Dispersion and Spectra

Step 1: Use the relationship between focal length and refractive index: $\frac{1}{f} = (\mu - 1)\left(\frac{1}{R_1} - \frac{1}{R_2}\right) = \frac{K}{f}$

Step 2: Calculate the difference in refractive indices: $\mu_v - \mu_r = K\left(\frac{1}{f_v} - \frac{1}{f_r}\right) = K\left(\frac{1}{86.4} - \frac{1}{90.0}\right) = K \times 4.6 \times 10^{-4} \text{ cm}^{-1}$

Step 3: Calculate the average refractive index minus 1: $\mu_y - 1 \approx \frac{K}{2}\left(\frac{1}{f_v} + \frac{1}{f_r}\right) = K \times 1.1 \times 10^{-2} \text{ cm}^{-1}$

Step 4: Calculate dispersive power: $\omega = \frac{\mu_v - \mu_r}{\mu_y - 1} = \frac{4.6 \times 10^{-4}}{1.1 \times 10^{-2}} = 0.042$

Significance: This value (0.042) indicates the lens material's ability to separate different wavelengths. A higher dispersive power means greater chromatic aberration, requiring correction in precision optical instruments.

Manifestations of Dispersion:

1. Wavelength-dependent refraction causing separation of colors
2. Different focal points for different wavelengths
3. Color fringing at boundaries of images

Problems in Imaging Applications:

1. Chromatic Aberration:

   • Longitudinal: Different wavelengths focus at different distances
   • Lateral: Different wavelengths focus at different heights from optical axis

2. Color Fringing: Colored edges around high-contrast boundaries

3. Reduced Resolution: Blurring due to overlapping color focal points

4. Reduced Contrast: Especially in systems requiring precise focus

5. Spectral Artifacts: False colors or missing wavelength information

Solutions include achromatic lenses (crown+flint combinations), apochromatic designs (correcting for three wavelengths), and reflective optics (mirrors that avoid dispersion entirely).

For a light ray passing symmetrically through a prism:

1. The angle of minimum deviation ($\delta$) occurs when the ray passes symmetrically through the prism

2. At this position, the angles of incidence and emergence are equal

3. The mathematical relationship is: $\mu = \frac{\sin\left(\frac{A+\delta}{2}\right)}{\sin\left(\frac{A}{2}\right)}$

Derivation:

• For symmetric passage, angle of incidence = angle of emergence
• From Snell's law and geometric considerations:
  • $\sin i = \mu \sin r$
  • $i + r = A/2$
  • Total deviation $\delta = 2(i-r)$
  • Solving for $\mu$ gives the formula above

This equation allows precise determination of refractive index by measuring the minimum deviation angle experimentally.

Line Absorption Spectrum:

• Appearance: Sharp, discrete dark lines on bright continuous background
• Origin: Atomic transitions where specific wavelengths are absorbed when electrons move from lower to higher energy states
• Examples: Fraunhofer lines in solar spectrum, stellar spectra
• Applications:
  • Identifying chemical elements in astronomical objects
  • Measuring relative velocities via Doppler shifts
  • Determining stellar compositions and temperatures

Band Absorption Spectrum:

• Appearance: Dark bands (groups of closely spaced lines) on bright continuous background
• Origin: Molecular transitions involving vibrational and rotational energy states
• Examples: Absorption by hydrogen gas at moderate temperature, organic compounds in solution
• Applications:
  • Identifying molecular compounds in samples
  • Atmospheric analysis and pollution monitoring
  • Biochemical analysis of organic solutions
  • Pharmaceutical quality control

Key Distinction: Line spectra reveal atomic composition while band spectra reveal molecular structure and bonding characteristics.

Principles of Dispersion Without Average Deviation:

1. When two prisms with angles $A$ and $A'$ are arranged in reverse orientation:

   • Net deviation: $\delta = (\mu_y - 1)A - (\mu'_y - 1)A'$
   • Angular dispersion: $\delta_v - \delta_r = (\mu_v - \mu_r)A - (\mu'_v - \mu'_r)A'$

2. For zero average deviation: $(\mu_y - 1)A = (\mu'_y - 1)A'$

3. For non-zero dispersion: $(\mu_v - \mu_r)A ≠ (\mu'_v - \mu'_r)A'$

Application in Achromatic Lens Design:

1. Combine crown glass (low dispersion) and flint glass (high dispersion) lenses
2. Make one lens converging and one diverging
3. Choose powers to satisfy: $P_{\text{crown}} + P_{\text{flint}} = P_{\text{desired}}$
4. Choose materials and powers so dispersion of one cancels the other
5. Result: Two wavelengths focus at exactly the same point

This principle allows lenses to form images without color fringing, critical for microscopes, telescopes, and precision optical instruments.

