Optical Freeform surfaces

Common optical surfaces include planes, spherical surfaces, rotationally symmetrical aspheric surfaces, and cylindrical surfaces. These types of surfaces do not fall under the category of freeform surfaces. According to the definition of freeform surfaces, a freeform surface is one that does not possess axial rotational symmetry or translational symmetry constraints. Therefore, one significant characteristic of freeform surfaces is their asymmetry; they are not symmetric with respect to any rotational axis, nor symmetric with respect to any cross-section.

Freeform surfaces were first applied in the field of illumination. With the development of single-point diamond turning technology, various molds can be manufactured for different types of curved surfaces. With these molds, optical surfaces meeting specific requirements can be produced through molding or injection techniques. Compared to traditional components, freeform surfaces offer more degrees of freedom in parameters, enabling more compact designs, larger optical fields, and reduced system weight, thus facilitating the functional expansion of optical systems. In the field of optical imaging, freeform surfaces have many applications in astronomical observation and space optical systems. Some smartphone lenses also incorporate freeform surface elements to correct aberrations.

Common Types of Optical Freeform Surfaces

Freeform surfaces used for optical imaging are typically not arbitrarily shaped but are defined and described using equations and parameters. In this section, we introduce the commonly encountered types of optical freeform surfaces in practice. 

A. Off-Axis Aspheric Surfaces

According to the definition of freeform surfaces, an off-axis aspheric surface, which is a section cut from a rotationally symmetric aspheric surface, falls within the category of freeform surfaces. The shape of an off-axis aspheric surface can be circular or square. Off-axis aspheric surfaces have an additional off-axis quantity or angle parameter based on the original aspheric surface equation, which can be processed through CNC grinding and polishing methods.

 B. Complex Surfaces (TORUS) 

Complex surfaces, also known as toroidal surfaces, resemble a section taken from a car tire. They are curved in both the X and Y directions, having two different radii of curvature in two mutually perpendicular cross-sections. In optical systems, complex surfaces have unique applications, such as serving as deformable optical elements in deformation systems or scanning elements in infrared thermal imaging systems. In extreme ultraviolet spectrometers, complex surfaces can be used as front mirrors to collect larger optical flux.   

A complex surface is expressed as follows: Let the radius of curvature in the horizontal X direction be Rx, the quadratic coefficient be Kx, and in the horizontal Y direction be Ry, with the quadratic coefficient as Ky. The expression for a complex surface can be represented as: 

C. XY Polynomial Freeform Surfaces 

XY polynomials are typically derived from aspheric surfaces by adding polynomial equations in x and y. The forms of these polynomial equations can be arbitrary, including linear, quadratic, cubic, and higher-order polynomials. The equations of such surfaces usually have multiple parameters to control, allowing for different shapes of surfaces by varying the parameter values.

D. Zernike Polynomial Freeform Surfaces 

In previous articles, we have detailed the concept of Zernike polynomials (Optical Element Surface Shape Errors (Part II): Zernike and Chebyshev Polynomial Decomposition). The basis functions of Zernike polynomials are continuous, orthogonal, and complete within the unit circle. Each term corresponds to a form of aberration in optical testing, and the existence of orthogonality ensures that the magnitudes of various aberration coefficients are independent of the number of terms used for fitting. These properties make it an ideal representation for freeform surfaces, widely applied in imaging optical design. A freeform surface of aperture D obtained by superimposing Zernike polynomials on a quadratic surface has the following expression for the height: 

The first term represents the quadratic surface part, k is the conic constant, c is the curvature, r is the square root of the sum of squares of x and y, and the second term represents the Zernike polynomial part, where Ai stands for the Zernike polynomial coefficients, Zi are the Zernike polynomials, ρ is the normalized radius (r/(D/2)), and φ is the azimuthal angle. 

E. Q Polynomial Freeform Surfaces 

Q polynomial freeform surfaces are a type proposed by Forbes from QED, developed from Forbes' rotationally symmetric Q polynomial surfaces. Its surface shape coefficients can directly characterize the height deviation gradient of the surface relative to the best-fitting sphere. This allows for tolerance analysis of freeform surfaces, enabling simultaneous evaluation of the difficulty of optical design and processing testing, thereby avoiding the cumbersome process of evaluations post-design. The expression for Q polynomials is as follows: 

F、Non-Uniform Rational B-Splines（NURBS）

NURBS surfaces describe surfaces by controlling vertex networks, basis functions, and the weights of points. It is a parametric method for surface description and is defined as the only mathematical method for defining the geometric shape of industrial products in the ISO standard STEP for data exchange of industrial products. Adjusting each control point of NURBS or its weight only affects the surface shape near that point, making NURBS a locally controllable freeform surface. The expression for such surfaces is relatively complex, as shown below: 

NURBS has excellent properties and has been successfully applied in the field of illumination. However, the large number of variables makes ray tracing extremely complicated, resulting in long tracing times and difficulties in optimization, leading to limited applications in the imaging field.