3) 50 mm Disks. Four 50 mm diameter, 2 mm thick disks were specially polished by Meller Optics to try to obtain strengthening disks. A previous polishing attempt by Meller produced the disks whose failure test results are presented in Fig. 12. Whereas these disks were superior to standard disks, they were not consistently strengthened. SEM inspection showed that some of the disks were scratched. Analysis of the polishing process indicated that the thinness of the disks resulted in a close spacing of the polishing fixtures that caused debris to enter the polishing interface and rescratch the disks. To avoid this problem 2 mm disks were polished.
Two disks were pressure tested in the standard hydraulic fixture, measuring the disk deflection with the apparatus illustrated in Fig. 5. Deflection is accurately measured to an accuracy of 0.01 mm, but O-ring compression is always a complicating factor. Both disks failed at a relatively low pressure: 2.9 and 1.9 MPa. Modeling predicted that standard strength sapphire would fail at approximately 3.4 MPa, so these disks showed no indication of strengthening. Possible causes of the low failure pressure include inherent weakness of R-plane disks, residual stress in the disks, and inadequate polish. Further work will have to be done to resolve this problem. The measured deflection was comparable with O-ring compression, so that true disk deflection was small, consistent with theory.
One of the disks was then inspected in the SEM; again showing problems with the polish at the edge of the disk. In this case there were no residual scratches as seen with the smaller disks, but there were many defects at the edge, decreasing with the distance away from the edge. The damaged region extended to between 2.5 to 5 mm from the edge. This accounted for the poor pressure results and forced TvU to begin its own internal polishing development.
Minimum Thickness Design As a result of the work of this program, a series of effects can be taken advantage of when designing minimum thickness sapphire windows. The first effect is polish strengthening of the sapphire. The second effect is that when the disks are thin enough so that their deflection under load is large compared with their thickness, membrane forces lead to a reduction in the maximum stress at the center of the disk relative to only bending forces. The disks can then be designed to be thinner to achieve the same central stress as was the case without membrane effects. The final effect is that some bending from loading of the part of the disk beyond the aperture is transmitted to the center of the disk reducing the maximum stress there, again allowing a thinner disk to be designed to achieve the same maximum stress in the disk. In terms of stress modeling, the boundary conditions on the disk are designed to minimize the peak stress in the disk, while the thickness of the disk is specified such that the peak stress in the disk is equal to the failure strength of the disk divided by some safety factor. Some care must be taken with the use of larger disks that have loading beyond the aperture, since the design-limiting peak stress may be at the aperture radius rather than the center of the disk.
To design a disk to tolerate a specific pressure, the primary concern is the maximum stress in the disk. The location of the maximum stress is usually either at the radius of the edge of the aperture upon which the disk is mounted or at the center of the disk. If the disk is not clamped in any way, the stress is low at the aperture radius and high at the center of the disk. This is rarely true in a practical case, since the disk must somehow be sealed to the aperture to allow it to be pressure loaded, and the seal provides some degree of clamping. The stress at the aperture edge is much more difficult to predict than the stress at the center of the disk.
One implication of the higher sensitivity of design thickness to failure stress and load for large deflection theory is that the design values for these parameters must be optimized as much as possible. This is difficult to do in terms of the failure strength of standard sapphire because this strength is statistical, as it is for most ceramics. However, since sapphire (and ceramics) are less sensitive to fatigue, disks can be tested to eliminate the weakest samples, and the surviving disks can be used based on their actual tested strength. In the case of a standard double window for a high power microwave tube the pressure loading on the window can be minimized by not operating the cooling liquid at pressures above one atm, and by designing a pressure relief system so that overpressure cannot occur. If the disks are pretested for strength and the cooling system cannot exceed 1 atm operating pressure, then common engineering safety factors can also be reduced, perhaps from 100% to 50% or even 25%. Since the windows are pretested they must possess at least some margin of safety themselves, which would supply some of the engineering safety factor beyond that specified.
The key point is that the factor of 3 extra strength gained from the polishing allows the sapphire to be thin enough at low pressures (5 atm) to function partially as a membrane rather than as a flexing disk. A flexing disk is in compression on one side and in tension on the other, whereas a membrane is in tension at the center throughout the disk. As a result, the membrane can distribute the stress over the whole thickness, greatly reducing the peak stress compared with a flexing disk and allowing a much thinner disk to be used for the same design pressure. No other material except diamond is strong enough to be in the membrane regime, making strengthened sapphire unique as a low-loss material for μ W windows. This says nothing about the additional reduction in absorption losses for sapphire at cryogenic temperatures, where it also becomes somewhat stronger.
