Retarder Types Comparison
Compare zero-order, multi-order, compound, achromatic and Fresnel-rhomb retarders: retardance against wavelength and tilt.
Retardance against wavelength (multi-order folded to one wave)
The compound zero-order curve (dotted) lies on the zero-order one: only d₁ − d₂ counts at normal incidence.
Retardance change against tilt at λ₀
Tilt in the plane holding the first plate's optic axis (the rhomb: in its plane of reflection). Tilting about the other axis changes the sign for the plates.
Comparison at λ₀ = 550 nm, target 0.250 waves
| Type | Plates | Retardance (waves) | Band within ±1 % (nm) | Change at 5° tilt (waves) | Note |
|---|---|---|---|---|---|
| Zero order | quartz 15.0 µm | 0.2500 | 545.1–555.1 | -0.0004 | Thin plate, often cemented to a window. |
| Multi-order (m = 10) | quartz 615.5 µm | 0.2500 | 549.9–550.1 | -0.0163 | One thick plate: right only near λ₀. |
| Compound zero order | quartz 615.5 − 600.5 µm, crossed | 0.2500 | 545.1–555.1 | -0.0321 | Zero-order dispersion, multi-order tilt sensitivity. |
| Achromatic | quartz 292.4 + MgF₂ 237.8 µm, crossed | 0.2524 | 495–625 | 0.0179 | Exact at λ₁ and λ₂. |
| Fresnel rhomb | N-BK7, 2 reflections at 55.35° | 0.2500 | 422–886 | -0.0144 | Total internal reflection: nearly achromatic; displaces the beam. |
Indices: quartz (Ghosh 1999), MgF₂ (Dodge 1984), N-BK7 (SCHOTT). Bands are searched from 300 to 2000 nm. The achromat's curve depends on Δn at the 10⁻⁵ level; real plates are trimmed after polishing.
Key Formulas
Γ = 2π·Δn(λ)·d/λ; order-m plate: d = (m + f)·λ₀/|Δn(λ₀)|
Crossed plates: Γ = 2π(Δn₁d₁ − Δn₂d₂)/λ; achromat: Γ(λ₁) = Γ(λ₂) = target
Tilted plate: Γ = (2πd/λ)(kz,e − kz,o)
Total internal reflection: tan(δ/2) = cos θ·√(sin²θ − 1/n²)/sin²θ
Model and references
Exact. The equations are exact within the stated assumptions, or are a standard's tabulated values. The results are as good as the inputs. About the tiers
Ideal plates, Γ = 2πΔn·d/λ (tilted: k₀d(k_z,e − k_z,o)), Δn(λ) from Sellmeier fits; rhomb phase from total internal reflection in N-BK7. Not modelled: temperature, reflections, coatings.
- Hecht E. (2002). Optics, 4th ed., §8.4 Birefringence and §8.7 Retarders. Addison-Wesley.
- Veiras F. E., Perez L. I., Garea M. T. (2010). Phase shift formulas in uniaxial media: an application to waveplates. Appl. Opt. 49, 2769. doi:10.1364/AO.49.002769
- Born M., Wolf E. (1999). Principles of Optics, 7th ed., §1.5.4 Total reflection (Fresnel rhomb). Cambridge University Press. doi:10.1017/CBO9781139644181.010
- Ghosh G. (1999). Dispersion-equation coefficients for the refractive index and birefringence of calcite and quartz crystals. Opt. Commun. 163, 95–102. doi:10.1016/S0030-4018(99)00091-7
- Dodge M. J. (1984). Refractive properties of magnesium fluoride. Appl. Opt. 23, 1980–1985. doi:10.1364/AO.23.001980
- SCHOTT optical glass catalog (Zemax AGF 2017-01-20): N-BK7 Sellmeier coefficients.