Kitabı oku: «Optical Engineering Science», sayfa 8

4.4 Refraction Due to Optical Components

4.4.1 Flat Plate

Equations (4.5a)–(4.5d) give the Gauss-Seidel aberration terms for a spherical reflector. However, for a flat surface, where 1/R = 0, the aberration is non zero.

(4.21)

If we now make the approximation that r₀/u∼NA₀ and express all wavefront errors in terms of the normalised pupil function, we obtain the following expressions.

(4.22)

In all expressions, the wavefront error is proportional to the object distance. Equation 4.22 only considers refraction at a single surface. For a flat plate whose thickness is vanishingly small, it is clear that refraction at the second (glass-air) boundary will produce a wavefront error that is equal and opposite to that induced at the first surface. Furthermore, it is also clear that the form of wavefront error contribution will be identical to Eq. (4.22), but reversed in sign. For a glass plate of finite thickness, t, the effective object distance, expressed as the object distance in air, will be given by u + t/n. Therefore, the relevant wavefront error contributions at the second surface are given by:

(4.23)

The total wavefront error is then simply given by the sum of the two contributions. This is expressed in standard format, as below:

(4.24)

The important conclusion here is that a flat plate will add to system aberration, unless the optical beam is collimated (object at infinite conjugate). This is of great practical significance in microscopy, as a thin flat plate, or ‘cover slip’ is often used to contain a specimen. A standard cover slip has a thickness, typically, of 0.17 mm. Examination of Eq. (4.24) suggests that this cover slip will add significantly to system aberration. In practice, it is the spherical aberration that is of the greatest concern, as θ₀ is generally much smaller than NA₀ in most practical applications. As a consequence, some microscope objectives are specifically designed for use with cover slips and have built in aberration that compensates for that of the cover slip. Naturally, a microscope objective designed for use with a cover slip will not produce satisfactory imaging when used without a cover slip.

Worked Example 4.2 Microscope Cover Slip

A microscope cover slip 0.17 mm thick is to be used with a microscope objective with a numerical aperture of 0.8. The refractive index of the cover slip is 1.5. What is the root mean square (rms) spherical aberration produced by the cover slip? The aberration is illustrated in Figure 4.7.

From Eq. (4.24):

Figure 4.7 Spherical aberration in cover slip.

Substituting the above values we get: Ksa = 0.003 22 mm or 3.2 μm.

The wavefront error (in microns) is thus given by:

where p is the normalised pupil function.

For reasons that will become apparent later, in practice, wavefront errors are usually expressed as a fraction of some standard wavelength, for example 589 nm. The above wavefront error represents about 0.4 × λ when expressed in this way. An rms wavefront error of about λ/14 is considered consistent with good image quality. This level of aberration is, therefore, significant and measures must be taken (within the objective) to correct for it.

4.4.2 Aberrations of a Thin Lens

We extend the treatment already outlined to analyse a thin lens. A thin lens can be considered as combination of two refractive surfaces, where the distance between the two surfaces is ignored. In practice, this is a reasonable assumption, provided the thickness is much less than the radii of the surfaces in question. Of course, the wavefront error produced by the two surfaces is simply the sum of the aberrations of the individual surfaces. A schematic for the analysis is shown in Figure 4.8.

The wavefront error contribution for the first surface is very easy to compute; it is simply that set out in Eqs. (4.5a)–(4.5d). To compute the contribution for the second surface, one can analyse this using the same methodology as in Section 4.2, but exploiting natural symmetry. That is to say, one can analyse the second surface by rotating the whole surface about the y axis, such that z → −z and x → −x. In this event, for the second surface, R → −R₂, u → v, θ → −θ. It is then simply a case of substituting these values into the formulae in Eqs. (4.5a)–(4.5d) and adding the wavefront error contribution of the first surface. The total wavefront error for the thin lens is then:

Figure 4.8 Aberration analysis for thin lens.

(4.25a)

(4.25b)

(4.25c)

(4.25d)

4.4.2.1 Conjugate Parameter and Lens Shape Parameter

In terms of gaining some insight into the behaviour of a thin lens, the formulae in Eqs. (4.25a)–(4.25d) are a little opaque. It would be somehow useful to express the aberrations of a thin lens directly in terms of its focusing power and some other parameters. The first of these other parameters is the so called conjugate parameter, t. The conjugate parameter is defined as below:

(4.26)

As we are dealing with a thin lens, we can use the thin lens formula to calculate the focal length, f, of the lens:

This, in turn, leads to expressions for u and v:

(4.27)

Figure 4.9 illustrates the conjugate parameter schematically. The infinite conjugate is represented by a conjugate parameter of ±1. If the conjugate parameter is +1, then the image is at infinity. Conversely, a conjugate parameter of −1 is associated with an object located at the infinite conjugate. In the symmetric scenario where object and image distances are identical, then the conjugate parameter is zero. As illustrated in Figure 4.9, where the conjugate parameter is greater than 1, then the object is real and the image is virtual. Finally, where the conjugate parameter is less than −1, then the object is virtual and the image is real.

