## Introduction

For the past two decades, the phenomena of extreme light guiding, bending and dispersing based on the principle of diffraction has attracted much attention. Examples are sharp angle light-bending at one wavelength by a photonic crystal structure1, ultra sensitive light-dispersion in a photonic crystal structure2 and 90-degree light-bending in a volume hologram3,4. Furthermore, diffractive optical elements (DOEs) can achieve more complex functionalities than traditional refractive optical elements, such as lenses, prisms or aspheres, could. Also, DOE is much lighter in their mass density. Hence, DOEs offer an appealing solution for the construction of next-generation, ultra-large > 20 m space telescope primaries5,6,7,8,9,10. However, diffractive optical elements may produce a large chromatism when illuminated with a broad-bandwidth white light source. Nonetheless, the high chromatism of a DOE may be advantageous for spectroscopic applications. As an example, a novel telescope, Dual Use Exoplanet Telescope (DUET)11, was designed to directly analyze spectra of an exoplanet with extremely high spectral resolution based on diffractive optics. DUET can simultaneously perform high resolution spectroscopy and has a low areal mass for space exploration. Figure 1 shows a schematic diagram of such an exoplanet telescope. The main component of the telescope is an annulus gossamer membrane holographic primary objective with large collection area, yet its mass and stowage allow it to be delivered on a single lifter. In the inset, we show a photo of a diffracted image of sunlight when passing through our Fresnel hologram objective.

In this paper, we show a successful design and construction of a Fresnel hologram to focus light in the transverse direction (x–y plane) and disperse it in a linear fashion along the optical axis (z-axis). Previously, a holographic optical element (HOE) has been proposed to function as lenses for optical imaging purpose and was modeled using ray tracing method12. In this study, our Fresnel hologram is designed to function not only as a focusing but also a dispersive element along an optical axis. Additionally, the computational method we used is based on wave-optics principle. While typical FZP (Fresnel Zone Plate)13 is used for beam focusing of a specific wavelength, our hologram is designed to dispersed broad wavelengths over a long distance on axis. This Fresnel hologram constitutes the primary diffractive optics in a DUET system10,11. The main function of the Fresnel hologram is to collect the collimated star-light source and focus it into the detector. When a broad-band light source passes through an annulus diffractive optic, the incident light with different wavelengths will be focused into different positions along the optical axis. It is called dispersive behavior and can be applied for high resolution spectroscopy applications. In this paper, we will focus our effort on designing the annulus Fresnel hologram, realizing this hologram and demonstrating its unusually large chromatic behavior.

## Design and simulation results of the Fresnel hologram

Our Fresnel hologram is based on the interference of two spherical waves, emerging from point sources UR (Reference) and UO (Object), on a hologram film located at z = 0 plane. Note that a typical Fresnel zone plate is produced using a plane-wave and a spherical wave, which may be viewed as a special case of our two spherical-waves configuration. Figure 2a shows a schematic diagram of the two-wave recording geometry. The reconstruction of the hologram is achieved by placing another point source UP at the same side of UR and UO (z < 0) and then searching for point image formation at the other side of the hologram, i.e. z > 0. The recording wavelength, λi, of UR and UO may be different from the reading one of UP, λ. When the propagating waves from UR, UO and UP satisfy the paraxial approximation14, formation of a point image can occur near or on the optical axis. Also, noted that the focal length of the point image is expected to be λ-dependent.

In this paper, we consider two special conditions: (1) both UR and UO are placed on the z-axis, leading to an on-axis hologram with cylindrical symmetry and annular shape of the interference pattern; (2) UP is placed quasi-infinitely away from the hologram, so it has a planar wave-front. Under this configuration, our hologram can image (or concentrate) a collimated, single λ beam onto a point on the axis. It can also image a broadband, collimated beam onto a series of points along the axis. Traditionally, when a broadband radiation of a given intensity is dispersed in space, it gains angular spread of λ but at the expense of losing its intensity flux. On the contrary, for our Fresnel hologram, a broadband radiation is dispersed along the optical axis but each wavelength of light is focus on its own focal point. This is an important feature of our hologram, leading to both super λ-resolution and super sensitive detection for an exoplanet telescope.

