Filter center-wavelength shift with angle of incidence

Enter an interference filter's center wavelength at normal incidence, its effective index and the angle of incidence. The calculator gives the shifted center wavelength, the tilt that moves the filter to a target wavelength, and, for a converging beam, the cone half-angle, the edge-ray shift and the mean shift across the pupil.

Inputs

Filter

Center wavelength at normal incidence λ₀

The filter's CWL (or edge position) at 0°. Allowed: 100 to 20000 nm.

Effective index of the filter n*

Depends on the coating design, not only on its materials: typically between the low and high layer indices. Ask the supplier for it, or fit it to measured curves at two angles. Allowed: 1 to 4.

Index of the medium the light arrives from n₀

1.00 for air. Use the cement or immersion index if the filter is bonded or immersed. Allowed: 1 to 2.5.

Beam geometry

Angle of incidence θ

Angle between the beam axis and the filter normal, in the incidence medium. Allowed: 0 to below 90 °.

Working f-number of a converging beam (0 = collimated) N

For a filter in a focused or diverging beam. 0 skips the cone results. Allowed: 0 to 100.

Tilt tuning (optional)

Target center wavelength (0 = skip) λ_t

The wavelength you want the filter to sit at after tilting. Tilt only moves a filter to shorter wavelengths. Allowed: 0 to 20000 nm.

Results

λ(θ), Center wavelength at this angle: 529.991nm
Δλ, Shift from normal incidence: −2.009nm
θ_t, Tilt that moves the filter to the target: 9.98°

Shift at this angle

Center wavelength at this angle λ(θ)	529.991	nm
Shift from normal incidence ΔλNegative: tilt always moves an interference filter to shorter wavelengths.	−2.009	nm
Blue shift as a share of λ_0	0.378	%
Limit of the shift as the angle approaches 90° λ(90°)No tilt can move the filter below this wavelength.	460.73	nm

Tilt to reach the target

Tilt that moves the filter to the target θ_t	9.98	°

Converging beam

Cone half-angle, arctan(1/(2N)) α	n/a	°
Center wavelength for the steepest ray, θ + α	n/a	nm
Center wavelength for the shallowest ray At θ − α, or at normal incidence (λ_0) when the cone contains the normal.	n/a	nm
Edge-ray shift from λ_0	n/a	nm
Spread of center wavelengths across the cone Specify a bandwidth wider than this, or move the filter to collimated space.	n/a	nm
Mean shift the cone adds (small-angle estimate) Δλ_mean−λ_0·α²/(4n*²) for a uniformly filled circular pupil. Exact for small angles at normal incidence; for a small tilt, add it to the tilt shift.	n/a	nm

Method and limits

Tilt: $\lambda(\theta)=\lambda_0\sqrt{1-(n_0\sin\theta/n^*)^2}$ . The passband always moves to shorter wavelengths, and a lower effective index $n^*$ moves it faster. Two filters with the same CWL but different coating designs can have different $n^*$ and so shift by different amounts.

Tilt tuning: the inverse, $\theta=\arcsin[(n^*/n_0)\sqrt{1-(\lambda_{\mathrm t}/\lambda_0)^2}]$ , gives the tilt that moves the filter to a target wavelength. Order a filter slightly long and tilt it into place.

Converging beam: the cone half-angle is $\alpha=\arctan(1/(2N))$ . The steepest and shallowest rays in the plane of tilt bound the spread of center wavelengths, from Eq. (7). The pupil-averaged extra shift, $-\lambda_0\alpha^2/(4{n^*}^2)$ , is the small-angle estimate for a uniformly filled circular pupil; in the guide's f/2 worked example it agrees with the exact pupil average to the two decimals shown.

The effective index is a property of the coating design. Ask the supplier for it, or fit it to curves measured at two angles.

Disclaimer

This calculator uses the single-effective-index model of a tilted interference filter. It is a good approximation at small and moderate angles for one polarization state, and only approximate at large angles, where the s and p passbands separate and the bandwidth changes. The cone results assume a uniformly filled circular pupil and ignore the change of bandwidth and peak transmission. The effective index depends on the coating design. Results are estimates for early design. Confirm them against the supplier's measured curves at your angles. Provided as is, without warranty.

