Optical Design and Stray Light Concepts and Principles

Abstract

To insure that an optical system performs to specifications, the optical engineer needs to fully consider several aspects of the design process. Each of these tasks can be aided with the use of software tools. The optical engineer needs to understand the strengths and limitations of the available software tools and how to best apply these programs to each design. There are several distinct steps in the implementation of an optical system: the first order optical layout, optimized design of the optical system, performing stray, scattered and ghost analysis, performing a tolerance analysis of the optical system and performing manufacturing analysis. Although each of these steps are often considered separately, and often require the use of several different software tools, it is imperative for the engineer to consider the entire process during each phase of system development so that issues arising from stray light or manufacturing tolerances do not force a redesign of the system. A thorough understanding of the optical design and analysis process as well as the proper use of available optical software tools are necessary to insure the optimum optical design for each specific application.

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

The design of an optical system is comprised of several distinct but interrelated steps. The initial phase in the optical design process is the design of the optical system. That is, establishing a set of optical parameters for each necessary optical component so that the designʼs performance level exceeds the design specifications. Another phase of the design process is to perform a tolerance analysis to verify that the system will still perform to specification when the limitations of material properties, manufacturing and assembly are considered. Another important phase is a separate analysis to consider the effects of stray light due to ghost reflections, scatter, and the design of proper baffling.

The optical designer is aided in each of these steps with computer programs specifically written for these tasks. The design and tolerancing of an optical system is generally performed using software referred to as sequential optical design programs or lens design codes. Stray light analysis is performed using a non-sequential, or unconstrained, optical analysis program. We will consider the capabilities and requirements of each of those optical programs.

The Design Process

The process of designing an optical system consists of establishing a set of optical components with optical properties, including materials (indices of refraction, dispersions), surface curvatures, element thicknesses and spacing, surface shape, etc., which in combination with system properties such as focal length, focal ratio, field of view, and others requirements, provide a design framework that will meet (or exceed) the required performance specification.

Although a lens design code is the most efficient method of determining the necessary parameter values, it is important for the designer to perform a first-order predesign before turning the process over to the computer. The predesign serves several purposes. First, it gives the designer another opportunity to consider all aspects of the design specifications closely to ensure that all necessary information and design specifications have been provided and to discuss any missing, incomplete or inconsistent requirements. Understanding the design requirements is a critical step in the design process. If any necessary specifications are uncertain, it is important to discuss any issues with the customer before the design begins. Any judgment call made by the optical designer can be overruled at any time, resulting in redesigns and costly delays. At this time the designer can also make a reasonable determination as to whether or not the desired system is physically realizable.

After determining that all necessary specification information is available, a first-order system layout can be determined (see Table 7.1 for a partial listing). As part of this analysis, it is important to consider any constraints placed on the design that may result in tolerance requirements that may exceed current fabrication limits. Although the full tolerance analysis will be performed after the design has been completed, it is at the designerʼs peril to ignore these aspects during the initial design phase. A similar warning applies to stray light issues as well.

Table 7.1 Basic optical system specifications and requirements

Some of the system parameters that must be clearly understood before beginning the design include focal length, aperture, focal ratio and field of view (including a clear understanding of how the customer defines this parameter), wavelength band. Another necessary parameter is the actual performance requirement. In other words, how well must the system perform under a given set of operating conditions. The performance specification can be based on an encircled or ensquared energy size, modulation transfer function (MTF) values as well as consideration of some optical aberrations, in particular distortion and field curvature. Additional non-optical constraints may be provided including total footprint, volume and weight. Environmental issues such as humidity, pressure, salinity and temperature can degrade the performance of an optical system and can also limit the choices of optical materials as well as surface shape. As with any engineering task, allowable or measurable cost and time constraints will place limitations on the design. Parameters that can significantly impact the cost include aspheric surfaces, certain optical materials, and components with tight fabrication tolerances.

The stop surface is a critical surface in an optical design. The position and size of the stop significantly affect the ability of the design to perform as required. An optical system consists of a series of lenses and/or mirrors. The finite size of these surfaces limits the amount of light that can pass through the optical system. There is one surface that will limit the angular extent of a light bundle entering the system from an on-axis point source. This surface is called the systemʼs aperture stop. The stop can be formed directly by an optical surface in the system or by a separate mechanical surface, or iris. The stop can be located anywhere in the optical system. Two other related surfaces are the entrance pupil and the exit pupil. The entrance pupil is defined as the image of the aperture stop as seen from object space. If the aperture stop is at, or in front of, the first optical surface, the entrance pupil and aperture stop are coincident. The exit pupil is the image of the aperture stop as seen from image space. If the aperture stop is at, or behind, the last optical surface, the exit pupil and aperture stop are coincident.

In specifying an optical system in a lens design code, it is necessary for the designer to input information concerning the size and location of the system aperture. Often the entrance pupil size is used, as this is the parameter used to specify the systemʼs focal ratio. When the entrance pupil is used to specify the aperture, the size and location of the actual stop surface can change during the design process. If the stop surface must be of a specific size, then the aperture is defined by the stop. In this case, the size of the entrance pupil will be determined by the magnification between the stop surface and the entrance pupil. Other aperture definitions may be available in the lens design program for use with other constraints. Figure 7.1 shows a typical optical system with an embedded stop and indicating the location of the entrance and exit pupils, which for this design are virtual surfaces.

Fig. 7.1

Aperture stop and pupils

The first-order optical design is used to provide a starting point for the actual design process. Information concerning the relative location of the system stop, the number of lenses or lens groups, and the effective power of each group is determined. First-order optics defines the properties of an aberration-free optical system. Many basic optical system parameters, such as focal length, focal ratio, magnification, and others, are derived from first-order, or paraxial, optics.