Comparison with Current Practice. The baseline case is an edge-mounted sapphire window with a thickness that has been calculated to be half the failure strength of standard sapphire, based on standard bending thin disk theory. Assume that the design strength of sapphire is the accepted commercial design strength of 420 MPa, and assume throughout the discussion that a safety factor of 2 is used. Take the design pressure to be 1 atm - supporting a standard atmosphere against vacuum. This is not the usual design condition for sapphire microwave windows, because they are usually cooled with a low microwave absorption pressurized fluid (5 atm). Pressurization is not necessary, however, and the calculations will show that the benefits of designing for a lower pressure far outweigh the engineering effort necessary to reduce the pressure. If the disk is assumed to be thin and edge mounted with a small deflection under load, then the standard thin flexing disk theory (Eq. 1) predicts a design thickness:
t = [1.22(2w)/σ f]0.5a
This increases the failure strength of the disk by approximately a factor of 3, and thus, by the standard theory, the disks can be made (3)1/2 times thinner and still support the same pressure with the same safety factor.
The failure testing results show very high failure pressures for such thin windows compared with current practice. A standard sapphire microwave window in a 7.5 cm aperture is about 2.5 mm thick [16], supporting 2 atm pressure with a safety factor of 2. This would imply a failure pressure of about 4 atm.
For a sapphire disk with a 44.6 mm aperture radius that was designed to fail at a stress of 800 MPa under a 2 atm pressure, the disk thickness predicted by large deflection theory would be approximately 0.1 mm thick. A standard 100 mm microwave window with a thickness of 2.7 mm can nominally survive almost 500 kW of power, so the new window should be able to survive many MW if it can be made and tested.
Based on the experience of CPI a comparison with state of the art microwave windows can be made. Their 110 GHz double disc sapphire window is expected to be useful to 500 to 600 kW CW. This window fixture uses 3/2 wavelength thick (2.72 mm) sapphire disks with a FC-75 pressure of about 4 atm absolute. An appropriate improvement based on this research would be a 0.1 mm thick or 1/20 wavelength thick, which will certainly be easier to frequency match.
Thermal Design. One of the weaknesses of sapphire is its sensitivity to thermal stress as a result of a combination of a thermal expansion coefficient and thermal conductivity that are both relatively high. Thermal loading of a microwave window is different from the usual application where the heating is external, because the heat is deposited throughout the bulk of the window, primarily at its center where the radial maxima of the electric field occurs for the primary microwave modes (e.g. a gaussian mode).
Thermal stress has been a key factor in limiting the use of sapphire in general, and in particular in the case of sapphire microwave windows. The power limitation on current double sapphire windows cooled with FC-75 is a result of the boiling of this coolant followed by local thermal stress fracture where the coolant boils. The heat is generated either by direct microwave heating of the sapphire or by local arcing.
Thermal Stress Testing. A series of thermal stress tests were performed on the thin sapphire as a result of other work presented at the SPIE conference on Window and Dome Technologies and Materials [17]. The work presented in this paper concerns thermal stress in missile domes as a result of aerodynamic heating. The military has sponsored a great deal of research in an effort to solve this problem. A simple test of thermal stress resistance that is often used is to quench a hot sample in water. The lowest sample temperature that still results in fracture after quenching is a measure of the resistance of a material to thermal stress. The conference report [17] includes this type of data for sapphire. For a 2.5 cm diameter, 2 mm sapphire thick disk with a standard polish quenching at 170°C results in no fracture, whereas quenching from 200°C does cause fracture, and quenching from 250°C causes much more severe fracture. This is a worst case scenario of thermal stress, because quenching causes the outside of the sapphire to be in tension and the inside to be in compression, so that the unpolished edge of the disk provides a weak surface from which cracks can propagate.
Thermal stress in sapphire has been the subject of a great deal of work at TvU. Tests have shown that failure caused by thermal stress varies greatly with the geometry of the piece as well as with the direction and magnitude of the temperature gradient (higher to lower temperature) relative to the piece geometry. Sapphire normally has a much higher compressive strength than tensile strength (except at moderate temperatures around 800°C), and a much lower surface strength than bulk strength. As a result, the same temperature gradients that would result in fracture caused by a surface in tension and internal compression would not harm the part if the gradients were reversed in sign to cause compression at the surface and tension in the interior of the part. Simple tests were done to confirm this and to confirm the standard quench tests.
The basic parameter that describes thermal stress is the product E α Δ T, where E is Young's Modulus, α is thermal diffusivity of the material, and Δ T is the temperature difference in the material. For sapphire E = 345 GPa, and α = 6 x 10-6, so if a failure strength of σ f = 400 MPa is assumed for sapphire, a temperature difference of approximately 200°C would result in thermal stress failure. This compares closely with the water quench tests discussed above.
The most serious problems that are caused by thermal gradients are usually the result of transient conditions such as quenching. A heat pulse applied to a surface cannot be instantaneously conducted away into the material, so, for a short time, the applied heat raises the surface temperature to unusually high values. After a period of time defined by the material's thermal properties, the thermal energy in the hot surface layer is conducted away and the surface temperature drops. The peak surface temperature occurs when the heat input to the surface is balanced by the heat conducted into the material. The higher the material thermal conductivity is the lower will be the maximum surface temperature, given a fixed heat input. The heat input to the surface is in turn controlled by the heat flux available at boundary, and the boundary conductance, h, at the surface. The transient thermal stress response at the surface of a plate in response to a heat input to its face is shown in Fig. 17. The peak thermal stress
Figure 17. Thermal stresses at the surface of a free plate heated symmetrically by an environment at Ta