Figure 4.9 Conjugate parameter.

Figure 4.10 Coddington lens shape parameter.

We have thus described object and image location in terms of a single parameter. By analogy, it is also useful to describe a lens in terms of its focal power and a single parameter that describes the shape of the lens. The lens, of course, is assumed to be defined by two spherical surfaces, with radii R₁ and R₂, defining the first and second surfaces respectively. The shape of a lens is defined by the so-called Coddington lens shape factor, s, which is defined as follows:

(4.28)

As before, the power of the lens may be expressed in terms of the lens radii:

where n is the lens refractive index.

As with the conjugate parameter and the object and image distances, the two lens radii can be expressed in terms of the lens power and the shape factor, s.

(4.29)

Figure 4.10 illustrates the lens shape parameter for a series of lenses with positive focal power. For a symmetric, bi-convex lens, the shape factor is zero. In the case of a plano-convex lens, the shape factor is 1 where the plane surface faces the image and is −1 where the plane surface faces the object. A shape factor of greater than 1 or less than −1 corresponds to a meniscus lens. Here, both radii have the same sense, i.e. they are either both positive or both negative. For a shape parameter of greater than 1, the surface with the greater curvature faces the object and for a shape parameter of less than −1, the surface with the greater curvature faces the image. Of course, this applies to lenses with positive power. For (diverging) lenses with negative power, then the sign of the shape factor is opposite to that described here.

4.4.2.2 General Formulae for Aberration of Thin Lenses

Having parameterised the object and image distances and the lens radii in terms of the conjugate parameter, shape parameter, and lens power, we can recast the expressions in Eqs. (4.25a)–(4.25d) in a more generic form. With a little algebraic manipulation, we obtain the following expressions for the Gauss-Seidel aberration of a lens with the stop at the lens surface:

(4.30a)

(4.30b)

(4.30c)

(4.30d)

Again, casting all expressions in the form set out in Chapter 3, as for the expressions for the mirror we have

(4.31a)

(4.31b)

(4.31c)

(4.31d)

Once again, the Petzval curvature is simply given by subtracting twice the KAS term in Eq. (4.31c) from the field curvature term in Eq. (4.31d). This gives:

(4.32)

That is to say, a single lens will produce a Petzval surface whose radius of curvature is equal to the lens focal length multiplied by its refractive index. Once again, the Petzval sum may be invoked to give the Petzval curvature for a system of lenses:

(4.33)

It is important here to re-iterate the fact that for a system of lenses, it is impossible to eliminate Petzval curvature where all lenses have positive focal lengths. For a system with positive focal power, i.e. with a positive effective focal length, there must be some elements with negative power if one wishes to ‘flatten the field’.

Before considering the aberration behaviour of simple lenses in a little more detail, it is worth reflecting on some attributes of the formulae in Eqs. (4.30a)–(4.30d). Both spherical aberration and coma are dependent upon the lens shape and conjugate parameters. In the case of spherical aberration there are second order terms present for both shape and conjugate parameters, whereas the behaviour for coma is linear. However, the important point to recognise is that the field curvature and astigmatism are independent of both lens shape and conjugate parameter and only depend upon the lens power. Once again, it must be emphasised that this analysis applies only to the situation where the stop is situated at the lens.

4.4.2.3 Aberration Behaviour of a Thin Lens at Infinite Conjugate

We will now look at a simple special case to apply to a thin lens with the stop at the lens. This is the common situation where a lens is being used to focus an object located at the infinite conjugate, such as a telescope objective or a lens focusing a parallel laser beam. From Eq. (4.26), the conjugate parameter, t, is equal to −1. Substituting t = −1 into Eq. (4.31a) gives the spherical aberration as:

(4.34)

The important point to note about Eq. (4.34) is that the spherical aberration can never be equal to zero and that for a positive lens, KSA is always negative. This means that the longitudinal aberration for a positive lens is also negative and that, for all single lenses, more marginal rays are brought to a focus closer to the lens. Whilst Eq. (4.34) asserts that the spherical aberration in this case can never be zero, its magnitude can be minimised for a specific lens shape. Inspection of Eq. (4.34) reveals that this condition is met where:

(4.35)

This optimum shape factor corresponds to the so-called ‘best form singlet’ and is generally available from optical component suppliers, particularly with regard to applications in the focusing of laser beams. For a refractive index of 1.5, the optimum shape factor is around 0.7. This is close in shape to a plano-convex lens. However, it is important to emphasise, that optimum focusing is obtained where the more steeply curved surface is facing the infinite conjugate. Generally, also, where a plano-convex lens is used to focus a collimated beam, the curved surface should face the infinite conjugate. This behaviour is shown in Figure 4.11, which emphasises the quadratic dependence of spherical aberration on lens shape factor.

