convolution of the densified sub-pupil on the entrance slit, ${E}_{2}\left(\tfrac{{x}_{2}}{{f}_{1}},\tfrac{{y}_{2}}{{f}_{1}}\right)$, with the Fourier transform of the dispersive element function, FT{M(x₄, y₄)}. Consequently, the position of the dispersive element between the collimator and the camera elements does not affect the dispersion property.

Finally, the above mathematical descriptions are extended to the case of n² densified sub-pupils, instead of each one individually. The electric field on the entrance slit, ${E}_{2}\left(\tfrac{{x}_{2}}{{f}_{1}},\tfrac{{y}_{2}}{{f}_{1}}\right)$, can be expressed as a summation of each densified sub-pupil:

$$\begin{eqnarray}&&{E}_{2}({x}_{2},{y}_{2})-\gt \displaystyle \sum _{i}^{{n}^{2}}{E}_{2,i}\left(\displaystyle \frac{{x}_{2}}{{f}_{1}},\displaystyle \frac{{y}_{2}}{{f}_{1}}\right).\end{eqnarray} \tag{ 14 }$$

As a result, an interference pattern is formed on the dispersion element:

$$\begin{eqnarray}&&{E}_{4}({x}_{4},{y}_{4})=-\lambda {f}_{1}\mathrm{exp}\left(\displaystyle \frac{4\pi {{if}}_{1}}{\lambda }+\displaystyle \frac{\pi i}{2}\right)\mathrm{FT}\left\{\displaystyle \sum _{i}^{{n}^{2}}{E}_{2,i}\left(\displaystyle \frac{{x}_{2}}{{f}_{1}},\displaystyle \frac{{y}_{2}}{{f}_{1}}\right)\right\}.\end{eqnarray} \tag{ 15 }$$

On the other hand, because each spectrum is separated on the detector according to Equation (13), the dispersive property is the same as for the case of a single sub-pupil. We continue to consider the case of a densified sub-pupil to understand the nature of this concept in the ensuing discussion.

On the basis of the analytical description of the wavefront propagation in Section 2.2, we derive the spectral resolution of this concept for the case where a grating is adopted as the dispersive element. When the pitch of the grating and the number of the cycles are g and N, respectively, the grating function, M(x₄, y₄), can be represented as

$$\begin{eqnarray}&&M({x}_{4},{y}_{4})=G({x}_{4},{y}_{4})\ast \left\{\displaystyle \sum _{k=0}^{N-1}\delta ({x}_{4}-{gk})\mathrm{exp}({ik}{\rm{\Phi }})\right\},\end{eqnarray} \tag{ 16 }$$

where G(x₄, y₄) is the sub-pupil function in each cycle and Φ is the phase difference between two adjacent cycles. Based on Equation (13), the dispersion property of this concept is characterized by the Fourier transform of the grating function, FT{M(x₄, y₄)}:

$$\begin{eqnarray}&&\mathrm{FT}\{M({x}_{4},{y}_{4})\}\\ &&\qquad ={\lambda }^{2}\delta \left(\displaystyle \frac{{y}_{6}}{{f}_{2}}\right)\frac{\mathrm{sin}\left(\tfrac{N\left(\tfrac{2\pi {{gx}}_{6}}{\lambda {f}_{2}}+{\rm{\Phi }}\right)}{2}\right)}{\mathrm{sin}\left(\tfrac{\left(\tfrac{2\pi {{gx}}_{6}}{\lambda {f}_{2}}+{\rm{\Phi }}\right)}{2}\right)}{\mathrm{FT}}^{-1}\{G({x}_{4},{y}_{4})\},\end{eqnarray} \tag{ 17 }$$

where FT⁻¹{G(x₄, y₄)} represents the efficiency of the grating. From the above equation, the maximum displacement of the spectrum element on the detector is determined by

$$\begin{eqnarray}&&\displaystyle \frac{N\left(\tfrac{2\pi {{gx}}_{6}}{\lambda {f}_{2}}+{\rm{\Phi }}\right)}{2}=m\pi ,\end{eqnarray} \tag{ 18 }$$

where m represents the order of the the dispersive element. As a result, the relation between the image displacement on the detector plane and the wavelength of the spectrum element is derived as follows:

$$\begin{eqnarray}&&\displaystyle \frac{g}{{f}_{2}}{{dx}}_{6}={md}\lambda .\end{eqnarray} \tag{ 19 }$$

