1. Introduction

In solar cell characterization, the short-circuit current $I_{\mathrm {sc}}$ is a fundamental quantity to be measured. Highly accurate $I_{\mathrm {sc}}$ values are a crucial part of solar cell calibrations, efficiency calculations and research. The spectral irradiance distribution of the incident radiation on the solar cell influences the generated $I_{\mathrm {sc}}$ and, thus, is a key parameter for accurate measurements. If the spectral irradiance distribution differs from the desired reference spectral irradiance $E^{\mathrm {ref}}_{\mathrm{\lambda}}{(\lambda )}$ distribution, a spectral mismatch correction factor (SMM) [1] must be applied to determine $I_{\mathrm {sc}}$ accurately. However, the determination of SMM is time-consuming and introduces further uncertainties. Therefore, it is highly favorable and essential to spectrally match the generated spectral irradiance to $E^{\mathrm {ref}}_{\mathrm{\lambda}}{(\lambda )}$ to reduce measurement effort and to achieve highly accurate measurements of $I_{\mathrm {sc}}$.

In solar cell characterization, the tabulated reference solar spectral distribution AM1.5g [2] is used as $E^{\mathrm {ref}}_{\mathrm{\lambda}}{(\lambda )} := E_{\mathrm{\lambda} }^{\mathrm {AM1.5g}}(\lambda )$. Even optimized solar simulators introduce SMM values resulting from spectral deviations between the generated simulator spectral irradiance $E^{\mathrm {sim}}_{\mathrm{\lambda}} {(\lambda )}$ and the AM1.5g, due to the limitations of artificial radiation sources (for example xenon arc or tungsten halogen lamps). Several works have addressed the problem of spectral mismatch by developing setups with a spectral shaping capability, where the spectral irradiance $E^{\mathrm {sim}}_{\mathrm{\lambda}} {(\lambda )}$ distribution is adjustable by combining several radiation sources [3]. Different LEDs with a bandwidth of approximately 40 nm to several hundreds nanometers have been combined to approach the reference spectral irradiance, but a SMM determination was still required [4,5].

Promising results have been achieved by using a supercontinuum laser as a radiation source and a Grating Light Valve (GLV) spatial light modulator as a spectral shaping tool [6,7]. The spectral irradiance $E^{\mathrm {sim}}_{\mathrm{\lambda}} {(\lambda )}$ has been adapted to the AM1.5g in the wavelength range from approximately 460 nm to 1200 nm [6] or from 400 nm to 1750 nm [7]. Due to the restrictions of the supercontinuum laser, a lack of radiation below 400 nm was observed. However, silicon (Si) solar cells can have a spectral responsivity from 300 nm to 1200 nm, thus a spectral mismatch is still present and further improvements are needed. In [6,7], the generated radiant power in the measurement plane (irradiance multiplied with the irradiated area) would be too low to achieve the required standard condition of an irradiance of 1000 W/m${^2}$ [2] for large solar cells with a size of approximately 400 cm${^2}$. Nevertheless, in [6,8,9] it has been shown that an accurate determination of the short-circuit current $I_{\mathrm {sc}}$ is feasible by measuring the slope of the short-circuit current versus irradiance relation. In this method, a steady bias radiation with an irradiance of up to 1000 W/m${^2}$ but with a less demanding spectral distribution and an additional chopped radiation with $E^{\mathrm {sim}}_{\mathrm{\lambda}} {(\lambda )}$ is incident to the solar cell [6,9]. It is required and beneficial that the irradiance of the chopped radiation is significantly lower than the irradiance of the steady bias radiation (in this work, the irradiance of the chopped radiation is approximately 0.1 W/m${^2}$). Thus, it is also highly promising to match $E^{\mathrm {sim}}_{\mathrm{\lambda}} {(\lambda )}$ to $E^{\mathrm {ref}}_{\mathrm{\lambda}} {(\lambda )}$ at a low irradiance level to determine accurate $I_{\mathrm {sc}}$ values of solar cells.