Coma for the infinite conjugate also depends upon the shape factor. However, in this instance, the dependence is linear. Once more, substituting t = −1 into Eq. (4.31b), we get:

(4.36)

Unlike in the case for spherical aberration, there exists a shape factor for which the coma is zero. This is simply given by:

(4.37)

For a refractive index of 1.5, this minimum condition is met for a shape factor of 0.8. This is similar, but not quite the same as the optimum for spherical aberration. Again, the most curved surface should face the infinite conjugate. Overall behaviour is illustrated in Figure 4.12.

Figure 4.11 Spherical aberration vs. shape parameter for a thin lens.

Figure 4.12 Coma vs lens shape for various conjugate parameters.

Once again, this specifically applies to the situation where the stop is at the lens surface. Of course, as stated previously, neither astigmatism nor field curvature are affected by shape or conjugate parameter.

Although it is impossible to reduce spherical aberration for a thin lens to zero at the infinite conjugate, it is possible for other conjugate values. In fact, the magnitude of the conjugate parameter must be greater than a certain specific value for this condition to be fulfilled. This magnitude is always greater than one for reasonable values of the refractive index and so either object or image must be virtual. It is easy to see from Eq. (4.31a) that this threshold value should be:

(4.38)

For n = 1.5, this threshold value is 4.58. That is to say for there to be a shape factor where the spherical aberration is reduced to zero, the conjugate parameter must either be less than −4.58 or greater than 4.58. Another point to note is that since spherical aberration exhibits a quadratic dependence on shape factor, where this condition is met, there are two values of the shape factor at which the spherical aberration is zero. This behaviour is set out in Figure 4.13 which shows spherical aberration as a function of shape factor for a number of difference conjugate parameters.

Worked Example 4.3 Best form Singlet

A thin lens is to be used to focus a Helium-Neon laser beam. The focal length of the lens is to be 20 mm and the lens is required to be ‘best form’ to minimise spherical aberration. The refractive index of the lens is 1.518 at the laser wavelength of 633 nm. Calculate the required shape factor and the radii of both lens surfaces. From Eq. (4.35) we have:

Figure 4.13 Spherical aberration vs shape factor for various conjugate parameter values.

The optimum shape factor is 0.742 and we can use this to calculate both radii given knowledge of the required focal length. Rearranging Eq. (4.29) we have:

This gives:

It is the surface with the greatest curvature, i.e. R1, that should face the infinite conjugate (the parallel laser beam).

4.4.2.4 Aplanatic Points for a Thin Lens

Just as in the case of a single surface, it is possible to find a conjugate and lens shape pair that produce neither spherical aberration nor coma. For reasons outlined previously, it is not possible to eliminate astigmatism or field curvature for a lens of finite power. If the spherical aberration is to be zero, it must be clear that for the aplanatic condition to apply, then either the object or the image must be virtual. Equations (4.31a) and (4.31b) provide two conditions that uniquely determine the two parameters, s and t. Firstly, the requirement for coma to be zero clearly relates s and t in the following way:

Setting the spherical aberration to zero and substituting for t we have the following expression given entirely in terms of s:

and

Finally this gives the solution for s as:

(4.39a)

Accordingly the solution for t is

(4.39b)

Of course, since the equation for spherical aberration gives quadratic terms in s and t, it is not surprising that two solutions exist. Furthermore, it is important to recognise that the sign of t is the opposite to that of s. Referring to Figure 4.10, it is clear that the form of the lens is that of a meniscus. The two solutions for s correspond to a meniscus lens that has been inverted. Of course, the same applies to the conjugate parameter, so, in effect, the two solutions are identical, except the whole system has been inverted, swapping the object for image and vice-versa.

An aplanatic meniscus lens is an important building block in an optical design, in that it confers additional focusing power without incurring further spherical aberration or coma. This principle is illustrated in Figure 4.14 which shows a meniscus lens with positive focal power.

It is instructive, at this point to quantify the increase in system focal power provided by an aplanatic meniscus lens. Effectively, as illustrated in Figure 4.14, it increases the system numerical aperture in (minus) the ratio of the object and image distance. For the positive meniscus lens in Figure 4.14, the conjugate parameter is negative and equal to −(n + 1)/(n − 1). From Eq. (4.27) the ratio of the object and image distances is given by:

As previously set out, the increase in numerical aperture of an aplanatic meniscus lens is equal to minus the ratio of the object and image distances. Therefore, the aplanatic meniscus lens increases the system power by a factor equal to the refractive index of the lens. This principle is of practical consequence in many system designs. Of course, if we reverse the sense of Figure 4.14 and substitute the image for the object and vice versa, then the numerical aperture is effectively reduced by a factor of n.