References

P. H. Lissberger and W. L. Wilcock, "Properties of all-dielectric interference filters. II. Filters in parallel beams of light incident obliquely and in convergent beams", J. Opt. Soc. Am. 49, 126–130 (1959), doi:10.1364/JOSA.49.000126.
H. A. Macleod, Thin-Film Optical Filters, 5th ed., CRC Press, 2017 (tilted coatings).
Companion guide: "Optical filter selection guide" (Sample 2), sections "Angle of incidence" and "Converging and diverging beams", Eq. (7) and (10).

Equations

Wavelength shift

\displaystyle \lambda(\theta) = \lambda_0\sqrt{1-\left(\frac{n_0\sin\theta}{n_{\mathrm{eff}}}\right)^2}

\displaystyle \Delta\lambda = \lambda(\theta)-\lambda_0

n₀ is the incident-medium index; nₑff is the coating's effective index. Trigonometric functions use radians.

Tilt to a target wavelength

\displaystyle \theta_{\mathrm t} = \arcsin\left[\frac{n_{\mathrm{eff}}}{n_0}\sqrt{1-\left(\frac{\lambda_{\mathrm t}}{\lambda_0}\right)^2}\right]

The target must be positive and no longer than λ₀, with an arcsine argument below 1.

Converging beam

\displaystyle \alpha = \arctan\left(\frac{1}{2N}\right)

\displaystyle \theta_{\mathrm{hi}} = \theta+\alpha

\displaystyle \theta_{\mathrm{lo}} = \max(\theta-\alpha,0)

\displaystyle \Delta\lambda_{\mathrm{edge}} = \lambda(\theta_{\mathrm{hi}})-\lambda_0

\displaystyle \text{Spread} = \lambda(\theta_{\mathrm{lo}})-\lambda(\theta_{\mathrm{hi}})

\displaystyle \Delta\lambda_{\mathrm{cone}} \approx -\frac{\lambda_0(n_0\alpha)^2}{4n_{\mathrm{eff}}^2}

N is the f-number. The last relation is the small-angle pupil-average estimate; cone bounds must remain within the model's angular domain.

Calculation steps and constants

require  f_num == 0 or f_num >= 0.5
th = theta * rad_per_deg
ns0 = n_0 * sin(th)
require  ns0 < n_eff
lam = lam0 * sqrt(1 - (ns0 / n_eff)^2)
dlam = lam - lam0
drop_pct = (lam0 - lam) / lam0 * pct
lam_min = if(n_eff > n_0, lam0 * sqrt(1 - (n_0 / n_eff)^2), 0 * lam0)
r_t = lam_t / lam0
arg_t = if(lam_t > 0 and r_t <= 1, n_eff / n_0 * sqrt(1 - r_t^2), 2)
tune_ok = lam_t > 0 and r_t <= 1 and arg_t < 1
theta_t = if(tune_ok, asin(arg_t) * deg_per_rad, 0)
alpha = if(f_num > 0, atan(1 / (2 * f_num)), 0)
th_hi = th + alpha
th_lo = max(th - alpha, 0)
ns_hi = n_0 * sin(th_hi)
cone_ok = f_num > 0 and th_hi < pi / 2 and ns_hi < n_eff
lam_hi = if(cone_ok, lam0 * sqrt(1 - (ns_hi / n_eff)^2), lam)
lam_lo = if(cone_ok, lam0 * sqrt(1 - (n_0 * sin(th_lo) / n_eff)^2), lam)
dlam_edge = lam_hi - lam0
spread = lam_lo - lam_hi
dlam_cone = -lam0 * (n_0 * alpha)^2 / (4 * n_eff^2)
th_hi_deg = th_hi * deg_per_rad
th_max = max(theta, theta_t, if(cone_ok, th_hi_deg, 0))

constants:
    rad_per_deg = 0.0174533 rad/°
    deg_per_rad = 57.2958 °/rad
    pct = 100 %
    theta_note = 15 °

Jezdan Gomez · Technical documents and tools for photonics

Filter AOI shift v1.0.0 · calc engine 1.0.0 · config 0a663b0cd4ba