We chose a laser-driven light source (LDLS) as a radiation source and a GLV as a spectral shaping tool. The LDLS contains a laser which heats a xenon plasma generating radiation over the investigated wavelength range from 300 nm to 1200 nm. The GLV limits the lowest wavelength to 355 nm. In general, the irradiance in the measurement plane is proportional to the radiance of the radiation source and the etendue (also called throughput) of the optical system [10]. The radiance of the LDLS is approximately 1-3 orders of magnitudes higher compared to light sources like xenon arc or tungsten halogen lamps, but it is usually still orders of magnitudes lower than that from a supercontinuum laser. In this work, the etendue of the GLV is limiting for the whole optical system and is determined by the optically active area of the GLV multiplied with the solid angle from which radiation is accepted. Even though the required irradiance can be low, we must use the largest possible etendue of the GLV to achieve a sufficient signal-to-noise ratio in spectral measurements or in solar cell characterization. Consequently, we present our developed optical layout in this paper in detail.

To sum up, this paper presents an experimental setup, which is capable of highly resolved spectral shaping in the wavelength range from approximately 355 nm to 1200 nm. First, we describe the GLV and the optical layout in detail. After that, we develop an algorithm for calculating the suitable adjustments for the GLV, which controls the spectral irradiance $E^{\mathrm {sim}}_{\mathrm{\lambda}} {(\lambda )}$ of the generated chopped radiation. Next, we characterize the optical properties and discuss the spectral resolution of our setup. Finally, we demonstrate the highly resolved spectral shaping capability of our setup by matching the spectral irradiance $E^{\mathrm {sim}}_{\mathrm{\lambda}} {(\lambda )}$ to the AM1.5g reference solar spectral irradiance.

2. Setup for spectral shaping

Figure 1 shows a schematic drawing of our experimental setup for spectral shaping.A laser-driven light source from Energetiq Technology, Inc. generates radiation in the wavelength range from 300 nm to 1200 nm, which is chosen for its high spectral radiance and broadband radiation. A transmissive diffraction grating from Wasatch Photonics spatially separates the wavelengths of the radiation. An essential component of the presented setup is a Grating Light Valve (GLV) spatial light modulator from Silicon Light Machines which is able to control the spectral irradiance [11]. Additionally, the GLV was used to generate chopped radiation. Followed by mixing optics, the spectral irradiance $E^{\mathrm {sim}}_{\mathrm{\lambda}} {(\lambda )}$, incident to the measurement plane, is uniformly distributed.

 

Fig. 1. Schematic drawing of the experimental setup for spectral shaping. An essential component of the setup is a Grating Light Valve (GLV).

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2.1 Adjustable diffraction grating

Figure 2 shows a detailed look at the GLV and its mechanism of influencing the spectral power of the reflected radiation.The following specifications are taken from Ref. [12]. The GLV is approximately 27.2 mm wide, 100 $\mathrm{\mu}$m long, and consists of 1088 parallel pixels, which can be actuated individually. One GLV pixel is shown in Fig. 2(a). A GLV pixel has a width of 25 $\mathrm{\mu}$m and consists of 6 ribbons, each 3 $\mathrm{\mu}$m wide. A thin air gap of around 650 nm exists between the ribbons and the underlying substrate. An applied electrical potential between the underlying substrate and every second ribbon deflects the ribbons. As a result, a square-well diffraction grating is created with the grating groove density $n_{\mathrm {GLV}} \approx 120$ l/mm. The irradiance of the diffracted radiation into the $\pm$1$^{\textrm{st}}$ orders generated by pixel $i$ with $0 \leq i < 1088$ is described by [12]

 

Fig. 2. a) A drawing of the GLV taken from [11]. Deflected ribbons create a phase diffraction grating. b) Light path of 0$^{\textrm{th}}$ order and $\pm$1$^{\textrm{st}}$ orders generated by the GLV. Aperture P3 blocks $\pm$1$^{\textrm{st}}$ orders. The GLV controls the spectral power in 0$^{\textrm{th}}$ order.

