Photometry

Foucault's method improved earlier techniques by:

• Allowing laboratory-scale measurements (no long distances required)
• Enabling speed of light measurement in different media (like water)

Key components:

1. Rotating mirror (high angular velocity)
2. Fixed mirror (at distance)
3. Light source and lens system

Working principle:

• Light reflects from rotating mirror to fixed mirror and back
• During travel time, rotating mirror turns slightly
• Returning beam reflects at different angle
• Angular displacement is proportional to light's travel time
• c = 8πLNr/θ where:
  • L = distance between mirrors
  • N = rotation rate
  • r = distance from rotating mirror to observation point
  • θ = observed angular displacement

Relationship between luminous flux and radiant flux:

• Radiant flux: total energy emitted per unit time (measured in watts)
• Luminous flux: measure of the perceived brightness power, weighted by human eye sensitivity
• Conversion depends on wavelength-specific luminosity function
• Mathematically: Φ = K₃ ∫ P(λ)V(λ)dλ where:
  • P(λ) is spectral power distribution
  • V(λ) is luminosity function
  • K₃ is a constant

Definition of lumen:

• 1 lumen equals the luminous flux of a 1/683 watt monochromatic light source of wavelength 555 nm
• Alternatively: 1 watt of 555 nm light produces 683 lumens
• 555 nm corresponds to peak sensitivity of human vision under normal lighting conditions
• Basis for all photometric measurements that relate to human visual perception

Lambert's Cosine Law states that for a perfectly diffused surface:

I = I₀ cos θ

Where:

• I = luminous intensity in a given direction
• I₀ = luminous intensity along the normal to the surface
• θ = angle between the direction and the normal

Implications:

1. Brightness appears uniform from all viewing angles
2. Apparent brightness of a surface is independent of viewing angle
3. Radiance decreases with cosine of emission angle
4. Total power emitted into hemisphere equals πI₀
5. Surfaces following this law are called "perfectly diffused" or "Lambertian"

Real-world examples:

• Matte paper (approximately Lambertian)
• Certain types of paint
• Diffuse reflection of light from rough surfaces
• Standard for designing illumination systems

Derivation of inverse square law:

1. For a point source with intensity I:

   • Solid angle subtended by area A at distance r: Ω = A·cos(θ)/r²
   • Luminous flux through this solid angle: Φ = I·Ω = I·A·cos(θ)/r²

2. Illuminance E is flux per unit area:

   • E = Φ/A = I·cos(θ)/r²

Physical significance:

• Illuminance decreases with square of distance from light source
• Doubling distance reduces illuminance to ¼ of original value
• Results from geometric spreading of light in three dimensions
• Conservation of energy: same energy spreads over area proportional to r²
• Applies to any isotropic point source (light, sound, radiation)

Applications:

• Lighting design calculations
• Photography exposure settings
• Astronomical brightness measurements
• Radiation safety standards

Working principle of Bunsen photometer:

1. Setup components:

   • Paper screen with oil spot (creates a translucent area)
   • Two light sources positioned on opposite sides
   • Movable screen on optical bench

2. Physical basis:

   • Oil spot appears darker than surroundings when viewed from more illuminated side
   • Oil spot appears brighter than surroundings when viewed from less illuminated side
   • Oil spot disappears when illuminance is equal on both sides

3. Measurement procedure:

   • Position screen between sources and adjust until oil spot "disappears"
   • At equilibrium, illuminance from both sources is equal: E₁ = E₂
   • Apply inverse square law: I₁/d₁² = I₂/d₂²
   • Therefore: I₁/I₂ = d₁²/d₂²

4. Practical considerations:

   • Screen must be perpendicular to line joining sources
   • Background lighting must be eliminated
   • Observer's position should be consistent
   • Multiple readings improve accuracy

This method directly applies the inverse square law to determine the ratio of intensities without requiring absolute measurements.

Relative luminosity and human visual perception:

1. Definition: Relative luminosity measures how efficiently different wavelengths produce brightness sensation relative to the most effective wavelength (555 nm).

