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We present a design methodology for volume hologram filters (VHFs) with telephoto objectives to improve contrast of solar–illuminated artificial satellites observed with a ground–based optical telescope and camera system operating in daytime. VHFs provide the ability to selectively suppress incoming light based on the range to the source, and are used to suppress the daylight background noise since signal (satellite) and noise (daylight scatterers) are located at different altitudes. We derive the overall signal–to–noise ratio (SNR) enhancement as the system metric, and balance main design parameters over two key performance considerations – daylight attenuation and spectral bandwidth – to optimize the functioning of VHFs. Overall SNR enhancement of 7.5 has been achieved. Usage of multi–pixel cameras can potentially further refine this system.
© 2013 Optical Society of America
1. Introduction
Observing solar–illuminated artificial satellites with ground–based telescopes in daytime is challenging due to the usually bright sky background. The majority of the background noise comes from Rayleigh–scattered sunlight below an altitude of 30 kilometers [1, 2]. The objects of interest, artificial satellites, are at a minimum distance of 200 km away from the sensor. Thus, to enhance the signal–to–noise ratio (SNR) of satellite detection, our goal is to design a system which only images sources located at least 200 km from the sensor, but eliminates the light from nearby atmospheric scatterers. In this paper, we use volume holograms, a good candidate to provide the ability to selectively filter incoming light based on the range to the source.
Volume holograms are three–dimensional (3D) gratings where the thickness along the optical axis is not negligible, and they therefore exhibit Bragg–like diffraction [3, 4]. They have been utilized in numerous applications, including signal processing [3], information storage [5, 6], and imaging [4]. Typically, a volume hologram is recorded by the interference of two plane waves, signal and reference beams, at a certain angle, resulting a 3D periodic refractive index distribution, i.e. a grating. When the hologram is probed by the same signal beam, it is Bragg–matched and the identical reference beam can be readout. However, when probed by different beams, Bragg condition is no longer satisfied thus the diffraction efficiency is dramatically reduced, which gives rise to superior signal selectivity [4]. Combined with imaging lenses, volume holograms can be adapted for depth–selective imaging, where only object located at certain depth satisfies the Bragg–matching condition thus is able to reach the detector. Other light sources only have negligible intensity passing through. This concept has been applied, for example, in microscopy to selectively imaging one or more planes at various depths inside a sample simultaneously [7].
Therefore, by including volume holograms in our satellite detection systems, we expect to be able to selectively image only the depth with artificial satellites (usually > 200 km), but not daylight (< 30 km). In this way, daylight noise is mitigated and satellite image contrast is enhanced.
This paper is organized as follows. Section 2 presents the detailed design of VHF systems incorporating telephoto objectives. Section 3 discusses two analyzing methods and relevant performance metrics, including two key design considerations: daylight noise attenuation and multispectral bandwidth. The overall SNR enhancement is also derived. Section 4 optimizes design parameters to achieve large SNR enhancement. Section 5 discusses the advantages of using multi–pixel cameras instead of simple single–element power meters.
2. VHF system design
Our design of the VHF system is illustrated in Fig. 1. The signal beam is from an object at the altitude of satellites, i.e. the sunlight reflected from the satellites, and the reference beam is at an angle θs with respect to the signal beam. The recorded hologram is then probed to produce an image. If the probe beam is from the object at the same altitude, the hologram is Bragg–matched and provides the maximum diffraction efficiency. Otherwise, for atmospheric scatterers, the beam after the objective lens is no longer a plane wave and is Bragg mis–matched, resulting in a reduction of diffraction intensity at the detector. In this way, the VHF provides depth selectivity based on the distance of the probe source (longitudinal detuning), and the intensity of the sky background is reduced.
Fig. 1 VHF design setup. As an example, this VHF is assumed to be used for Iridium satellite detection. Inset: a typical diffraction efficiency plot with respect to longitudinal defocus δ.