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When all GLV pixels are deactivated, the GLV reflects all radiation into the 0$^{\textrm{th}}$ order with the maximal spectral irradiance $E^{\mathrm {bias,max,0 th}}_{\mathrm{\lambda}}{(\lambda )}$, and the GLV acts like a mirror as shown in Fig. 2(b). A lens L merges all wavelengths and the radiation passes an appropriate aperture P3. By activating GLV pixels, the GLV diffracts a part of the radiation into $\pm$1$^{\textrm{st}}$ orders with the spectral irradiance $E^{\mathrm {\pm 1 st}}_{\mathrm{\lambda}} {(\lambda )}$, which is blocked by P3 after being collected by L. Thus, the GLV attenuated the radiation of the 0$^{\textrm{th}}$ order. The spectral irradiance into the 0$^{\textrm{th}}$ order can then be written as

The rectangular function $\mathrm {rect}(f_0)$ takes the values 0 and 1, and models the feature of our GLV to switch quickly between $E^{\mathrm {bias,max,0 th}}_{\mathrm{\lambda}}{(\lambda )}$ and $E^{\mathrm {0 th}}_{\mathrm{\lambda}} {(\lambda )}$ with a frequency $f_0$ from 30.8 mHz up to 13.5 kHz. In solar cell characterization, we will determine the current response of a solar cell with the lock-in technique, which is able to recover a signal with a defined frequency. Signals with other frequencies are suppressed. We will take advantage of this GLV feature and, thus, the lock-in amplifier signal $R_{\mathrm {lock-in}}$ will be proportional to the spectral irradiance $E^{\mathrm {\pm 1 st}}_{\mathrm{\lambda}} {(\lambda )}$, which is modulated with $\mathrm {rect}(f_0)$:

The width of aperture P3 is matched to the beam width in the 0$^{\textrm{th}}$ order as shown in Fig. 2(b). A full separation of the 0$^{\textrm{th}}$ and $\pm$1$^{\textrm{st}}$ orders is feasible, when the light paths of the 0$^{\textrm{th}}$ and $\pm$1$^{\textrm{st}}$ orders do not overlap. P3 should be as wide as possible for maximizing etendue and spectral power. The basic grating equation is given by [13]

2.2 Optical system and bandpass

In our optical system, we must balance the need for a sufficient high radiant power with the need for high spectral resolution in the spectral shaping. Our optical system ensures that the solid angle from which the GLV accepts radiation is as large as possible, which increases the available radiant power. At the same time, it allows us to increase the spectral resolution assuming a required minimal irradiance. Additionally, the amount of optics is kept to a minimum to reduce transmission and reflection losses.

In Section 1 in Fig. 3, the optical path to the GLV is illustrated, which mimics a typical optical path in a spectrograph [13].The setup consists of the radiation source S, mirrors M, apertures P and the grating G. The focal lengths $f_{\mathrm {M1}}$, $f_{\mathrm {M2}}$, $f_{\mathrm {M3}}$, $f_{\mathrm {M4}}$ are 101.6 mm long. In a spectrograph, a linearly expanded detector would be placed at the GLV’s position measuring the spectral irradiance of the source. The aperture P2 ensures that the maximal numerical aperture of the GLV in the x-z-plane is used (Eq. (6)). In the y-z-plane, the numerical aperture is limited by the size of the mirrors and increasing optical aberrations. A diameter of 50 mm was chosen for all mirrors.

 

Fig. 3. Schematic drawing of light path in x-z-plane and y-z-plane. The setup consists of light source S, mirrors M, apertures P, grating G, Grating Light Valve (GLV), lens L, and cylindrical lens C. The distances are determined by the focal lengths $f_i$.

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Referring to spectrograph theory, aperture P1 acts as an entrance slit, and the size of the GLV pixels determines the exit slit dimensions [13]. Both slits influence the bandpass of the optical system, which is an important parameter to determine its resolution capability. To calculate the bandpass, we need the reciprocal linear dispersion, which describes the spatial extent $dx$ of a spectral interval $d\lambda$ in the focal field. The tilt angle of the GLV $\alpha '$ has to be considered, where $\alpha ' = 0^{\circ }$ means that the GLV is perpendicular to the incident rays. The reciprocal linear dispersion is given by [13]

Section 2 in Fig. 3 illustrates the collection of the shaped radiation after the GLV. In the x-z-plane, a lens L ($f_{\mathrm {L}}=60$ mm) images the grating G plane to the aperture P3, which blocks higher diffraction orders generated from the GLV (Fig. 2). Consequently, all wavelengths paths are merged at plane P3. In the y-z-plane, an additional cylindrical lens C is used to image the GLV plane to the P3 plane. The cylindrical lens C ensures the transfer of all radiation into the mixing optics.