The first-order predesign locates the aperture stop, as well as the pupil and image locations in the optical design by tracing rays through the system. In particular, two specific rays, the marginal ray and the chief ray are used. The marginal ray is a ray that exits from an on-axis object point and travels through the top of the pupil or aperture stop before re-crossing the axis at the image. The marginal ray is used to determine the location of the image plane, the effective focal length and the focal ratio (F/#). The chief ray is a ray that exits from the top of the object and crosses the optical axis at the aperture stop or pupil locations. The chief ray determines the location of the pupil planes, the field of view and the image height. Figure 7.2 shows the marginal and chief rays for the same optical system.

Fig. 7.2

System marginal and chief rays

After performing the predesign to determine the general structure of the system, to include the number of elements, use of lenses and/or mirrors, relative stop location, and the basic system shape, the design of the actual system can be started. The synthesis of an optical system starts from the basic layout determined in the predesign. Using this as the basis, the designer uses the lens design program to determine the optical parameters necessary to meet the required performance specification. The lens design program will then take the information provided by the designer to find the combination of parameter values that best meet the performance goals. During this phase, the designer will need to determine whether the initial basic design had enough free parameters, called variables, to provide an acceptable solution. If not, the process begins anew.

The selection of the initial optical design form is a key role for the optical designer. The lens design program can only search for solutions within the solution space defined by the base design. Often a starting-point design is based on a prior known design used for a similar problem. In some instances an off-the-shelf solution, such as a camera lens or a microscope objective, might be the best option. Patent or literature searches can provide many useful ideas. These can provide information concerning the merits of reflective or refractive designs, the number of optical elements, the relative stop position and other important characteristics. Experience and intuition can also be employed in the predesign.

Design Parameters

The process of determining the optimal parameter values has become the domain of the lens design program. These programs, when used on a standard computer, can easily trace millions of rays through an optical system in seconds. Provided that the system has been properly defined and a proper set of constraints exist, the program can systematically adjust the parameter values until it reaches a solution that best matches the design goals. At this point, the designer needs to evaluate the information provided by the program to determine if the performance goals have actually been met. If not, a determination needs to be made as to whether the goals as defined to the program were sufficient and reasonable. If there were problems in the definition, then the goals need to be redefined before attempting further optimization. If the system and goals were properly defined and a solution was still not found, this indicates that there were an insufficient number of degrees of freedom in the design for a solution to be found. In this case the basic design needs to be reconsidered to allow for additional free parameters. This could also indicate that the design goals may need to be reevaluated.

In designing an optical system, the first and most frequent question asked should be: does this make sense? Design goals must be physically unrealizable. The optical designer needs to be able to recognize these issues and to offer alternative solutions. The designer must also be able to recognize when design goals compromise the performance of the system.

To use an optical design program, it is necessary to provide the program with sufficient information to understand the system under design. Lens design programs are called sequential ray traces because they trace geometrical rays through an optical system in a predefined sequence. These programs are based on a surface, rather than a component, model. Each surface defines a transition from one optical space to the next. Each surface has an object side and an image side. If we consider a simple optical system consisting of a singlet lens, there are four optical surfaces:

• 1. The object surface

• 2. The front lens surface

• 3. The rear lens surface

• 4. The image surface

Additionally, in this model, either the front or rear lens surface must be designated as the stop surface.

Each ray that is traced through the system starts at the object surface and is then traced sequentially through the front lens surface, the rear lens surface and on to the image surface. Rays cannot go through surfaces in any other order such as 1 to 3 to 2 to 4; this would require non-sequential ray tracing.

The optical design program simply applies Snellʼs Law to each ray on a sequential surface-by-surface basis. Some of the information the program will consider or provide includes the exact ray path, the effects of reflection and refraction, wavefront phase, aberrations and image quality. Some information concerning polarization effects may also be provided.

Equally important is the information that is ignored during sequential geometrical ray tracing, such as surface and volume scatter, if the system or any particular ray is physically realizable, or if there is any edge diffraction or other non-geometrical propagation of the wavefront.

The following information needs to be available to the optical design program before a system can be analyzed:

• 1. System aperture type and size

• 2. Surface information (sphere, asphere, others)

• 3. Number of surfaces

• 4. Which surface is the aperture stop

• 5. Wavelength band and weighting

• 6. Field of view

Additionally, optimization requires:

• 1. Variable parameters

• 2. Merit function

Optical design is an interaction between the designer and the computer program. The designer provides the program with information about the initial design form and the performance goals. An optical design program is constrained to work within the design space provided. The program can make significant changes to the values associated with any variable parameter, but it cannot add additional variables, or add new parameters, such as additional lenses or aspheric coefficients. The designer needs to use the analysis provided by the design program to determine if any changes need to be made to the design and what the most effective changes are.

As previously discussed, the system aperture is used to determine the size of the entrance pupil, which determines how large a light bundle enters the optical system.

Surface information includes the curvature, thickness or distance to the next surface, the optical material the ray is entering, and possibly other information including aspheric or other shape coefficients.

Wavelength information consists of one or more wavelengths. During optimization and analysis, rays are traced for all defined wavelengths. Wavelengths may be weighted to indicate importance. A common use of weighting would be to apply the photopic or scotopic curves for visual optical systems or an appropriate detector response curve. Rays are only traced at the defined wavelengths, not as a continuum. Additionally, one wavelength is designated as the primary wavelength. This wavelength is used to calculate wavelength-dependent system properties, such as the focal length.

The field of view is also defined as a collection of specific points. Field points can also be importance weighted. Field points serve as source locations for the rays to be traced. Rays from each of the defined wavelengths will be launched from each of the defined field points to perform analysis and optimization. The field of view can be defined in terms of angles or heights in either object or image space. The angle definitions are necessary for infinite conjugates; either angles or heights can be used for finite object distances. For rotationally symmetric optical system, it is only necessary to trace rays from half of the field.