A theoretical equation for calculating the diffraction efficiency of a VHF at various longitudinal defocus positions was derived in Ref. [8], yielding
where a is the aperture radius, L is the thickness of the hologram, n is the refractive index of the hologram, λf is the recording wavelength, f is the focal length of the front lens, δ is the longitudinal defocus of the probe source along the optical axis with respect to the position of recording signal source, ρ and ϕ are the radial and angular integration variables in the unit circle domain of integration, respectively, and Id is the diffracted optical intensity. The diffracted intensity Id has its peak value I0 when no longitudinal defocus exists, i.e. at Bragg–matched readout δ = 0. When the probe source point moves away from the Bragg–matched position, readout efficiency is reduced. A typical plot relating diffraction efficiency and defocus δ is shown in the inset of Fig. 1. From this we could characterize the full–width at half–maximum (FWHM) of longitudinal defocus as [8] where G is a constant. To better eliminate daylight, the diffraction efficiency should be minimized. This means that the defocus of daylight scatterers δ, i.e. the distance between signal satellite and atmospheric scatterers, should be at the very least greater than ΔzFWHM. However, notice that ΔzFWHM increases proportional to the square of lens focal length f. Since the satellites to be detected are at least 200 km away from the first lens, f2 term results in a ΔzFWHM value much larger than δ (which is only comparable to f). Thus, this simple VHF geometry involving a single objective lens will not perform adequately.In order for the VHF to function, we need to minimize ΔzFWHM by decreasing the lens focal length f, i.e. the distance between the satellites and the first collecting lens (see Fig. 1). However, since the VHF has to be on the ground, and satellites have fixed orbits, the imaging distance could only be “effectively” reduced. A telephoto objective [8, 9] is a good candidate for this purpose.
A typical telephoto is comprised of two lenses, one positive and the other negative. As can be seen from Fig. 2, a telephoto effectively reduces the working distance from the front focal length (FFL = f) to the effective focal length (EFL), according to
where a is the aperture radius of the front lens, and r is the effective aperture radius. With such a telephoto as objective, the FWHM of the longitudinal defocus becomes [8] Comparing with Eq. (2), effectively ΔzFWHM has been reduced by a factor of EFL/FFL. For example, when the focal lengths of these two lenses are chosen as 2.5 m and −2.5 mm, ΔzFWHM is reduced to 1/1000 of its original value with only a single lens objective. Therefore, longitudinal defocus of daylight scatterers δ is able to be larger than ΔzFWHM, effectively mitigating the daylight noise on the detector.Fig. 2 (top) VHF setup using a telephoto as objective, and (bottom) its effective configuration. PP1: first principal plane.
3. Analyzing methods and performance metrics
Without loss of generality, in this paper we assume that the VHF system is designed for detection of Iridium satellites which are located at an altitude of 780 km. To explicitly calculate the daylight rejection, two methods are used. One is based on an analytical solution shown in Eq. (1), which is similar to Eq. (40) of Ref. [8]. The other uses ZEMAX® to calculate the defocused wavefronts illuminated on the volume hologram, and MATLAB® to simulate the recording/probing process [10]. The ZEMAX model used here is the same as the VHF setup shown in the top part of Fig. 2. From now on, we refer to these two methods as “Analytical” and “MATLAB+ZEMAX” method, respectively. The “Analytical” method is faster but more approximate, while the “MATLAB+ZEMAX” method yields more accurate results at lager computational cost. Though not shown here, we have verified that both methods give consistent results, when the longitudinal defocus is small enough so that most of the diffracted intensity is incident upon the finite–size detector modeled in the MATLAB+ZEMAX method.
3.1. Daylight attenuation
Equipped with these two methods, we are now able to calculate the amount of attenuation of daylight in the VHF setup for satellite detection with a telephoto objective. Again here we assume that the majority of the daylight is scattered from the atmosphere no higher than 30 km above the ground [1]. Diffraction efficiency of atmospheric scatterers is illustrated in Fig. 3, where we normalized the efficiency to the Bragg–matched readout with a probe beam from Iridium satellites. The results from the two methods do not match but they both show large attenuation. The daylight noise level has been lowered to at least 0.17 of its original value. The MATLAB+ZEMAX method even shows that 99% of the daylight has been eliminated. The discrepancy between these two sets of results is mainly due to finite detector size used. When probed by atmospheric scatterers, the longitudinal defocus is so large that the image at the detector is no longer a focused spot. Instead, the image is so spread–out that a large portion of the diffracted intensity does not even reach the detector. The analytical approach takes all the diffracted intensity into account while the MATLAB+ZEMAX method only sums over the intensity incident upon the detector.