2. Characteristics:

   • Peaks at 555 nm (yellow-green) = 1.0
   • Falls to ~0.1 at 450 nm (blue) and 650 nm (red)
   • Nearly zero beyond 400 nm and 700 nm
   • Based on average human photopic (daylight) vision

3. Physiological basis:

   • Corresponds to sensitivity of cone cells in retina
   • Evolution optimized human vision for sunlight spectrum
   • Different distribution for scotopic (night) vision (peaks ~507 nm)

4. Practical significance:

   • Explains why equal-energy yellow light appears brighter than red or blue
   • Forms basis for photometric measurements (lumens, candela)
   • Critical for lighting design and efficiency calculations
   • Explains why green laser pointers appear brighter than red ones of equal power

5. Mathematical representation: V(λ) function (CIE standard luminosity function)

• Relative luminosity: The ratio of brightness-producing capacity of light at a given wavelength compared to light at 555 nm (where it equals 1.0)
• Dimensionless quantity ranging from 0 to 1.0
• Determined by physiological response of human photoreceptors

To calculate luminous flux of a monochromatic source:

1. Identify the wavelength of the source
2. Find the corresponding relative luminosity value
3. Multiply the radiant flux (in watts) by the relative luminosity value
4. Multiply by the conversion factor 685 lumen/watt

Formula: Luminous flux (lumen) = Radiant flux (watt) × Relative luminosity × 685 lumen/watt

Example: For a 10W source at 600 nm with relative luminosity of 0.6: Luminous flux = 10W × 0.6 × 685 lumen/watt = 4,110 lumen

Luminous intensity (I) is the measure of the luminous flux emitted by a source in a particular direction per unit solid angle.

Mathematical definition: $I = \frac{dF}{d\omega}$

Where:

• I is the luminous intensity (measured in candela or lumen/steradian)
• dF is the luminous flux emitted within a small solid angle
• dω is the solid angle (measured in steradians)

For an ideal point source that emits uniformly in all directions with total luminous flux F: $I = \frac{F}{4\pi}$

Luminous intensity is one of the seven base units in the SI system, with the standard unit being the candela (cd).

The inverse square law in photometry states that the illuminance (E) at a point is inversely proportional to the square of the distance from the light source.

Key relationships:

• For a point source with luminous intensity I, the illuminance E at distance r is: $E = \frac{I}{r^2} \cos\theta$ where θ is the angle between the light ray and surface normal

• This law derives directly from the solid angle geometry:

  • The solid angle subtended by a surface area A at distance r is: dω = (A·cosθ)/r²
  • As light spreads out, the same luminous flux covers larger area at greater distances
  • The solid angle remains constant, but the illuminated area increases with r²

Physical explanation:

• Light energy spreads out spherically from a point source
• At twice the distance, the same light flux spreads over four times the area
• At three times the distance, it spreads over nine times the area

Applications include calculating required lighting levels at different distances and determining light source placement in architectural and stage lighting design.

Solid Angle as the Core Relationship

Solid angle (ω) measured in steradians (sr) connects luminous intensity and flux through:

1. Fundamental Definitions:

   • Solid angle: A 3D angular measure equal to area intercepted on a sphere divided by radius squared (ω = A/r²)
   • Luminous intensity (I): Flux per unit solid angle [candela, cd]
   • Luminous flux (Φ): Total light energy emitted per unit time [lumens, lm]

2. Mathematical Relationship:

   • Intensity is the differential of flux with respect to solid angle: I = dΦ/dω
   • Flux through any solid angle: Φ = I × ω

3. For Point Sources:

   • With uniform intensity in all directions: Φ_total = I × 4π sr
   • With non-uniform intensity: Φ = ∫I(ω)dω (integrated over all directions)

Practical Applications:

• Illuminance calculations (the flux incident on a surface)
• Inverse square law modeling
• Light distribution pattern analysis
• Spotlight and beam angle specifications

Note: This relationship is why the candela (cd) is defined as 1 lumen per steradian, highlighting how solid angle serves as the conversion factor between these fundamental photometric quantities.