Environmental Stability

Water filling the pores of a film of packing density p raises its index (n = p·n_s + (1 − p)·n_v, n_v = 1 + 0.333 f), temperature moves it by dn/dT and the thickness by α; a level-L surface carries N(≥x) = 10^(0.926(log²L − log²x)) particles per 0.1 m²; an ageing test at T_test stands for AF = (RH_test/RH_use)ⁿ·exp[(Eₐ/k_B)(1/T_use − 1/T_test)] times as long in use.

Model: Textbook approximation

Spectral shift: temperature and moisture

n_H, dry (e.g. TiO₂)n_L, dry (e.g. SiO₂)n_substraten_incidentNumber of pairs (N)Design λ₀ (nm)Packing density p_HPacking density p_L

H layer mixing law

L layer mixing law

Pore volume filled with water (%)Temperature (°C)dn_H/dT (10⁻⁶/K)dn_L/dT (10⁻⁶/K)Expansion α_H (10⁻⁶/K)Expansion α_L (10⁻⁶/K)

The indices are those of the dry film (empty pores, as in vacuum) at 25°C, where each layer is a quarter wave at λ₀. Packing densities are most often 0.8–0.9 for evaporated films and near 1 for ion-assisted and sputtered ones (Macleod 1988). The L defaults are bulk fused silica (Crystran: dn/dT 12.9 × 10⁻⁶/K, α 0.55 × 10⁻⁶/K); the H defaults are placeholders, so enter your film's values. How much of the pore volume fills depends on the humidity, the pore sizes and the time since venting.

Solid-column index: n_s,H = 2.5882, n_s,L = 1.5294 (the index of the dense material the pores are mixed into; values far from the bulk material's index mean the packing density does not fit the film index).

Band-centre shift, moisture

15.32 nm

n_H 2.4000, n_L 1.4999 with 100 % filled

Band-centre shift, 75 °C

0.357 nm

ΔT = 50 K from 25 °C, dry

Combined shift

15.68 nm

Δλ/λ₀ = 2.851 %

Particle contamination (IEST-STD-CC1246)

Cleanliness level (µm)Contaminated surfaces in the path

Obscured area per surface

0.0253 %

Particles 1 µm to the level, face-on spheres

Geometric transmission, 2 surfaces

99.9494 %

(1 − obscuration)^surfaces at normal incidence

Size ≥ (µm)	Particles per 0.1 m²
1	480,901
2	396,408
5	169,688
15	25,190
25	7,455
50	1,021
100	95
250	2

log₁₀ N = 0.926 (log₁₀² L − log₁₀² x): N particles of size ≥ x per 0.1 m², one at the level L. Particles much larger than the wavelength also diffract about as much light again into small angles, so the loss to a well-collimated, aperture-limited beam can approach twice the obscuration. Ma, Fong & Lee's elongated particle shape (a cylinder with hemispherical ends) covers about a quarter less area than spheres.

Accelerated ageing (Arrhenius–Peck)

Use temperature (°C)Test temperature (°C)Activation energy Eₐ (eV)Test duration (h)Use humidity (% RH)Test humidity (% RH)Humidity exponent n (0 = Arrhenius only)

Acceleration factor

96.00

AF = (RH_test/RH_use)ⁿ · exp[(Eₐ/k_B)(1/T_use − 1/T_test)]

Equivalent time in use

9.600e+4 h

11.0 years

One activation energy must govern the failure at both temperatures; Eₐ and n come from your own life tests. Peck fitted them to humidity failures of epoxy-packaged circuits, not to optical coatings.

Model and references

Textbook approximation. A standard approximation from the literature (for example paraxial, plane-wave or Gaussian-beam), valid within the range the page states. About the tiers

Coherent (HL)^N stack by transfer matrix, quarter waves when dry at 25 °C. Water enters through Macleod's linear or Bragg–Pippard mixing law, temperature through dn/dT and free-film expansion (no substrate constraint, stress, hysteresis or dispersion); the H-layer thermal defaults are placeholders. Particles: the CC1246 0.926-slope distribution from 1 µm as face-on spheres (exact for that distribution). Ageing: one activation energy.

Macleod H. A. (1988). Moisture adsorption in optical coatings. NASA conference paper, NTRS 19890003956 (eqs. 3–5). ntrs.nasa.gov
Macleod H. A., Richmond D. (1976). Moisture penetration patterns in thin films. Thin Solid Films 37, 163–169. doi:10.1016/0040-6090(76)90179-6
Macleod H. A. (2010). Thin-Film Optical Filters, 4th ed., ch. 12 (factors affecting layer and coating properties). CRC Press. doi:10.1201/9781420073034
Crystran, fused silica (SiO₂) data sheet: dn/dT 12.9 × 10⁻⁶/K, expansion 0.55 × 10⁻⁶/K. crystran.com
IEST-STD-CC1246E (2013). Product Cleanliness Levels – Applications, Requirements, and Determination. Institute of Environmental Sciences and Technology.
Ma P. T., Fong M. C., Lee A. L. (1990). Surface particle obscuration and BRDF predictions. Proc. SPIE 1165, 381. doi:10.1117/12.962866
Peck D. S. (1986). Comprehensive model for humidity testing correlation. 24th International Reliability Physics Symposium, 44–50. Fit for epoxy-packaged circuits: n 2.66, E_a 0.79 eV (nearly equivalent: 2.50 with 0.81 eV, 3.00 with 0.77 eV). doi:10.1109/IRPS.1986.362110

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