We now present a brief derivation of the theoretical framework of image formation. A more detail account of it is discussed in the Method section. From Fig. 2a, the reference wave UR is on (xR, yR, zR) and the objective wave UO on (xO, yO, zO). The two recording waves propagate along the optical axis and interfere at z = 0 plate. The intensity distribution of the interference pattern of the two spherical waves can be written as:

$$I\left( {x,y,z} \right) = \left| {U_{R} + U{}_{O}} \right|^{2} \approx \left| {A_{{\text{R}}} \frac{{e^{{ik\sqrt {(x - x_{R} )^{2} + (y - y_{R} )^{2} + \left( {z - z_{R} } \right)^{2} } }} }}{{\left| {z_{R} } \right|}} + A_{{\text{O}}} \frac{{e^{{ik\sqrt {(x - x_{o} )^{2} + (y - y_{o} )^{2} + \left( {z - z_{O} } \right)^{2} } }} }}{{\left| {z_{O} } \right|}}} \right|^{2}$$
(1)

Here, AR and AO are the amplitude of the reference and objective beams, respectively. k is the recording beam wave vector, k = 2π/λ and λi is its wavelength. Under the Paraxial approximation. UR and UO may be re-written as :

\begin{aligned} U_{R} & = \left( {A_{R} /z_{R} } \right)e^{{ik\sqrt {(x - x_{R} )^{2} + (y - y_{R} )^{2} + (z - z_{R} )^{2} } }} \approx \left( {A_{R} /z_{R} } \right)e^{{ik\phi_{R} }} \\ U_{O} & = \left( {A_{O} /z_{O} } \right)e^{{ik\sqrt {(x - x_{O} )^{2} + (y - y_{O} )^{2} + (z - z_{O} )^{2} } }} \approx \left( {A_{O} /z_{O} } \right)e^{{ik\phi_{O} }} \\ \phi_{R} & \equiv - \frac{\pi }{{\lambda_{i} z_{R} }}\left( {x^{2} + y^{2} - 2xx_{R} - 2yy_{R} } \right) \\ \phi_{O} & \equiv - \frac{\pi }{{\lambda_{i} z_{O} }}\left( {x^{2} + y^{2} - 2xx_{O} - 2yy_{O} } \right) \\ \end{aligned}
(2)

To simplify Eq. (1) and also to achieve the maximum contrast of the interference pattern, the amplitude of the reference and objective beams are set to AR/|zR| = AO/|zO| = 1. Thus, the intensity distribution of the interference pattern becomes:

$$I\left( {x,y,z} \right) = \left| {e^{{ik\sqrt {x^{2} + y^{2} + \left( {z - z_{R} } \right)^{2} } }} + e^{{ik\sqrt {x^{2} + y^{2} + \left( {z - z_{O} } \right)^{2} } }} } \right|^{2}$$
(3)

To illustrate the annular feature of the hologram, we compute the intensity distribution of the interference pattern using Eq. (3). The results are shown in the inset of Fig. 2b. Here, λi = 515 nm and the two spherical waves are located at zR = − 72 cm and zO = − 25.2 cm, respectively. The interference pattern at the origin (x, y) = (0, 0) is shown on the upper left over an area 2 × 2 mm2. The blue color represents the minimum intensity and the yellow the maximum intensity. It exhibits an annular-like grating structure, consistent with the axial symmetry of the recording and reading configuration. Here the radius of first blue circle is 380 μm and the grating pitch between the first and second blue circle is Λ = 260 μm. Additionally, the interference patterns at (x, y) = (30 mm, 0) and (50 mm, 0) are shown on the lower right corner over an area 50 × 100 µm2. They both display a shorter grating pitch. The grating pitch is reduced to Λ = 6.7 µm at (30 mm, 0) and further reduced to Λ = 4 µm at (50 mm, 0). Theoretically, the grating pitch of the intensity distribution pattern is a function of the radial position r = (x2 + y2)1/2. Figure 2b plots the grating pitch as a function r as the blue color curve. The grating pitch decreases rapidly as r is increased. For instance, grating pitch decreases from Λ = 201, 20, 6.7 to 4 µm as the radius is increased from r = 1, 10, 30 to 50 mm, respectively. Note that the decreasing Λ from the origin to the outer regions serves to bring the incoming plane wave to focus much like the traditional Fresnel zone plate13. This provides for a mechanism for the hologram to strongly concentrate light to a “point” bounded only by the diffraction limit.