Variables are any parameters in the system that can be adjusted during the optimization process. These include radii, thickness, refractive indices, Abbe numbers, conic constants, aspheric coefficients and others, depending on the type of surface. In some cases, wavelength and field of view can also be appropriate variables. During the optimization process, all of the variables are adjusted. It is important for the designer to limit the ability of the program to change the variables freely to insure a successful design. Common constraints to apply include a minimum and maximum lens thickness.

Although optical design codes can be used to analyze existing designs, the most significant reason to use a design code is optimization. This is the designing part of the process. As indicated earlier, for the program to perform optimization, the designer needs to have provided a set of variable parameters and to have defined a merit function. All lens design programs have multiple algorithms for optimization. The designer needs to determine which, if any, of the native optimization routines is appropriate for a particular design, and if one of these routines is not appropriate, the program needs to allow the designer to create a suitable merit function. Merit functions native to the design program are designed to maximize the image quality of the design. Typically they include procedures to minimize the spot size or wavefront error on either a root-mean-square (RMS) or peak-to-valley (PTV) basis. The spot size or wavefront error will be weighted over all the defined fields and wavelengths. Additionally, it may be necessary to select a point of reference on the image surface for the optimization; usually either the intercept coordinates of the primary-wavelength chief ray or the weighted center position over all the wavelengths, the centroid. The proper merit function depends on the required performance level of the system. For systems that will operate at or near the diffraction limit, wavefront-based optimizations are more appropriate. For designs that do not need to perform at the diffraction limit, the spot-size minimization should be used. In cases where a user-specified merit function is required, such as a merit function based on MTF constraints, it is often more efficient (faster) to begin with a default merit function until the design is nearly finished before switching to the necessary merit function for the final iterations.

The most common optimization algorithms use a damped least-squares process. Let the merit function be defined as:

(7.1)

In this case there are m items that are being considered or targeted. The contribution of each of these targets to the total merit function is determined by the difference between the actual value and the desired value:

(7.2)

In the ideal case, the actual value exactly equals the targeted value, so the contribution of that item to the total merit function would be zero. The goal of the optimization is to determine the set of parameter values that will drive the total merit function value to zero.

In these equations, each target or operand is given equal weight. In practice, applying weighting factors to each targeted value is necessary for optimal results. In this case the merit function is defined as:

(7.3)

The values of each of the variables form an m-dimensional vector X. The goal of the optimization is to find the value for X for which Φ is a minimum. Φ is minimized in a least-squares sense, where the possible movements in solution space are computed by determining the direction where the derivative matrix of X is minimized. Careful sampling of solution space by the design program is necessary to insure that the minimum can be isolated.

There are some inherent problems with damped least-squares optimizations algorithms. The first issue is that the optimization follows a downhill path. That is, the algorithm locates a direction (new location in solution space) where the total merit function is lower than the current merit function and changes the vector X to those coordinates. This process continues until movement in any direction around the final coordinate location results in an increase in the total merit function value. At this point the optimization stops. This value is a local minimum, not necessarily the global minimum. The algorithm cannot search further through the solution space for a better minimum. The solution found depends on the initial starting point. By changing some of the initial parameter values, the optimization could find another local minimum that may be better or worse than that previously found. Figure 7.3 shows a simple two-dimensional (2-D) model of solution space. A, B and C represent possible starting points, with W, X, Y and Z the local minimum values, Z being the global minimum in this space. An optimization starting from point B will most likely find the global minimum value. An optimization from starting point A will stop at the local minimum Y. The starting point C can finish at either W or Z, depending on the direction of the first step.

Fig. 7.3

Local and global minima

The second problem that can occur is called stagnation. This occurs when the derivatives of each of the targets with respect to all of the variables is so ill-conditioned that a suitable next step cannot be determined. In this case, the optimization can stop even before reaching a local minimum.

Most optical design programs have global optimization techniques available that can be effectively used to find alternate, better solutions. However, even with global optimization it can never be assumed that the solution is the true global minimum.

The key for the optical engineer is to understand that the solution found by the optimizer might not be the best solution. However, it is not necessary to find the best solution, only to find a suitable design form that meets the performance specifications. If not, then the optimization must continue either by changing the initial parametric values or by adding additional degrees of freedom in terms of additional variable parameters.

In constructing a merit function, the optical designer must perform several tasks. The first is the selection of the proper form of the merit function. As indicated earlier, RMS or PTV minimizations of the spot (blur) size or wavefront departure from spherical relative to either the weighted centroid or the primary-wavelength chief ray are the most common choices. Indeed, an RMS spot size relative to the centroid is often the most appropriate merit function for a design. Even for situations where the final design should be diffraction-limited, this form of optimization is often the best starting point. Figures 7.4a and 7.4b show the blurs for a double Gauss lens design. Although the physical distribution of the rays has not changed, the RMS and geometric values calculated differ for the off-axis field points.

Fig. 7.4

(a) Spot diagram centered on the centroid (b) Spot diagram centered on the primary-wavelength chief ray

The RMS spot size is a measure of image quality based on tracing geometrical rays through the optical system. Geometric rays propagate through the optical system according to Snellʼs law, ignoring the effects of diffraction from edges and apertures. The RMS spot size is calculated by tracing a selection of rays of each defined wavelength from each defined field point. The number of rays propagated is adjusted to consider any applied weighting factors. To calculate the RMS spot size relative to centroid, it is first necessary to locate the centroid position. This is the average image position determined by tracing a number of rays:

(7.4)

The RMS spot size is then determined using

(7.5)

In attempting to optimize the design, the result will be a reduction in the aberration in the design usually without specifically targeting aberrations. It is also important for the designer to consider what is actually important in the design. The optimal result, where the total blur size is reduced, generally indicates optimal image quality. In using a native merit function, it is important for the engineer to understand what is and is not considered in the merit function. For example, distortion is not typically considered directly in optimization. If specific constrains are placed on distortion in the design specifications, it will be necessary for the designer to modify the merit function to include this constraint. Similar requirements may also apply to field curvature requirements.