Mixing optics in section 3 randomize the optical paths of each wavelength, and the spectral irradiance in the measurement plane is uniformly distributed.

2.3 Algorithm for spectral shaping

The GLV offers 1088 independently addressable pixels, each with 1024 levels. This enables a huge amount of 1024$^{1088}$ possible combinations and, thus, possible spectral irradiance $E^{\mathrm {sim}}_{\mathrm{\lambda}} {(\lambda )}$ distributions. Therefore, we require an efficient algorithm for finding the appropriate GLV adjustments for a desired reference spectral irradiance $E^{\mathrm {ref}}_{\mathrm{\lambda}} {(\lambda )}$ distribution.

The spectral irradiance $E^{\mathrm {sim}}_{\mathrm{\lambda}}{(\lambda )} \in \mathbb {R}^m$ distribution is composed of the irradiance $E^{\mathrm {sim}}(\lambda _k) \in \mathbb {R}$ elements with $0 \leq k < m$ and $m=$ (number of data points in $E^{\mathrm {sim}}_{\mathrm{\lambda}} {(\lambda )}$). Every $E^{\mathrm {sim}}_{\mathrm{\lambda}} {(\lambda )}$ can be interpreted as a superposition of the spectral irradiance $E^{\mathrm {sim}}_{\lambda ,i}(\lambda , l_i) \in \mathbb {R}^m$ distributions, generated by each single GLV pixel $i$ with $0 \leq i < n=1088 \in \mathbb {N}$ at level $l_i$ with $0 \leq l_i < 1024 \in \mathbb {N}$. The spectral irradiance $E^{\mathrm {sim}}_{\mathrm{\lambda} ,i}(\lambda , l_i)$ distribution is composed of the irradiance $E^{\mathrm {\pm 1 st}}_i(\mathrm{\lambda} _k, l_i) \in \mathbb {R}$ elements, which are again described with Eq. (1) and Eq. (2):

In our setup, each single GLV pixel $i$ generates highly monochromatic $E^{\mathrm {sim}}_{\mathrm{\lambda} ,i}(\lambda , l_{i})$ distributions (see Fig. 5), so that most of the irradiance $E^{\mathrm {\pm 1 st}}_{i}({\lambda _{k}}, {l_{i}})$ elements are zero, except for close elements around a center wavelength $\lambda _{i}^{\mathrm {center}}$. In this case, the relative distributions of $E^{\mathrm {sim}}_{\mathrm{\lambda} ,i}(\lambda , l_{i})$ remain constant at different levels $l_i$ and Eq. (9) is approximated by

A matrix $E_{\mathrm {GLV}}^{\mathrm {M}} \in \mathbb {R}^{m \times n}$ is now constructed by consecutively filling column $i$ with $E_{\mathrm{\lambda} , i}^{\mathrm {mono,max}}(\lambda )$, and Eq. (10) can be written as a multiplication of matrix $E_{\mathrm {GLV}}^{\mathrm {M}}$ and vector $x \in \mathbb {R}^{n}$, where $x$ is filled with the elements $x_i(l_i)$:

When a reference spectral irradiance $E^{\mathrm {ref}}_{\mathrm{\lambda}} {(\lambda )}$ is given as a target for $E^{\mathrm {sim}}_{\mathrm{\lambda}} {(\lambda )}$, we can calculate the entries of vector $x$ as well. We can see Eq. (12) as an overdetermined system of linear equations with solution $x$ and with constraints given by $0 \leq x \leq 1 \in \mathbb {R}^{n}$. In this case, we must solve an optimization problem. Here, we identify the best solution $x$ by solving

Finally, we transfer the solution $x$ to the corresponding GLV levels $l_i$, a so called Programmable Look Up Table (PLUT) $\in \mathbb {N}^{n}$ with $n=1088$ under consideration of Eq. (11) and Eq. (2). The algorithm is then complete to generate a PLUT for a given $E_{\mathrm {ref}}(\mathrm{\lambda} )$.

