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?
 A: 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.
A: 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.
A: 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.