Because the minimum sampling interval of each spectrally resolved component on the detector plane corresponds to the pupil size, d_p, as shown in Equation (13), the spectral resolution of this concept is

$$\begin{eqnarray}&&\displaystyle \frac{\lambda }{d\lambda }=\displaystyle \frac{m}{2}\displaystyle \frac{2\lambda {f}_{2}/{d}_{{\rm{p}}}}{g}.\end{eqnarray} \tag{ 20 }$$

Because the number of the grating cycles included in the Fraunhofer pattern, N_PSF, is

$$\begin{eqnarray}&&{N}_{\mathrm{PSF}}=\displaystyle \frac{2\lambda {f}_{2}}{{{gd}}_{{\rm{p}}}},\end{eqnarray} \tag{ 21 }$$

the spectral resolution is characterized by m and N_PSF. Based on the above consideration, an increase is required in the number of divisions of the pupil along the pupil, n, and the densification of each pupil, η for higher spectral resolution. Since the pupil size, d_p, and the pitch of the grating, g, cannot be reduced down to λ, the maximum spectral resolution, R_max, is

$$\begin{eqnarray}&&{R}_{\mathrm{max}}\equiv {\left\{\displaystyle \frac{\lambda }{d\lambda }\right\}}_{\mathrm{max}}\lt m\displaystyle \frac{{f}_{1}}{\lambda }.\end{eqnarray} \tag{ 22 }$$

To investigate whether the above analytical descriptions on wavefront propagation are satisfied with actual parameters, we derive one-dimensional electric fields on the dispersive element and the detector plane based on the Fresnel diffraction equation. By assuming that aplanatic lenses are applied to the collimator and camera optics, the phase function given by passing through a lens of focal length f is defined as

$$\begin{eqnarray}&&U(x,y)=\mathrm{exp}\left(-\displaystyle \frac{2\pi i({r}^{\prime }-f)}{\lambda }\right),\end{eqnarray} \tag{ 23 }$$

where (x, y) is the coordinate system on the lens and r' is the distance between the focal point of the lens and the position on the lens, (x, y). The r' is approximated by

$$\begin{eqnarray}&&{r}^{\prime }\simeq f+\displaystyle \frac{{x}^{2}+{y}^{2}}{2f}.\end{eqnarray} \tag{ 24 }$$

By using the Fresnel diffraction equation and Equations (22) and (23), the complex amplitudes on the various planes from P3 to P6 can be analytically calculated. Figure 2 shows the overall view of the optical system and the absolute values of the complex amplitudes on the entrance slit of the spectrometer, on the dispersive element, and on the detector plane. In this calculation, the parameters are set as follows. A single densified sub-pupil with diameter of 0.4 mm is put on the entrance slit for simplicity. The source is monochromatic light at a lambda of 0.01 mm. The focal lengths of both the collimator and the camera optics have the same parameter of 100 mm. The dispersive element is placed at distance of 100 mm from the collimator lens. The left side of Figure 2 shows the modulus of the complex amplitude on the entrance slit. Since the diameter of the densified sub-pupil is much smaller than the focal length of the collimator lens, the Fraunhofer diffraction pattern of the electric field on the entrance pupil is formed as shown in Equation (3). The center of Figure 2 shows the modulus of the complex amplitude on the dispersive element. The Fraunhofer diffraction pattern is kept in the propagation from P3 to P4 as a parallel light, as explained in Section 2.2. Consequently, as described in Equations (20) and (21), the spectral resolution of this method is determined by the grating cycles included in the Fraunhofer pattern, N_PSF. The right side of Figure 2 shows the absolute value of the complex amplitude on the detector plane. The densified sub-pupil on the entrance slit is re-imaged on the detector but the sub-pupil is slightly affected by spatial filtering due to the finite aperture of the optics. Thus, the mathematical description derived thus far is satisfied without any approximations. Note that the sizes of the collimator lens, the grating, and the camera lens should be 10 times larger than Airy disk formed by each sub-pupil to avoid the spatial filtering of the optical elements.