Additionally, the optical designer may need to add constraints to target specific system properties such as effective focal length, focal ratio or magnification. Boundary limitations also must be considered. Boundary constraints placed on lens components can assure that the lens is physically viable. The center thickness of the lens should be one tenth to one sixth of the lens diameter. Also for lenses with convex faces, the edge thickness needs to be sufficient to prevent chipping and to provide a solid surface for mounting. Often lenses will be oversized and squared up to prevent problems. However, if lenses are oversized during manufacture, it is necessary for the designer to insure that all light that could pass through the oversized lens is actually blocked by the lens housing, or the image quality may be degraded by excess light entering the system. Additional constraints that need to be considered and may be appropriate for the merit function include the total length of the system, the weight of any and all components, glass properties and grades, and transmission.

After the design program has reached a minimum solution, several graphical and numerical analytical tools are available to help evaluate the design. These tools provide information to help the designer understand the limitations or dominant aberrations present in the current design form. By understanding the system limitations, the designer can introduce the necessary degrees of freedom to correct for that limit. In terms of aberration correction, particular forms of variables may be effective for controlling some aberrations and be ineffective at correcting other aberrations. Introducing ineffective degrees of freedom can lead to stagnation of the optimization process.

Some of the tools include geometric-ray-based spot diagrams, encircled energy plots, ray fan and optical path difference (OPD) fan plots, wavefront maps, as well as plots of field curvature, distortion, longitudinal aberration and lateral color. Additional tools may be available that include edge diffraction effects. These tools should be used as the systemʼs performance nears the diffraction limit. These tools include diffraction-based encircled energy plots, MTF plots and point spread function (PSF) plots.

Ray or OPD fan plots show the relative location of linear cross sections of rays through the entrance pupil. Fan plots are generated in the tangential and sagittal direction for each of the defined field points. Each wavelength is plotted separately. Figure 7.5 shows the tangential fan, which is directed along the y-axis of the pupil. Figure 7.6 shows the sagittal fan, which is directed along the x-axis of the pupil. The coordinate origin of the fan plots is the intercept of the primary-wavelength chief ray. Figure 7.7 shows the ray fan plot for a double Gauss design. The other points indicate how far each of the other rays landed from the chief-ray intercept in either the tangential or sagittal direction. The OPD plot (Fig. 7.8) indicates the difference in optical path traveled for each ray relative to the total optical path of the chief ray.

Fig. 7.5

Tangential ray fan

Fig. 7.6

Sagittal ray fan

Fig. 7.7

Ray fan plot

Fig. 7.8

Optical path difference fan plot

Although the fan plots are limited to showing information about a pair of linear ray distributions, each is very useful in determining the many first- and third-order aberrations present in the design. Each of these aberrations has a characteristic appearance on the fan plot. By recognizing these traits, the designer can determine which aberrations are limiting the designʼs performance.

As a systemʼs performance level nears the diffraction limit, it becomes necessary to consider the effects of diffraction from apertures in the performance evaluation. Light has properties of both particles and waves. Rays are used to model particle-like behavior; diffraction and interference characterize wave-like behavior. At optical wavelengths light is a wave phenomenon, and the approximations used in geometric optics may not be sufficient to explain the images formed in such systems. To understand these effects, it is necessary to consider the effects of diffraction from edges and apertures. Although diffraction occurs in all optical systems, it is only necessary to consider diffraction effects when the scale of the image blur due to diffraction is on the same scale as the blur size due to geometric aberrations.

Diffraction theory says that the image formed by a converging wavefront is simply the Fourier transform of the complex wavefront in the exit pupil of the optical system. An important consequence of this is called compact support. This tells us that a signal band-limited in the spatial domain cannot be limited in its Fourier domain. As the wavefront passes through a physical aperture, some of the wavefront is clipped off. Because the wavefront is now limited in the spatial domain, it cannot be limited in the Fourier domain. Energy is spread out over all angular space. Due to this spreading of energy, point objects such as stars cannot form point images. The size of the image blur due to diffraction is a function of the focal ratio and wavelength of the system.

(7.6)

Summing the plane waves incident on an optical system a long distance from the aperture gives the far-field diffraction pattern. In the common case of a planar wavefront incident on a circular aperture, the energy distribution is referred to as an Airy pattern. The irradiance distribution on a circular aperture is

(7.7)

where A ₀ is the incident amplitude and π is an area normalization term. Integrating over the distribution yields

(7.8)

with k being 2π/λ, r the aperture radial size and z the image distance and J ₁ is a first-order Bessel function. The first root of J ₁(x) occurs at 1.22λ(F/#). The Bessel function, or Airy disk pattern, describes a pattern of alternating bright and dark zones, with the location of the dark zones determined from the successive roots of the Bessel function. Figure 7.9 shows a cross section of the diffraction image. Figure 7.10 shows a 2-D image. Both figures are plotted on a logarithmic scale to allow further bands to be seen. The central core of the Airy pattern is useful in determining if a design is approaching the diffraction limit. Consider the spot diagrams shown in Fig. 7.11. The circle indicates the Airy disk. In the spot on the left, the geometric blur is contained within the Airy disk, and the effects of diffraction need to be considered; the size of the image blur will be larger than that calculated using only geometric analysis. For the system on the right, geometrical aberrations dominate the performance and the effects of diffraction can be ignored.

Fig. 7.9

Cross-sectional plot of the Airy function

Fig. 7.10

2-D plot of the Airy function

Fig. 7.11

Comparison of the blur size to the Airy disk

When performing diffraction-based calculations in a lens design code, it is important to understand the process being applied. Ray tracing programs propagate rays through the optical systems. Although each ray represents the normal to the wavefront, the actual wavefront is not propagated. Therefore some form of approximation is often used in diffractive calculations. The optical designer is responsible for determining the validity of the diffraction calculations for any given system.

