For wavelengths ranging from 390 to 720 nm, the eye sensitivity function V(X) is greater than 10-3. Although the human eye is sensitive to light with wavelengths < 390 nm and > 720 nm, the sensitivity at these wavelengths is extremely low. Therefore, the wavelength range 390 nm < X < 720 nm can be considered the visible wavelength range. The relationship between color and wavelength within the visible wavelength range is given in Table 16.5. This relationship is valid for monochromatic or near-monochromatic light sources such as LEDs. Note that color is, to some extent, a subjective quantity. Also note that the transition between different colors is continuous.

16.1 Luminous efficacy and luminous efficiency The luminous flux, Фьш, is obtained from the radiometric light power using the equation

ф lum = 683 W f V(X) P(X) dX

W jx

where P(X) is the power spectral density, i. e. the light power emitted per unit wavelength, and the prefactor 683 lm/W is a normalization factor. The optical power emitted by a light source is then given by

P = f P(X)dX.

High-performance single-chip visible-spectrum LEDs can have a luminous flux of about 10­100 lm at an injection current of 100-1 000 mA.

The luminous efficacy of optical radiation (also called the luminosity function), measured in units of lumens per watt of optical power, is the conversion efficiency from optical power to luminous flux. The luminous efficacy is defined as

For strictly monochromatic light sources (AX ^ 0), the luminous efficacy is equal to the eye sensitivity function V(X) multiplied by 683 lm/W. However, for multicolor light sources and especially for white light sources, the luminous efficacy needs to be calculated by integration over all wavelengths. The luminous efficacy is shown on the right-hand ordinate of Fig. 16.4.

The luminous efficiency of a light source, also measured in units of lm/W, is the luminous

flux of the light source divided by the electrical input power.

Luminous efficiency = Фlum / (IV) (16.4)

where the product (I V) is the electrical input power of the device. Note that in the lighting community, luminous efficiency is often referred to as luminous efficacy of the source.

Inspection of Eqs. (16.3) and (16.4) reveals that the luminous efficiency is the product of the luminous efficacy and the electrical-to-optical power conversion efficiency. The luminous efficiency of common light sources is given in Table 16.6.

The luminous efficiency is a highly relevant figure of merit for visible-spectrum LEDs. It is a measure of the perceived light power normalized to the electrical power expended to operate the LED. For light sources with a perfect electrical-power-to-optical-power conversion, the luminous source efficiency is equal to the luminous efficacy of radiation.

Exercise: Luminous efficacy and luminous efficiency of LEDs. Consider a red and an amber LED emitting at 625 and 590 nm, respectively. For simplicity, assume that the emission spectra are monochromatic (AX ^ 0). What is the luminous efficacy of the two light sources? Calculate the luminous efficiency of the LEDs, assuming that the red and amber LEDs have an external quantum efficiency of 50%. Assume that the LED voltage is given by V = Eg / e = hv / e.

Assume next that the LED spectra are thermally broadened and have a gaussian lineshape with a linewidth of 1.8 kT. Again calculate the luminous efficacy and luminous efficiency of the two light sources. How accurate are the results obtained with the approximation of monochromaticity?

Some LED structures attain excellent power efficiency by using small light-emitting areas (current injection in a small area of chip) and advanced light-output-coupling structures (see, for example, Schmid et al., 2002). However, such devices have low luminance because only a small fraction of the chip area is injected with current. Table 16.7 summarizes frequently used figures of merit for light-emitting diodes.

16.2 Brightness and linearity of human vision Although the term brightness is frequently used, it lacks a standardized scientific definition. The frequent usage is due to the fact that the general public can more easily relate to the term brightness than to photometric terms such as luminance or luminous intensity. Brightness is an attribute of visual perception and is frequently used as synonym for luminance and (incorrectly) for the radiometric term radiance.

To quantify the brightness of a source, it is useful to differentiate between point and surface area sources. For point sources, brightness (in the photopic vision regime) can be approximated by the luminous intensity (measured in cd). For surface sources, brightness (in the photopic vision regime) can be approximated by the luminance (measured in cd/m2). However, due to the lack of a formal standardized definition of the term brightness, it is frequently avoided in technical publications.

Standard CIE photometry assumes human vision to be linear within the photopic regime. It is clear that an isotropically emitting blue point source and an isotropically emitting red point source each having a luminous flux of, e. g., 5 lm, have the same luminous intensity. Assuming linearity of photopic vision, both sources still have the same luminous intensity as the luminous fluxes of the sources are increased from 5 to, e. g., 5000 lm.

However, if the luminous fluxes of the two sources are reduced so that the mesopic or scotopic vision regime is entered, the blue source will appear brighter than the red source due to the shift of the eye sensitivity function to shorter wavelengths in the scotopic regime.

It is important to keep in mind that the linearity of human vision within the photopic regime is an approximation. Linearity clearly simplifies photometry. However, human subjects may feel discrepancies between the experience of brightness and measured luminance of a light source, especially for colored light sources if the luminous flux is changed over orders of magnitude.