Aspherical lenses - perfect analytical shape

It's known that single spherical lens cannot focus parallel beam of monochromatic light into single (diffraction-limited) point, so it has to have aspherical shape to achieve that.

Is perfect analytical lens shape is known that is able to focus light into a single point (again, light is monochromatic)? Is there a universal perfect solution with conic factor, or higher-order components are always required?

• its a parabola; f(x)=a*x^2+bx+c Dec 23, 2010 at 14:43
• point-to-point focusing in the geometric optics limit would be a hyperbola. Dec 23, 2010 at 21:22

The last part of your question is the easiest to answer, so I'll get to that first. The best book on the fundamentals of optical design is "Modern Optical Engineering" by Warren J. Smith. It is not specific to aspheric optics, but does cover them in addition to the rest of geometrical optics and lens design. It is probably the single most common reference book among optical engineers.

Now, the rest of your question is a bit complicated, and needs a little bit of background, so bear with me for a moment. As has been mentioned, even an ideal lens will produce a focal spot of some minimum size, determined by the ratio of the lens focal length to its aperture (this quantity is called the "f-number, or $f/\#$") and the wavelength of the light. This is what optical engineers call the diffraction limited spot size. For a circular aperture, the diameter of the diffraction limited spot size will be $$2.44 \times \lambda \times f/\#$$ where $\lambda$ is the wavelength.

So as the $f/\#$ decreases (as the lens gets "faster") the diffraction limited spot will become smaller. However, any aberrations in the lens will also become more significant! This means that a very slow lens (one with a long focal length, relative to its aperture) can produce a diffraction limited spot even though it may have some aberration relative to an ideal lens, while a very fast lens will need to have a slightly aspheric shape to achieve diffraction limited performance.

This is important to understand because it means that, in some cases, a spherical lens can indeed focus light as close to a point as is physically possible, even though a sphere isn't the ideal shape.

So what is that ideal shape? Well again, it depends on a few things. For both lenses and mirrors, the ideal shape will change depending on the distance from the object plane to the lens, and from the lens to the image plane. In the case you've asked about, where the incoming light is collimated, optical engineers would say that the object plane is at infinity. In this case, as some other people have pointed out, the ideal shape for a mirror is indeed a parabola. However, for a lens this is not the case. As it turns out, the ideal shape for a lens to focus a collimated beam of light to a point is to have the first surface of the lens (the one the light hits first) be elliptical, and the back surface be hyperbolic.

Lens designers usually specify the shape of a lens surface with the following equation: $$Z = \frac{C r^2}{1 + \sqrt{1-(1+\kappa) C^2 r^2}}$$ where $Z$ is the "sag" of the lens surface, or its departure from a plane tangent to the lens surface at the center of the lens, $r$ is the radial distance from the center of the lens, $c$ is the curvature of the lens (the reciprocal of its radius of curvature) and $\kappa$ is called the "conic constant." It is the value of $\kappa$ which determines what sort of conic section describes the surface:

• $\kappa > 0$ Oblate ellipse
• $\kappa = 0$ Sphere
• $0 > \kappa > -1$ Prolate Ellipse
• $\kappa = -1$ Parabola
• $-1 > \kappa$ Hyperbola

On a related note, it is more than just the conic constant that can be adjusted to control aberrations. Even with purely spherical surfaces, the relative curvature of the front and back lens surface can be varied, while keeping the effective focal length constant. Adjusting this is more common than adding aspeheric surfaces to a lens, because aspheric surfaces are expensive to manufacture. Many optical supply companies even offer off-the-shelf optics with an ideal bending ratio for a given application. These are often sold as "best form" lenses.

• I see, that was an awesome answer :-) So I guess ring illumination is something which should produce minimal possible spot? Jan 11, 2011 at 0:18
• @Bars: ring illumination is something typically seen in a phase contrast microscope. Is that where you heard of it? The reason for ring illumination would actually make a very good question on its own, but it's completely unrelated to the focusing ability of a lens. In fact it can't even be explained by purely geometrical optics. Jan 11, 2011 at 0:58
• That's from optical lithography. Doesn't it offer reduced abberations as we use only small piece of lens AND gives good diffraction limit? Jan 11, 2011 at 2:31
• @bars: Not really. The illumination pattern doesn't change the aberrations introduced by the lens, and the diffraction limited spot size can't be reduced by using ring illumination. But I'm not really an expert on the state of the art in lithography these days. Jan 11, 2011 at 14:40
• BTW I've found out how expensive is to make custom aspheric lens - ~2500 euro for d=20mm, so I can make 200 custom sperical lenses for the cost of 1 custom aspherical one :-) Jul 7, 2011 at 10:02

Thorlabs has a little information on aspheric lens design on their product page (scroll down below the product pictures and click on "Lens Formula"). Note that no traditional lens can focus light down to a single point; the minimum size is subject to the diffraction limit and is in the order of the wavelength.

• Surely. wavelength & na. Dec 23, 2010 at 16:00
• "the minimum size is subject to the diffraction limit and is in the order of the wavelength." - unless you're working with meta-materials (negative index of refraction materials) and the traditional criteria of optics don't apply anymore.
– user346
Jan 9, 2011 at 6:58
• You are of course absolutely right, but I don't think the OP will find any books on aspherical metamaterials and their manufacturing. ;-) Jan 9, 2011 at 16:30

The problem you are talking about is called the spherical aberration. Spherical lenses are much easier to make, while from geometrical point of view the ideal focusing surface is a parabola. Since the light on optical instruments goes close to optical axis, one uses the paraxial approximation where sphere and parabola are the same up to the quadratic term.

There is a large variety of aspheric lenses, with parabolic lenses among them. But it is usually simpler to use combinations of lenses to deal with spherical and other aberrations.