# 2.2 Spectral Sensitivity of an Integral Detector

Within the normal range of operation of an integral detector, the relation between the input signal (the spectral radiometric quantity to be measured X_{λ}(λ)) entering the detector and its corresponding output signal Y has to fulfil the following condition of **linearity**:

Let Y_{1} be the detector’s response to the input signal X_{1}_{λ}(λ) and Y_{2} the detector’s response to the input signal X_{2}_{λ}(λ). The detector’s response to the superimposed input signal X_{1}_{λ}(λ) + X_{2}_{λ}(λ) is given by Y_{1} + Y_{2}. Moreover, the detector’s response is proportional to the input signal and therefore, the response to the input signal a × X_{1}_{λ}(λ) is given by a × Y_{1} (where a denotes an arbitrary positive number). A detector might possibly show a certain **dark signal Y _{0}** (usually dark current or dark voltage), which is a nonzero output signal even when the detector is not exposed to any radiation at all. In this case, Y, Y

_{1}and Y

_{2}have to be substituted with Y - Y

_{0}, Y

_{1}- Y

_{0}and Y

_{2}- Y

_{0}.

Deviations from this behavior are called **nonlinearities** and cause measurement errors. It is however possible to experimentally determine the nonlinearities of a detector and correct them. An example of a nonlinearity effect is the saturation of a detector’s output signal at high radiation levels, which represents the upper limit of a detector’s range of operation.

When nonlinearity effects can be neglected, the detector’s output signal under arbitrary polychromatic radiation can be regarded as a superposition of the detector’s output signals under monochromatic radiation. This leads to the concept of spectral sensitivity.

In detail, the CIE defines a detector’s **spectral sensitivity (also: spectral responsivity) s(λ)** as

s(λ) = 1 × dY X _{λ}(λ)dλ

where X_{λ}(λ) denotes the spectral radiometric quantity defining the detector’s input signal and dY denotes the (differential) increase of the output signal caused by the input radiation in the (differential) wavelength interval between λ and λ + dλ. When linear behavior of the detector can be assumed, the detector’s signal Y is given by

Y = ∫ X _{λ}(λ) × s(λ) dλλ

Often, the spectral sensitivity function s(λ) is described by the product of a reference value s_{m} and the relative** spectral sensitivity s _{r}(λ)** is:

s

_{r}(λ) = s_{m}× s_{r}(λ)

In many cases, s_{m} is given by the maximum of s(λ), thus s_{r}(λ) is normalized to a value of 1 in its maximum. Another possibility is the normalization of s_{r}(λ) to a total wavelength integral value 1, which is achieved by the definition of

s

_{m}=_{λ }∫ s(λ) × dλ

In terms of relative spectral sensitivity, the detector’s output signal Y is given by

Y = s _{m}×∫ X _{λ}(λ) × s_{r}(λ) dλλ

This integral relation is equivalent to the definition of photopic quantities where the detector’s relative spectral sensitivity s_{r}(λ) corresponds to the CIE spectral luminous efficiency function V(λ) and s_{m} corresponds to K_{m} = 683 lm/W. Similarly, the calculation of effective radiation doses relevant for certain biological reactions is based on a corresponding relation containing the respective biological action spectrum. For instance, the erythemal action spectrum is used for definition of **Sunburn Unit**, which is used for quantification of erythemally active solar UV irradiance (UV radiation).

This correspondence allows for the direct determination of photopic quantities or biologically active radiation using a specially designed integral detector. In particular, the detector’s relative spectral sensitivity s_{r}(λ) has to be matched closely to the CIE spectral luminous efficiency function V(λ) or to the respective action spectrum. For the determination of chromaticity coordinates or correlated color temperature, it is necessary to simultaneously use three detectors with their spectral sensitivities that are specially adapted to the color matching functions defined by the CIE 1931 standard colorimetric observer.

Gigahertz Optik uses different combinations of photodiodes and filters to achieve proper spectral sensitivities for detectors used in photometry, radiometry and colorimetry.

### Monochromatic radiometry

For radiometric characterization of monochromatic or near monochromatic radiation of a known wavelength, a detector’s spectral sensitivity does not necessarily have to match a certain predefined shape. A photodiode can therefore be used without any spectral correction filters as long it is sensitive at the respective wavelength.

*Fig. 1: Sensitivity ranges of various types of photodiodes*

Typical tasks of monochromatic radiometry are the laser power measurements, characterization of LEDs with near monochromatic light output and power measurements in fiber optical telecommunication. Gigahertz Optik offers

- laser power meters equipped with a flat field detector (for lasers with collimated beams) or an integrating sphere (for lasers with noncollimating beams and LEDs)
integrating spheres equipped with small area photodiodes whose low capacitance results in a detector time constant in the order of nanoseconds. These detectors are thus perfectly ideal for laser pulse analysis with high time resolution

detectors equipped with integrating spheres that have a unique baffle design for measurements in fiber optics telecommunication. Additional adapters for standard fiber optic connectors are available

### Polychromatic radiometry

The determination of total radiation power over a certain spectral range requires the detector’s spectral sensitivity function to closely match a rectangular shape.

*Fig. 2: Spectral sensitivity of Gigahertz Optik’s RW-3702 visible 400 – 800 nm irradiance detector closely matches the ideal rectangular shape.*

Gigahertz Optik offers absolutely calibrated irradiance and radiant power meters equipped with a cosine diffuser or an integrating sphere whose spectral sensitivity is optimized for UVA, UVB, UVC as well as visible (VIS) and near infrared (NIR) ranges.

### Photometry

For photometric measurements, the detector’s relative spectral sensitivity s_{r}(λ) has to match the CIE spectral luminous efficiency function V(λ) as close as possible. In order to quantify a detector’s inevitable spectral mismatch, the CIE recommends the evaluation index f_{1}', which is defined by

f _{1}' =_{λ }∫ | s^{*}_{r}(λ) - V(λ) | dλ_{λ }∫ V(λ) dλ

where s^{*}_{r}(λ) is given by

s ^{*}_{r}(λ) =_{λ }∫ S_{A}(λ) V(λ) dλ× s _{r}(λ)_{λ }∫ S_{A}(λ) s_{r}(λ) dλ

where S_{A}(λ) is the spectral distribution of the CIE Standard Illuminant A, which is the recommended photometric calibration source. High quality photometric detectors show a value of f_{1}' below 3 %, whereas a value of f_{1}' above 8 % isconsidered as poor quality. The DIN 5032, part 7 requires a spectral mismatch of f_{1}' ≤ 3 % for “Class A” instruments and f_{1}' ≤ 6 % for “Class B” instruments. Gigahertz Optik offers high quality illuminance, luminance and luminous flux detectors corresponding to Class A level (f_{1}' = 3 %) and, as an economical alternative, detectors meeting class B level (f_{1}' = 5 %). Furthermore, the BTS Technology allows for an online correction of the spectral mismatch factor.

### Colorimetry

For the determination of a color stimulus X, Y and Z values as defined by the CIE 1931 standard colorimetric observer, the same stimulus has to be measured by three different detectors, whose spectral sensitivity functions have to be adapted to the CIE 1931 XYZ color matching functions. However, as the x(λ) color matching function consists of two separate sensitivity regions, the X value is often determined by two detectors. In this case, all four detectors are needed for the determination of the X, Y and Z stimulus values. Since the color matching function y(λ) is identical to the CIE spectral luminous efficiency function V(λ), the respective detector can be calibrated definitely for simultaneous photometric measurements.

*Fig. 3: Spectral sensitivity functions used for colorimetric measurements with Gigahertz-Optik’s CT-3701 High Precision Color Meter.*