Thanks @MichaelH and @Jeff.Wilde. You gave me a different perspective on this problem. I like the idea of using parabasal rays. I also find paraxial ray tracing to be extremely valuable and its interesting to see how paraxial ray tracing subtly integrates with real ray tracing into OpticStudio.

I went down a rabbit hole when I was figuring out where the object principal plane of a thick plano-convex lens is. I’ve been taught to find the principal planes like its described at RP Photonics, that is:

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Plano-convex and plano-concave lenses have one principal plane that intersects the optical axis, at the edge of the curved surface, and the other plane buried inside the glass.

Aspheres that are not axi-symmetrical (rotationally symmetrical) can be used to change the magnification of vertical and horizontal images in imaging optics. Also, in illumination and focusing optics, light emitted from a point light source can be projected in the form of a line. In this way, aspheres that are not axisymmetric (rotationally symmetric) can achieve new functions that cannot be achieved with spherical lenses alone.

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Principal planes (along with all the Cardinal Points) are defined using Gaussian optics.  From the Field Guide To Geometric Optics:

However, when rays parallel to the optical axis enter through the plane face of the lens, the ingoing and outgoing rays will meet exactly along the convex face, which is not a plane.

Although the time required for transfer and molding is shorter than for the spherical polishing process, manufacturing of precision aspheric molds (which incurs cost) in advance are necessary. For this reason, consideration of whether or not to use aspheric lens prior to production, based on the estimated total cost of the production volume is necessary.

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The method of manufacturing aspherical lenses by transferring and molding the aspherical shape of the mold onto the lens requires three technologies: ultra-precision mold processing technology, ultra-precision transfer and molding technology, and precision measurement and evaluation technology for these surface shapes.

I think you’re on the right track with real vs paraxial rays.  When I encounter a basic question like this, I always like to go back to the fundamentals, namely how are the principal planes truly defined (not simply how are they calculated).

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With fewer lenses, it is possible to reduce lens materials, processing costs, and assembly man-hours, leading to overall cost reductions.

From a calculation standpoint, I believe all the Cardinal Points in OpticStudio use paraxial ray tracing, so the curved surface is actually a plane.  Another way to think of it is you calculate the Principal Plane using parabasal rays that are so close to the optical axis the local curvature of the lens looks like a vertical plane.

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I agree with Michael.  The principal planes (and more generally the cardinal planes), are paraxial entities.  In a paraxial world, all lens surfaces are planes that have ray-bending optical power.  The planes reside at locations calculated relative to where the lens surfaces intersect the optical axis.  Paraxial ray tracing is very valuable; for example, the Seidel aberrations are computed by tracing the paraxial marginal and chief rays.  In Zemax, rays don’t have to be small-angle to be considered paraxial.  Instead, rays that are traced using linearized paraxial math (i.e., linearized Snell’s Law, again with lens surfaces represented as planes), are called paraxial.

Spherical surfaces are characterized by the fact that the radius of curvature is the same at all positions on the sphere, and this leads to the fact that they are easy to polish and high precision can be obtained. On the other hand, aspheric lenses require the radius of curvature to be made different depending on the position, which requires precision mold processing and technology to precisely transfer and mold the aspheric shape.

Axi-symmetric aspheres include rotational parabolas, rotational hyperbolic surfaces, rotational elliptic surfaces, and rotational quadric surfaces. In imaging optics, the use of such axisymmetric aspheres increases the degree of freedom in shape and makes it possible to suppress aberrations that would be difficult with spherical lenses alone.

Lenses ② to ④ are lenses with continuous, smooth, non-spherical lens surfaces and are called aspherical lenses in a broad sense. ② is a lens with an aspheric surface that is axi-symmetric (rotationally symmetric) with respect to the optical axis of the lens, and is often used in imaging optical systems. Lenses ③ to ④ are aspheric lenses that do not have axisymmetry (rotational symmetry) with respect to the optical axis of the lens, and are mainly used in lighting and focusing optical systems.

On the other hand, in the case of ⑥, the refractive index inside the lens is not homogeneous but distributed, and in the case of ⑦, light is focused and diverged by using the diffraction effect on the surface instead of refraction.

I can accept this definition, but I was curious, from a teaching perspective, to hear if someone can provide an explanation that would be compatible with the “traditional” explanation: find the plane where the in and out rays bend. I know we are at a boundary between paraxial and real optics and it might not make sense to ask this question, but I could imagine a student asking about this and all I could answer for now is that in this instance we choose the vertex of the convex surface as the principal plane and its a special case.