Although there are situations where geometric or diffractive calculations must be applied, it is important to remember that the two can also be related: if geometric optics predicts a high level of performance for a design, than the system will perform close to its diffraction limit. If geometric optics predicts the blur size is smaller than the Airy disk, then the actual image blur will be near the size of the Airy disk. Also, if the geometric blur is smaller than the Airy disk, there is little value in attempting to reduce geometric aberrations further.

The design of an optical system requires the optical designer to develop a first-order design to insure that all necessary specifications have been provided and make sense. Then the basic system prescription including system parameters such as focal length, aperture, field of view and wavelength data, as well as the basic sequential optical layout with the desired number and types of surfaces and materials, needs to be defined in the lens design program. To perform optimization, variable parameters and an appropriate and well-constrained merit function must be defined. After allowing the optical design program to perform optimization, the designer needs to consider the appropriate geometrical or diffractive analysis available to insure that the performance goals are truly met. This ends the first and possibly easiest part of optical design.

The next step in designing the optical system is to perform a tolerance analysis. Tolerancing is one of the most complex aspects of optical design and engineering. Tolerance analysis is a statistical process during which changes or perturbations are introduced into the optical design to determine the performance level of an actual design manufactured and assembled to a set of manufacturing tolerances. To perform an optical tolerance is to accept reality. No optical surface will be polished to perfect curvature and figure. No mechanical mount will be perfectly machined. No component will be perfectly positioned. All these, as well as many other error sources will serve to degrade the performance of the assembled system. It is up to the optical engineer to define the fabrication limits properly for each optical and mechanical task. This needs to be done with a consideration for the system performance as well as an understanding of the cost impact of an overly constrained design.

There are several steps to performing a tolerance analysis.

• 1. Establish a tolerance budget,

• 2. Perform a test plate fitting,

• 3. Define a set of tolerance ranges that are easily within fabrication limits,

• 4. Define compensators that may be used to limit performance degradation,

• 5. Select an appropriate figure of merit for tolerancing,

• 6. Evaluate the tolerances to estimate the impact of each perturbation,

• 7. Generate random designs for statistical analysis,

• 8. Revise the tolerance ranges as necessary.

The error budget is the total system degradation that is available. If the basic design just met the performance requirement, then it is unlikely that a manufactured system would work to specification, as a system will not be made exactly to the prescription. The available error needs to be shared among all the factors that can impact the system including materials, manufacturing, assembly, environmental, and the residual design error. Table 7.2 lists some of the parameters that should be considered.

Table 7.2 Possible tolerance defects

Test plate fitting refers to the process of matching the surfaces of all optical surfaces to a set of test plates available to the optician. The curvature of an optical surface is determined interferometrically by comparing the surface being polished to a surface of known curvature. If the design can be matched to the available test plates, the cost both in terms of fabrication as well as time will be significantly reduced. If matching all surfaces to test plates significantly degrades the performance, the optical engineer needs to determine if the cost of custom test plates is worth the cost and delay or if a system redesign should begin.

Setting the initial tolerance ranges is a balance: not so tight as to have a cost impact but not so loose that assembly is difficult. Understanding the capabilities of the optical fabrication shop as well as the mechanical fabrication shop is a good starting point for the parameter ranges. Start with readily achievable tolerances and then selectively tighten only those ranges that result in out-of-specification performance.

A compensator is a system parameter that can be adjusted to counteract the effect of an error in another parameter. The ability to use compensators can significantly loosen the parameter ranges, which can be useful in reducing costs. A common compensator is the image distance: the image location is set for each system based on the performance of that system. Compensators represent degrees of freedom, usually mechanical adjustments that need to be available and made during the assembly of the optical design. The cost impact of the compensator needs to be considered before including it in the analysis. A $500 kinematic mount cannot be included with each $7 optical system.

The tolerance figure of merit is usually similar to the final merit function used during the design phase. As each parameter is perturbed within the tolerance range, the compensator value is optimized to minimize the figure of merit. The difference between the original design merit function and the tolerance figure of merit describes the impact of each perturbation on the system performance.

During the tolerance process, each parameter is perturbed individually. This allows the optical engineer to determine which parameters are most sensitive. Parameters that perturb the design outside a portion of the error budget available for fabrication errors need to be more tightly constrained. Tightening constraints adds significantly to the manufacturing costs. Additionally there are some manufacturing limits that cannot be exceeded no matter what the cost. Designs that require such constraints should be deemed unbuildable and the engineer should return to the design process, with careful consideration as to why the design failed.

By generating a series of perturbed lenses, it is possible to determine what the performance level of any manufactured system would be. For these systems, each parameter is perturbed to a value within a specified statistical distribution of the nominal value. The accuracy of this analysis is directly related to the provision by the optical designer of the necessary information to the tolerance algorithm regarding the appropriate distribution. If such information is not available, a Gaussian distribution is normally assumed. This is not always the best choice. Consider the thickness of an optical window. It is unlikely that the window will be polished to the thin side. As soon as the component thickness is within the allowed thickness + Δ, the polishing will stop. The fabricator will not continue to polish to get closer to the nominal value.

After analyzing the statistical information about the random designs, further tightening of the parameter ranges may be necessary to achieve the required pass rate. Again a careful balance needs to be achieved between the success rate (the percentage of the systems that will meet the required performance level) and the cost to increase it.

An optical design is not complete until a tolerance analysis has been carried out. The lens design needs to be manufacturable with some realistic range of parameter specifications to be useful to the customer. The best design is not necessarily the design that best matches the design specification. It is the design that can be built to best match the design specification.

Now that a finalized optical design has been established, it may be necessary to perform a stray light analysis to simulate the real working environment more completely and insure that the limitations imposed by sequential ray tracing have not hidden significant design flaws.

Stray Light Design Analysis

Years ago a stray light analysis was often an afterthought, if considered about at all. Today it is known that the stray light analysis should be considered along with the conception of the optical design in the very earliest stages of the design study. It might even play a major role in the selection of the starting optical design.