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This method principally liberates the photometry from the low-order aberrations such as the telescope pointing jitter and any deformation of the primary mirror, as discussed in Section 2.1. However, the telescope pointing jitter affects the photometric stability in real systems for the following reason. Since the field of view should be limited by a field stop, which is a circular mask put on the image plane, to avoid contamination of background stars, the field stop occults the wings of the point-spread function (PSF) formed the target star. The partial occultation of the target star leads to degradation of the photometric stability under the pointing jitter of the telescope, θ_jitter, during the transit observation. We estimated the degradation of the photometric stability due to the partial occultation of the PSF formed on the image plane with the field stop through numerical simulations. Figure 3 shows the photometric stabilities as a function of the radius of the field stop for three different values of θ_jitter = (0.1λ/D, 0.5λ/D, 1λ/D). The photometric stabilities for all values of the pointing jitter are more improved as the mask radius increases. The upper limit of the photometric stability, ΔL_mask, approximately obeys the following equation:

$$\begin{eqnarray}&&{\rm{\Delta }}{L}_{\mathrm{mask}}\leqslant \alpha ({\theta }_{\mathrm{jitter}}){\theta }_{\mathrm{mask}}^{-2.3},\end{eqnarray} \tag{ 25 }$$

where θ_mask is the mask radius in unit of λ/D and α represents a constant as a function of the pointing jitter, θ_jitter. α takes 0.0035 and 0.05 for the pointing jitters of 0.1λ/D and more than 0.5λ/D, respectively. Based on this estimation, the radius of the field stop should be larger than 12.5λ/D for θ_jitter = 0.1λ/D and 40 for θ_jitter > 0.5λ/D to achieve a photometric stability down to 10 ppm. In other words, large pointing jitter easily leads to the contamination of background stars because of the requirement of a field stop with a large radius. Figure 4 shows the cumulative number density of galactic stars in the N band for various galactic coordinates, (b, l), based on the calculation procedure derived by Konishi et al. (2015). The dashed black line in Figure 4 indicates the number density corresponding to that which a galactic star is at least contaminated within a field of view of 10 arcsec in radius. In other words, it is difficult to achieve a high photometric stability down to 10 ppm for the galactic plane, b = 0, in case of the field of view of 10 arcsec in radius. Thus, as the pointing jitter increases, the observable region is more restricted.

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The photometric stability of this pupil densified spectroscopy is limited by image movement on the detector, which has inter- and intra-pixel sensitivity variations, given that the telescope defocus is not changed during the transit observation due to the thermally stable environment provided by the cryogenic telescope. We first consider the impact of the intra-pixel sensitivity variation on the photometric stability. When the position of the central gravity of a sub-pupil is (x_c, y_c) on the detector plane, the observed luminosity of the sub-pupil, L(x_c, y_c), is simply written as

$$\begin{eqnarray}&&L({x}_{{\rm{c}}},{y}_{{\rm{c}}})=\int {{dx}}_{6}\int {{dy}}_{6}i({x}_{6}+{x}_{{\rm{c}}},{y}_{6}+{y}_{{\rm{c}}})\xi ({x}_{6},{y}_{6}),\end{eqnarray} \tag{ 26 }$$

where i(x₆, y₆) and ξ(x₆, y₆) are the intensity profile of the sub-pupil on the detector and the pixel sensitivity with the intra-pixel sensitivity variation as a function of the position of the detector plane, respectively. In a case where the sub-pupil moves on the detector plane by (δx_c, δy_c), the luminosity can be rewritten as:

