Tolerance is a critical factor impacting the performance and cost of an optical system.  Optical components usually require much tighter tolerances than that commonly associated with mechanical components.  As a result, special equipment and techniques are used in the manufacturing and measuring of the optical tolerances. 

Part 1: Optical element tolerances

Jump to part 2: Modulation transfer function (MTF)

Surface accuracy

When attempting to specify how closely an optical surface conforms to its intended shape, a measure of surface accuracy is needed. Surface accuracy can be determined by interferometric techniques. Traditional techniques involve comparing the actual surface to a the test plate gage. In this approach, surface accuracy is measured by counting the number of rings or fringes and examining the regularity of the fringe. The accuracy of the fit between the lens and the test gage (as shown below) is described by the number of fringes seen when the gage is in contact with the lens. Test plates are made flat or spherical to within small fractions of a fringe. Modern techniques for measuring surface accuracy utilize phase measuring interferometry with advanced computer data analysis software.   During manufacture, a precision component is frequently compared with a test plate that has an accurate polished surface that is the inverse of the surface under test. When the two surfaces are brought together and viewed in nearly monochromatic light, Newton’s rings (interference fringes caused by the near-surface). The number of rings indicates the difference in radius between the surfaces. This is known as power or sometimes as figure. It is measured in rings that are equivalent to half wavelengths. Beyond their number, the rings may exhibit distortion that indicates non-uniform shape differences. The distortion may be local to one small area, or it may be in the form of noncircular fringes over the whole aperture. All such non-uniformities are known collectively as irregularity. 

Surface flatness

Surface flatness is simply surface accuracy with respect to a plane reference surface. It is used extensively in mirror and optical flat specifications.

Centration

The mechanical axis and optical axis exactly coincide in a perfectly centered lens. For a simple lens, the optical axis is defined as a straight line that joins the centers of lens curvature. For a plano-convex or plano-concave lens, the optical axis is the line through the center of curvature and perpendicular to the plano surface. The mechanical axis is determined by the way in which the lens will be mounted during use. There are typically two types of mounting configurations, edge mounting and surface mounting. With edge mounting, the mechanical axis is the centerline of the lens mechanical edge. Surface mounting uses one surface of the lens as the primary stability for lens tip and then encompasses the lens diameter for centering. The mechanical axis for this type of mounting is a line perpendicular to the mounting surface and centered on the entrapment diameter. Ideally, the optical and mechanical axes coincide. The tolerance on centration is the allowable amount of radial separation of these two axes, measured at the focal point of the lens. The centration angle is equal to the inverse tangent of the allowable radial separation divided by the focal length. Centration error is measured by rotating the lens on its mechanical axis and observing the orbit of the focal point. To determine the centration error, the radius of this orbit is divided by the lens focal length and then converted to an angle.

Surface quality

Cosmetic surface quality describes the level of defects that can be visually noted on the surface of an optical component. Specifically, it defines state of polish, freedom from scratches and digs, and edge treatment of components. These factors are important, not only because they affect the appearance of the component, but also because they scatter light, which adversely affects performance. Scattering can be particularly important in laser applications because of the intensity of the incident illumination. Unwanted diffraction patterns caused by scratches can lead to degraded system performance, and scattering of high-energy laser radiation can cause component damage. Over specifying cosmetic surface quality, on the other hand, can be costly. The most common and widely accepted convention for specifying surface quality is the U.S. Military Surface Quality Specification, MIL-0-13830A, Amendment 3.  IMPORTANT: Surface quality can be impacted if improper cleaning is used. 

As stated above, all optics in this catalog are referenced to MIL-PRF-13830B standards. These standards include scratches, digs, grayness, edge chips, and cemented interfaces. It is important to note that inspection of polished optical surfaces for scratches is accomplished by visual comparison to scratch standards. Thus, it is not the actual width of the scratch that is ascertained, but the appearance of the scratch as compared to these standards. A part is rejected if any scratches exceed the maximum size allowed. Digs, on the other hand, specified by actual defect size, can be measured quantitatively. Because of the subjective nature of this examination, it is critical to use trained inspectors who operate under standardized conditions in order to achieve consistent results.

The scratch-and-dig designation for a component or assembly is specified by two numbers. The first defines allowable maximum scratch visibility, and the second refers to allowable maximum dig diameter, separated by a hyphen; for example, 80–50 represents a commonly acceptable cosmetic standard. 60–40 represents an acceptable standard for most scientific research and commercial applications. 10–5 represents a precise standard for very demanding laser applications.

SCRATCHES

A scratch is defined as any marking or tearing of a polished optical surface. In principle, scratch numbers refer to the width of the reference scratch in ten thousandths of a millimeter. For example, an 80 scratch is equivalent to an 8-µm standard scratch. However, this equivalence is determined strictly by visual comparison, and the appearance of a scratch can depend upon the component material and the presence of any coatings. Therefore, a scratch on the test optic that appears equivalent to the 80 standard scratch is not necessarily 8 µm wide. If maximum visibility scratches are present (e.g., several 60 scratches on a 60–40 lens), their combined lengths cannot exceed half of the part diameter. Even with some maximum visibility scratches present, MIL-0-13830A still allows many combinations of smaller scratch sizes and lengths on the polished surface.