The focal-point of our hologram, i.e. (x3, y3, z3), may be derived analytically as well. Again, its full derivation is discussed in the Method section. Briefly, the derivation principle is to match the phase factor from the object (ϕO) and recording beam ϕR to the phase factor of the diffracted beam ϕ3. As a result, the focal point (x3, y3, z3) of the image is a function of zR, zO, λ, λi, and given by:

\begin{aligned} x_{3} & = y_{3} = 0; \\ \frac{1}{{z_{3} }} & = \left( {\frac{1}{{z_{R} }} - \frac{1}{{z_{O} }}} \right)\frac{\lambda }{{\lambda_{i} }} \, \;\;\;or\;\;\; \, z_{3} \lambda = \left( {\frac{1}{{z_{R} }} - \frac{1}{{z_{O} }}} \right)^{ - 1} \lambda_{i} \\ \end{aligned}
(4)

This derivation shows that, for our hologram design, the image is located on-axis (x3 = y3 = 0) and the product of (z3λ) is a constant. In our experimental case, when zR = − 72 cm, zO = − 25.2 cm, λi = 515 nm, (z3λ) = 19,966 cm-nm. Note that z3 is a positive number. This means that the diffracted output is focused onto the z3 position forming a real image. On contrary, if z3 is a negative number, the diffracted output becomes a divergent wave that originates from − |z3| position. Hence, both the recording parameters, zR and zO, provides for a new design degree-of-freedom for producing either a convergent or divergent diffracted beam. The combinational use of two spherical waves also allow us to design a large value of z3 (when zRzO), and thus can produce a highly dispersive optical element. The position z3 of the diffracted spherical wave, measured from the hologram plane, is called the focal-length of the hologram.

Because z3 is inversely proportional to the reading λ, our hologram has a significant chromatic dispersion property when illuminated with a broadband light source. When λ = λi = 515 nm, z30 = 38.8 cm and is the reference focal-length of the image. When $$\lambda \ne \lambda_{i}$$, the diffracted wave is focused onto a different position along z-axis such that $$z_{3} \ne z_{30}$$. For instance, when λ is varied from λ = 400 nm to 700 nm, the corresponding focal length is shifted from z3 = 49.9 cm to 28.5 cm. This result shows that the hologram can achieve a spectral dispersion of 400 nm to 700 nm wavelength over an on-axis distance of 21.4 cm. An important feature of our hologram is its ability to achieve a long tuning length for a super high spectral resolution. It originates from an unusual on-axis dispersive behavior of the Fresnel hologram.

To independently confirm the chromatic dispersion of our hologram, we use commercially available software, Zemax®15, to simulate the performance of our hologram. The simulated result is illustrated in Fig. 3. In this simulation, the hologram has a circular size with a diameter D = 10 cm. It is modeled in accordance with our recording parameters, i.e. the two point sources are at zR = − 72 cm and zO = − 25.2 cm and the recording λi = 515 nm. Also, its grating period varies as a function of r as shown in Fig. 2b. The illuminating light is chosen to be collimated and contains three different colors, i.e., λ = 600 nm (red), 515 nm (green) and 400 nm (blue). The result shows that the three colors of light are been focused onto three different spots along the z-axis. We obtain z3(λ = 600 nm) = 333 mm, z3(λ = 515 nm) = 389 mm, and z3(λ = 400 nm) = 500 mm. We found that the shift of the focal length versus λ is consistent with that predicted from Eq. (4). This result numerically demonstrates the unusually large chromatic dispersion produced by our hologram. Figure 3 also plots the intensity distribution of the three wavelengths at their respective focal plane and along the x-axis. The spatial intensity distributions for all three colors are sharp. We found the diameter of the focus points to be less than 5 µm. Although, we note that Zemax® is based on ray tracing method and the minimum focus spot should be set by the diffraction limit. Nonetheless, the intensity distribution plots illustrate the unusual light concentration capability of our hologram. This simulation shows that our hologram can achieve strong concentration of a single color and also on-axis stretching of multi-colors all at the same time.