$$\begin{eqnarray}L({x}_{{\rm{c}}}+\delta {x}_{{\rm{c}}},{y}_{{\rm{c}}}+\delta {y}_{{\rm{c}}}) & = & [i({x}_{6},{y}_{6})\ast \xi ({x}_{6},{y}_{6})]\\ & & \times ({x}_{{\rm{c}}}+\delta {x}_{{\rm{c}}},{y}_{{\rm{c}}}+\delta {y}_{{\rm{c}}}).\end{eqnarray} \tag{ 27 }$$

Here, on the basis of the sub-pixel sensitivity measurements of near-infrared detectors by Barron et al. (2007), the intra-pixel sensitivity is characterized by a convolution of a box function with a width of the pixel size, p, and the lateral charge diffusion factor. We approximate the lateral charge diffusion factor as a two-dimensional Gaussian function for simplicity. Assuming that all pixels have the same intra-pixel sensitivity, we can give the intra-pixel sensitivity variation as

$$\begin{eqnarray}\xi ({x}_{6},{y}_{6}) & = & \displaystyle \sum _{k,l}\delta ({x}_{6}-{kp},{y}_{6}-{lp})\ast \left[\mathrm{exp}\left(-\displaystyle \frac{{x}^{2}+{y}^{2}}{{(0.1p)}^{2}}\right)\right.\\ & & \ast \quad \left.\left(\mathrm{rect}\left(\displaystyle \frac{x}{p}\right)\mathrm{rect}\left(\displaystyle \frac{y}{p}\right)\right)\right]({x}_{6},{y}_{6}),\end{eqnarray} \tag{ 28 }$$

where p and (k, l) represent the pixel pitch of the detector array and the two-dimensional pixel number, respectively, and the rect function is defined as follows:

$$\begin{eqnarray}\left\{\begin{array}{rcl}\mathrm{rect}(x) & = & 1\ \ (| x| \leqslant 1)\\ \mathrm{rect}(x) & = & 0\ \ (\mathrm{otherwise}).\end{array}\right.\end{eqnarray} \tag{ 29 }$$

By using Equations (28) and (29) the luminosity becomes

$$\begin{eqnarray}&&L({x}_{{\rm{c}}}+\delta {x}_{{\rm{c}}},{y}_{{\rm{c}}}+\delta {y}_{{\rm{c}}})=\displaystyle \sum _{k,l}{i}^{\prime }({kp}+\delta {x}_{{\rm{c}}},{lp}+\delta {y}_{{\rm{c}}}),\end{eqnarray} \tag{ 30 }$$

where

$$\begin{eqnarray}{i}^{\prime }({x}_{6},{y}_{6}) & = & i({x}_{6},{y}_{6})\ast \left[\mathrm{exp}\left(-\displaystyle \frac{{x}^{2}+{y}^{2}}{{(0.1p)}^{2}}\right)\right.\\ & & \ast \;\left.\left(\mathrm{rect}\left(\displaystyle \frac{x}{p}\right)\mathrm{rect}\left(\displaystyle \frac{y}{p}\right)\right)\right]({x}_{6},{y}_{6}).\end{eqnarray} \tag{ 31 }$$

When the diameter of the sub-pupil on the detector, $\tfrac{\eta D}{n}$, is much larger than the pixel pitch, p, the intensity profile of i'(x₆, y₆) is constant except for both edges of the sub-pupil. In other words, the intra-pixel sensitivity variation has little influence on the luminosity. Otherwise, the intensity profile of i'(x₆, y₆) is affected by the intra-pixel sensitivity. On the basis of the above considerations, the photometric variation caused by an image movement on a detector with the intra-pixel sensitivity variation, ΔL_intra, is finally described as

$$\begin{eqnarray}{\rm{\Delta }}{L}_{\mathrm{intra}} & \equiv & \displaystyle \frac{L({x}_{6}+\delta {x}_{6},{y}_{6}+\delta {y}_{6})-L({x}_{6},{y}_{6})}{L({x}_{6},{y}_{6})}\\ & = & \displaystyle \frac{{\displaystyle \sum }_{k,l}{i}^{\prime }({kp}+\delta x,{lp}+\delta y)}{{\displaystyle \sum }_{k,l}{i}^{\prime }({kp},{lp})}-1.\end{eqnarray} \tag{ 32 }$$

