Optical density and blocking for a stack of filters

Enter the optical density of up to three filters at the wavelength you must block, how much of the blocked light each one reflects, and your power budget. The calculator gives the stack's OD in both layouts: with the reflections between filters removed (tilted, baffled or absorbing elements), where the ODs add, and with the filters parallel, where light bounces between them and the ODs do not add. It compares both with the OD your power budget needs.

Inputs

Filter 1 (first in the beam)

Filter 1 optical density at this wavelength OD₁

The filter's own OD at the wavelength to block, OD = −log10 T, from the measured curve. Allowed: 0 to 15.

Filter 1 share of the blocked light that it reflects ρ₁

100 % for an interference (reflective) blocker that absorbs nothing. A few per cent for absorbing glass, where only the surfaces reflect (about 4 % for uncoated glass of index 1.5). In between for a hybrid. Allowed: 0 to 100 %.

Filter 2

Filter 2 optical density at this wavelength OD₂

0 means no filter. Allowed: 0 to 15.

Filter 2 share of the blocked light that it reflects ρ₂

100 % for a reflective blocker; a few per cent for absorbing glass. Allowed: 0 to 100 %.

Filter 3 (optional: OD 0 means no third filter)

Filter 3 optical density at this wavelength OD₃

Leave at 0 if there is no third filter. Allowed: 0 to 15.

Filter 3 share of the blocked light that it reflects ρ₃

100 % for a reflective blocker; a few per cent for absorbing glass. Allowed: 0 to 100 %.

Power budget at the wavelength to block

Power reaching the stack at this wavelength P_in

For a laser line, the line's power at the filters. Allowed: above 0 to 10000000 mW.

Power that may pass the stack P_allowed

The most leaked power the rest of the system can tolerate. 1 nW = 10⁻⁶ mW. Allowed: above 0 to 1000000000000 nW.

Highest OD the supplier's measurement can verify (0 = no check) OD_meas

A quoted OD can be no better than the dynamic range of the instrument that measured it. Ask the supplier, and enter it here to flag any filter quoted beyond it. Allowed: 0 to 15.

Results

OD_sup, Stack OD, reflections between filters removed (tilted, baffled or absorbing): 10.00
OD_par, Stack OD, filters parallel (light bounces between them): 5.30
OD_req, Required OD, log10(P_in / P_allowed): 8.70

If these filters sit parallel to each other, light reflected between them reaches the output: the stack gives OD 5.30, not OD 10.00. Tilt one filter so its reflection leaves the beam into a dump, put an absorbing element between them, or space and baffle them. Tilting shifts the passband, so check the shift with the angle-of-incidence calculator.

The stack meets the required OD 8.70 only if the reflections between filters are removed (OD 10.00). Mounted parallel it reaches OD 5.30.

Stack OD in the two layouts

Stack OD, reflections between filters removed (tilted, baffled or absorbing) OD_supThe ODs add. Valid only when light reflected by one filter cannot get back into the beam.	10.00
Stack OD, filters parallel (light bounces between them) OD_parIncoherent multiple reflections between parallel filters. Use this value unless the layout removes those reflections.	5.30
OD lost to reflections between parallel filters ΔOD	4.70

Against the power budget

Required OD, log10(P_in / P_allowed) OD_req	8.70
Margin over the requirement, reflections removed Negative means the stack falls short.	1.30
Margin over the requirement, filters parallel	−3.40
Power that passes, reflections removed P_out	0.0500	nW
Power that passes, filters parallel P_out	2.50 × 10³	nW
OD each filter needs if all are equal and reflections are removed	4.35

Method and limits

Each filter's transmittance is $T=10^{-\mathrm{OD}}$ . Its reflectance is $R=\rho(1-T)$ , where ρ is the share of the blocked light it reflects: close to 100 % for an interference blocker that absorbs nothing, a few per cent for absorbing glass.

Reflections removed: when light reflected by one filter cannot re-enter the beam (one filter tilted with its reflection sent to a dump, an absorbing element between reflective ones, or spacing and baffles), the transmittances multiply and the ODs add.

Filters parallel: light reflected back and forth between two filters adds to what gets through, $T_{12}=\frac{T_1T_2}{1-R_1R_2}$ . For two reflective blockers that absorb nothing, $R\approx1-T$ , so the denominator is tiny: two OD 4 filters give about OD 4.3, not OD 8. With a third filter the same rule is applied again, using the reflectance of the first pair as seen from the third filter.