## Experimental discussion

We have established an optical system to realize the Fresnel hologram and experimentally demonstrate the chromatic behavior of the hologram. Figure 4a shows a schematic diagram of the optical setup system. The laser we used for hologram recording is a collimated, diode-pumped-solid-state (DPSS) laser with a single longitudinal mode output. It emits light at λ = 515 nm and has a long coherent length of about one-meter. The laser light is first split into two beams, where one is the reference and the other the object beam. The half-wave (λ/2) plate is used to adjust the relative intensity of the reference and objective beams. Both beams are focused through 40X objective lenses and then passed through pinholes of 15 µm diameter. This procedure produces point light source and, therefore, the desirable propagating spherical waves. The pinholes for the reference and object beams are located at zR = − 72 cm and zO = − 25.2 cm away from the hologram plane, respectively. In order to record an on-axis hologram, an additional one-inch mirror is used to guide and align the object beam such that it propagates along the same optical axis as the reference beam. Finally, a Litiholo holographic film of 4" × 5" size is placed at the desired exposure position to record the hologram. Figure 4b shows a photo of the optical recording system for our Fresnel hologram. Note the size of a realizable hologram is limited by the power and finite coherent length of a laser. Using our current optical setup system, the hologram might be scaled up to about one-meter in diameter.

Once the hologram is recorded, a high resolution microscope is used to verify if the resultant interference pattern follows the design prediction shown in Fig. 2b. Figure 5a shows a schematic drawing of the hologram and coordinates where microscope images were taken. Figure 5b shows a photo of the 4" × 5" hologram film after been exposed to ~ 40 mJ/cm2 recording energy. Figure 5c–e show microscope images of the hologram at point r1 = (0, 25 mm), r2 = (40 mm, 0) and r3 = (50 mm, 0), respectively. In all three images, clear grating structures were observed that illustrate a successful interference of the reference and object beams. The grating period is systematically reduced from Λ ~ 9.1, 5.6 to 4.1 µm as the coordinates radius is increased from r = 25, 40 to 50 mm, respectively. The decreasing trend of Λ toward outward region of the hologram is consistent with the trend predicted in Fig. 2b. It is noted that the theoretical simulation predicts a grating period of Λ ~ 8.0 and 4.0 µm at r = 25 and 50 mm, respectively. There are two reasons that might cause a slight discrepancy between the observed and predicted grating period. One is due to a non-ideal optical alignment between the two recording beams. For instance, from our calculation, if the two beams are not perfectly aligned along the same direction and off by 0.1°, it will result in a deviation of grating period by ~ 1.1 µm at r = 25 mm. Conversely, it may be stated that the upper bound of our alignment error is within 0.2°. The other is a possible measurement error of finding the exact central position of the almost transparent hologram under the microscope. Nonetheless, the observation of clear grating structures and the agreement of grating periods between experiment and theory throughout the hologram film indicate that we have successfully recorded a Fresnel hologram.

The recorded hologram is expected to function simultaneously as a chromatic and a focusing element. Figure 6 experimentally illustrates the chromatic behavior of the hologram. A white LED light (covering λ = 450–650 nm spectral range) is used to illuminate the hologram. The LED light serves as a point source, which is expanded and collimated using a Cassegrain reflecting telescope16 in a reverse input configuration and normally incident onto the hologram. The transmitted wave is then diffracted and imaged onto different positions on the z-axis in accordance to their specific λ. Figure 6a shows a photo of the reconstructed image of the hologram. Firstly, the white LED light is well focused and dispersed along the optical axis, where a ruler is carefully placed near the axis. The finite spread of light along the y-direction is due to a slight mis-alignment of the ruler to the z-axis and also the scattering of light by the ruler. The actual spot size is much smaller and is defined by the diffraction limit of the hologram size, see Fig. 3b. Secondly, the white LED light is dispersed into a series of vivid colors, ranging from red, orange, yellow, green, blue and purple. The observation of bright color (i.e. strong intensity) for a dispersed light is in sharp contrast to those of typical dispersive elements, such as a prism, where light is dispersed, spread in all three dimensions and its intensity flux greatly weakened. Finally, consistent with the prediction of Eq. (8), the shorter wavelength (λ) light has a longer focal point (z3).