What is a stray light analysis? Stray light analysis is the calculation of how much unwanted radiation reaches the image plane/detector plane. The fine distinction being made here is that in some designs a detector is placed at the exit pupil plane where no traditional image is formed. In this chapter they will be used interchangeably. In a stray light analysis one must consider all the elements in the optical design. It must consider each surface as an independent element that will scatter directly to the image plane or that might create significant non-sequential stray light contributions due to ghost images or total internal reflections off any of its surfaces. In a lens system, all possible combinations of lens and mirror surfaces need to be considered. Each ground or polished edge of the lens elements are potential sources of stray light propagation, and each aperture edge of an optical element creates diffraction. All the mechanical structures that hold the elements in place are potential stray light propagators. Then there are often baffles, the main cylindrical tube-like structures that are used to either enclose the optical system or to baffle the stray light from reaching deep into the system. On these baffles, there are often vane structures to suppress the propagation of the unwanted light further, and those vanes usually have tapered edges, and as fine as they may get, they can sources of scattered stray light and diffracted stray light. These vanes and baffle surfaces are usually coated with paint or light-absorbing material that have a complicated bidirectional scattering function. There is no such thing as a Lambertian black coating at any wavelength. In the infrared there are also potential sources of thermally emitted stray light from the system itself.

In spite of all this complexity, significant stray light suppression design can be accomplished without doing any numerical calculations. Sooner or later one will eventually need a software program to quantify the performance and then a system performance measure to confirm the results of the analysis and fabrication. There are computer-aided design (CAD) tools available that greatly assist in inputting very complicated systems into stray light analysis codes.

Why do a stray light analysis? One needs to perform a stray light analysis and then a measurement when stray light might be a problem in systems that:

• 1. are in compromising environments that have one or more strong stray light sources

• 2. observe faint objects

• 3. make precise measurements

• 4. require high contrast

• 5. propagate high-power laser energy, when even small percentages of stray light can damage the system

One carries out the stray light analysis because in many systems it is necessary to assure that the system goals are achieved or even achievable. Stray light analysis usually improves system performance by a factor of 1000 and sometimes by a factor of 100000 (e.g., in the Hubble telescope).

The adverse effects of stray light in a system are:

• 1. It can cause such severe problems that the design will not reach its desired optical performance;

• 2. It reduces the contrast on the image plane;

• 3. It obscures faint signals or creates false ones;

• 4. It produces false artifacts across the image plane that cause false alarms;

• 5. It causes magnitudinal errors in radiometric measurements;

• 6. It damages fragile optical components;

• 7. and stray light can burn out detectors.

Much like in the optical design, the design of stray light suppression in the system starts in the first-order optical design, and with initial specifications determined most often by the purpose, mission and environment of the instrument. The required basic thought analysis of a stray light designer must consider the items the designer considered in Tables 7.1 and 7.2 and some more given in Table 7.3. While stray light design is considered by some to be an esoteric art, it is not. Most good stray light design aspects of a system can be incorporated, in concept, without much precalculation. There are about eight basic principles in stray light design that simplify the process. They are explained in the text. They are:

• 1. Stray light analysis should be incorporated in the very earliest stages of a preliminary design study. It should play a role in the selection of the starting optical design.

• 2. Start from the detector.

• 3. Determine critical objects, i.e., objects seen from the image plane.

• 4. Move it or block it.

• 5. Trace rays out of the system from the image/detector plane.

• 6. Trace rays into the system from various field positions to determine the “illuminated” objects.

• 7. Only the aperture stop should not be oversized. All other elements have extra imaging surface from any given field point on the image surface. Be concerned about concessions made to oversizing elements for the convenience of manufacturability.

• 8. One wants and needs to know more than just how much stray light reaches the detector. One needs to know what path it took and the propagation method (scatter, total internal reflection (TIR), diffraction, etc.) in order to improve performance.

The concept is to think out the system before any stray light analysis is performed. It is encouraged to learn and appreciate the concepts so that, when one gets to the computer analysis, it will run faster and will probably produce a more accurate answer. Why? If one can limit the number of critical paths along which the unwanted stray light propagates there will be fewer interactions that can be miscalculated. Fewer interactions mean that the analyst will probably pay more attention to the fine details, and in the end the software will run faster because there are fewer calculations. Once you know the paths you do not have to make a Monte Carlo zoo of the analysis. After we look at the basic mathematical calculation that needs to be performed we will come back to this point.

Table 7.3 Basic optical system stray light specifications

The Basic Equation of Radiation Transfer

The fundamental equation relating differential power transferred from one section on an object, be it a baffle or an optical element, or even a diffracting edge, to another object is determined by

(7.9)

where dΦ _c is the differential power transferred, L _s(θ ₀, ϕ ₀) is the bidirectional radiance of the source section; dA _s and dA _c are the elemental areas of the source and collector; θ _s and θ _c are the angles that the line of sight from the source to the collector makes with their respective normals. This equation can be rewritten as three factors, which helps simplify the reduction of scattered radiation. With E(θ _i, ϕ _i) being the incident irradiance incident on the scattering surface, the three terms are:

(7.10)

The first term is the scatter function, known as the bidirectional reflectance distribution function (BRDF):

(7.11)

The second term is the power on the propagating surface section

(7.12)

The final term has to do with the geometry. It represents the projected solid angle (PSA) of the collector as seen from the scattering surface, i.e., the source.

(7.13)

dΩ _sc is the solid angle of the collector section as seen from the source. Another term used often is the geometrical configuration factor (GCF) introduced by thermal analysis engineers over a century ago. GCF = PSA/π because the radiance of an emitting surface was assumed to be Lambertian (it includes the BRDF term).

(7.14)

In words, the power propagated from one surface area section to another is equal to the power from the surface that is scattering the radiation times the solid angle the collector subtends as seen from the source times the scattering characteristics of the source. So stray light boils down to the repetitive use of the multiplication of just three numbers: the power on the scattering surface, the scatter value for the specific input and output of the scattering surface, and the projected solid angle of the collector. When conceptually designing a system, (7.14) is the equation to keep in mind.