3. Results and discussion

3.1 Setup characterization

The algorithm for finding the appropriate GLV levels $l_i$ for a given reference spectral irradiance $E^{\mathrm {ref}}_{\mathrm{\lambda}} {(\lambda )}$ distribution requires matrix $E_{\mathrm {GLV}}^{\mathrm {M}}$, and the normalized irradiance-to-level relation $x_i(l_i)$ according to section 2.3. Both are determined by an automated measurement procedure with a setup-integrated spectrometer from Avantes. This spectrometer provided a sufficiently high signal-to-noise ratio in $E_{\mathrm{\lambda} , i}^{\mathrm {mono,max}}(\lambda )$ measurements within the wavelength range of $420 \mathrm {\ nm} < \lambda < 1100 \mathrm {\ nm}$. Outside of this wavelength range, data was extrapolated and modelled with fitting parameters, which are shown in this section.

3.1.1 Grating light valve characterization

First, we determined the normalized irradiance-to-level relation $x_{i}(l_{i})$ for every pixel $i$. We measured $x_i(l_i)$ for every 10$^{\textrm{th}}$ pixel $i$ at 45 different GLV levels $l_i$. The measured data was fitted with Eq. (11) and Eq. (2) to determine the maximal deflection $d_{\mathrm {max}}$, the exponent $\gamma$ and the factor $I_i^{0}$. The center wavelengths of $E_{\mathrm{\lambda} , i}^{\mathrm {mono,max}}(\lambda )$ defined $\lambda _{i}^{\mathrm {center}}$. Furthermore, the levels $l_{i,\mathrm {maxPower}}$ were determined, where the maximal spectral irradiance $E_{\mathrm{\lambda} , i}^{\mathrm {mono,max}}(\lambda )$ distributions were achieved. Missing data in $x_i(l_i)$ was then modelled with these fitting parameters.

Figure 4(a) shows the measured and modelled normalized irradiance-to-level relation $x_i(l_i)$ for one GLV pixel.It can be modelled accurately, which is verified by low residuals. Furthermore, $x_i(l_i)$ is wavelength-dependent. This leads to unique curves for every GLV pixel $i$ as seen in Fig. 4(b). Figure 4(c) shows the fitting parameter maximal deflection $d_{\mathrm {max}}$, which varies between approximately 180 nm and 215 nm. Figure 4(d) shows the exponent $\gamma$, which takes values between 2.4 and 2.8. Due to the normalization of $x_i(l_i)$, the fitting factor $I^{0}_{i}$ is approximately equal to unity for pixels $i \lesssim 570$ as shown in Fig. 4(e), where $d_{\mathrm {max}}$ can reach $\lambda /4$, and the $\sin ^{2}$ function in Eq. (11) reaches unity. For pixels $i \gtrsim 570$, the deflection of the GLV is too short for the incident wavelengths, which yields $I^{0}_{i}$ values greater than unity. Consistent with that, the levels $l_{i,\mathrm {maxPower}}$ also reach their limits of level 1023 for pixels $i > 570$ as seen in Fig. 4(f). Due to the high accuracy of the models, missing data could be interpolated leading to a reduced amount of required measurement data and, thus, measurement time for full characterization of the GLV.

 

Fig. 4. a) Example for measured and modelled normalized irradiance $x_i(l_i)$ depending on the level $l_i$ of one active GLV pixel $i$. The relation is nonlinear and could be modelled accurately. b) Modelled normalized irradiance $x_i(l_i)$ of every 20$^{\textrm{th}}$ pixel. $x_i(l_i)$ is also wavelength-dependent leading to unique curves for every pixel. c) Fitting parameter maximal deflection $d_{\mathrm {max}}$. d) Fitting parameter $\gamma$. e) Fitting parameter $I^{0}$. f) GLV level $l_{i,\mathrm {maxPower}}$, where the maximal spectral irradiance $E^{\mathrm {sim,max}}_{\mathrm{\lambda} ,i}(\lambda )$ is achieved.

