Lens Theory

Singlet Lenses - Nomenclature

Below is a common diagram used to define quantities for the analysis of a singlet lens. The quantities defined can also be extended to multi-element systems by procedures outlined in optical design textbooks.

Ray A is incident from left to right and parallel to the optical axis. Ray A is refracted to point F2 on axis, the back, second, or secondary focal point.

Ray B is incident from right to left and parallel to the optical axis. Ray B is refracted to point F1 on axis, the front, first, or primary focal point.

Suppose the lens is "well-corrected" That means all rays A parallel to the axis are focused to a single point F2, regardless of their distance from the axis. It is clear then that the family of rays refracted to F2 intersect the family of progenitor rays A in a sphere centred at F2. This sphere, or, in general, surface of intersection of progenitor and refracted rays is called the back, second, or secondary principal surface. The intersection H2 of this surface with the optical axis is called the back, second, or secondary principal point.

Similarly, rays B incident from the right, and their refracted rays impinging on F1, intersect to form a front, first, or primary principal surface, which intersects the optical axis at H1, the front, first, or primary principal point.

In the paraxial approximation, all ray angles are small. This approximation permits the modelling of the principal surfaces as principal planes. The distances from the front and back

principal points to their respective focal points are equal and given by f , the focal length or effective focal length. The front and back focal distances FFD and BFD, respectively, are measured from the surface intersections V1 and V2 to the respective focii. V1 and V2 are called the front and back vertices.

To calculate these quantities above we need sign conventions.

1. Distance increases towards the right.
2. A surface has positive curvature if its centre of curvature is to the right of the vertex of the surface. In other words, if you have to move in the positive direction from the vertex of a surface to reach its centre of curvature, the curvature is positive.
3. Angles increase from zero, parallel to the optical axis, to positive values in a counterclockwise direction from the optical axis.
4. V1H1 and H2V2 are positive if the principal points are to the right and left, respectively, of their corresponding vertices. Note that the principal points can fall outside of the lens.

The formulas below now permit the calculation of the effective focal length, front and back focal distances, and positions of the principal points of a singlet lens in the paraxial approximation. The distance tc below is the centre thickness of the lens V1V2.

Performance and Proper Orientation of Singlet Lenses for Minimum Spot Size

High quality singlet lenses are of particular interest in laser focusing and beam handling applications because of their low cost, high damage threshold, and availability of a large variety of standard parts. We offer a large selection of BK7, UV grade fused silica, and SF11 singlets that can satisfy most monochromatic focusing requirements. Can singlet lenses provide diffraction limited performance in the focusing of collimated beams? Examine the figure below.

Shape Factor

For singlet lenses, the smallest beam diameter dMS at plane MS can be computed from third order aberration theory. The result is:

dMS = fblur

where blur is the "full angle angular blur" of the lens for collimated input light, given by:

blur = (d0/f)³[ n² - (2n+1)K + (n+2)K² / n ] / 32(n-1)²

where d0 is the input beam diameter and K is the shape factor of the lens given by:

K = R2 / ( R2 -R1)

Note that the lens orientation is contained in K. The angular blur is plotted on the next page as a function of shape factor for three typical glasses used in our singlets.

diff = 2.44 / d0

ddiff = fdiff = 2.44 f / d0

This is the diameter of the first dark ring of the Airy pattern in the focal plane. For the above example, ddiff = 4.8 µm at a wavelength of 632.8 nm. Comparing the aberration limited geometrical optic spot size fblur to the diffraction limited spot size fdiff suggests a criterion for diffraction limited operation of a singlet lens. Compute the focal length as a function beam diameter where fblur = fdiff.
If fblur > fdiff the spot size is limited by aberration.

If fblur < fdiff the spot size is essentially diffraction limited. The curves on the following page depict this computation for properly oriented plano convex BK7 and fused silica singlets. To use the curves, find the beam diameter and desired focal length point on the graph. If this point is above the curve for intended wavelength, plano convex singlet lens performance is essentially diffraction limited. If this point is on or below curve, consider using a doublet to decrease aberrated spot size.

Additional Notes

1. In designing optical systems for low divergence laser beams, the paraxial approximation is excellent as far as the positioning of the elements is concerned. However, due to diffraction, the magnifications encountered may be completely different than those calculated from geometric optics. Use the familiar ABCD matrix formalism. If you treat the lenses as thin lenses, use distances between principal points to determine the element positioning. If you use the actual radii of curvature, indices of refraction, and centre thicknesses in a more detailed calculation, your calculation will already implicitly take into account the principal planes.
2. Formulas for Gaussian beam waists are readily derived using standard ABCD matrix formalism. Do not take the beam diameter results too seriously, however, unless you are working with cw HeNe, ion, Ti:Sapphire, Nd:YAG, or dye lasers with published and verifiable TEMoo waists and locations. In particular, "near-Gaussian" pulsed solid state lasers are many times diffraction limited as may be verified by comparing the manufacturers beam diameter and full angle divergence specifications.
3. Single-stage excimer and copper vapour lasers are even more divergent, even when using unstable resonator optics. (Here, the beam divergence actually varies during the pulse, with the brightness increasing with round trip number.) For these lasers, the divergence limited spot size is given by
   df · , where is the measured full angle divergence. It is this spot size that should be computed and compared to the spot size due to spherical aberration in the choice of a focusing lens.