Fig. 3 Diffraction efficiency of atmospheric scatterers (daylight) for our VHF setup, calculated using (a) analytical method and (b) MATLAB+ZEMAX method. The following parameters are used: λf = 632.8 nm, L = 1.0 mm, a = 1.0 m, θs = 5°, f1 = 2.5 m, f2 = −2.5 mm, FFL = 780 km, EFL = 780 m. Note that these parameters are generic and have not yet been optimized.
3.2. Multispectral bandwidth
Besides longitudinal detuning, volume holograms also act as spectral filters, since the diffraction efficiency decreases when the probe beam wavelength deviates from the recording wavelength [11]. In our design, the hologram is recorded by a single wavelength; however, the sunlight reflected from the satellites and the daylight are both broadband sources, ranging from ultra–violet (UV) to infrared (IR). Among all the probe wavelengths, only certain combinations of probe angle and wavelength are Bragg–matched, resulting in reduced diffraction efficiency.
To characterize the relationship between diffraction efficiency and probe wavelengths, we again use the two methods mentioned above, except that now we are scanning through wavelength instead of longitudinal defocus, and the equation used in the analytical method should be derived as
where μ = λp/λf, the ratio between probe and recording wavelengths. The multispectral performance of the VHF setup shown in Fig. 3 has been calculated and plotted in Fig. 4. Note again that the difference between these two plots is due to the finite size of the detector discussed above. It can be observed from the figure that the FWHM of bandwidth around the recording wavelength λf is only 0.22λf when no longitudinal defocus exists, which means that the majority of probe wavelengths are Bragg–mismatched and the readout efficiency of the signal probe beam is very low. Therefore, multispectral performance should be seriously considered in our design in order to achieve satisfactory performance. This multispectral bandwidth problem is perfectly appropriate for noise (daytime sky scatterers) attenuation; however, it also reduces the readout efficiency of the signal (sunlight reflected from artificial satellites). From Fig. 4, we also notice that larger longitudinal defocus corresponds to wider bandwidth. For atmospheric daylight scatterers with altitudes smaller than 30 km, the multispectral curve is almost flat (and low).Fig. 4 Multispectral performance of the VHF system for probe source at different altitudes, calculated using (a) analytical method and (b) MATLAB+ZEMAX method, for the VHF setup shown in Fig. 3.
3.3. Overall SNR enhancement
The SNR enhancement is the metric we use to characterize the performance of the overall VHF system. We assume that all other noise sources, such as electronic noise, as well as the received power from the satellite are negligible compared with the received power from the sky background . Without the VHF system, the SNR across a small bandwidth centered at wavelength λ is
where h is the Planck constant and c is the light velocity. With the VHF system, the satellite probe signal and daylight noise have diffraction efficiency of ηs and ηn, respectively. The photons of daylight hitting the detector may be modeled as a Poisson process whose standard deviation is a square–root of the expectation value. Therefore, the amount of noise photons has been reduced to ηn of its original value, and the noise level is reduced by . The SNR of the VHF system is then given by Therefore, readout efficiency of signal ηs has to be larger than daylight noise in order to make the VHF system increase SNR versus a conventional imaging system. The potential reduction in signal efficiency due to the spectral selectivity of the VHF therefore becomes a serious design issue. We thus need to design a VHF system with the largest bandwidth possible.To be more precise, other important factors need to be taken into consideration, such as the spectra of daylight and sunlight. Now we proceed to derive the explicit expressions of ηs and ηn. First, without loss of generality, we assume that the detector has uniform sensitivity and is only sensitive along the visible spectrum. For detectors with other sensitivity performance, the only modification we need is to multiply the sensitivity function during all the integrations below.
