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.

Net angular dispersion (δv-δr):

• Formula: (δv-δr) = (μy-1)ωA - (μ'y-1)ω'A'
• Where μy is average refractive index of first prism
• ω and ω' are dispersive powers of the prisms
• A and A' are the refracting angles

Net average deviation (δy):

• Formula: δy = (μy-1)A - (μ'y-1)A'
• For zero average deviation: (μy-1)A = (μ'y-1)A'
• For zero dispersion: (μv-μr)A = (μ'v-μ'r)A'

Application: Used in the design of achromatic prisms and compound lenses that minimize chromatic aberration while maintaining desired optical properties.

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.

The principle involves:

• Two prisms with refracting angles A and A' placed with reversed orientations
• Different dispersive powers (ω and ω') for the two prisms
• When (μy-1)A = (μ'y-1)A', the average deviation (δy) becomes zero
• The net angular dispersion equals (μy-1)A(ω-ω')
• This arrangement separates colors without changing the average direction of light
• Example: A crown glass prism combined with a flint glass prism of smaller angle can achieve this effect, used in achromatic prism systems

Note: This is the foundational principle behind achromatic optical devices that correct chromatic aberration.

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.

When white light passes through a spectrometer:

1. The collimator creates parallel light rays that strike the prism uniformly
2. The prism disperses these rays according to wavelength (refractive index varies with wavelength)
3. Violet light bends more than red light due to its shorter wavelength
4. Each wavelength exits the prism at a different angle

The telescope must be repositioned because:

• Each wavelength travels along a unique path after dispersion
• The telescope can only receive rays traveling along its optical axis
• To observe different colors, the telescope must be rotated to the specific angle where that wavelength's rays are traveling
• This angular separation is what allows precise measurement of spectral lines and identification of elements based on their characteristic emission/absorption patterns

This demonstrates Snell's law and the wavelength dependence of refractive index in action.

Pure vs. Impure Spectrum:

• Pure spectrum: Each wavelength occupies one specific spatial position without overlap; colors appear sharp with distinct boundaries
• Impure spectrum: Wavelengths overlap spatially; colors appear diffused with blended boundaries

How a spectrometer ensures spectral purity:

1. Narrow slit: Restricts the width of the incoming light beam, preventing multiple light paths that would cause spatial overlap

2. Collimation: The collimator lens converts divergent light from the slit into parallel rays, ensuring all rays of a given wavelength strike the prism at the same angle

3. Proper focus: The telescope focuses parallel rays of each wavelength to a single point in its focal plane

4. Precision rotation: The ability to precisely rotate the telescope allows isolation and measurement of specific wavelengths

If the slit were too wide or the collimation imperfect, different points of the slit would produce overlapping spectra, resulting in an impure spectrum with diffused color impressions.

A spectrometer for minimum deviation measurement consists of:

1. Collimator (C):

   • Contains an adjustable slit (S) at one end and an achromatic lens at the other
   • Converts divergent light from the source into a parallel beam
   • The slit must be placed at the focal point of the collimator lens

2. Prism on rotating table:

   • Mounted on an adjustable platform that can rotate precisely
   • Disperses the light according to wavelength
   • Must be positioned so the refracting edge is vertical

3. Telescope (T):

   • Receives the refracted light and forms an image of the slit
   • Can rotate around the same vertical axis as the prism table
   • Contains crosshairs for precise alignment with the refracted image

4. Measurement scales:

   • Angular scales with vernier readings to measure rotation angles
   • Used to determine the difference between direct view and minimum deviation positions

The procedure requires coordinated rotation of both the prism and telescope to find the exact position where deviation is minimized, allowing precise determination of optical properties.