Required OD: log10 of the incident power over the power that may pass. The page compares both layouts with it. Design to the parallel value unless the mechanical layout is known to remove the reflections.

Disclaimer

This calculator treats each filter as a single element with one transmittance and one reflectance at the wavelength entered, the same from both sides, and adds reflections between parallel filters incoherently (no interference between filters). It ignores scatter, reflections from other surfaces in the system, the change of OD with angle and wavelength, and light that goes around the filters. Real layouts fall between the two cases shown. Results are estimates for early design. Confirm them against the supplier's measured curves and by measurement in your system. Provided as is, without warranty.

References

H. A. Macleod, Thin-Film Optical Filters, 5th ed., CRC Press, 2017 (incoherent combination of reflections between separate components; specification of filters).
Companion guide: "Optical filter selection guide" (Sample 2), section "Optical density and blocking", Eq. (1) to (4).

Equations

Individual filters

\displaystyle T_i = 10^{-\mathrm{OD}_i}

\displaystyle R_i = \frac{\rho_i}{100}(1-T_i)

\displaystyle \mathrm{OD}_{\mathrm{sup}} = \sum_{i=1}^{3}\mathrm{OD}_i

ρ is the reflected share of blocked light, entered as a percentage. With reflections removed, optical densities add.

Parallel filters

\displaystyle D_{12} = 1-R_1R_2

\displaystyle T_{12} = \frac{T_1T_2}{D_{12}}

\displaystyle R_{12} = R_2+\frac{T_2^2R_1}{D_{12}}

\displaystyle D_{123} = 1-R_{12}R_3

\displaystyle \mathrm{OD}_{\mathrm{par}} = \mathrm{OD}_{\mathrm{sup}}+\log_{10}D_{12}+\log_{10}D_{123}

R₁₂ is the pair's reflectance as seen from the third filter. The implementation evaluates equivalent loss terms to avoid cancellation at high optical density.

Power budget

\displaystyle \mathrm{OD}_{\mathrm{req}} = \log_{10}\left(\frac{P_{\mathrm{in}}}{P_{\mathrm{allowed}}}\right)

\displaystyle P_{\mathrm{leak}} = P_{\mathrm{in}}\,10^{-\mathrm{OD}_{\mathrm{stack}}}

\displaystyle \text{Margin} = \mathrm{OD}_{\mathrm{stack}}-\mathrm{OD}_{\mathrm{req}}

Use the same power units in the ratio. Leakage and margin are evaluated for both stack arrangements.

Calculation steps and constants

f_1 = rho_1 / pct
f_2 = rho_2 / pct
f_3 = rho_3 / pct
T_1 = 10^(-OD_1)
T_2 = 10^(-OD_2)
T_3 = 10^(-OD_3)
R_1 = f_1 * (1 - T_1)
R_2 = f_2 * (1 - T_2)
R_3 = f_3 * (1 - T_3)
L_1 = (1 - f_1) + f_1 * T_1
L_2 = (1 - f_2) + f_2 * T_2
L_3 = (1 - f_3) + f_3 * T_3
A_2 = (1 - f_2) * (1 - T_2)
OD_sup = OD_1 + OD_2 + OD_3
D_12 = L_1 + R_1 * L_2
T_12 = T_1 * T_2 / D_12
R_12 = R_2 + T_2^2 * R_1 / D_12
L_12 = (L_1 * L_2 + R_1 * A_2 * (L_2 + T_2)) / D_12
D_123 = L_12 + R_12 * L_3
OD_loss = -(log10(D_12) + log10(D_123))
OD_par = OD_sup - OD_loss
OD_req = log10(P_in * nW_per_mW / P_allowed)
leak_sup = P_in * nW_per_mW * 10^(-OD_sup)
leak_par = P_in * nW_per_mW * 10^(-OD_par)
margin_sup = OD_sup - OD_req
margin_par = OD_par - OD_req
n_el = (OD_1 > 0) + (OD_2 > 0) + (OD_3 > 0)
OD_each = if(n_el > 0, OD_req / max(n_el, 1), 0)
OD_top = max(OD_1, OD_2, OD_3)

constants:
    pct = 100 %
    nW_per_mW = 1000000 nW/mW
    gap_warn = 0.1

Jezdan Gomez · Technical documents and tools for photonics

Filter stack OD v1.0.0 · calc engine 1.0.0 · config 6f0ee704f746