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3.1.2 Spectral characterization

Next, we determined the matrix $E_{\mathrm {GLV}}^{\mathrm {M}}$. It is filled with the spectral irradiance $E_{\mathrm{\lambda} , i}^{\mathrm {mono,max}}(\lambda )$ distributions generated by the individually activated GLV pixels $i$ with the levels $l_{i,\mathrm {maxPower}}$. The upper plot in Fig. 5 displays $E_{\mathrm{\lambda} , i}^{\mathrm {mono,max}}(\lambda )$ of every 20$^{\textrm{th}}$ pixel $i$.Additionally, the measured spectral irradiance $E^{\mathrm {sim,max}}_{\mathrm{\lambda}} {(\lambda )}$ is shown as a black curve, which is the superposition of all $E_{\mathrm{\lambda} , i}^{\mathrm {mono,max}}(\lambda )$ distributions or, equally, is a multiplication of matrix $E_{\mathrm {GLV}}^{\mathrm {M}}$ and vector $x$, where $x$ is filled with ones. The radiant power of $E^{\mathrm {sim,max}}_{\mathrm{\lambda}} {(\lambda )}$ was determined to be approximately $3 \mathrm {\ mW}$.

 

Fig. 5. Spectral characterization of the optical setup. The upper plot shows the spectral irradiance $E_{\mathrm{\lambda} , i}^{\mathrm {mono,max}}(\lambda )$ distributions for every 20$^{\textrm{th}}$ GLV pixel $i$ and the measured maximal achievable spectral irradiance $E^{\mathrm {sim,max}}_{\mathrm{\lambda}} {(\lambda )}$ representing the sum of all $E_{\mathrm{\lambda} , i}^{\mathrm {mono,max}}(\lambda )$. Radiation from approximately 355 nm to 1200 nm can be shaped. Every possible $E^{\mathrm {sim}}_{\mathrm{\lambda}} {(\lambda )}$ results from a linear combination of $E_{\mathrm{\lambda} , i}^{\mathrm {mono,max}}(\lambda )$. Full-Width-at-Half-Maximum (FWHM) values of all $E_{\mathrm{\lambda} , i}^{\mathrm {mono,max}}(\lambda )$ remain below 15 nm and increase only slightly as shown in the lower plot.

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The lower plot in Fig. 5 shows the bandwidths of $E_{\mathrm{\lambda} , i}^{\mathrm {mono,max}}(\lambda )$ for every pixel $i$. The bandwidths, represented by FWHM, are between approximately $7 \mathrm {\ nm}$ and $15 \mathrm {\ nm}$ and show the spectral resolution capability of our system. The rather low FWHM values allow the sum of the quadratic deviations between $E^{\mathrm {ref}}_{\mathrm{\lambda}} {(\lambda )}$ and $E^{\mathrm {sim}}_{\mathrm{\lambda}} {(\lambda )}$ (Eq. (13)) to remain low, even for target spectral irradiance distributions with distinct spectral features which have low bandwidths. According to Eq. (8), a bandpass with $\mathrm {FWHM} = 7.3 \mathrm {\ nm}$ results in an effective slit width of $\omega _{\mathrm {eff}} \approx 200$ $\mathrm{\mu}$m. It matches perfectly with the pinhole size of $200$ $\mathrm{\mu}$m, which we used as an entrance aperture. In the wavelength range from approximately $400 \mathrm {\ nm}$ to $800 \mathrm {\ nm}$, the FWHM values remain rather constant at $7.0 \mathrm {\ nm}$ to $7.8 \mathrm {\ nm}$, showing that optical aberrations do not increase the effective slit width $\omega _{\mathrm {eff}}$. As seen by the increasing FWHM values for wavelengths above $800 \mathrm {\ nm}$, optical aberrations significantly broaden the effective slit width $\omega _{\mathrm {eff}}$. In this wavelength range, a smaller entrance slit would reduce $\omega _{\mathrm {eff}}$ and the FWHM values only slightly due to the present and significant optical aberrations. Hence, a pinhole size of $200$ $\mathrm{\mu}$m balances the need for high spectral resolution and high spectral irradiance.