While the concentration capability of our hologram is characterized by a tight spatial extend of the focused spot in the transverse plane (xy) shown in Fig. 3b, its chromatic behavior is quantified by a long spatial extend along the longitudinal axis (z-axis). Here we perform a spectroscopic study of the spectral resolution of the focused light at a series of focal points along the z-axis. A fiber coupled optical spectrometer (Ocean Optics USB4000) with a fiber core diameter of d = 400 µm is utilized to measure the spectrum. The fiber, shown in the far left of Fig. 6a, is to be translated along the z-axis to perform spectroscopic measurement. Figure 6b shows a series of measured spectra taken at a range of z-position from z = 28 cm to 47 cm. Firstly, the measured spectra show stronger intensities at λ ~ 455 nm (blue color) and λ ~ 550 nm (yellow) region. This is due to spectral distribution of the primary and secondary emission of the LED light used in our testing17. Secondly, the measured spectra displayed well-defined peak wavelength, which are shifted systematically toward longer-λ for shorter z. The full-width-half-maximum (FWHM) of the individual peak is ΔλFWHM ~ 7.5 nm at λ ~ 450 nm and increased by about two times to ΔλFWHM ~ 15 nm at λ ~ 650 nm. The relatively large FWHM is mainly due to the finite size of the fiber core diameter d = 400 µm. It does not represent the intrinsic spectral resolution of the hologram, which is to be addressed in a following paragraph.

Thirdly, the measured peak is shifted from λ = 440 nm to 705 nm as the focal length is decreased from z = 470 mm to 280 mm. So, white light of Δλ = 265 nm bandwidth is well dispersed into a spatial extend of Δz = 190 mm. Figure 7b summarizes the measured (red dots) and predicted (blue curve) focal length as a function of λ. The measured focal length shows a decreasing trend as λ is increased. Recall from Eq. (4) that $$z_{3} \lambda = z_{30} \lambda_{i}$$, or

$$z_{3} \left( \lambda \right) = \left( {\frac{{z_{30} }}{\lambda }} \right)\lambda_{i}$$
(5)

where $$z_{30} = \left( {{1 \mathord{\left/ {\vphantom {1 {z_{R} }}} \right. \kern-\nulldelimiterspace} {z_{R} }} - {1 \mathord{\left/ {\vphantom {1 {z_{o} }}} \right. \kern-\nulldelimiterspace} {z_{o} }}} \right)^{ - 1}$$. So, from Eq. (5), the chromatic behavior, z3(λ), of our hologram is characterized by the pre-factor "z30". When "z30" is increased by recording design, the incoming starlight will be more dispersed and the hologram is more suited for achieving super-resolution purpose. When "z30" is reduced, the diffracted λ of the incoming starlight is more compressed (focused) along the z-axis and the hologram is more suited for high sensitivity application. Returning to experimental data, when the reading-λ equals the recording λi = 515 nm, z3 = z30 = 388 mm (the reference). The measured focal length value is z30 = 385 mm, which is in good agreement with the predicted result. Note also that z3 should follow the (1/λ) dependence. And, indeed, our measured focal length accurately reproduces the predicted values (the blue curve).