The above equation can be applied both to the photometric stability of this densified pupil spectroscopy and also to that of a conventional spectroscopy.

Next, we analytically describe the impact of an image movement on a detector with an inter-pixel sensitivity variation. Assuming that the travel distance of the image is pixel level, the luminosity of a sub-pupil can simply be written as a summation of the luminosity per pixel instead of an integration of pixels. When the sub-pupil extends across N pixels along the two axes, the luminosity of the sub-pupil positioned at pixel number of (l, m) is

$$\begin{eqnarray}&&{L}_{l,m}={i}_{\mathrm{pixel}}\displaystyle \sum _{l=-\displaystyle \frac{N}{2}}^{\displaystyle \frac{N}{2}}\;\displaystyle \sum _{m=-\displaystyle \frac{N}{2}}^{\displaystyle \frac{N}{2}}{\xi }_{l,m},\end{eqnarray} \tag{ 33 }$$

where i_pixel and ξ_{l, m} represent the luminosity per one pixel and the pixelized inter-pixel sensitivity variation described as a function of the pixel number, (l, m), respectively. The photometric variation limited by the image motion on the inter-pixel sensitivity variation, ΔL_inter, is

$$\begin{eqnarray}{\rm{\Delta }}{L}_{\mathrm{inter}} & \equiv & \displaystyle \frac{{L}_{l+\delta l,m+\delta m}-{L}_{l,m}}{{L}_{l,m}}\\ & \simeq & \displaystyle \frac{{\displaystyle \sum }_{l=-\displaystyle \frac{N}{2}}^{\displaystyle \frac{N}{2}}{\displaystyle \sum }_{m=-\displaystyle \frac{N}{2}}^{\displaystyle \frac{N}{2}}({\xi }_{l+\delta l,m+\delta m}-{\xi }_{l,m})}{{N}^{4}}.\end{eqnarray} \tag{ 34 }$$

Using the operating variance of Equation (34), $V({\rm{\Delta }}{L}_{\mathrm{inter}})$, the standard deviation of the photometric stability is calculated as

$$\begin{eqnarray}&&{\sigma }_{\mathrm{inter}}\equiv \sqrt{V({\rm{\Delta }}{L}_{\mathrm{inter}})}=\displaystyle \frac{\sqrt{2{N}_{\mathrm{motion}}}{\sigma }_{\xi }}{{N}^{2}},\end{eqnarray} \tag{ 35 }$$

where N_motion, and σ_ξ are, respectively, the number of pixels newly irradiated or not newly irradiated by the image motion and the standard deviation of the inter-pixel sensitivity variation. The above equation can be extended to the case of sub-pixel movement of the image and then N_motion becomes not an integer but a real number. As N increases, the impact of the inter-pixel sensitivity on the photometric stability decreases.

In conclusion, the photometric stability limited by an image motion on the detector due to the telescope pointing jitter can be characterized by the sampling number of the sub-pupil, N, the number of the pixels newly irradiated or not newly irradiated by the image motion, N_motion, and the inter-pixel sensitivity variation. The size of the sub-pupil on the detector should be much higher than the pixel pitch, p, in order to suppress the instrumental systematic noises caused by the image motion on the intra- and the inter-pixel sensitivity variations. In contrast, the signal from the transiting planet is dominated by the detector noise such as dark current and readout noise. The sampling number of the sub-pupil on the detector should be optimized by a balance between the instrumental systematic noise and the detector noise.

