Calibration chain design based on integrating sphere transfer radiometer for SI-traceable on-orbit spectral radiometric calibration and its uncertainty analysis^*

In the purposes of establishing the SI-traceable on-orbit calibration transfer chain based on spectral radiation standard for satellite remote sensors, CIOMP consults some mature technology for transfer radiometers used in other metrology research institutions, such as SXR (sea WiFS transfer radiometer)^[10,11] designed by NASA, and PTR (polarize transfer radiometer) designed by ESA.^[12] CIOMP developed an ISTR system with an innovative transfer technology that is suitable for the absolute radiometric calibration taking the Sun as an invariable source.^[13] The ISTR system takes ISTR as the core equipment and it integrates the functions of integral sphere light source and spectral standard calibration transfer. It also achieves the conversion from cryogenic radiometer benchmark to radiance standard with the assistance of FBM, and it realizes the goal of calibrating and measuring other on-orbit remote sensors. The components of the system are shown in Fig. 1.

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ISTR is composed of three components, an integral sphere shell with a dual-aperture field stop, a filter radiometer (FR), and a photoelectric measurement module (PMM). Therefore, it can work as a secondary standard^[14] or a filter spectral radiance working standard. ISTR can achieve the traceability from trap detectors to the SI-traceable cryogenic radiometer benchmark through using FBM output narrow-band monochromatic light whose power is calibrated by the cryogenic on-orbit radiometer.^[15] ISTR can also complete the inner calibration for filter radiometer. Coupling with the dual-aperture field stop, it can implement the conversion between power and radiance. Through this conversion, ISTR can accomplish the calibration standard transferring from the above reference standard to payloads such as imaging spectrometers. The whole process is shown in Fig. 2. In the following parts of this paper, the structure, parameters, and implementation process for every component of the ISTR system are introduced in detail.

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The function of the FBM in the ISTR system is outputting monochromatic light whose power is calibrated by a cryogenic radiometer through moving the bended fiber on the rail. It can transfer the power benchmark to ISTR to achieve the radiance standard calibration by FBM. The transfer accuracy is mainly limited by the low power level of the monochromatic light when using sunlight as the spectral on-orbit radiometric calibration source. Currently, most monochromators adopt the light dispersing methods of grating or interferometry, which can obtain high-quality monochromatic light with an accurate central wavelength and an extremely narrow bandwidth.^[6] However, in consideration of the limitation of the minimum detectable power of the cryogenic radiometer and the measurement accuracy of 0.5%, the output power of FBM should be larger than 1 μW. Meanwhile, the output power scales between FBM and SDP should be as similar as possible for improving the measurement accuracy of the radiance reflected by SDP. Therefore, FBM uses the double-grating type, which is suitable for an on-orbit ISTR system, to get monochromatic light with an accurate central wavelength and a narrow bandwidth but lower power. To increase the output power, it adopts gratings with high efficiency, parabolic mirrors with high reflectance, and fibers with high coupling efficiency.

FBM is composed of three parts, a pre-convergent parabolic mirror used to collect the incident solar light, the double-grating monochromator for light dispersing, and a fiber system to export the monochromatic light. The monochromator adopts four off-axis parabolic mirrors and two gratings, the optical configuration is central symmetry about the central slit, as shown in Fig. 3. It has three channels with the same optical path but different gratings to cover the range of 320–640 nm, 640–1250 nm, and 1250–2500 nm with corresponding resolution of 1 nm, 2 nm, and 6 nm, respectively. The pre-convergent off-axis parabolic mirror uses Schott Zerodur as a base with the surface coating AlSiO. The core diameter and outer diameter of the optical fiber are 550 μm and 600 μm, respectively. The fiber length is 1 m and the fiber numerical aperture is NA = 0.22. FBM is characterized by a narrow bandwidth with high system transmittance and satisfies the demand of minimum light power larger than 1 μW for the high-accuracy measurement.