The multispectral performance of the diffraction efficiency (Id/I0) of the VHF system, denoted here as qsatellite(λ), has been illustrated in Fig. 4. The spectrum of the signal beam from the artificial satellites is similar to the spectrum of sunlight, since satellites directly reflect the light from the sun. We denote it as psun(λ), plotted in Fig. 5(c). The overall diffraction efficiency for signal (satellite) is then calculated as
Calculation of daylight noise efficiency is more complicated. First of all, we have demonstrated in Fig. 4 that the multispectral curve for daylight noise scatterers is almost flat. Thus, it makes sense to assume that the multispectral response of the VHF does not contribute to the daylight attenuation, leaving only longitudinal defocus to consider. Another interesting phenomenon to consider is the varying daylight radiance from atmospheric scatterers at different altitudes, denoted as rdaylight(z). As illustrated in Fig. 5(a), the radiance decays exponentially for increasing altitudes of scatterers. Also, the contribution of scatterers higher than 30 km is negligible. If we use qdaylight(z) to symbolize the diffraction efficiency of daylight due to longitudinal detuning (see Fig. 3 for example), the overall diffraction efficiency for noise (daylight) is then calculated as
To be more precise, we should remove the assumption of constant spectral performance for noise (see Fig. 4); and the overall diffraction efficiency becomesFig. 5 (a) Radiance of daylight scattering at different altitudes, (b) spectral radiance of the sky background at ground level, and (c) solar spectral irradiance. All figures are calculated using MODTRAN 4 [12] assuming 23 km ground visibility and rural extinction haze model. The angle was 10 degrees east of zenith at 3:00 PM local time on June 21 at 45 degrees latitude (mid–latitude summer atmospheric model).
4. Design optimization
In the previous section, we discussed two main design considerations: daylight attenuation and multispectral bandwidth, together with the expression of the VHF system performance metric: SNR enhancement. Now we proceed to relate key design parameters to the system performance, and design an optimized system.
The key design parameters for our system have been listed in Table 1, together with their relationship to our two main performance considerations. This relationship was derived from Eqs. (1) and (5), and has also been verified with the MATLAB+ZEMAX method. It can be seen that two parameters, aperture radius and EFL, are unrelated to bandwidth; while the other four parameters have exactly opposite influence on daylight attenuation and bandwidth. Therefore, we need to find a balance depending on our design requirements.
Table 1. Requirements on design parameters for better system performance.
(a) Aperture radius and EFL
We start from two parameters unrelated to the multispectral performance to achieve better daylight attenuation. From Table 1, a larger aperture radius and smaller EFL are needed. Smaller EFL means a smaller r/a value since the FFL, i.e. altitude of satellites, is fixed. Thus, considering practical implementation, we chose the following parameters a = 1.0 m, r = 1.0 mm, f1 = 2.5 m, f2 = −2.5 mm, FFL = 780 km, EFL = 780 m, where the working distance has been increased 1000 times without deteriorating the depth selectivity.
(b) Wavelength and hologram refractive index
These two parameters are limited by the availability of laser sources and materials, and thus are actually not free to adjust. The sky background light in daytime is bluer than the solar spectrum reflected from satellites (see Fig. 5(b)&(c)). In order to achieve better daylight mitigation, the VHF should be operated at longer wavelengths. Thus, we use red light with wavelength λf = 632.8 nm. As for the selection of hologram material, the hologram should not be erasable at readout wavelength so here we use phenanthrenquinone–doped poly(methyl methacrylate) (PQ–PMMA), a material that is particularly sensitive to wavelengths between 450 nm and 500 nm [13, 14]. A two–lambda approach can be used to record the hologram at 488 nm but actually designed for readout at 632.8 nm [14, 15]. At readout wavelength, the refractive index of PQ–PMMA is 1.49.
(c) Hologram thickness and recording angle
These two parameters should be optimized to achieve a large system SNR. Using Eqs. (8) and (10), diffraction efficiency of signal and noise, as well as the overall system SNR enhancement have been calculated with respect to different hologram thickness and recording angle, illustrated in Fig. 6. Results show that θs should be small enough for large SNR. On the other hand, it should be large enough for easy system assembly. We here choose θs = 5°. At this recording angle, the largest overall SNR is realized when hologram thickness is L = 0.5 mm. For this optimized setup, ηs = 0.74, ηn = 0.11, and resulting overall system SNR enhancement is 2.2.