All software programs implement some variation of this calculation, even the ray-based programs. In ray-based programs, the scattered ray is weighted by the power on the scattering surface, the BRDF for the directions of the incoming and outgoing ray direction and some form of weighted solid angle. The mathematics of a stray light analysis does not appear to be an overwhelming calculation; it is just the product of three numbers and the BRDF does not seem to be that hard to determine if one has measured data. All of which is correct. So why does stray light analysis seem so challenging? For one thing, in a typical analysis the equation is calculated 100 million times. No one has the time to do it by hand in detail.

Stray Radiation Paths

Since it is only the third term in (7.14) that can be reduced to zero, it should receive attention first

(7.15)

How this factor can go to zero is hard to understand at first. The two cosine values can reduce the PSA to zero but seldom can this level of tilt be reached. Usually this is done on lens mounting structures, by tilting them completely out of the way. The diffraction effect remains, but is usually much lower than at the high angle of incidence of the forward-scattering path. For dA _c the finite area of the collector is always present, so this seldom goes to zero. The PSA can be made to go to zero by moving the collector out of the scattering surfaceʼs field of view, in which case it is blocked. In some cases this can be done by a field or aperture stop. Alternatively, this can be achieved by placing vanes on a baffle surface so that the direct forward-scatter path is blocked and two scatterings are present in the path path (each absorbing maybe 99% of the energy with a very good black) and one additional PSA reduction by 90% along the path from the front surface of one vane to the back side of the preceding vane. This then results in a reduction by a factor of about 100000 in the propagated energy along the path.

This is a crucial point in a stray light analysis. Most analysts make the mistake of working on the BRDF term first. They want the blackest black or the lowest-scatter optical surface without knowing if it will make any measurable difference. A stray light analysis will pay for itself in the long run.

Start from the Detector

Having explained the apparent possibilities for decreasing the PSA_sc term, we now consider another concept, by considering the start from the detector and the move it or block it concepts. By placing baffles, stops, and apertures into a system in the correct places, many critical objects will be removed from the view of the detector. In other cases propagation paths from the directly illuminated surfaces to the critical objects can be blocked. The beautiful part of this is that one does not need to do the calculation. We all know what zero times something is zero. It is not unusual for an experienced stray light person to reduce the necessary calculations for a software program greatly without reducing the reliability of the result.

What is needed is a logical approach that first blocks off as many direct paths for unwanted energy to the detector as possible and then reduce the list of illuminated objects. Finally the PSA_sc for the remaining paths from the illuminated objects to the critical objects is minimized.

Below several stray light paths are pointed out in detail. During the design phase the analyst designs out and reduces the number of critical objects that the detector can see. Then the direct stray light paths from the stray light sources to the illuminated objects are reduced using aperture stops, Lyot stops and intermediate field stops. Every effort is made to use these apertures to block the direct field of view from the image plane completely. Therefore the PSA_sc in the above transfer goes to zero, maximizing stray light suppression for these paths.

The next step is to make sure that there is no section of an illuminated object that is both illuminated and seen (i.e., is also a critical object) by the detector. These are single-scatter paths from the stray light source to the detector and must be solved before proceeding. The good news is that, having done this, one more or less knows all the key stray light propagation paths without having done a calculation.

The analysis should not start from stray light sources as, even if some parts of the system are well lit, the emitted photons are not important if they are strongly attenuated before reaching the detector. Unfortunately many engineers first want to determine the destination of the stray light from each source, leading them to choose the best optics and the blackest blacks. This is not the right way to proceed, and the following two points should be borne in mind:

• 1. Only objects seen by the detector can contribute stray light;

• 2. The PSA is the only term that can go to zero.

Therefore the approach is first to determine the critical objects that can be seen from the image plane and list all objects that it can see either directly or through the various optics, be they reflective or refraction. This will be covered in detail shortly. The next step is to start moving the critical objects outside the field of view (FOV) or block them with baffles, apertures stops, field stops, or vanes. The PSA should be made to go to zero before a calculation is considered, with the number of critical objects that can be seen as low as possible.

The next step is to trace the propagation of stray light into the system and determine the illuminated objects. Having done the first step we now know where we do not want any of the direct energy to arrive, i.e., the critical objects, because it will create a single-scatter path to the detector. The significant stray light propagation paths will be defined as the paths:

• from the stray light sources of unwanted radiation to illuminated objects,

• then from these illuminated objects to critical objects,

• then from the critical objects to the detector.

This approach simplifies the analysis immensely and directs attention to the most productive solutions.

With computer software, one can then quantify the power propagated along these paths, which will then reveal which paths are the most significant. The analyst then gains the further advantage of working on the other two terms of (7.14). It is only at the end of the analysis that the analyst considers if the coating will make a difference. Its effect will usually be by less than a factor of five, maybe ten, for any single scatter between a great black and a mediocre black. Blacks are not the secret to good stray light design.

The Reverse Ray Trace

The purpose of a reverse ray trace is to determine what the detector can see. It is only from objects that the detector can see that there can be any direct contribution to stray light. An object in this sense can be a diffracting edge and any object seen in reflection or imaged by and through the optical system. This is not an easy calculation for most optical design programs as there is seldom a single aperture that defines the reverse aperture stop. Usually there are several apertures that define the limiting size of the ray bundle, so it can be difficult to use those codes.

Take for example a simple two-mirror Cassegrain reflective telescope. The on-axis incoming beam is limited by the aperture of the primary mirror, it is then reflected towards the secondary mirror where the circular beam footprint is then reflected back towards the primary, and often through a hole in the primary to an on-axis beam spot on the image plane behind the primary mirror.

