Radiance

The formula for calculating radiance involves two key parameters: radiant intensity and projected solid angle. Radiant intensity represents the power emitted or reflected by a point source within a specific solid angle. The projected solid angle refers to the area over which this radiant flux is distributed.

The formula for radiance is given by:

Where:

• L is the radiance
• dφ/dA is the radiant flux through an infinitesimal area dA
• dω is the solid angle subtended by the infinitesimal area dA
• θ is the angle between the normal to the surface and the direction of the radiance

This formula accounts for the fact that radiance varies with the angle of observation, with the cosine term (cos θ) representing the geometric relationship between the surface normal and the direction of observation.

Radiance represents the radiant flux per unit area and per unit solid angle. The units of radiant flux, area, and solid angles are Watt (W), meter squared (m²), and steradian (sr), respectively. Therefore, the unit of radiance is watts per meter squared per steradian or W/(m²ˑsr).

Radiance has found numerous applications in various fields. One of the prominent areas where it is extensively used is astronomy. Astronomers can gain insights into their composition, temperature, and other physical properties by analyzing the electromagnetic radiation emitted by celestial bodies.

Radiance plays a crucial role in remote sensing applications. Remote sensing involves collecting data about the Earth’s surface from a distance using sensors mounted on satellites or aircraft. Radiometric measurements enable scientists to gather information about land cover, vegetation health, atmospheric conditions, etc. This data is invaluable for environmental monitoring, urban planning, agriculture management, and disaster response.

Radiance is also used in medical imaging techniques such as X-rays and positron emission tomography (PET) scans to diagnose diseases and monitor treatment progress. Radiometric measurements are also employed in industrial processes like thermal imaging for detecting heat loss or identifying faulty equipment.

Spectral radiance refers to the amount of radiant power emitted, transmitted, or reflected by a surface per unit solid angle, per unit projected area, and per unit wavelength interval. In simpler terms, it quantifies the amount of light energy emitted or reflected by an object at different wavelengths.

Spectroradiometers are commonly used to measure spectral radiance accurately. These devices are designed to capture and analyze light across various wavelengths. Spectroradiometers consist of optical sensors that detect light at different wavelengths and convert it into electrical signals for further analysis.

Understanding spectral radiance is crucial in various fields, such as remote sensing, astronomy, photometry, and colorimetry. By analyzing the spectral radiance of an object or scene, scientists can gain valuable insights into its composition, temperature, reflectivity, and other essential characteristics.

Problem 1: A surface has a radiant flux of 500 W/m². Calculate the radiance of the surface if the surface area is 2 m² and the solid angle is π steradians. Assume cos θ = 1/π.

Solution:

Given:

• Radiant flux: dφ/dA= 500 W/m²
• Surface area: A = 2 m²
• Solid angle: dω = π steradians
• cos θ = 1/π

Therefore,

\[ L = \frac{500 \, \text{W/m}²}{2 \, \text{m}² \cdot π \cdot \frac{1}{\pi}} \]

\[ => L = \frac{500}{2} = 250 \, \text{W/(m}² \cdot \text{steradian)} \]

Problem 2: A surface has a radiance of 150 W/(m²·sr) with a solid angle of 0.5 steradian and a surface area of 3 m². Calculate the radiant flux. Assume cos θ = 1/π.

Solution:

\[ L = \frac{d\Phi}{dA \cdot d\omega \cdot \cos(\theta)} \]

Rearranging the formula to solve for radiant flux:

\[ \frac{d\Phi}{dA} = L \cdot d\omega \cdot \cos(\theta) \]

Given:

• Radiance: L = 150 W/(m²·sr)
• Solid angle: dω = 0.5 steradian
•  Surface area A = 3 m²
• cos θ = 1/π

\[ \frac{d\Phi}{dA} = 150 \, \text{W/(m}² \cdot \text{sr)} \cdot 0.5 \cdot \frac{1}{\pi} \]

\[ \frac{d\Phi}{dA} \approx 23.91 \, \text{W/m}² \]

Problem 3: A surface with a radiance of 180 W/(m²·sr) is observed at an angle where cos θ = 0.8. If the solid angle dω is 0.4 steradian, calculate the radiant flux for a surface area of 4 m}².

Solution:

\[ \frac{d\Phi}{dA} = L \cdot d\omega \cdot \cos(\theta) \]

Given:

• Radiance: L = 180 W/(m²·sr)
• Solid angle: dω = 0.4 steradian
• cos θ = 0.8
• Surface area: A = 4 m²

\[ \frac{d\Phi}{dA} = 180 \, \text{W/(m}² \cdot \text{sr)} \cdot 0.4 \cdot 0.8 \]

\[ \frac{d\Phi}{dA} = 57.6 \, \text{W/m}² \]