Fig. 6 (a) Signal diffraction efficiency, (b) noise diffraction efficiency, and (c) total system SNR enhancement with respect to hologram thicknesses and recording angles.
The above SNR enhancement assumes that the detector is infinitely large. If we assume that the size of the detector is 1 mm–by–1 mm, using MATLAB+ZEMAX approach, we now have ηs = 0.74, ηn = 0.0097, and total SNR enhancement . This value can be potentially increased by implementing multi–pixel cameras (see Sec. 5).
5. Discussions on multi–pixel cameras
All the above discussion assumes that the detector is a simple single–element power meter, where only the readout intensity or power (i.e. a number) is measured. However, an area detector with multiple pixels, such as a CCD, could be used to better facilitate our judgment of whether a true signal or a false alarm has been detected, and to estimate the distance to the satellite. The number of pixels used per detector is based on many practical issues, such as size of CCD, average power hitting each pixel, detector noise, etc. In this section, we discuss two advantages of using multi–pixel cameras.
5.1. Increased SNR
When the volume hologram is probed by a signal beam from a satellite, the majority of the irradiance covers several adjacent pixels (see bottom right plot of Fig. 7(b)) of the multi–pixel camera; when probed by daylight, the diffracted intensity is more uniformly distributed on all the pixels. The noise variance of a measurement over a group of pixels increases approximately linearly with the number of pixels, and for each pixel, the individual noise level should be divided by the number of pixels. In this application the signal levels are high enough that CCD readout noise should always be negligible. Therefore, for the measurement of small signal and background levels the SNR is assumed to increase by the square root of the number of pixels. Thus the actual noise is much reduced compared with a single power meter detector. In this way better SNR is achieved.
Fig. 7 (a) Diffraction efficiency for satellites at different altitudes, when the system is solely designed for detection of Iridium satellites. Diffraction efficiency (Id/I0) is normalized to the readout intensity when the hologram is probed by Iridium satellites. (b) Turbulence–free point spread functions at multi–pixel cameras for satellites at different orbit heights. From left to right, top to bottom: Sputnik-1 (215 km), International Space Station (340 km), Hubble Space Telescope (595 km), and Iridium (780 km). Color shading denotes the normalized intensity. Note that different axes are used for these four pattern plots.
5.2. Estimating the distance to the target satellite
The VHF system is designed, without loss of generality, for detection of Iridium satellites located in orbit with altitude of 780 km. The exact same setup can also be used for detecting other satellites at other altitudes. The diffraction efficiency with respect to satellites at different orbit heights has been plotted in Fig 7(a). It can be observed that for satellites higher than Iridium, especially those in medium earth orbit (MEO) and high earth orbit (HEO), the readout efficiency can be more than 98%. However, for low earth orbit (LEO) satellites, the readout efficiency is not satisfactory. But in these cases, multi–pixel cameras can help. Multi–pixel cameras produce an image of the defocused VHF readout rather than just a simple readout intensity. Satellites from different orbits produce patterns with different sizes and shapes. In Fig. 7(b), we have plotted the diffraction patterns for four different satellite distances. From the different defocus images, the distance to the satellite can be estimated, by template–matching, for example.
6. Conclusion
This study presents volume hologram filter designs to mitigate the daytime sky background noise in artificial satellites detection and imaging, by utilizing the longitudinal depth selectivity of volume holograms. We use a telephoto objective to enhance the working distance, adapting this system for observing objects at distances on the order of hundreds of kilometers. Key design parameters have been optimized to achieve an overall SNR enhancement of 7.5, with larger daylight attenuation and wider spectral bandwidth. Advantages of introducing multi–pixel cameras have also been discussed.
Acknowledgments
This work is sponsored by the Assistant Secretary of Defense for Research and Engineering under Air Force Contract #FA8721-05-C-0002. Opinions, interpretations, conclusions and recommendations are those of the authors and are not necessarily endorsed by the United States Government. Hanhong Gao and Professor George Barbastathis also receive support from Singapore’s National Research Foundation through the Singapore–MIT Alliance for Research and Technology (SMART) Centre.
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