Thanks for your gracious comments.  I haven’t thought about putting together a reference list organized by topic.  Maybe that’s something I’ll work on in the future.

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Gaussian optics treats imaging as a mapping from object space into image space.  It is a special case of a collinear transformation application to rotationally symmetric systems, and it maps points to points, lines to lines and planes to planes.

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Speaking of references, I think most optical engineers know about the SPIE Field Guides.  They are very handy, concise handbooks chocked full of good information.  For example, the principal planes/points for a thick lens are succinctly described in the SPIE Field Guide to Geometrical Optics by J. E. Greivenkamp.

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An aspherical lens is a lens whose lens surface is not spherical. By using lenses with aspherical surfaces, which offer a high degree of freedom in design, it becomes possible to reduce aberrations that could not be fully corrected with spherical lenses alone.

Therefore, when you take a real system and Gaussian reduce it (so you can use concepts like Lens Maker’s equation, reduced distances, magnification), there are no curved surfaces but only points, lines, and planes.  This is where thin lens and paraxial optics comes from.

In addition, when axi-symmetric aspheres are used in illumination and focusing optics, it is possible to achieve uniform illumination distribution and increase the degree of freedom in ray control.

At Optical Design Technology Navigator, we use state-of -the-art ultra-precision processing machines to process aspheric surfaces on a sub-micron order, transfer these aspheric surfaces using molding technology that incorporates a high level of know-how, and then transfer these aspheric surfaces into an ultra-precision 3D mold.

From this description it is clear that the principal planes (at P and P’) need not coincide with the physical lens surfaces.  However, for plano lenses (with one of the two surfaces being a flat), then one of the principal planes does coincide with a lens surface.  Here’s a nice figure from Fundamentals of Optics, 4th Ed. by Jenkins & White:

A spherical glass lens is processed by grinding one surface at a time, but grinding and polishing an aspherical lens one surface at a time would be very expensive. For this reason, aspheric shapes are generally processed into molds, which are then transferred and molded onto glass or plastic.

Aspherical surfaces are classified into two categories: axi-symmetric aspherical surfaces, which have axial symmetry (rotational symmetry) with respect to the lens optical axis, and aspherical surfaces, which do not have axial symmetry. Each type of aspheric surface has its own characteristics.

There are many different types of lenses. They can be broadly classified as the following according to the principle of light focusing and divergence and the type of surface.

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In this section, we will explain the features, advantages / disadvantages, and manufacturing methods of aspheric lenses.

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Something that would be really nice from a teaching perspective is if OpticStudio had the ability to show Paraxial Layouts…then the calculation would become visually obvious.

In this way, aspherical lenses make it possible to reduce the size and weight of products, and even to cut costs. However, the production of aspherical lenses requires a very high level of manufacturing technology.

TOYOTEC, operator of the Optical Design Technology Navigator, is an all-around optical manufacturer with proficiency in optical, mechanical, and electronical technology. We can design and develop products from scratch based on our customers’ needs, and provides integrated support from design to productization. In addition to manufacturing aspheric lenses, we offer one-stop manufacturing services from ultra-precision machining of lens cores to the design and assembly of lens units, including systems and peripheral components.

On a side note, @Jeff.Wilde have you ever considered sharing a curated reading list for optical design/engineering? You always have the right references and you point people to the exact chapter/section that they need. Your knowledge of the literature on this subject is a literal treasure trove for the community.

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Therefore, I believe the principal plane is defined as the plane coincident with the convex surface vertex. At least, this is what I’ve taken away when I used OpticStudio (surface 1 is the convex surface).

In this section, we will introduce the advantages and disadvantages of axi-symmetric (rotationally symmetric) aspheres in imaging optics. There are three major advantages.

This is because when we perceive an object as white, we are seeing all the colors on the visible light spectrum bouncing off the object and hitting our eye, ...

Lenses ① to ⑤ are lenses that have a focusing and diverging effect solely due to refraction on the lens surface. Of these, lenses ① to ④ have a continuous smooth surface, while lens ⑤ has a lens surface that is divided into discontinuous zones.

In imaging optics, multiple spherical lenses are used in combination to reduce aberrations such as image blur and distortion. By using aspherical lenses, it is possible to reduce the number of lenses while maintaining the same performance. For example, you can achieve the same performance of an 5-spherical-lens optical system with a total of 4 lenses using 2 spherical lenses and 2 aspherical lenses.