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# Properties of Light

The light that we see everyday is only a fraction of the total energy emitted by the sun incident on the earth. Sunlight is a form of "electromagnetic radiation" and the visible light that we see is a small subset of the electromagnetic spectrum shown at the right.The electromagnetic spectrum describes light as a wave which has a particular wavelength. The description of light as a wave first gained acceptance in the early 1800's when experiments by Thomas Young, François Arago, and Augustin Jean Fresnel showed interference effects in light beams, indicating that light is made of waves. By the late 1860's light was viewed as part of the electromagnetic spectrum. However, in the late 1800's a problem with the wave-based view of light became apparent when experiments measuring the spectrum of wavelengths from heated objects could not be explained using the wave-based equations of light. This discrepancy was resolved by the works of [1] in 1900, and [2] in 1905. Planck proposed that the total energy of light is made up of indistinguishable energy elements, or a quanta of energy. Einstein, while examining the photoelectric effect (the release of electrons from certain metals and semiconductors when struck by light), correctly distinguished the values of these quantum energy elements. For their work in this area Planck and Einstein won the Nobel prize for physics in 1918 and 1921, respectively and based on this work, light may be viewed as consisting of "packets" or particles of energy, called photons.

Today, quantum-mechanics explains both the observations of the wave nature and the particle nature of light. In quantum mechanics, a photon, like all other quantum-mechanical particles such as electrons, protons etc, is most accurately pictured as a "wave-packet". A wave packet is defined as a collection of waves which may interact in such a way that the wave-packet may either appear spatially localized (in a similar fashion as a square wave which results from the addition of an infinite number of sine waves), or may alternately appear simply as a wave. In the cases where the wave-packet is spatially localized, it acts as a particle. Therefore, depending on the situation, a photon may appear as either a wave or as a particle and this concept is called "wave-particle duality". A wave-packet, or photon is pictured as below.

A complete physical description of the properties of light requires a quantum-mechanical analysis of light, since light is a type of quantum-mechanical particle called a photon. For photovoltaic applications, this level of detail is seldom required and therefore only a few sentences on the quantum nature of light are given here. However, in some situations (fortunately, rarely encountered in PV systems), light may behave in a manner which seems to defy common sense, based on the simple explanations given here. The term "common sense" refers to our own observations and cannot be relied on to observe the quantum-mechanical effects because these occur under conditions outside the range of human observation.

A photon is characterized by either a wavelength, denoted by λ or equivalently an energy, denoted by °E. There is an inverse relationship between the energy of a photon (E) and the wavelength of the light (λ) given by the equation:

where h is Planck's constant and c is the speed of light. The value of these and other commonly used constants is given in the constants page.

h°= 6.626 × 10^{-34} joule·s

c°= 2.998 × 10^{8°} m/s

By multiplying to get a single expression, °hc° = 1.99 × 10^{-25} joules-m

The above inverse relationship means that light consisting of high energy photons (such as "blue" light) has a short wavelength. Light consisting of low energy photons (such as "red" light) has a long wavelength.

When dealing with "particles" such as photons or electrons, a commonly used unit of energy is the electron-volt (eV) rather than the joule (J). An electron volt is the energy required to raise an electron through 1 volt, thus a photon with an energy of 1 eV = 1.602 × 10^{-19}°J.

Therefore, we can rewrite the above constant for °hc° in terms of eV:

hc° = (1.99 × 10^{-25} joules-m) × (1ev/1.602 × 10^{-19} joules) = 1.24 × 10^{-6}°° eV-m

Further, we need to have the units be in µm (the units for λ):

hc° = (1.24 × 10^{-6} eV-m) × (10^{6}°µm/ m) = 1.24 eV-µm

By expressing the equation for photon energy in terms of eV and µm we arrive at a commonly used expression which relates the energy and wavelength of a photon, as shown in the following equation:

The exact value of 1 × 10^{6}(hc/q) is 1.2398 but the approximation 1.24 is sufficient for most purposes.

**Photon Flux**

The photon flux is defined as the number of photons per second per unit area:

The photon flux is important in determining the number of electrons which are generated, and hence the current produced from a solar cell. As the photon flux does not give information about the energy (or wavelength) of the photons, the energy or wavelength of the photons in the light source must also be specified. At a given wavelength, the combination of the photon wavelength or energy and the photon flux at that wavelength can be used to calculate the power density for photons at the particular wavelength. The power density is calculated by multiplying the photon flux by the energy of a single photon. Since the photon flux gives the number of photons striking a surface in a given time, multiplying by the energy of the photons comprising the photon flux gives the energy striking a surface per unit time, which is equivalent to a power density. To determine the power density in units of W/m², the energy of the photons must be in Joules. The equation is:

using SI units

for wavelength in *μm*

for energy in eV where Φ is the photon flux and q is the value of the electronic charge 1.6 ·10^{-19}

**Spectral Irradiance**

The spectral irradiance as a function of photon wavelength (or energy), denoted by F, is the most common way of characterising a light source. It gives the power density at a particular wavelength. The units of spectral irradiance are in Wm^{-2}µm^{-1}. The Wm^{-2} term is the power density at the wavelength λ(µm). Therefore, the m^{-2} refers to the surface area of the light emitter and the µm^{-1} refers to the wavelength of interest.

In the analysis of solar cells, the photon flux is often needed as well as the spectral irradiance. The spectral irradiance can be determined from the photon flux by converting the photon flux at a given wavelength to W/m^{2} as shown in the section on Photon Flux. The result is then divided by the given wavelength, as shown in the equation below.

in SI units

where in SI units:

F(λ) is the spectral irradiance in Wm^{-2}μm^{-1};

Φ is the photon flux in # photons m^{-2}sec^{-1};

where:

F(λ) is the spectral irradiance in Wm^{-2}µm^{-1};

Φ° is the photon flux in # photons m^{-2}sec^{-1};

E° and λ° are the energy and wavelength of the photon in eV and µm respectively; and q is a constant of 1.6° · 10^{-19}

The spectral irradiance of artificial light sources (left axis) compared to the spectral irradiance from the sun (right axis).

**Radiant Power Density**

The total power density emitted from a light source can be calculated by integrating the spectral irradiance over all wavelengths or energies

where:

However, a closed form equation for the spectral irradiance for a light source often does not exist. Instead the measured spectral irradiance must be multiplied by a wavelength range over which it was measured, and then calculated over all wavelengths. The following equation can be used to calculate the total power density emitted from a light source.

△λ° is the wavelength.

Calculating the total power density from a source requires integrating over the spectrum by calculating the area of each element and then summing them together.

Measured spectra are typically not smooth as they contain emission and absorption lines. The wavelength spacing is usually not uniform to allow for more data points in the rapidly changing parts of the spectrum. The spectral width is calculated from the mid-points between two the adjacent wavelengths.

Power in each segment is then:

Summing all the segments gives the total power H as in the equation above.