In this section, we present an optimal design for the pupil densified spectroscopy concept. The diameter of the telescope primary is set to 2.5 m. The observation wavelength range is set in the range 10–20 μm for observation of thermal emissions from cooler planets less than 1000 K. The spectral resolution at 10 μm is 100 for measurement of the atmospheric compositions. Owing to the relation between the sampling number and the instrumental systematic errors, the optical system is designed such that the sampling number of the spectrally resolved sub-pupil is eight on the detector. The overall view of the optical system combined with the telescope is shown in Figure 5. The field of view is restricted with an aperture mask at the telescope's focal point, as described in Section 2.5. Given that the pointing jitter of the telescope is 0.1 arcsec rms in total, which approximately corresponds to requirement on the pointing stabilities of SPICA³ , the radius of the aperture mask is set to 10 arcsec for achieving the high photometric stability down to 10 ppm based on Equation (25). Based on the number density of the galactic stars shown in Figure 4, the observable region for this system is limited to high galactic latitude of $| b| \gt 30$. The beam diameter collimated by the first two lenses after the telescope focus, D, is 10 mm. The collimated beam is divided into 20 sub-pupils by a mask positioned on the output pupil. The figure of the mask is same as that shown in Figure 1. After the two microlens arrays densify each sub-pupil with a densification factor of 10, densified sub-pupils with a diameter of 200 μm are formed on the output pupil, corresponding to the entrance slit of the spectrometer. The light beam of one sub-pupil positioned at the center of the entrance slit is drawn in the spectrometer of Figure 4. After the beam collimated by the two lenses with an focal length of 75 mm is reflected by a reflection grating with 20 cycles per 1 mm, the five lenses with an effective focal length of 75 mm form the spectrum of the sub-pupil on the detector with a pixel format of 1000 × 1000 and a pixel pitch of 25 μm. The detector applied to this system is a Si:As Impurity Band Conductor array for the mid-infrared wavelength range. The total of the optical throughput, including in the detector quantum efficiency and the grating efficiency, is set to 30% for the latter calculation. Figure 6 shows the footprint on the detector plane. Figure 7 shows the spot diagrams on the detector plane for various positions and wavelengths of the sub-pupils. All of the lenses in this optical system are designed such that the image enlargements due to the aberrations for all sub-pupils at 10–20 μm are much smaller than the size of each sub-pupil formed on the detector plane. As a result, the spectra acquired on the detector are expected to be stabilized against pointing jitter.

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On the basis on the derived analytical expression on photometric stability, we estimate the photometric stability of this optical system on a cryogenic telescope cooled to less than 10 K, in which the thermal background from the telescope is negligible. Then, we compare the estimated performance with the depth of the secondary eclipse of a temperate super-Earth with a radius two times that of Earth around various spectral types of stars. We also consider transmission spectroscopy of the super-Earth with a small atmospheric scale height like Earth. Table 1 shows the average and worst values of the image motions on the detector in the case of a telescope pointing jitter of 0.1 arcsec rms in total. As predicted by the theory underlying this concept, the image motion on the detector due to the pointing jitter is extremely suppressed. Based on Equations (32) and (35), the photometric stability of this system can be calculated. However, the flat-fielding accuracy of the Si:As detector is unknown in the era of SPICA and CALISTO. Deming et al. (2009) applied to the flat-fielding error in Spitzer/IRAC of 0.4% to calculate the photometric accuracy in the Mid-infrared Instrument (MIRI) of the James Webb Space Telescope (JWST), assuming that the flat-fielding error in JWST/MIRI is not improved from the Spitzer Space Telescope. On the other hand, according to the Explanatory Supplement to the Wide-field Infrared Survey Exploler (WISE) all-sky data release products compiled by Cutri et al. (2013), the flat-fielding accuracy of the Si:As detector is 0.07%. In this paper, we use 0.07% as the flat-fielding error. Table 2 shows the mean and worst values of the instrumental systematic error caused by the pointing jitter in case of the flat-fielding accuracy of 0.07%. Since the systematic noise error due to the image motion on the intra-pixel sensitivity variation is down to a level of 10⁻⁸ due to the large sampling number of the image, the systematic noise error is dominated by the inter-pixel sensitivity variation. Thus, the photometric stability is better than 10 ppm.