From an off-axis position the incoming beam is again limited by the same aperture of the primary mirror, where it is again reflected towards the secondary mirror where the circular beam footprint is offset from the center of the secondary mirror and reflected back towards the primary to an off-axis beam spot on the image plane. So the secondary has to be bigger than the beam footprint from any given field position to account for the field of view (FOV) of the sensor. Other than diffraction and aberrational effects, this has little impact on the incoming beam. To the first order, it is geometrically similar, although it may be slightly elliptical.

However, looking out of the system it is dramatically different. All sorts of stray light concepts are involved. Consider the secondary as the aperture stop of the system in reverse; i.e., the full surface intercepts the ray bundle from the on-axis position that is reflected towards the primary mirror. However, when the ray bundle from an off-axis position on the detector reaches the plane of the rim of the primary mirror, it is bigger on one side than the diameter of the aperture of the primary. This is where the primary also becomes the limiting aperture and gives design programs a challenge. Nevertheless, it is because the instantaneous footprint of the incoming beam from an off-axis stray light source position needed a bigger secondary mirror aperture to accommodate the beam. The same extra mirror surface needed to accommodate the incoming beam allows the detector to see more in the reverse direction, which is how it sees the baffles, mirror mounts, struts, and (in other systems) the ground edges of lenses, etc. All of these are critical objects and are therefore more sources of scattered and diffracted light to the detector.

The amount of baffle seen increases as the point is moved to an off-axis position on the detector. If there were a baffle one would see much more of the baffle, which could be directly illuminated by a distant stray light source. A source that does not put any direct power on the primary mirror, but does to a black surface with a higher BRDF value is not good.

This concept applies to almost all optical systems. As briefly referred to above, the struts that support the secondary mirror in a typical two-mirror telescope are seen in double reflection off the secondary and them off the primary. As one moves off-axis there is a view to the edges of the struts if they are not tapered out of the field of view. It this case the forward scatter to the detector can be quite high.

These reverse ray traces are not easy in a conventional optical design package because, as the trace starts from the off-axis image plane position, there are often multiple apertures that limit the edge rays in the reverse direction. Once discipline is learned it is often easy, maybe even easier than using a program, to conceptualize what would be seen from the various field positions on the detector.

The concept is this: all apertures that are not the aperture stop or conjugate to it will be oversized for any given field position and therefore are windows to other surfaces and surface areas; you see critical objects. Once the real hard stop is encountered the detector should not see any more surfaces beyond the ray bundle size. Note that central obscurations and strut-like objects in the field of view will be seen.

Field Stops and Lyot Stops

Field stops have not yet been discussed, but they too have an important role in stray light suppression. If one exists, it is placed at the position of an intermediate image position. Ray bundles traced from an out of for position into the system will be intercepted by this field stop; hence there should not be any directly illuminated objects passed by an intermediate field stop. Field stops limit the number of illuminated objects. Caution should be exercised in systems that have mirrors, as a beam focused out of field onto a field stop might be seen in the center of a reflection off a mirror, and would then be a bright point source in the field of view of the detector.

A Lyot stop is placed at the image of the aperture stop, often at the exit pupil. Lyot stops are often really the stop of the system as they are usually slightly undersized to account for aberrations and diffraction effects. Because they follow the defined aperture stop they are closer to the image plane in the series of optical elements and baffles that follow. Therefore the Lyot stop will limit the number of critical objects seen by the detector.

In detail the concept is more complicated and a software tool is needed to help speed up the calculation. For instance, the image at the intermediate field stop is often not of as high a quality as the final image. That is because the optical parameters are varied to control the final image quality. Therefore aberration will blur the image and increase the spot size, and some stray light might get through the aperture. The same goes for the aperture stop and its Lyot stop. An effect known as pupil aberration may also occur; if bad enough it will allow extra critical objects to be seen. Usually only a small portion of some new critical object will be seen. However, even if it is only a small piece that is directly illuminated, it could make orders of magnitude difference in the stray light background noise.

After the system is designed and analyzed a stray light system test is performed. One does not just believe the analysis because computers assume a perfect system, a perfectly clean system, a perfectly and completely well-coated system whereas the system will not be ideal. Then why do the stray light analysis at all? Why not just test it? If it does not work, then it can be fixed. However, this approach is not as easy as it first sounds, if you do not know the propagation paths then you do not know what to fix. A trial-and-error approach can be very time consuming and very expensive. It would be unfortunate if you found out that the system was not at fault but that the stray light test was the culprit.

Two comments from experience. In about 20 different stray light system tests on real built systems the test environment created a problem in 100% of the cases; no exceptions. In some cases the system to the pleasure of the designers exceeded performance goals but that was not correct, and several other times the system test indicated a failure when the failure was in the design of the chamber. It has gone both ways. Fortunately in most of the 20 cases the chamber was pre-evaluated first and a fix to the test procedure was made saving more time and money. In the other cases the program was halted until the fault in the chamber test was found via a stray light analysis. This happens often.

Table 7.4 gives a comparison between a stray light analysis and a system stray light test.

Table 7.4 Comparison of the advantages and disadvantages of a stray light system test and a computer-generated stray light analysis. It shows that every disadvantage of a stray light test is an advantage of a system analysis, and every disadvantage of a stray light analysis is an advantage of a stray light test

Every stray light analysis should be backed up with a stray light test. The chamber itself should have a performance analysis performed on it before it is used in any particular configuration. The two methods complement each other, the strengths of one cover the weaknesses of the other. Neither one alone should be considered sufficient.

Conclusion

During the process of the design and analysis of an optical system, it is necessary for the optical designer or engineer to consider many factors that will impact on the performance of the system in operation. These include the actual system design, usually defined using geometrical ray tracing, the effects of diffraction, manufacturing and material limitations, as well as the effects of stray and scattered light. Software tools are available to help perform these tasks, but it is up to the engineer to understand the strengths and limitations of these programs and to insure that a complete analysis of the system is performed before committing the design to hardware.