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Considering the solar light is not ideally collimated (with a divergence angle of 0.5°), the pre-convergent mirror is designed to image at the entrance of the monochromator and fill the slit, then we can obtain

where Φ(λ)_enter is the spectral power entered into FBM, E(λ)_Sun is the incident sunlight spectral irradiance at the central wavelength λ. Through diameter D and focus f_pre of the pre-convergent parabolic mirror, we can know that the first Airy diameter D_Airy is lower than 0.02 mm which is narrower than the slit width w = 0.2 mm at the wavelength range from 300 nm to 2500 nm, l = 4 mm is the slit length. Thus diffraction effects can be ignored and we obtain

where Φ_out represents the spectral output power of FBM at the central wavelength λ with the bandwidth Δλ, r is the reflectance of the m parabolic mirrors in FBM, ρ is the diffraction efficiency of the two gratings, τ_c1 is the fiber coupling efficiency,^[16] τ_c2 is the fiber transmittance, and τ is the transmittance of the convergent system.

The spectral irradiance of the Sun distributes irregularly in the range of 0.3–2.5 μm, E(λ)_Sun at 15 typical central wavelengths of the monochromator as shown in Table 1, where the minimum incident spectral power of the Sun in the range of 0.3–1.1 μm is E₃₀₀ = 0.351 W·m⁻²·nm⁻¹ and in the range of 1.0–2.5 μm is E₂₅₀₀ =0.052 W·m⁻²·nm⁻¹. Synthesizing the properties of every channel component in FBM, the values of the parameters in Eq. (3) are as follows: m = 4, r = 0.9, τ = 0.9, ρ = 0.2, τ_c1 = 0.5, τ_c2 = 0.85, Δλ = 1–6 nm, D = 100 mm. Then, Φ_out is in the range of 1.5–13.5 μW as shown in Fig. 4. The output power of channel 1 (central wavelength is 300 nm) is minimum but still more than 1 μW, which satisfies the measuring dynamic range (1 μW–1 mW) of the high-responsivity cavity in the cryogenic radiometer under 50 K temperature. It can ensure the uncertainty of the FBM output power less than 0.08%.

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ISTR receives the monochromatic light output from FBM and then the power primary standard traced to cryogenic radiometer is converted to radiance working standard by using a dual-aperture field stop. Relying on the Lambertian property of the integral sphere, the uniform output radiance of ISTR is

where Φ_out is the FBM output light power and A_s is the entire sphere inner surface with reflectance ρ. The port fraction f is given by

where A_i is the sphere input port area and A_e is the sum area of all exits. To acquire the steady output condition of smaller reflection numbers for better special uniformity, lower ρ and larger f are necessary. Considering the requirement of the export light power and the spatial uniformity of ISTR,^[17] a larger sphere diameter with smaller port fraction will improve the radiance spatial uniformity performance. Therefore, ISTR diameter D > 50 mm, the requirement for all the channels is ρ > 0.94, f < 0.05.

Combining the above analysis of the parameters with the actual limitation of the payload in space, the final design of ISTR adopts the sphere diameter of 80 mm with the input port diameter of 10 mm. A field stop composed of two precision apertures is used as a power-to-radiance standard converter in front of the input port when the light fills the field stop. The two apertures are 20 mm apart with diameters of 16 mm and 10 mm respectively. The equation of the standard conversion process is

where ϕ_i(λ) is the received spectral power of the input port at the wavelength λ, and N_S(λ) is the radiance of the source under test. The diameter of the aperture in the field stop near the light source is d₁ = 16 mm, the other one close to the input port is d₂ = 10 mm, and the distance between them is l = 20 mm. According to Eq. (6), the spectral radiance responsivity to the source under test is R_SN(λ) which can be expressed by the spectral power responsivity R_Sϕ(λ) as follows:

where R_Sϕ(λ) can be calibrated by the FBM output monochromatic light power which has been calibrated. To achieve the transfer from power benchmark to radiance standard with high accuracy, the area precision of each aperture needs to meet the demand of 0.03%.