To create dispersion without average deviation:

Design approach:

1. Use two prisms with reversed orientations (refracting angles in opposite directions)
2. Choose two different materials with different dispersive powers (ω and ω')
3. Adjust the prism angles (A and A') to satisfy: (μy - 1)A = (μ'y - 1)A'
4. The net angular dispersion will be: δv - δr = (μy - 1)A(ω - ω')

Requirements: • The materials must have different dispersive powers (ω ≠ ω') • The prism angles must be calculated precisely to cancel average deviation • The material with higher dispersive power typically has smaller prism angle

Applications: • Direct vision spectroscopes • Dispersing elements in spectrometers that maintain beam direction • Chromatic correction in optical systems • Prism systems where wavelength separation is needed without beam deflection

This principle is foundational for achromatic and apochromatic lens design.

Violet light undergoes greater deviation because:

• Refractive index decreases as wavelength increases (μ = μ₀ + A/λ²)
• Violet has shorter wavelength (~380-450 nm) than red (~620-750 nm)
• For a thin prism with small angle A, deviation δ = (μ-1)A
• Therefore, δᵥᵢₒₗₑₜ > δᵣₑₐ

The angular dispersion equals: δᵥᵢₒₗₑₜ - δᵣₑₐ = (μᵥ - μᵣ)A

The dispersive power (ω) quantifies this effect: ω = (μᵥ - μᵣ)/(μᵧ - 1)

Example: For flint glass with μᵣ = 1.613, μᵧ = 1.620, μᵥ = 1.632: ω = (1.632 - 1.613)/(1.620 - 1) = 0.019/0.62 = 0.0306

The different focal positions result from wavelength-dependent refraction, explained by:

1. Refractive Index Variation: The refractive index (μ) varies with wavelength according to Cauchy's equation: μ = μ₀ + A/λ² where A is Cauchy's constant and λ is wavelength

2. Dispersion Mechanism:

   • Violet light (shorter wavelength) has higher refractive index than red light (longer wavelength)
   • Higher refractive index causes greater deviation when passing through the dispersing element
   • This creates angular separation between different wavelengths

3. Focal Position Effect:

   • The focusing lens collects parallel rays of each wavelength at different points in its focal plane
   • Violet light focuses closer to the lens than red light due to its greater deviation
   • The separation between focal points depends on the dispersive power of the material

The dispersive power (ω) quantifies this effect: ω = (μᵥ - μᵣ)/(μy - 1), where μᵥ, μᵣ, and μy are the refractive indices for violet, red, and yellow light respectively.

This wavelength-dependent focusing forms the basis for both spectrum analysis and chromatic aberration in optical systems.

• Atomic energy structure:

  • Only electronic transitions contribute significantly
  • Energy levels determined primarily by electron configuration
  • Transitions follow strict selection rules between discrete states

• Molecular energy structure:

  • Total energy = Electronic + Vibrational + Rotational energy
  • Vibrational levels are spaced closer than electronic levels (roughly 0.1-0.5 eV)
  • Rotational levels are even more closely spaced (roughly 0.001-0.01 eV)
  • Each electronic state has associated vibrational and rotational sublevels

• Spectroscopic implications:

  • Atomic spectra: Used for elemental identification and abundance measurements
  • Molecular spectra: Used for molecular structure determination, chemical analysis
  • Analytical technique selection depends on target information:
    • Atomic absorption for metal concentration (e.g., mercury in water)
    • Molecular spectroscopy for organic compound identification (e.g., IR spectroscopy for functional groups)

Note: The complexity of molecular spectra requires higher resolution instruments but provides more structural information than atomic spectra.

The refractive index determination using minimum deviation requires:

Formula: μ = sin[(A + δₘᵢₙ)/2] / sin(A/2) Where:

• μ = refractive index of the prism material
• A = apex angle of the prism
• δₘᵢₙ = angle of minimum deviation

Required conditions:

1. Light must be monochromatic (single wavelength)
2. The prism must be positioned at exactly the minimum deviation angle
3. The refracting edge of the prism must be perpendicular to the plane of the incident and refracted rays

Procedure steps:

1. Measure the apex angle (A) of the prism
2. Rotate the prism and telescope to find the position where moving the prism slightly in either direction increases the deviation
3. Record the angle between the incident beam and deviated beam at this position (δₘᵢₙ)
4. Apply the formula to calculate the refractive index

Note: This method is wavelength-dependent. For dispersive materials, repeating the process with different wavelengths allows plotting of a dispersion curve (μ vs. λ), which characterizes the optical properties of the material.

Accuracy considerations: The minimum deviation position is a turning point, making it precisely determinable even with small measurement uncertainties, yielding highly accurate refractive index values.