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Normalized Parameter Relating the Spherical Aberration of a Thin Lens to the Diffraction Limit and Fiber Coupling Efficiency Matt Young, Physics Department, Colorado School of Mines, Golden, CO.

Abstract In this paper, we calculate the transverse spherical aberration TA of a thin lens and defines a normalized aberration Y equal to TA divided by the theoretical resolution limit. As a rule of thumb, (a) a thin lens that suffers only from spherical aberration may be considered effectively diffraction-limited as long as Y < 1.6. Similarly, (b) the coupling efficiency of a Gaussian beam to a single-mode fiber may be high even when Y > 1.6, and, specifically, (c) the lens need be diffraction-limited only over a radius approximately equal to the radius (to the 1/e-point) of the Gaussian beam.

Background We begin with the standard third-order aberration theory as formulated by Jenk­ins and White.1 Their Eq. (9h) shows that the transverse spherical aberration ta of a thin lens is

where

Here, V is the paraxial image distance, h is the height (distance from the axis) of the incoming ray at the point where it hits the lens, n is the index of refraction of the glass, ƒ ' is the focal length of the lens, p = (l' - l)/(l' + l) is the position factor, q = (r2 + r1)/(r2 - r1) is the shape factor, r1 is the radius of curvature of the first sur­face of the lens, and r2 is the radius of cur-

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vature of the second surface. In the nota­tion of Jenkins and White, the object dis­tance l is positive if the object is real, l' is positive if the image is real, and r1 and r2 are positive if the lens is double convex.

When there is spherical aberration, a ray does not intercept the paraxial image plane on the axis but rather at a distance to from the axis (Figure 1). to is greatest for a mar­ginal ray, and we call that extreme value TA. TA is a measure of the spherical aberration of the lens. Roughly, spherical aberration is significant if the aberration TA of the mar­ginal ray exceeds the theoretical resolution limit RL = 0.61λ/NA of the lens, where λ is the wavelength of the light, and NA is the effective numerical aperture of the lens.

We may use Eq. (415) of Martin to relate to to the wavefront aberration, that is, to the deviation of the wavefront from a sphere.2 The result is that

where a(h) or α(θ) is the wavefront aberra­tion (in units of length) due to spherical aberration and θ = h/l'. When h equals the radius of the lens, θ equals the effective numerical aperture of the lens.

A lens that displays no aberrations is diffraction-limited. However, a lens is commonly said to be diffraction-limited as long as the maximum wavefront aberra­tion is less than λ/4. Hereafter, we will call such a lens "effectively diffraction-limited."

Normalized aberration Instead of talking about the transverse aberration in units of length, we may

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define a normalized aberration Y = TAJRL, where TA means the transverse aberration of the marginal ray. Y may be expressed in terms of ƒ ' and λ as

where Φ = 1/(2 · NA) is the effective F-number of the lens. The condition Y= 1.6 is equivalent to the condition that the maximum wavefront aberration equals λ/4. Thus, a lens is effectively diffraction-limited as long as Y≤ 1.6.

For given effective F-number and wavelength, Y is pro­portional to ƒ'; that is, the longer the focal length of the lens, the less likely it is to be effectively diffraction-limited. This is so because the theoretical resolution limit RL depends only on the effective F-number, whereas the aberration TA is pure­

ly geometrical and scales in proportion to the focal length, provided that the effective numerical aperture or F-num-ber is held constant.

Table 1 lists Y for several planoconvex and symmetrical, double-convex lenses both with the object at

Figure 1 . A planoconvex lens displaying spherical aberration. The vert ical line to the right of the lens represents the paraxial image plane. The longitudi­nal aberration LA is assumed to be very much smaller than the image distance l'. The transverse aberration is TA, and the radius at which the ray hits the lens is h. n is the index of refraction of the lens, and λ is the wavelength of the light.

