Optical lens design

# Optical lens design

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Encyclopedia
Optical lens design refers to the calculation of lens
Lens (optics)
A lens is an optical device with perfect or approximate axial symmetry which transmits and refracts light, converging or diverging the beam. A simple lens consists of a single optical element...

construction parameters (variables) that will meet a set of performance requirements and constraints, including cost and schedule limitations.

Construction parameters include surface profile types (spherical, aspheric, holographic, diffractive
Diffraction
Diffraction refers to various phenomena which occur when a wave encounters an obstacle. Italian scientist Francesco Maria Grimaldi coined the word "diffraction" and was the first to record accurate observations of the phenomenon in 1665...

, etc.), and the parameters for each surface type such as radius of curvature
Radius of curvature has specific meaning and sign convention in optical design. A spherical lens or mirror surface has a center of curvature located in either along or decentered from the system local optical axis. The vertex of the lens surface is located on the local optical axis...

, distance to the next surface, glass type and optionally tilt and decenter.

## Design requirements

Performance requirements can include:
1. Optical
Optics
Optics is the branch of physics which involves the behavior and properties of light, including its interactions with matter and the construction of instruments that use or detect it. Optics usually describes the behavior of visible, ultraviolet, and infrared light...

performance, i.e., image quality: quantified by encircled energy
Encircled energy
The optics term encircled energy refers to a measure of concentration of energy in an optical image, or projected laser at a given range. If a single star is brought to its sharpest focus by a lens giving the smallest image possible with that given lens , calculation of the encircled energy of the...

, modulation transfer function, Strehl ratio
Strehl ratio
The Strehl ratio, named after the German physicist and mathematician Karl Strehl , is a measure for the optical quality of telescopes and other imaging instruments...

, ghost reflection control, and pupil performance (size, location and aberration control); the choice of the image quality metric is application specific.
2. Physical requirements such as weight
Weight
In science and engineering, the weight of an object is the force on the object due to gravity. Its magnitude , often denoted by an italic letter W, is the product of the mass m of the object and the magnitude of the local gravitational acceleration g; thus:...

, static volume
Volume
Volume is the quantity of three-dimensional space enclosed by some closed boundary, for example, the space that a substance or shape occupies or contains....

, dynamic volume, center of gravity
Center of gravity
In physics, a center of gravity of a material body is a point that may be used for a summary description of gravitational interactions. In a uniform gravitational field, the center of mass serves as the center of gravity...

and overall configuration requirements.
3. Environmental requirements: ranges for temperature
Temperature
Temperature is a physical property of matter that quantitatively expresses the common notions of hot and cold. Objects of low temperature are cold, while various degrees of higher temperatures are referred to as warm or hot...

, pressure
Pressure
Pressure is the force per unit area applied in a direction perpendicular to the surface of an object. Gauge pressure is the pressure relative to the local atmospheric or ambient pressure.- Definition :...

, vibration
Oscillation
Oscillation is the repetitive variation, typically in time, of some measure about a central value or between two or more different states. Familiar examples include a swinging pendulum and AC power. The term vibration is sometimes used more narrowly to mean a mechanical oscillation but sometimes...

and electromagnetic shielding
Electromagnetic shielding
Electromagnetic shielding is the process of reducing the electromagnetic field in a space by blocking the field with barriers made of conductive and/or magnetic materials. Shielding is typically applied to enclosures to isolate electrical devices from the 'outside world' and to cables to isolate...

.

Design constraints can include realistic lens element center and edge thicknesses, minimum and maximum air-spaces between lenses, maximum constraints on entrance and exit angles, physically realizable glass index of refraction and dispersion
Dispersion (optics)
In optics, dispersion is the phenomenon in which the phase velocity of a wave depends on its frequency, or alternatively when the group velocity depends on the frequency.Media having such a property are termed dispersive media...

properties.

Manufacturing costs and delivery schedules are also a major part of optical design. The price of an optical glass blank of given dimensions can vary by a factor of fifty or more, depending on the size, glass type, index homogeneity
Refractive index
In optics the refractive index or index of refraction of a substance or medium is a measure of the speed of light in that medium. It is expressed as a ratio of the speed of light in vacuum relative to that in the considered medium....

quality, and availability, with BK7 usually being the cheapest. Costs for larger and/or thicker optical blanks of a given material, above 100mm to 150mm or so, usually increase faster than what would be proportional
Proportionality (mathematics)
In mathematics, two variable quantities are proportional if one of them is always the product of the other and a constant quantity, called the coefficient of proportionality or proportionality constant. In other words, are proportional if the ratio \tfrac yx is constant. We also say that one...

to just the increase in physical volume. This is primarily due to increased blank annealing
Annealing (glass)
Annealing is a process of slowly cooling glass to relieve internal stresses after it was formed. The process may be carried out in a temperature-controlled kiln known as a Lehr. Glass which has not been annealed is liable to crack or shatter when subjected to a relatively small temperature change...

time required to achieve acceptable index homogeneity and internal stress birefringence
Birefringence
Birefringence, or double refraction, is the decomposition of a ray of light into two rays when it passes through certain anisotropic materials, such as crystals of calcite or boron nitride. The effect was first described by the Danish scientist Rasmus Bartholin in 1669, who saw it in calcite...

levels throughout the blank volume. Availability of glass blanks is driven by how frequently a particular glass type is mixed and poured by a given manufacturer, and can seriously affect manufacturing cost and schedule.

