Windows thickness

[Phillip J. MacQueen]

Dear Alain,

  There are several formulae for use in determining a window thickness.  I've
used them for many years and am happy with the results.  The equations are
given below, followed by an actual example.

  S=(4*Fa/(K*P))*(t/D)**2

  sag=(3*P*(v**2+4*v-5)/(16*E))*(r**4)/(t**3)

  R=(sag**2 + r**2)/(2*sag)

  tb=D*sqrt(S*K*P/(4*Fa))

where

  S=safety factor                    See text below
  Fa=Tensile strength                Fa=48.92 MPa for fused silica
  K=constant                         K=1.125 for unclamped edges of the window
  P=pressure differential            P=101 KPa for 1 atmosphere.
    across the window
  t=thickness of window
  D=unsupported diameter of window
  r=D/2
  sag=deflection of centre of 
    window under load P
  v=Poisson's ratio (Greek nu)       v=0.17 for fused silica
  E=Youngs's modulus                 E=69.66 GPa for fused silica
  R=radius of curvature of the
    deformed surface
  tb=burst thickness                 The critical thickness at which the
                                     window might can burst from light impacts.

The use of the four equations is basically a matter of iterating until you
are happy with the input values.  The safety factor is the most important
thing.  The best advice I've gathered, and believe, is use a safety factor
below 3 (three) at you own peril.  Given the value of CCDs, I like to add a
safety factor to the safety factor, and usually aim for about S=4.
  Another consideration is optical performance.  If a fast focal ratio beam is
passing through the window, or if the field angle from the optics illuminating
the window is large, you want to keep the window thickness to a low value to
minimize spherochromatism and lateral chromatic aberration.


As an example, the detail of one of my implementations are:

Material:             Fused silica
Unsupported diameter: D=106 mm
Thickness:            t=5 mm
Safety factor:        S=3.83
Central deflection:   sag=71 um
Radius of curvature:  R=19.05 m
Clear diameter:       Dc=D-sqrt(2)*B=111.1 mm
Diameter:             Dw=112.5 mm
Bevel:                B=1.0 mm
Sag over Dc           sag_Dc=81 um

The windows that were made for this case were ground and figured to have a
meniscus shape with each surface having a radius of curvature of R.  They are
installed convex to the atmosphere, and under vacuum, pull down flat with the
window material largely being under compression.

You also asked about window materials.  Don't use sapphire, despite it's
wonderful strength, because it seems to be extremely difficult to get material
which isn't radioactive.

Regarding references for these equation, I'm not sure without quite a bit of
hunting around.  Some of this comes from the ESCO catalogue, some from
mechanical engineering text books, and some from colleagues at various times.


Cheers, 

Phillip MacQueen
pjm at Wairau.as.utexas.edu
McDonald Observatory
The University of Texas at Austin
Austin, Texas 78712, U.S.A.