3.2 Approximated AM1.5g reference spectral irradiance

We have chosen the AM1.5g reference spectral irradiance $E_{\mathrm{\lambda} }^{\mathrm {AM1.5g}}(\lambda )$ as a target $E_{\mathrm{\lambda} }^{\mathrm {ref}}(\lambda )$ to show the ability of the presented setup for highly resolved spectral shaping. A high-quality multi-channel spectrometer from Zeiss with a spectral resolution of approximately 3 nm was selected for measuring the spectral irradiance $E_{\mathrm{\lambda} }^{\mathrm {sim}}(\lambda )$ (superscript indicates the device solar simulator). The irradiated area in the measurement plane had a size of approximately 50 cm${^2}$. A further increase is feasible. The spectral irradiance data of the spectrometer are highly accurate due to a calibration with a radiation standard lamp calibrated at the Physikalisch-Technische Bundesanstalt (PTB) in Germany. Considering the spectral resolution of the spectrometer, every reference spectral irradiance $E_{\mathrm{\lambda} }^{\mathrm {ref}}(\lambda )$ should only include measurable spectral features with a resolution greater than 3 nm, because features with lower bandwidths cannot be resolved. For this reason, the AM1.5g reference spectral irradiance $E_{\mathrm{\lambda} }^{\mathrm {AM1.5g}}(\lambda )$ data was smoothed by a convolution with a Gaussian function with a FWHM of 3 nm before being used as a target $E_{\mathrm{\lambda} }^{\mathrm {ref}}(\lambda )$.

The upper plot in Fig. 6 shows the smoothed AM1.5g reference spectral irradiance $E_{\mathrm{\lambda} }^{\mathrm {AM1.5g,3nm}}(\lambda )$ as a black curve and the measured $E_{\mathrm{\lambda} }^{\mathrm {sim}}(\lambda )$ as a red curve.Apparently, the measured $E_{\mathrm{\lambda} }^{\mathrm {sim}}(\lambda )$ matches excellently with $E_{\mathrm{\lambda} }^{\mathrm {AM1.5g,3nm}}(\lambda )$. We optimized the spectral match by minimizing the sum of the squared deviations (Eq. (13)). Thus, residuals are also minimized and are used to evaluate the spectral match. The lower plot in Fig. 6 displays the residuals as a red curve and the measurement noise as a blue curve. The measurement noise curve shows the upper and lower bound of the twofold standard deviation of 100 spectral measurements and usually remains below 0.5 to 3 percent of $E_{\mathrm{\lambda} }^{\mathrm {sim}}(\lambda )$. Slightly larger residuals compared to the measurement noise are observed around the deep oxygen absorption line at $\lambda = 759.6 \mathrm {\ nm}$. Here, the optical resolution of our system limits the spectral matching. A lack of shaped spectral irradiance was observed for $\lambda < 355 \mathrm {\ nm}$. Radiation in this wavelength range could not be shaped due to restrictions of the GLV. But overall, most of the residuals are smaller than the measurement noise. Further improvements in spectral matching would require a significant reduction of spectral measurement uncertainties and a broader spectral efficiency of the GLV.

 

Fig. 6. Upper plot: Slightly smoothed AM1.5g reference spectral irradiance $E_{\mathrm{\lambda} }^{\mathrm {AM1.5g,3nm}}(\lambda )$ and measured spectral irradiance $E_{\mathrm{\lambda} }^{\mathrm {sim}}(\lambda )$. Lower plot: Residuals (= $E_{\mathrm{\lambda} }^{\mathrm {sim}}(\lambda ) - E_{\mathrm{\lambda} }^{\mathrm {AM1.5g,3nm}}(\lambda )$) and spectral measurement noise represented by the twofold standard deviations of 100 $E_{\mathrm{\lambda} }^{\mathrm {sim}}(\lambda )$ measurements. The residuals are mostly within the measurement noise indicating the excellent spectral match.