In the following, we discuss spectral resolution and starlight detection sensitivity of our Fresnel hologram. The spectral resolution Δλ of our hologram is determined by the detection length Δz of a detector of choice along the z-axis. Let us assume that Δz << z30 = 388 mm and rewrite Eq. (5) in a differential form, we find: $$\Delta \lambda \cong - \left( {\frac{\Delta z}{{z_{30} }}} \right)\left[ {\frac{{\lambda^{2} }}{{\lambda_{i} }}} \right]$$. The formula indicates that the spectral resolution depends on the intrinsic property of the hologram (λiz30), the detection length Δz, as well as λ2. This explains our earlier observation in Fig. 6b that the spectral linewidth ΔλFWHM is wider for longer-λ peaks. Also, when λ is increased from 450 to 650 nm, the theory predicts a (650/450)2 ~ 2 × increase of linewidth which agrees with the observation shown in Fig. 5b. As a specific example, when λ = λi = 515 nm and z30 = 388 mm, we have $$\Delta \lambda \cong - \Delta z \times \left( {{{\lambda_{i} } \mathord{\left/ {\vphantom {{\lambda_{i} } {z_{30} }}} \right. \kern-\nulldelimiterspace} {z_{30} }}} \right) = - \Delta z \times 1.33 \times 10^{ - 3}$$ nm/µm. When Δz = 50 µm, 5 µm, and 0.5 µm, the spectral resolution of the hologram is |Δλ| = 0.0665, 0.00665 and 0.000665 nm, respectively. And, the corresponding spectral resolvance (λλ) is 7.74 × 103, 7.74 × 104, and 7.74 × 105, respectively. A resolvance of 7740 to 77,400 is in the medium to high resolution range and is a good range for determining stellar properties and radial velocities of stars in the Milky Way galaxy. A resolvance of 774,000 is in the super-resolution range for the detection of exoplanets by the radial velocity method. Noted that, to extract a selective bandwidth of light over a length Δz, a secondary optics can be used to diffract light away from the z-axis for an easy and full spectral analysis. A schematic of the approach by a diffracted optical element is illustrated in the inset of Fig. 7a.

Alternatively, a fiber or detector may be placed facing the z to collect starlight, such as shown in the photo of Fig. 6a. In this case, the minimum detector size d is set by the diameter of the Airy disk13 of our Fresnel hologram, i.e., d ≥ dAiry. For our Fresnel hologram, its diameter is D = 10 cm and the corresponding diameter of Airy disk is $$d_{Airy} = 2.44 \times \frac{{\lambda z_{3} }}{D} = 2.44 \times \frac{{\lambda_{i} z_{30} }}{D} = 4.8$$ µm. Interestingly, since dAiry depends on intrinsic parameters such as λi and z30 and D, its value remains the same for all wavelengths or it is Δλ-independent. Additionally, if we increase the diameter of our hologram to D' = 1 m, the corresponding Airy disk is reduced to $$d^{\prime}_{Airy} = 2.44 \times \frac{{\lambda_{i} z_{30} }}{{D^{\prime}}} = \frac{{1.22\lambda_{i} }}{N.A.}\sim 1.22\lambda_{i} \approx 0.5$$ µm. Here, the N.A. (Numerical Aperture) of holographic lens is close to 1 and the recording wavelength λi is 405 nm. It is this diffraction limit that determines the minimum d of a photodetector to be used and, therefore, the minimum Δz for starlight collection.

According to the geometry shown in Fig. 7a, the amount of light collected over a diameter d (≡ Δy) is equivalent to those light distributed over a length Δz along the z axis. If the light incident angle is θ, we have $$\Delta z = \frac{\Delta y}{{2 \times \tan \theta }}$$. Here, $$\theta = \tan^{ - 1} \left( {\frac{{\text{R}}}{{z_{30} }}} \right) = \tan^{ - 1} \left( {\frac{20}{{388}}} \right) = 3.0^{o}$$. We may now re-examine the experimental configuration and spectral linewidth shown in Fig. 6a,b, respectively. Since the fiber has a diameter d = Δy = 400 μm, the corresponding detecting length $$\Delta z = \frac{\Delta y}{{2 \times \tan \theta }} = \frac{{400 \times 10^{ - 3} }}{2 \times 0.05} = 4$$ mm. By Eq. (5), the predicted spectral linewidth $$\left| {\Delta \lambda } \right| \cong \Delta z \times 1.33 \times 10^{ - 3} \;{{{\text{nm}}} \mathord{\left/ {\vphantom {{{\text{nm}}} {\mu {\text{m}}}}} \right. \kern-\nulldelimiterspace} {\mu {\text{m}}}} \cong 5.3\;{\text{nm}}$$ at z30 = 388 mm. This finding agrees with the measured results shown in Fig. 6b within ~ 2 nm.