∞ and at unit magnifi­cation. The index of refraction of the lenses is 1.5, and the wave­length is 633 nm. The curved surface of the planoconvex lens faces

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the long conjugate; if the lens is inserted in the opposite orientation, each entry in the table must be multiplied by 4. In the lower half of the table, l' is used in place of ƒ, and "NA" and "F-no." mean "effective numerical aperture" and "effective F-number."3 The numbers in Table 1 may be scaled to other wave­lengths or focal lengths, but not indexes of refraction, by using Eq. (4). Table 1 quantifies spherical aberration only, not off-axis aberrations such as coma. It therefore pertains only to imaging point objects or laser beams on the axis of the lens, and to collimating a beam.

The planoconvex lens shows less aberration with the curved surface fac­ing the long conjugate because the rays undergo two small refractions instead of one large refraction. More specifical­ly, consider the case where the flat side faces the long conjugate. A given ray passes undeviated through the flat sur­face and is deviated by an angle δ at the second surface. The paraxial approxi­mation assumes that sinδ = δ, but in

fact sinδ differs from δ by an increment that is proportional to δ3, so the transverse aberration must also be proportional to δ3. Suppose now that the planoconvex lens is used in the opposite orientation, with the curved surface facing the long conjugate. Since the focal length of the lens is the same either way, a given ray is deviated by approximately δ/2 at each surface. But sin(δ/2) differs from δ/2 by an increment proportional to (δ/2)3, and there are two refractions. Thus, the deviation from the paraxial approximation is proportional to 2(δ/2)3 = δ3/4, which is 4 times less than δ3.

Use of Table 1. Suppose, for example, that we want to image a point at unit magnification, with an effective numerical aper­ture of 0.06 (effective F-number of 8), and with an image dis­tance l' = 2 cm. According to Table 1(d) for a double-convex lens, Y = 7 > 1.6, so the image projected by a double-convex lens will not be effectively diffraction-limited. (If we use a planoconvex lens at 1:1, by the way, Ywill be about 15, not 7.) Let us try two planoconvex lenses back to back. The focal length of each lens will have to be 2 cm. According to Table 1(a), for a planoconvex lens with one conjugate at ∞, Y= 0.9. Since there are two lenses in series, we have to double this number: 2 · Y= 1.8 ≅ 1.6, so the two-lens system is very close to being effectively diffraction-limited. The two-lens system is an improvement over the single-lens system for much the same reason that the planoconvex lens has a preferred orienta­tion: the four small refractions of the two-lens system more nearly obey the paraxial approximation than the two larger refractions of the double-convex lens.

Figure 2 shows in graphical form the data of Table 1(a) and (d) for a lens whose focal length is 1 cm. For a lens that has any other focal length, multiply the value of Y determined from the graph by the focal length of the lens in centimeters.

Finally, we can partially correct for spherical aberration

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by defocusing slightly.4 If the defocusing equals one-half the longitudinal aberration LA = TA/NA, then the wavefront aberration due to defocusing is

and the maximum aberration in the plane of best focus is reduced by a factor of 4 below that in the paraxial image plane. Table 1 is therefore conservative, and each entry can be divided by 4 for a less conservative, possibly more realistic esti­mate of Y. Similarly, any value of Y obtained from Figure 2 may be divided by 4 for a less conservative estimate.

Gaussian beams We often use single-element lenses to focus beams into and out of single-mode fibers, many of which display approxi­mately Gaussian beams. The lenses have relatively high numerical apertures and surprisingly high values of Y, yet they allow fairly efficient coupling into the fibers. Why? Pri­marily because the beam substantially underfills the lens, so the rays near the margin carry relatively little or no power.

More specifically, we can use Eqs. (2) and (3) to calculate the point spread function in the presence of spherical aberra­tion and defocusing. Briefly, the wavefront aberration due to spherical aberration is given by Eq. (3) and that due to defo­cusing by Eq. (5). The defocusing required to optimize the focusing of a Gaussian beam is less than that given by Eq. (5), since the rays near the margin do not necessarily contribute, so we write the net aberration as p(θ) = s(θ) - α · d(θ), where α is an adjustable parameter less than 1. The field just after the lens (strictly, in the exit pupil of the lens) is

where θ0 = Λ/ΠW0 is the numerical aperture of the fiber and w0 is its mode-field radius. Here, we assume that the beam that is focused onto the end of the fiber exactly matches the mode of the fiber, except for the aberrations of the lens.

