Cone Half-Angle (CHA) is the maximum angle formed between the optical axis and the marginal ray (the outermost ray) of a converging or diverging cone of light. It defines the angular limit of a light beam as it enters or exits an optical component.
In geometric optics, it serves as a fundamental measure of the "steepness" or "spread" of a focused beam. If you visualize a focused beam of light as a cone, the CHA is the angle measured from the center of that cone out to its edge.
CHA is rarely used in isolation; it is mathematically tied to the two most common industry metrics for light-gathering ability: Numerical Aperture (NA) and f-number (f/#).
Relationship to Other Optical Metrics
CHA is rarely used in isolation; it is mathematically tied to the two most common industry metrics for light-gathering ability: Numerical Aperture (NA) and f-number (f/#).
1. Numerical Aperture (NA)
Numerical Aperture is the standard metric for microscope objectives and optical fibers. It is derived directly from the CHA.
- Formula: NA = n * sin(θ)
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Where:
- NA = Numerical Aperture
- n = Refractive index of the medium (approx. 1.0 for air)
- θ= Cone Half-Angle
Calculation: To find the CHA from a known NA in air:
- theta = arcsin(NA)
2. f-number (f/#)
In camera lenses and photography, the "speed" of a lens is described by the f-number.
- Approximation: f/# ≈ 1 / (2 * tanθ)
Distinction: CHA vs. Angle of Incidence (AOI)
It is critical to distinguish between Cone Half-Angle and Angle of Incidence, particularly when selecting optical filters, as they describe different lighting conditions.
- AOI (Angle of Incidence): Refers to collimated light (parallel rays) striking a surface at a single, specific angle relative to the normal. For example, tilting a mirror at 45° creates a 45° AOI for all rays.
- CHA (Cone Half-Angle): Refers to non-collimated light (a cone). In this scenario, the optical component simultaneously experiences rays ranging from 0° (center) all the way up to the maximum theta (edge).
Impact on Optical Filters
When thin-film interference filters (such as bandpass filters) are used within a cone of light (non-zero CHA), the variation in ray angles causes performance degradation compared to collimated light.
- Blue Shift: The center wavelength (CWL) of the filter shifts toward shorter wavelengths.
- Bandwidth Broadening: Because the cone contains rays at many different angles, the resulting spectrum is a weighted average of multiple shifted spectra. This "smearing" effect widens the passband and reduces peak transmission.
Rule of Thumb: If the CHA exceeds 5° to 7°, specialized filters designed for high-NA environments are often required to prevent signal loss.
Practical Examples
High-Power Microscope Objective
Microscope objectives often have very high NAs, resulting in steep Cone Half-Angles.
- Component: 40x Objective
- Specification: NA = 0.65
- Medium: Air (n ≈ 1.0)
Calculation:
- theta = arcsin(0.65)
- CHA ≈ 40.5°
Significance: A standard optical filter placed in this light path would experience extreme blue shift because rays are hitting it at angles up to 40.5°.