ISTR has two exit ports with the same diameters of 6 mm, which are connected with an FR and a PMM, respectively. For improving the spatial uniformity of the output radiance, a baffle is mounted between the incident plane and the exit ports. The baffle can prevent the first reflected light outputting from ISTR directly and increase the reflection number of the incident light in ISTR. The inner surface of ISTR is coated with Spectralon^@ material and its reflectance is greater than 0.99 in the range of 400–1500 nm and greater than 0.95 in the range of 250–2500 nm. The schematic of the ISTR configuration is shown in Fig. 5.

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PMM is mounted in front of one of the integral sphere exit ports, it is composed of Silicon and InGaAs trap detectors and a signal control circuit.^[18] Going through the integrating sphere, the FBM output spectral power turns to a uniform weak radiance signal. Then, the trap detector receives it and works as a secondary standard traced to the cryogenic radiometer primary power standard to calibrate the FR. The incident spectral power exported from the integral sphere into the trap detector is ϕ_det(λ)

where E_det(λ) is the spectral irradiance incident on a trap detector whose active area A_det = 2.0 mm×2.4 mm. 2ω = 10° is the field of view from detector to the exit port, α is the corresponding solid angle, and L_sphere(λ) is the export radiance of the integral sphere. As the optical fiber diameter is 3 mm and NA = 0.22, FBM output light can propagate completely into ISTR. Combining ISTR's relative parameters with Eqs. (3)–(5), and (8), we can obtain the value of ϕ_det(λ) ranging from 0.3 nw to 2.6 nw corresponding to the wavelength of 300–2500 nm. Since the calibration uncertainty of the ISTR system is 0.3%, the noise equivalent power of the detector needs to be lower than 10⁻¹³ W scale. While, the noise power P_n (condition of B = 1000 Hz) is given as

The weak radiation signal is easy to be submerged in the noise if received by PMM directly, especially in the near infrared spectrum of 1200–2500 nm. Because the primary mission of on-orbit ISTR is to measure the radiance reflected by SDP with high precision, it needs to keep the power used to calibrate the detector similar to the power measured by the detector. The spectral power entering ISTR directly from SDP will be only 10⁻¹⁰ W scale due to the lack of concentrating light like FBM, thus, we cannot simply enhance the output power of FBM to solve the signal measuring problem above.^[19] A light modulating system (for both PMM and FR) should be introduced to make the spectral radiance measurement accuracy 0.3%.

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The light modulator locates in front of PPM and carries out high-accuracy electronic measurement with digital phase-sensitive detection technique, the whole control principle is shown in Fig. 6. With this additional measuring system, B in Eq. (9) reduces to 0.1 Hz and P_n depresses to 10⁻¹⁴ W scale. Comparing with the 10⁻¹⁰-W incident spectral power into PMM from FBM or SDP, it brings only 10⁻⁴ scale to the measurement uncertainty. Because the maximum spectral power ϕ_det(λ) is about 10⁻⁹ W scale, it allows the ISTR system to have a dynamic range of 10⁵. While the light modulator system has frequency drift caused by the rotational error of the chopper wheel, it will make a minor change to the light power incident on PMM. This change is the major uncertainty source of this measuring system and it is analyzed in detail as follows.

The input signal is shaped to be a rectangular wave through the SR830 with a high fidelity sine wave as the reference signal. Then, the synthetic source does phase lock based on the reference signal, which is

where R(t) is the synthetic reference signal with the modulation frequency ω₀, and Δω is the frequency drift caused by the rotational error of the chopper wheel, and V_S is the amplitude of the reference signal.

This measuring method modulates the input optical signal with the chopper's frequency and make it able to be expressed in the following rectangular wave:

where V_s is the amplitude of the input optical signal, ω₀ is the modulation frequency, and θ is the equivalent phase corresponding to the input light position. Multiplying X(t) with the reference signal R(t), we obtain

where U_out is the multiplier output. When U_out passes through the low-pass filter, the sum-frequency parts of the first item and the difference frequency parts with n > 1 of the second item in Eq. (12) are all filtered out. Therefore, the final output signal is

In practice, the value of phase θ is constant, the measuring accuracy of is mainly affected by the frequency drift Δω and phase-locked amplifier time t. Because Δω and t are both small amounts, the measuring relative uncertainty of can be approximated as

For the light modulator system above, the light chopper selects a three-hole cutting disc and the frequency is set to be 100 Hz with peak-to-peak phase accuracy of 0.1%. It means that there is a 0.2π-phase error generated per second. Depending on the 0.2-s amplifier time, the relative uncertainty δU calculated from Eq. (14) is 0.26%. Combined with the 10⁻⁴ scale uncertainty introduced by the noise signal of the detector itself, the whole measuring uncertainty of the PMM with the digital phase-sensitive detection technique is about 0.28%.