Because the lens has a finite aperture, it truncates the beam; the net transmittance of the lens is

The field u(r) at the focal point of the lens is given, in the paraxial approximation, by the Hankel transform

of the field U(θ) at the exit pupil of the lens.5 Here, k = 2π/λ, and J0 is the Bessel function of order 0. The amplitude u(r) represents the point spread function of the lens whose numer­ical aperture is NA when it is illuminated by a Gaussian beam.

Coupling into a fiber The efficiency of coupling into the fiber is the square of an overlap integral multiplied by the net transmittance T of the lens:

Figure 2 . The normalized aberration Y of a 1 cm (focal length) lens as a function of effect ive F-number or numerical aperture for the cases corre­sponding to Table 1(a) and (d), that is, to the optimum configurations for unit magnif ication and for one infinite conjugate.

Figure 3 . The point spread function of a diffraction-limited Gaussian beam compared wi th that of a Gaussian beam that is focused by a thin, planocon­vex lens wi th significant spherical aberration. The numerical aperture of the lens is 1.5 t imes that of the beam. The focal length of the lens is 12.5 mm, and the wavelength of the light is 1.3 μm. The image has been optimized by defocusing.

where g(r) = exp[-(r/w0)2] is the Gaussian function that approximates the mode field of the fiber and * means com­plex conjugate (irrelevant in this case since g(r) is real).6 The Gaussian approximation applies to many single-mode fibers designed for communications but by no means to all.

The defocusing required to attain maximum coupling into a fiber is not obvious, but it is simple to vary the parameter α until a maximum is achieved. Table 2 and Figure 3 are the results of such calculations for a specific lens and a specific fiber. The lens is a planoconvex silica lens whose index of refraction is 1.47 and whose focal length is 12.5 mm. The fiber is a common step-index fiber whose mode-field radius is 4.5 μm. The numer­ical aperture of the lens varies from 1 to 2.5 times the numerical aperture of the fiber. The wavelength of the light is 1.3 μm, and the beam is assumed to be collimated when it hits the lens.

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engineering & laboratory notes

Figure 3 is representative and compares u(r) for the diffraction-limited condition with the condition that the numerical aperture of the lens is 1.5 times the numerical aperture of the fiber. The defocusing has been optimized by varying the parameter α. The transverse aberration is 43 μm, and Y= 7.5, or several times 1.6. Even so, the rela­tive amplitude of the aberrated beam falls below 0.05 within 15 μm of the center of the beam. The transmit-tance of the beam is 0.98, and the coupling efficiency to the fiber is 0.84.

Table 2 shows that the coupling efficiency may be nearly 90%, even though the normalized aberration Y is many times the value of 1.6. Further, the diameter of the lens must be only 1.5 to 2 times the diameter of the beam, as indicated by the relative numerical aperture in column 1. The lens, however, evidently has to be effectively diffraction-limited

only over a diameter that corresponds approxi­mately to the numerical aperture of the fiber. The coupling efficiency of 80 to 90% is roughly consistent with what we have found when we use a cat's eye: a system in which the light from a fiber is collimated by a planoconvex lens and directed back through the lens and into the fiber by a plane mirror.

Many thanks to Kent Rochford of the Opto­electronics Division of the National Institute of Standards and Technology and Kenn Arnett of

Perdix for their perceptive readings of the manuscript. Kent Rochford suggested the graph, Figure 2. Much of this manu­script was prepared when I was a physicist at the National Institute of Standards and Technology and is not subject to copyright.

References 1. F.A. Jenkins and H.E. White, Fundamentals of Optics, 4th ed.,

McGraw-Hill, New York, 1976. 2. L.C. Martin, Technical Optics, 1 , Pitman, London, 1959, p. 114. 3. Table 1 differs from the tables that I have distributed in annual

short courses during the past 20 years. Here, I use image dis­tance l' and effective F-number l'/D, whereas, in the short courses, I used focal length f and F-number f'/D, even for imaging at unit magnification.

4. Warren M. Smith, Modern Optical Engineering, 2nd ed. , McGraw-Hill, New York, 1990.

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