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Figure 7 displays the corresponding GLV levels $l_i$ and the normalized irradiance $x_i(l_i)$ values to the measured spectral irradiance $E_{\mathrm{\lambda} }^{\mathrm {sim}}(\lambda )$ for all pixels $i$. GLV pixels with $i> 1000$ are on arbitrary levels, because radiation on these pixels has a wavelength above the investigated wavelength range from 300 nm to 1200 nm. Radiation on GLV pixels $50 \lesssim i \lesssim 150$ corresponds to wavelengths from 355 nm to around 450 nm. These GLV pixels are on levels generating almost the maximal achievable spectral irradiance $E_{\mathrm{\lambda} }^{\mathrm {sim,max}}(\lambda )$ in this wavelength range and, therefore, limit the overall achievable radiant power for $E_{\mathrm{\lambda} }^{\mathrm {sim}}(\lambda )$ with $E_{\mathrm{\lambda} }^{\mathrm {AM1.5g,3nm}}(\lambda )$ as a reference (see also Fig. 5(a)). The GLV levels $l_i$ show a discontinuous structure having sequential GLV pixels with highly different levels. Hence, the high spatial resolution of the GLV with its 1088 pixels, each 25 $\mathrm{\mu}$m wide, is essential for the presented highly resolved spectral shaping capability.

 

Fig. 7. Final solution for GLV levels $l_i$ and normalized irradiance values $x_i$ for generating $E_{\mathrm{\lambda} }^{\mathrm {sim}}(\lambda )$ matching a slightly smoothed AM1.5g reference spectral irradiance. A discontinuous structure with highly different levels is required.

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Due to the limited optical power of approximately 3 mW, applications are restricted to differential measurements with the lock-in technique. It should be highlighted that besides the AM1.5g, our optical setup can also generate other highly resolved spectral irradiance distributions within the range of the maximal spectral irradiance distribution $E_{\mathrm{\lambda} }^{\mathrm {sim,max}}(\lambda )$ shown in Fig. 5.

4. Conclusion and outlook

We presented an optical setup which is capable of generating highly resolved spectral irradiance $E_{\mathrm{\lambda} }^{\mathrm {sim}}(\lambda )$ distributions in the wavelength range from 355 nm to 1200 nm. A laser-driven light source is used, combined with a Grating Light Valve (GLV) as a spectral shaping tool. The GLV consists of 1088 pixels, which can be actuated individually with 1024 adjustable height levels. Each pixel generates quasi-monochromatic spectral irradiance $E_{\mathrm{\lambda} , i}^{\mathrm {mono,max}}(\lambda )$ distributions with a FWHM of approximately 7 nm to 15 nm. Any linear combination of these $E_{\mathrm{\lambda} , i}^{\mathrm {mono,max}}(\lambda )$ distributions is feasible. Thus, a huge amount of $E_{\mathrm{\lambda} }^{\mathrm {sim}}(\lambda )$ distributions can be generated. An efficient algorithm has been derived to calculate the appropriate GLV adjustments for any given target reference spectral irradiance $E_{\mathrm{\lambda} }^{\mathrm {ref}}(\lambda )$. We have achieved an excellent match between an AM1.5g reference spectral irradiance $E_{\mathrm{\lambda} }^{\mathrm {AM1.5g,3nm}}(\lambda )$ distribution, slightly smoothed to consider the limited bandwidth of the spectrometer, and the measured spectral irradiance $E_{\mathrm{\lambda} }^{\mathrm {sim}}(\lambda )$ in the wavelength range from approximately 355 nm to 1200 nm. Remaining spectral deviations from the reference spectral irradiance $E_{\mathrm{\lambda} }^{\mathrm {AM1.5g,3nm}}(\lambda )$ distribution are mostly within the spectral measurement noise, which is typically below 0.5 to 3 percent of $E_{\mathrm{\lambda} }^{\mathrm {sim}}(\lambda )$.

Future improvements of the optical setup might be obtained by a GLV with a broader and higher spectral efficiency. This would overcome the relatively low spectral power in the ultra-violet range and, thus, increase the overall radiant power. Furthermore, a radiation source with a higher radiance would be favorable, which would increase the available radiant power and would allow lower optical aberrations. In further steps, the generated chopped radiation with $E_{\mathrm{\lambda} }^{\mathrm {sim}}(\lambda )$ will be applied in highly accurate solar cell characterization.