## Process

Lenses can first be designed using paraxial theory to position images and pupils, then real surfaces inserted and optimized. Paraxial theory can be skipped in simpler cases and the lens directly optimized using real surfaces. Lenses are first designed using average index of refraction and dispersion
Dispersion (optics)
In optics, dispersion is the phenomenon in which the phase velocity of a wave depends on its frequency, or alternatively when the group velocity depends on the frequency.Media having such a property are termed dispersive media...

(see Abbe number
Abbe number
In physics and optics, the Abbe number, also known as the V-number or constringence of a transparent material, is a measure of the material's dispersion in relation to the refractive index...

) properties published in the glass manufacturer's catalog and though glass model calculations. However, the properties of the real glass blanks will vary from this ideal; index of refraction values can vary by as much as 0.0003 or more from catalog values, and dispersion can either remain about the same or vary slightly. These changes in index and dispersion can sometimes be enough to affect the lens focus location and imaging performance in highly corrected systems.

The lens blank manufacturing process is as follows:
1. The glass batch ingredients for a desired glass type are mixed together in a powder state,
2. the powder mixture is melted together in a furnace,
3. the fluid is further mixed while molten to maximize batch homogeneity,
4. poured into lens blanks and
5. annealed
Annealing (glass)
Annealing is a process of slowly cooling glass to relieve internal stresses after it was formed. The process may be carried out in a temperature-controlled kiln known as a Lehr. Glass which has not been annealed is liable to crack or shatter when subjected to a relatively small temperature change...

according to empirically determined time-temperature schedules.

The glass blank pedigree, or "melt data", can be determined for a given glass batch by making small precision prisms
Prism (optics)
In optics, a prism is a transparent optical element with flat, polished surfaces that refract light. The exact angles between the surfaces depend on the application. The traditional geometrical shape is that of a triangular prism with a triangular base and rectangular sides, and in colloquial use...

from various locations in the batch and measuring their index of refraction on a spectrometer
Spectrometer
A spectrometer is an instrument used to measure properties of light over a specific portion of the electromagnetic spectrum, typically used in spectroscopic analysis to identify materials. The variable measured is most often the light's intensity but could also, for instance, be the polarization...

, typically at five or more wavelengths. Lens design programs have curve fitting
Curve fitting
Curve fitting is the process of constructing a curve, or mathematical function, that has the best fit to a series of data points, possibly subject to constraints. Curve fitting can involve either interpolation, where an exact fit to the data is required, or smoothing, in which a "smooth" function...

routines that can fit the melt data to a selected dispersion curve
Sellmeier equation
The Sellmeier equation is an empirical relationship between refractive index and wavelength for a particular transparent medium. The equation is used to determine the dispersion of light in the medium....

, from which the index of refraction at any wavelength within the fitted wavelength range can be calculated. A re-optimization, or "melt re-comp", can then be performed on the lens design using measured index of refraction data where available. When manufactured, the resulting lens performance will more closely match the desired requirements than if average glass catalog values for index of refraction were assumed.

Delivery schedules are impacted by glass and mirror blank availability and lead times to acquire, the amount of tooling a shop must fabricate prior to starting on a project, the manufacturing tolerances on the parts (tighter tolerances mean longer fab times), the complexity of any optical coatings that must be applied to the finished parts, further complexities in mounting or bonding lens elements into cells and in the overall lens system assembly, and any post-assembly alignment and quality control testing and tooling required. Tooling costs and delivery schedules can be reduced by using existing tooling at any given shop wherever possible, and by maximizing manufacturing tolerances to the extent possible.

## Lens optimization

Optical design is partly a science because ray paths and wavefront
Wavefront
In physics, a wavefront is the locus of points having the same phase. Since infrared, optical, x-ray and gamma-ray frequencies are so high, the temporal component of electromagnetic waves is usually ignored at these wavelengths, and it is only the phase of the spatial oscillation that is described...

structure can be very accurately calculated anywhere along the propagation
Wave propagation
Wave propagation is any of the ways in which waves travel.With respect to the direction of the oscillation relative to the propagation direction, we can distinguish between longitudinal wave and transverse waves....

path through the lens. Glass and coating optical properties can be measured and modeled with sufficient precision for use in lenses. If tolerances are included during the design, parts can usually be manufactured accurately enough that the resulting lens assembly performs acceptably close to the paper design.