Separately, the Fresnel hologram may be utilized for a sensitive starlight detection. We will estimate the minimum detectable power of a micro-PMT detector and then examine the star light flux of alpha Lyrae, the 5th brightest star in the night sky. A typical micro-PMT has a noise level Vnoise = 6 mV for a detector area A = 3 × 3mm218. Its detectivity is such that a 1pW will lead to a signal level of Vsignal = 15 mV. This gives a signal-to-noise ratio of 15 mV/6 mV = 2.5. Let us assume that this is an acceptable S/N ratio and 1pW is the minimum detectable power of such a PMT detector. For our hologram detecting system, incoming light may be concentrated to Adet ~ 500 × 500 μm2, and so the PMT detector size may be reduced accordingly. Given that a detector's noise voltage, such as the Johnson and shot noises, is proportional to the square root of the optically active area19, its noise voltage becomes V′noise = 1 mV when its area is reduced by a factor 36 times to Adet ~ 500 × 500 µm2. So, the minimum detectable voltage is also reduced and given by V′signal = V′noise × 2.5 = 2.5 mV. The minimum detectable power of a reduced area micro-PMT is $${{1\;{\text{pW}}} \mathord{\left/ {\vphantom {{1\;{\text{pW}}} {\sqrt {36} }}} \right. \kern-\nulldelimiterspace} {\sqrt {36} }} \approx 1.6 \times 10^{ - 13} \;{\text{W}}$$. If we further assume that our 10 × 10 cm2 Fresnel hologram as a 30% diffraction efficiency, the minimum detectable power of a reduced area PMT is $${\text{P}}_{\min } \approx 5.3 \times 10^{ - 13} \;{\text{W}}$$.

We now examine the star light flux. A star with apparent magnitude V = 0 and zero color which is approximately the apparent magnitude and color of alpha Lyrae, the 5th brightest star in the night sky, has a flux of 37.4 × 10–12 W/m2/nm20,21 over the wavelength range of the V filter. The V filter has a spectral width of 88 nm and is centered at λ = 545 nm. Therefore, for a hologram that is 10 × 10 cm2, the flux of a 0th magnitude stars with zero color is 37.4 × 10–14 W/nm. From Eq. (5), the spectral width for a detector size of d = 500 µm and Δz = 5 mm is Δλ = 6.65 nm. So, the starlight power is (37.4 × 10–14 W/nm) × (6.65 nm) = 2.48 × 10–12 W. It is 4.5 times larger than the minimum detectable power of our Fresnel hologram with 30% diffraction efficiency. As a result, the Fresnel hologram holds great promise for a sensitive detection for space exoplanet telescope.

## Conclusion

In summary, we present the design of an unique annular Fresnel hologram based on the interference of two spherical waves. Through an analytical design, we found that the focal-length of our hologram can be tailored by tuning the radius-of-curvature of the two spherical waves. The grating period of the Fresnel hologram decreases monotonically from the center to the outer region and brings the incoming star-light to focus. Experimentally, we have successfully recorded our Fresnel hologram based on spherically symmetrical waves, with a focal-length by design. Furthermore, we have demonstrated a large on-axis dispersive behavior of an incoming, collimated light wave. This work represents the first successful implementation of a scale model optical system for realizing a telescope Fresnel hologram, which can achieve a spectral dispersion of λ = 440–705 nm over an on-axis distance of 190 mm. Therefore, our proposed DUET can simultaneously perform high resolution spectroscopy, high detection sensitivity and have a low areal mass for space exploration.

## Methods

### Reconstruction of the on-axis hologram with two spherical recording waves

Let us consider a plane wave UP illuminating the hologram at normal incidence and with a wavelength λ, Fig. 2a. The hologram is recorded using two spherical waves of wavelength λi, placed at zR and zO, respectively. The recorded, constructive interference regions attenuate the illuminating (reading) light, so the transmittance of the hologram22,23 is:

$$t_{A} (x,y,z = 0) = t_{0} - \kappa \times I\left( {x,y,z = 0} \right)$$
(6)

