Athermal bond thickness for axisymmetric optical elements

Tutorial by Eric Frater

Introduction

• Motivations– Survival of optics– Survival of bond– Performance of optics

• Concerns– Thermal stress• Radial stress• Shear stresses

– Glass distortion

rz r 𝜃

Example design

• Cell: Aluminum 6061-T6• Optic: Schott N-BK7• Adhesives:

– MG chemicals RTV 566– 3M 2216 B/A (gray)

• Design:– Bond provides constraint– Uniform and continuous

bond-line– Zero-strain in materials at

nominal bonding temp

Material constants

MATERIAL α(ppm/°C) Poisson ratio, ν E (Gpa)

N-BK7 7.1 .21 826061-T6 24 .33 692216 B/A (gray)[1] 102 ~.43 69RTV 566 200 ~.499 ~.003

αcr cαbαor o

Quick note: 2216 B/A and RTV 566 very different adhesives. As seen in the table, RTV compliance highly dependent on aspect ratio of bond.

Subscript notation:

“c”: cell“b”: bond“o”: optic

Required:αb > αc > αo orαb = αc = αo

[1] Yoder, Paul R. Mounting Optics in Optical Instruments

Bayar equation

• Consider positive ΔT in example design• Assume bond only radially constrained • Require:

ΔT

z r𝜃 r

Example: h= 2.75mm (2216 B/A), h= 1.22mm (RTV 566)

ΔT(Bayar equation[2])

Note: This vastly over-predicts thickness, neglects ν [2] Bayar, Mete. “Lens Barrel Optomechanical

Design Principles”

Radial strain and Hooke’s Law

• From Bayar equation (valid in all cases):

• Radial stress from Hooke’s Law:

Athermalizing:– Define εr and εθ – Set radial stress equal to zero (pre-factor drops out)– Solve for athermal bond thickness h

Van Bezooijen equation

• Assume bond is perfectly constrained in r, z, θ

• Solving for σr=0, . ΔT

z r𝜃 r

ΔT

Example: h= 1.03mm (2216 B/A), h= 0.40mm (RTV 566)

(van Bezooijen equation[3])

Note: This under-predicts thickness, neglects axial bulging of bond [3] Van Bezooijen, Roel. “Soft Retained AST Optics”

Modified van Bezooijen equation

• Assume bond is perfectly constrained in r, θ and unconstrained in z

• Solving for σr=0, . ΔT

z r𝜃 r

ΔT

Example: h= 1.50mm (2216 B/A), h= 0.60mm (RTV 566)

(modified van Bezooijen equation[4])

Note: This over-predicts thickness, allows excessive axial bulging [4] Monti, Christpher L. “Athermal bonded mounts:

Incorporating aspect ratio into a closed-form solution”

Aspect ratio

• Aspect ratio and axial constraint:– Part of bond expands freely

in z– Middle section is perfectly

constrained in z– Modifies the axial strain

z r𝜀𝑧=∆𝑇 (1− h𝐿 )(𝛼𝑏−

𝛼𝑜+𝛼𝑐

2 ) h=𝑟𝑜(𝛼𝑐−𝛼𝑜)

𝛼𝑏−𝛼𝑐+𝜈1−𝜈 (2− h𝐿 )(𝛼𝑏−

𝛼𝑜+𝛼𝑐

2 ).

Varies from 1-2 between limits of van Bezooijen eq.’s

𝑅𝑎𝑠𝑝𝑒𝑐𝑡=𝐿h

Unconstrained in z

if h=L

Closed-form aspect ratio approximation

ΔT

z r𝜃 r

ΔT

Example: h= 1.13mm (2216 B/A), h= 0.41mm (RTV 566)

(Aspect ratio approximation[4])

Note: Provides a best-guess for h in closed-form[4] Monti, Christpher L. “Athermal bonded mounts: Incorporating aspect ratio into a closed-form solution”

Conclusions

• Bayar equation– Good conceptual starting

point– Tends to vastly over-

estimate h– Applicable to highly

segmented bonds

THICKNESS EQUATION 2216 B/A RTV 566

Bayar 2.75 mm 1.22 mm

van Bezooijen 1.03 mm 0.40 mm

Modified van Bezooijen 1.50 mm 0.60 mm

Aspect ratio approximation 1.13 mm 0.41 mm

• Van Bezooijen equation– Takes all strains into account– Much more accurate than Bayar eq.– Under-predicts h due to bulk effects

• Aspect ratio approximation– Approximates varying bulk effects due to aspect ratio of bond– Matches empirical FEA-derived corrections to van Bezooijen

eq. well for >4 and ν [5] Michels, Gregory, and Keith Doyle. “Finite Element Modeling of Nearly Incompressible Bonds”

References

1. Yoder, Paul R. Mounting Optics in Optical Instruments, 2nd ed. SPIE Press Monograph Vol. PM181 (2008), p. 732.

2. Bayar, Mete. “Lens Barrel Optomechanical Design Principles”, Optical Engineering. Vol. 20 No. 2 (April 1981)

3. Van Bezooijen, Roel. “Soft Retained AST Optics” Lockheed Martin Technical Memo

4. Monti, Christpher L. “Athermal bonded mounts: Incorporating aspect ratio into a closed-form solution”, SPIE 6665, 666503 (2007)

5. Michels, Gregory, and Keith Doyle. “Finite Element Modeling of Nearly Incompressible Bonds”, SPIE 4771, 287 (2002)