Next, we compare the photometric stability at 10 μm with the photon noise, the detector noise, and exozodiacal light with the depths of the secondary eclipses of super-Earths around stars with various effective temperatures from 3200 to 5500 K. The parameters used for this calculation are compiled in Tables 3 and 4. In order to estimate the depths of the secondary eclipses around various type of stars, we apply a relation between the star radius and the effective temperature, which was measured by Boyajian et al. (2012) with a stellar interferometer, as shown in Table 4. The effective temperature of the super-Earth is fixed to 300 K. Given that the bond albedo of the planet is 0.3, the semimajor axis of the planet with an effective temperature of 300 K can be calculated. The transit durations of the planets around various type of stars are also estimated in the case of the impact parameter of b = 0.0 based on the following relationship:

$$\begin{eqnarray}&&{T}_{\mathrm{dur}}\simeq \displaystyle \frac{{{PR}}_{*}}{\pi a},\end{eqnarray} \tag{ 36 }$$

where P and a are the period and the semimajor axis of the planet, respectively, and R_* is the stellar radius. The semimajor axes and the transit durations of the planet with an effective temperature of 300 K around various types of stars are compiled in Table 4. The spectra of both the central star and the planet are taken to be blackbodies. The distance of this system is fixed at 10 pc because such a transiting super-Earth in the M dwarf habitable zone exists within about 10 pc at high confidence level, according to Dressing & Charboneau (2015). In terms of the detector noise, we refer to the measurement values of a Si:As detector for the MIRI of the JWST in Ressler et al. (2008). The dark current and the readout noise are set to 0.17e⁻ s⁻¹ pixel⁻¹ and 14e⁻, respectively. The exozodiacal lights around G- and A-type stars are recently modeled by Kennedy et al. (2015) for observations of the Large Binocular Telescope Interferometer based on the COBE/DIRBE observational results. According to this paper, the flux of the zodiacal light around a solar-type star is 100 μJy. The total integration time equals the multiplication of the transit duration and the number of the eclipses for three years, which is the mission lifetime of SPICA. Figure 8 compares this system performance with a requirement on 1σ detection of the secondary eclipse of a temperate super-Earth and its atmosphere of 20 km scale height through the transmission spectroscopy around various types of stars at 10 μm. We also consider the performance of the conventional spectrometer, which has 8 pixel sampling per one spectral channel in the same manner as the new system. Due to the high stability against the pointing jitter in this system, the photometric stability can be much improved from that of the conventional one. The systematic noise is also comparable with the random noise in this observation. Comparing to the depths of the secondary eclipses of the temperate super-Earths around various types of the stars, this system potentially provides us with an opportunity of spectral characterization of the emissions from the temperate super-Earths around nearby late-type stars. In addition, this system can resolve the atmospheric thickness of 20 km, which corresponds to the thicknesses of the ozone and the carbon dioxide layers in the Earth atmosphere (Kaltenegger & Traub 2009) through the transmission spectroscopy with a spectral resolution of 20. According to Barstow et al. (2016), since the terrestrial planets with Venus-type and Earth-type around a M5 star may be distinguished through the investigation of the presence or absence of additional absorption of the ozone molecules around 10 μm. However, because the frequency spectrum of the pointing jitter is unknown at this time, only the upper limit on the partial absorption of the target star with the field stop is derived. The final performance on the systematic noise error is expected to settle into the gray shaded area shown in Figure 8. We also note that because all systematic noises such as the detector gain variability and calibration error are not treated in this calculation, the above prediction on the science output is tentative. As the next step, we will investigate how to de-correlate the other systematic errors from the multiple spectra observed on the detector.

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