Optical design is also partly an art, though, as the multi-dimensional design space within which a constrained lens design is free to roam is literally beyond human imagination if more than a few construction parameters are free to vary. The number, type and placement of optical elements are partly driven by physical requirements, but are also often based on previous similar designs obtained from published data, patents and textbooks. Skill and intuition in lens design are acquired over years of experience spanning hundreds to thousands of different lens design projects, preferably leading to additional experiences (and headaches) dealing with fabricating and aligning systems.

As an example of the complexity of lens-design space, a simple two-element air-spaced lens has nine variables (four radii of curvature, two thicknesses, one airspace thickness, and two glass types). Even for this simplest case, the design space is thus nine-dimensional, and local or global solutions within this space can at least be imagined as smaller or larger bubbles in a sponge-like 9-D foam-scape. A complex multi-configuration lens corrected over a wide spectral band and field of view, at multiple zoomed focal lengths and over a realistic temperature range, can have an extremely complex design volume, having over a hundred dimensions.

Lens optimization techniques that can navigate this multi-dimensional space and proceed to local minima
Maxima and minima
In mathematics, the maximum and minimum of a function, known collectively as extrema , are the largest and smallest value that the function takes at a point either within a given neighborhood or on the function domain in its entirety .More generally, the...

have been studied since the 1940s, beginning with early work by James G. Baker, and later by D. Feder, Wynne, Glatzel, D. Grey and others. Prior to the advent of digital computers, lens design was an agonizingly slow hand-calculation process requiring high-precision trigonometric and logarithmic
Logarithmic
Logarithmic can refer to:* Logarithm, a transcendental function in mathematics* Logarithmic scale, the use of the logarithmic function to describe measurements* Logarithmic growth* Logarithmic distribution, a discrete probability distribution...

tables, reams of paper, plotting 2-D cuts through the multi-dimensional space, and significant patience and understudying from previous masters. Tracing a single ray through a given lens surface could take more than an hour of painstaking calculations and checks, and a lens designer could not design more than a very few complex, high-performance anastigmatic objectives in an entire lifetime.

Modern desktop computers can now raytrace
Ray tracing (physics)
In physics, ray tracing is a method for calculating the path of waves or particles through a system with regions of varying propagation velocity, absorption characteristics, and reflecting surfaces. Under these circumstances, wavefronts may bend, change direction, or reflect off surfaces,...

tens to hundreds of millions of rays per second through a lens, and perform hundreds to thousands of optimization cycles per second, rapidly exploring the n-dimensional design volume including hill-climbing in and out of local minima in the search for the best solution.

However, even with lightning-fast optimizers, seasoned experience is still needed to guide solution trajectories through unacceptably shallow local minima and achieve the desired performance requirements. Experience in the mechanical and physical properties of glass, metals, optical coatings and bonding materials is also needed, especially in systems required to give high sustained performance over wide temperature ranges and harsh environmental conditions.

• Optical engineering
Optical engineering
Optical engineering is the field of study that focuses on applications of optics. Optical engineers design components of optical instruments such as lenses, microscopes, telescopes, and other equipment that utilizes the properties of light. Other devices include optical sensors and measurement...

• Fabrication and testing (optical components)
Fabrication and testing (optical components)
Optical fabrication and testing spans an enormous range of manufacturing procedures and optical test configurations.The manufacture of a conventional spherical lens typically begins with the generation of the optic's rough shape by grinding a glass blank. This can be done, for example, with ring...

• OSLO
Optics Software for Layout and Optimization
Optics Software for Layout and Optimization is an optical design program originally developed at the University of Rochester in the 1970s. The first commercial version was produced in 1976 by Sinclair Optics. Since then, OSLO has been rewritten several times as computer technology has advanced...

• Zemax
Zemax
Zemax is a widely-used optical design program sold by Zemax Development Corporation of Bellevue, Washington . It is used for the design and analysis of optical systems...

• Code V
CODE V
Code V is a commercial software package developed by Optical Research Associates. It is used for the design and analysis of optical imaging and optical communication systems...

• HEXAGON
HEXAGON (optical software)
The Hughes Executable Application for General-Purpose Optical Analysis is an in-house optical engineering software program developed by Hughes Aircraft Company to allow them to design and analyze a wide variety of optical lens systems...

• Ray tracing
Ray tracing (physics)
In physics, ray tracing is a method for calculating the path of waves or particles through a system with regions of varying propagation velocity, absorption characteristics, and reflecting surfaces. Under these circumstances, wavefronts may bend, change direction, or reflect off surfaces,...

• Ray transfer matrix analysis
Ray transfer matrix analysis
Ray transfer matrix analysis is a type of ray tracing technique used in the design of some optical systems, particularly lasers...

• Photographic lens design
Photographic lens design
The design of photographic lenses for use in still or cine cameras is intended to produce a lens that yields the most acceptable rendition of the subject being photographed within a range of constraints that include cost, weight and materials...

• Stray light
Stray light
Stray light is light in an optical system, which was not intended in the design. The light may be from the intended source, but follow paths other than intended, or it may be from a source other than the intended source...