The optical phase accumulated along a path is computed from the optical path length (OPL): φ = (2π/λ₀) · OPL. For a plate of thickness d at oblique incidence, the transmitted angle θt follows Snell’s law: n₀ sin(θi) = n sin(θt).
The path through the plate is longer by 1/cos(θt), so: OPLm = n·d / cos(θt). A useful reference is the same thickness in the surrounding medium: OPLref = n₀·d / cos(θi). The relative phase delay is: Δφ = (2π/λ₀) · (OPLm − OPLref).
Relative time delay is computed from Δt = (OPLm − OPLref)/c. If a group index ng is provided, group delay uses tg = ngd/(c cosθt).
- Enter the vacuum wavelength and select its unit.
- Enter plate thickness or segment length and its unit.
- Set refractive indices for the material and reference medium.
- Add an incidence angle to model tilted plates or beams.
- Optionally enter a group index to estimate group delay.
- Click Calculate Phase Delay to view results above the form.
- Use the CSV and PDF buttons to export your computed results.
| λ₀ (nm) | d (mm) | n | n₀ | θi (deg) | Δφ (rad) | Δt (ps) |
|---|---|---|---|---|---|---|
| 632.8 | 1.0 | 1.50 | 1.00 | 0 | ≈ 4.963e+3 | ≈ 1.667e+0 |
| 1550 | 2.0 | 1.45 | 1.00 | 10 | ≈ 3.641e+3 | ≈ 2.999e+0 |
| 532 | 0.5 | 1.33 | 1.00 | 30 | ≈ 1.922e+3 | ≈ 5.917e-1 |
1. What phase delay means in optics
Phase delay is the phase advance a light field accumulates while crossing an optical element. It scales with optical path length, so thin optics can still add large phase at short wavelengths.
2. Optical path length and refractive index
Optical path length is OPL = n·L, combining refractive index and ray distance. In a tilted plate, refraction sets the internal angle and the distance becomes d/cos(θt). Use an index value matched to your wavelength to reduce dispersion error. If you evaluate multiple wavelengths, update n each time because Δφ changes roughly as 1/λ when n is constant.
3. Relative phase vs absolute phase
Absolute phase is the total phase inside the material. Relative phase delay compares the plate to an equal-thickness segment in the reference medium. Experiments usually measure this difference between optical arms.
4. Why incidence angle changes the result
Tilting the optic increases internal path length and therefore OPL. Snell’s law links θi and θt, so changing θi changes Δφ. The effect grows with thickness and with shorter wavelengths.
5. Time delay from path difference
Time delay follows Δt = ΔOPL/c. Millimeter-scale plates often add picoseconds of delay compared with air, which matters for pulse overlap, synchronization, and timing scans. For example, a 1 mm plate with n≈1.50 relative to air produces about 1.67 ps of extra delay at normal incidence.
6. Group delay for broadband pulses
Broadband pulses travel with group velocity, so timing is better estimated with ng than n. If provided, the calculator reports relative group delay using the same refraction geometry. This is useful for windows in ultrafast setups, fiber links, and dispersion-managed experiments.
7. Practical ranges and material choices
Typical indices: water ≈ 1.33, fused silica ≈ 1.45, many glasses ≈ 1.50–1.52. At 532 nm the phase per millimeter is higher than at 1550 nm. Use the tool to size compensators, spacers, and wedges, and to match interferometer arms. Higher-index crystals can introduce larger delays, so accurate material data improves alignment and balance.
8. Reporting and exporting results
Record wavelength, thickness, indices, and angles with Δφ and Δt. CSV exports fit spreadsheets and scripts, and PDF exports fit lab notes and reviews. Saving exports with your sample ID and date makes later troubleshooting much faster.
Convert radians to cycles by dividing by 2π. Many instruments report phase modulo 2π, but unwrapped phase is helpful when tuning thickness or angle.
1) Should I use vacuum or in-air wavelength?
Use vacuum wavelength for consistent phase calculations. If you only know in-air wavelength, the difference is small in most labs, but vacuum values align better with refractive-index tables.
2) What is the difference between n and ng?
n sets phase velocity and phase accumulation. ng sets group velocity and pulse timing. Use n for continuous-wave phase, and ng when estimating delays of broadband pulses.
3) Why does the calculator warn about total internal reflection?
Total internal reflection occurs when Snell’s law yields no real transmitted angle. It happens for large incidence angles when light goes from higher to lower index. Reduce angle or adjust indices to proceed.
4) How do I interpret “relative phase delay”?
Relative delay compares the plate to an equal-thickness segment in the reference medium. It is the phase difference you would see in an interferometer if one arm includes the plate and the other does not.
5) Are the phase values wrapped to 0–2π?
No. The reported phase values are unwrapped and can exceed 2π. For a wrapped phase, take the value modulo 2π in your analysis, depending on how your instrument reports phase.
6) What thickness should I enter for a tilted plate?
Enter the physical thickness normal to the surfaces. The calculator accounts for the longer internal path using the transmitted angle, so you do not need to manually increase thickness for tilt.
7) How accurate are results if n varies with wavelength?
Accuracy depends on using the correct n at your wavelength. For dispersive materials, update n (and ng if available) for each wavelength to avoid systematic phase and timing errors.