Here, t0 and κ are both constants and their values dependent on the holographic material chosen. Accordingly, the transmitted wave propagating through the hologram, Ut(x, y, z = 0), is given by:

\begin{aligned} U_{t} (x,y,z = 0) & = U_{P} \times t_{A} (x,y,z = 0) = U_{P} \times [t_{0} - \kappa \times I(x,y)] \\ & = U_{P} \times [t_{0} - \kappa \left| {U_{R} + U_{O} } \right|^{2} ] \\ & = U_{P} (t_{0} - \kappa \left| {U_{R} } \right|^{2} ) - \kappa U_{P} \left| {U_{O} } \right|^{2} - \kappa U_{P} U_{R}^{*} U_{O} - \kappa U_{P} U_{R} U_{O}^{*} \\ & = U_{1} + U_{2} + U_{3} + U_{4} \\ \end{aligned}
(7)

Here, U1 and U2 are both constants of (x, y). They are plane waves that directly transmit through the hologram. U3 and U4 are complex conjugate terms. They represent the two diffracted spherical waves. U3 is a real image and U4 a virtual image. U3 may be further computed to be:

\begin{aligned} U_{3} (x,y,z = 0) & = - \kappa U_{P} U_{R}^{*} U_{O} = - \kappa A_{P} A_{R}^{*} A_{O} e^{{i\left( {\phi_{A} + \phi_{O} - \phi_{R} } \right)}} = A_{3} e^{{i\phi_{3} }} \\ \phi_{3} \equiv \phi_{P} + \phi_{O} - \phi_{R} & = - \frac{\pi }{\lambda }\left[ {\left( {x^{2} + y^{2} } \right)\left( {\frac{1}{{z_{O} }}\frac{\lambda }{{\lambda_{i} }} - \frac{1}{{z_{R} }}\frac{\lambda }{{\lambda_{i} }}} \right) - 2x\left( {\frac{{x_{O} }}{{z_{O} }}\frac{\lambda }{{\lambda_{i} }} - \frac{{x_{R} }}{{z_{R} }}\frac{\lambda }{{\lambda_{i} }}} \right) - 2y\left( {\frac{{y_{O} }}{{z_{O} }}\frac{\lambda }{{\lambda_{i} }} - \frac{{y_{R} }}{{z_{R} }}\frac{\lambda }{{\lambda_{i} }}} \right)} \right] \\ \end{aligned}
(8)

When (xR, yR) = (0, 0) and (xO, yO) = (0, 0), ϕ3 can be simplified as:

$$\therefore \phi_{3} \equiv \phi_{P} + \phi_{O} - \phi_{R} = - \frac{\pi }{\lambda }\left[ {\left( {x^{2} + y^{2} } \right)\left( {\frac{1}{{z_{O} }}\frac{\lambda }{{\lambda_{i} }} - \frac{1}{{z_{R} }}\frac{\lambda }{{\lambda_{i} }}} \right)} \right]$$
(9)

Let us demand that the wave represented by U3 is to produce a point image at z > 0, so it must be a spherical wave originated from a point source at (x3, y3, z3). Similar to Eq. (2), U3 may also be written as:

\begin{aligned} U_{3} & = \left( {A_{3} /z_{3} } \right)e^{{ik\sqrt {(x - x_{3} )^{2} + (y - y_{3} )^{2} + (z - z_{3} )^{2} } }} \approx \left( {A_{3} /z_{3} } \right)e^{{ik\phi_{3} }} \\ \phi_{3} & \equiv - \frac{\pi }{{\lambda z_{3} }}\left( {x^{2} + y^{2} - 2xx_{3} - 2yy_{3} } \right) \\ \end{aligned}
(10)

For the first order diffraction, we match and equate the phase factors ϕ3, between Eqs. (9) and (10). As a result, the focal point (x3, y3, z3) of the image is given by:

$$\begin{gathered} x_{3} = y_{3} = 0; \hfill \\ \frac{1}{{z_{3} }} = \left( {\frac{1}{{z_{R} }} - \frac{1}{{z_{O} }}} \right)\frac{\lambda }{{\lambda_{i} }}{\text{ or }}z_{3} \lambda = \left( {\frac{1}{{z_{R} }} - \frac{1}{{z_{O} }}} \right)^{ - 1} \lambda_{i} \hfill \\ \end{gathered}$$
(11)