Lighting Series Part 4 – Irradiance and Radiant Intensity

In this post from Lighting Series, we are going to talk about Irradiance and Radiant Intensity. The previous post in this series is Lighting Series Part 3 – Radiance. In the next post, we are going to talk about Photometry.

The SI System (Système International d’unites) defines six radiometric units used for describing the radiation coupling between a light source and an optical system.  The most commonly used units are: Radiant Flux, Irradiance, and Radiance.

radiometric_units

What is Irradiance?

Irradiance E  is the radiometric term for flux per unit area dA measured at a point x on the surface (E = dφ / dA). It is for light arriving at a surface.

For example, if 50 watts of radiant power is incident on a surface that has an area of 1.25m^2, the irradiance at each surface point is 40 watts/m^2 (assuming the incident power is uniformly distributed over the surface).

irradiance2.png

This can be computed by integrating the incident or incoming radiance Li over a hemisphere Ω

E = ∫Ω (Li * cos θ * dω)

where cos θ * dω is the projected solid angle.

Example of projected solid angle:
projected_solid_angle

The SI unit for irradiance is watts per square meter [W/m^2] .

The quantity called just irradiance is a one-sided, surface-oriented property. If we are talking about a point x in a volume, one can discuss the irradiance at that point, by specifying a surface normal n and defining the irradiance E(x, n) as the irradiance that would fall on a small bit of surface oriented facing the direction n.

irradiance.png

Radiosity B, also called Radiant Emittance M or Radiant Exitance, is the flux per unit area dA that leaves the surface (B = dφ / dA). This can be computed by integrating the outgoing radiance Lo over a hemisphere Ω

B = ∫Ω (Lo * cos θ * dω)

For example, consider a light source, of area 0.1m^2, that emits 100 watts. Assuming that the power is emitted uniformly over the area of the light source, the radiant exitance of the light is 1000 W/m^2 at each point of its surface.

Irradiance and Radiance

Irradiance is a useful measure for applications where power must be delivered to large areas. For example, illuminating a classroom or a football field is primarily a question of delivering a certain number of watts per square meter. This can be achieved by using a single high power source. However, since irradiance does not depend on solid angle, multiple sources can be combined, illuminating the walls or the field from different angles.

The irradiance of a source is not the most useful measure when designing an efficient optical coupling system that collects radiation from a source, and then delivers the radiation into an optical instrument. Such optical instruments will have a limited entrance aperture and a limited acceptance solid angle. In such cases, it is the radiance of the source (its brightness) that is most useful.

You can see in the following picture that the element of the Earth’s surface dS receives an irradiance E from the upper half space and acts for the sensor as a source of radiance L along a direction Θ.

irradiance_radiance.png

Radiance is defined as “directional” and irradiance is “hemispheric”.

  • The sensor receives energy radiated by the source dS along a specific direction. Radiance is therefore directional.
  • Irradiance of the Earth’s surface in the visible range is caused by the Sun. Through the atmosphere scatters sunlight, visible radiation reaches us not only from the  direction of the Sun but also from all directions in the upper hemisphere. This is why we can see clearly along a shady street. Consequently, solar irradiance is the sum of all direct and diffuse irradiance and is therefore hemispheric.

What is Radiant Intensity?

Radiance is a very useful way of characterizing light transport between surface elements, but it is difficult to describe the energy distribution of a point light source with radiance because the point singularity at the source. This can be done with another quantity called radiant intensity.

The energy distribution from a point light source expands outward from the center. A small beam is defined by a differental solid angle in a given direction. The flux in a small beam dW is defined to be equal to

dΦ = I * dΩ

where I is the radiant intensity of the point light source with units of power per unit solid angle . Then we can define the radiant intensity as the power emitted by a point source A  per solid angle unit dΩ.

I =  /

radiant_intensity

Radiant intensity describes the directional distribution of light, but it is only meaningful for point light sources. The radiant intensity in a given direction is equal to the irradiance at a point on the unit sphere centered at the source.

Summary

The following diagram shows the relationships between radiant flux, radiance, irradiance, and radiosity:

relationship_radiometric_units.png

and between all the radiometric units we talked about in this and previous articles

diagram_comparison

 

In the next post, we are going to talk about Photometry.

References

Real-Time Rendering, 3rd Edition

Physically Based Rendering, 3rd Edition: From theory to implementation

Fundamentals of Computer Graphics, 4th Edition

Essential Mathematics for Games and Interactive Applications, 3rd Edition

Introduction to Computer Graphics: A Practical Learning Approach, 1st Edition

Computer Graphics: Principles and Practices, 3rd Edition

Foundations of 3D Computer Graphics

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