The Diffraction Limited Spot Size with Perfect Focusing

Purpose

The purpose of this Insights article is to give the reader a brief introduction to the principles behind diffraction-limited focusing. The reader is assumed to be somewhat familiar with the concepts of diffraction theory and the Huygens sources that are used to compute a diffraction pattern.

Perfect Focusing by paraboloidal mirror

When a collimated beam is brought to a perfect focus by a paraboloidal mirror,  the focusing is essentially perfect in the sense that when ray-tracing is used with the angle of incidence equal to the angle of reflection,  all of the parallel on-axis rays converge precisely to the focal point.  Even a simple diffraction theory-based calculation shows that the path distance to the focal point is identical for each of the rays.  (You may recall from algebra classes that one definition of a parabola is the locus of points that is equidistant from a focus and a line called the directrix.  Using this,  you can readily show that the path lengths to the focal point are identical for each of the rays at the focal point.  This means that constructive interference will occur at the focal point from a diffraction theory standpoint.)

What are the focused spot size and focused beam intensity in the case of perfect focusing? 

The question arises,  what is the intensity of the light at the focal point for a perfectly focused beam?  Does the intensity approach infinity for a perfect parabola?  The answer is supplied by diffraction theory.  The diffraction integrals can be computed for the optical system that includes the focusing mirror.  We won’t give the complete integral formulation here, where the incident beam across the surface of the focusing mirror is broken up into individual infinitesimal Huygens sources,  but will simply present the results.  The focused spot size,  found from diffraction theory,  is approximately ## \Delta x= \frac{\lambda f}{b} ## where ## b ## is the diameter of the beam and/or the diameter of the focusing mirror (the smaller of the two).  Meanwhile, the focused spot area ## A_f =(\Delta x)^2=\frac{\lambda^2 f^2}{A} ##,  where ## A ## is the area of the beam. This result is consistent with energy principles as follows:  If the electric field amplitude ## e_f ## at the focus computed from diffraction theory is the sum of infinitesimal Huygens sources, it should be proportional to the area ## A ## of the beam.  Since the electromagnetic field irradiance (watts/m^2) is proportional to the square of the electric field amplitude,  the electromagnetic field irradiance(watts/m^2) at the focus, ## E_f ##, is proportional to ## A^2 ##. The focused intensity/irradiance computed from diffraction theory,(at the peak of the diffraction pattern in the focal plane),  is ## E_f=\frac{E_o A^2}{\lambda^2 f^2} ## where ## E_o ## is the irradiance (watts/m^2) of the incident beam.  Meanwhile, the focused spot size area ## A_f ## is inversely proportional to ## A ##.  The focused power is the product of the intensity times the spot size area, ## P=(\frac{E_o  A^2}{\lambda^2 f^2})(\frac{\lambda^2f^2}{A})=E_o A  ## so that we have energy conservation.

(Note: The letter ## e ## above was used for electric field strength,  but in this second use of the letter ## E ## where we have capitalized it,  it represents the irradiance (watts/m^2.) The irradiance ## E ## is computed from the electric field amplitude ## e ## by the Poynting vector formula.## E=\frac{1}{2} \sqrt{\frac{\epsilon_o}{\mu_o}} e^2 ## (m.k.s. units))

(Notice that the diffraction-limited spot size becomes smaller as the beam area becomes larger.  To get a very small focused spot,  you need to have a beam and optics that are of sufficient size=you can’t expect to achieve a one or two-micron spot size by focusing a pencil ray-sized beam).

(It should be noted in the above that the focused spot does not have uniform intensity across the entire spot,  but instead simply peaks in the center of the spot.  The area of the focused spot ## A_f ## is thereby an “effective” area,  but it makes for useful calculations in checking for the conservation of energy.)

Additional detail

When bringing a (nearly) collimated beam such as a laser beam to a focus,  more detailed calculations show there is a region called the beam waist where the focused ## \Delta x_w ## turns out to be just slightly smaller than the ## \Delta x ## given above and the beam waist occurs at a point a short distance from the focal point.  An additional item is if the focus is not in the focal plane, it can be because the beam originated at a finite distance from the focusing optic,  and thereby the point of focus, (basically the image of the point source), will be out of the focal plane, with the focused distance ## m ## given by ## \frac{1}{f}=\frac{1}{b}+\frac{1}{m} ##,  where ## b ## is the distance from the source to the focusing mirror. (Here we’re using the letter ## b ##  differently from above where it represented the width of the beam.)  In any case,  the size of the focused spot will be near that given by the above formula.

Comparison with spherical lenses and mirrors=the case of imperfect focusing:

In less-than-perfect focusing,  for example in the case of a spherical mirror and/or lens, the focusing quality doesn’t quite achieve the diffraction limit.  In these cases,  it is the lens aberrations (such as spherical aberration) that determine the focused spot size from which,  by energy conservation,  the focused intensity can be computed. Unlike the case for an optic with perfect focusing,  the spot size from lens aberrations is likely to remain the same size or grow larger as the beam/optics becomes larger,  while for perfect focusing,  as we saw above,  it decreases in size.  In cases where lens aberrations cause the focused spot to be considerably larger than the diffraction limit of perfect focusing,  ray tracing methods can be used to compute the focused spot size.

Conclusion:

The reader should recognize that diffraction principles determine the focused spot size and intensity that occurs when parallel rays (a collimated beam) are brought to focus by an optic that has perfect focusing.  The focused intensity is proportional to the second power of the area of the incident beam, and the area of the focused spot is inversely proportional to the area of the incident beam, conserving energy in the process.