The filter radiometer will be utilized to measure the spectral radiance emitted from SDP in space and is the main task of the ISTR system operating on-orbit. Through measuring a series of spectral power at different central wavelengths with a narrow bandwidth by FR, the curve of the relative spectral response function (SRF) can be calculated by a smooth interpolation fitting method.^[20] According to the predictability of relative spectral responsivity, the calibration for the whole solar reflective spectrum could be accomplished. FR is connected with the other exit port of the integral sphere and receives the uniform output radiance. FR finally selects 15 spectral channels with interference filters of OptoSigma-VPF series, covering 300–1000 nm, 1000–1600 nm, and 1600–2500 nm with corresponding spectral resolution of 3 nm, 5 nm, and 10 nm in sequence. FR uses the same Silicon trap detector with PMM for the 11 spectral channels covering 300–1100 nm spectrum under 300 K working temperature, and the same InGaAs trap detector for the channels over 1100–2500 nm under cryogenic environment of 240 K controlled by the mechanical refrigerator. The distribution of these 15 channels is determined upon the smoothness and complexity of the SRF curve of the Sun (shown in Fig. 4). The spectral radiance responsivity of every channel is calibrated by the secondary standard PMM. The calibration process is expressed as follows:

where the value of is calibrated by cryogenic radiometer and represented as ϕ_FBM(λ₀) below. S_sFBM (Δλ,λ₀) is the output signal (unit: V) of the PMM response to the spectral channel at wavelength λ₀ with bandwidth Δλ, R_sϕ(Δλ,λ₀) is the average responsivity to the spectral power of the(λ₀, Δλ) channel by the chain consisted of the integral sphere and PMM. As Δλ is small for FBM, R_sϕ(Δλ,λ₀) can be represented approximately by R_sϕ(λ₀) and the traceable result is

During the shutdown period of cryogenic radiometer, the process of using PMM to calibrate FR is

In Eq. (15), ϕ_i(λ₀) represents the output spectral power of FBM(λ₀, Δλ) channel uncalibrated by a cryogenic radiometer, S_si(Δλ,λ₀) is the output signal (V) of the secondary standard trap detector in PPM response to ϕ_i(λ₀). In Eq. (18), S_fi(Δλ,λ₀) represents the output signal (V) of FR response to ϕ_i(λ₀), and R_fϕ(λ₀) is the average responsivity of the chain composed of the integral sphere and FR to the spectral power of the(λ₀, Δλ) channel. Since the two exit ports of the integral sphere are positional symmetric to the incident plane with the same diameter of 6 mm,R_fϕ(λ₀) just has one more influencing factor introduced by the filters than R_sϕ(λ₀). However, the bandwidth of FBM is narrower than that of the FR, hence in the calibration process, just the transmission of the filters contributes to the result of calibration, which means that the effect introduced is linear. Therefore, the power response coefficient of the FR can be obtained by R_sϕ(λ) calibrated by the SI-traceable ϕ_FBM(λ).

Combining Eq. (17) with Eq. (7), we can obtain the converted radiance response coefficient

As the stability of the detectors in the cryogenic environment is extremely high and the precision of the uniformity of the integral sphere output radiance can be 0.1%, the accuracy of the radiometric standard transfer can reach to the level of 0.3% in the traceability calibration process above. The FR will measure the spectral radiance emitted from the SDP through the digital phase sensitive detection technology described in Section 2.4. The main parameters and technical indicators of the whole system are summarized in Table 2.