DIGS

A dig is a pit or small crater on the polished optical surface. Digs are defined by their diameters, which are the actual sizes of the digs in hundredths of a millimeter. The diameter of an irregularly shaped dig is 1/2#(length plus width): 50 dig = 0.5 mm in diameter 40 dig = 0.4 mm in diameter 30 dig = 0.3 mm in diameter 20 dig = 0.2 mm in diameter 10 dig = 0.1 mm in diameter. The permissible number of maximum-size digs shall be one per each 20 mm of diameter (or fraction thereof) on any single surface. The sum of the diameters of all digs, as estimated by the inspector, shall not exceed twice the diameter of the maximum size specified per any 20-mm diameter. Digs less than 25 micrometers are ignored.

EDGE CHIPS

Lens edge chips are allowed only outside the clear aperture of the lens. The clear aperture is 90% of the lens diameter unless otherwise specified. Chips smaller than 0.5 mm are ignored, and those larger than 0.5 mm are ground so that there is no shine to the chip. The sum of the widths of chips larger than 0.5 mm cannot exceed 30% of the lens perimeter. Prism edge chips outside the clear aperture are allowed. If the prism leg dimension is 25.4 mm or less, chips may extend inward 1.0 mm from the edge. If the leg dimension is larger than 25.4 mm, chips may extend inward 2.0 mm from the edge. Chips smaller than 0.5 mm are ignored, and those larger than 0.5 mm must be stoned or ground, leaving no shine to the chip. The sum of the widths of chips larger than 0.5 mm cannot exceed 30% of the length of the edge on which they occur.

CEMENTED INTERFACES

Because a cemented interface is considered a lens surface, specified surface quality standards apply. Edge separation at a cemented interface cannot extend into the element more than half the distance to the element clear aperture up to a maximum of 1.0 mm. The sum of edge separations deeper than 0.5 mm cannot exceed 10% of the element perimeter.

BEVELS

Although bevels are not specified in MIL-0-13830A, our standard shop practice specifies that element edges are beveled to a face width of 0.25 to 0.5 mm at an angle of 45°±15°. Edges meeting at angles of 135° or larger are not beveled.

COATING DEFECTS

Defects caused by an optical element coating, such as scratches, voids, pinholes, dust, or stains, are considered with the scratch and-dig specification for that element. Coating defects are allowed if their size is within the stated scratch-and-dig tolerance. Coating defects are counted separately form substrate defects.

Part 2: Modulation Transfer Function (MTF)

The modulation transfer function (MTF) is a quantitative measure of image quality. MTF describes the ability of a lens or system to transfer object contrast to the image. Consider a sine-wave chart in the form of a positive transparency in which transmittance varies in one dimension. Assume that the transparency is viewed against a uniformly illuminated background. The maximum and minimum transmittances are Tmax and Tmin, respectively. A lens system under test forms a real image of the sine-wave chart, and the spatial frequency (u) of the image is measured in cycles per millimeter. Corresponding to the transmittances Tmax and Tmin are the image irradiances Imax and Imin. The contrast or modulation of the chart and image are defined, respectively, as

where Mc is the modulation of the chart and Mi is the modulation of the image. The modulation transfer function of the optical system at spatial frequency u is then defined to be

MTF curves can be either polychromatic or monochromatic. Polychromatic curves show the effect of any chromatic aberration that may be present. For a well-corrected achromatic system, polychromatic MTF can be computed by weighted averaging of monochromatic MTFs at a single image surface. MTF can also be measured by a variety of commercially available instruments.

The MTF curve for a perfect imaging lens is only limited by the laws of diffraction (diffraction limited performance).  For such a system, the theoretical MTF is calculated as follows:

Calculate diffraction MTF using MTF wizard

To achieve this level of performance,  the optical design must be free of any aberrations, and the manufacturing process must maintain very tight tolerances.  This requires a large number of lens elements with compensating aberrations. For most commercial applications, the lens MTF is far from diffraction limit.  The following diagram shows the design MTF of a practical lens vs. the diffraction limit (the black line is the diffraction limit, blue line is the MTF on-axis and green lines off-axis at 60 deg at tangential and saggital target orientations):

 

Since a lens MTF varies with the field angle, it is often more useful to exam the MTF at specific spatial frequencies vs. field position.  This plot shows the consistency of lens performance across the entire image plane.  It is useful to select two (a high and low) spatial frequencies.  The low frequency MTF value represents the overall contrast of the image, and high frequency MTF represents the capability of the lens to produce details.  An example of such a plot is as follows (green lines are MTF values at 20 cycle/mm and the blue lines at 60 cycle/mm.  The solid lines are for saggital target orientation. Dash lines for tangential orientation):

Simulate impact of MTF to a line pair target