Figure 4.14 Aplanatic meniscus lens.

Worked Example 4.4 Microscope Objective – Hyperhemisphere Plus Meniscus Lens

We now wish to add some power to the microscope objective hyperhemisphere set out in Worked Example 4.1. We are to do so with an extra meniscus lens situated at the vertex of the hyperhemisphere with a negligible separation. As with the hyperhemisphere, the meniscus lens is in the aplanatic arrangement. The meniscus lens is made of the same material as the hyperhemisphere, that is with a refractive index of 1.6. All properties of the hyperhemisphere are as set out in Worked Example 4.1.

What are the radii of curvature of the meniscus lens and what is the location of the (virtual) image for the combined system? The system is as illustrated below.

We know from Worked Example 4.1 that the original image distance produced by the hyperhemisphere is −23.4 mm. The object distance for the meniscus lens is thus 23.4 mm. From Eq. (4.39a) we have:

There remains the question of the choice of the sign for the conjugate parameter. If one refers to Figure 4.14, it is clear that the sense of the object and image location is reversed. In this case, therefore, the value of t is equal to +4.33 and the numerical aperture of the system is reduced by a factor of 1.6 (the refractive index). In that case, the image distance must be equal to minus 1.6 times the object distance. That is to say:

We can calculate the focal length of the lens from:

Therefore the focal length of the meniscus lens is 62.4 mm. If the conjugate parameter is +4.33, then the shape factor must be −(2n + 1), or −4.2 (note the sign). It is a simple matter to calculate the radii of the two surfaces from Eq. (4.29):

Finally, this gives R₁ as −23.4 mm and R₂ as −14.4 mm. The signs should be noted. This follows the convention that positive displacement follows the direction from object to image space.

If the microscope objective is ultimately to provide a collimated output – i.e. with the image at the infinite conjugate, the remainder of the optics must have a focal length of 37.44 mm (i.e. 23.4 × 1.6). This exercise illustrates the utility of relatively simple building blocks in more complex optical designs. This revised system has a focal length of 9 mm. However, the ‘remainder’ optics have a focal length of 37.4 mm, or only a quarter of the overall system power. Spherical aberration increases as the fourth power of the numerical aperture, so the ‘slower’ ‘remainder’ will intrinsically give rise to much less aberration and, as a consequence, much easier to design. The hyperhemisphere and meniscus lens combination confer much greater optical power to the system without any penalty in terms of spherical aberration and coma. Of course, in practice, the picture is complicated by chromatic aberration caused by variations in refractive properties of optical materials with wavelength. Nevertheless, the underlying principles outlined are very useful.

4.5 The Effect of Pupil Position on Element Aberration

In all previous analysis, it is assumed that the stop is located at the optical surface in question. This is a useful starting proposition. However, in practice, this is most usually not the case. With the stop located at a spherical surface, by definition, the chief ray will pass directly through the vertex of that surface. If, however, the surface is at some distance from the stop, then the chief ray will, in general, intersect the surface at some displacement from the surface vertex. This displacement is, in the first approximation, proportional to the field angle of the object in question. The general concept is illustrated in Figure 4.15.

Instead of the stop being located at the surface in question, the stop is displaced by a distance, s, from the surface. The chief ray, passing through the centre of the stop defines the field angle, θ. In addition, the pupil co-ordinates defined at the stop are denoted by r_x and r_y. However, if the stop were located at the optical surface, then the field angle would be θ′, as opposed to θ. In addition, the pupil co-ordinates would be given by rx′ and ry′. Computing the revised third order aberrations proceeds upon the following lines. All the previous analysis, e.g. as per Eqs. (4.31a)–(4.31d), has enabled us to express all aberrations as an OPD in terms of θ′, rx′, and ry′. It is clear that to calculate the aberrations for the new stop locations, one must do so in terms of the new parameters θ, rx, and ry. This is done by effecting a simple linear transformation between the two sets of parameters. Referring to Figure 4.15, it is easy to see:

(4.40a)

(4.40b)

Figure 4.15 Impact of stop movement.

(4.40c)

The effective size of the pupil at the optic is magnified by a quantity M_p and the pupil offset set out in Eq. (4.40b) is directly related to the eccentricity parameter, E, described in Chapter 2. Indeed, the product of the eccentricity parameter and the Lagrange invariant, H is simply equal to the ratio of the marginal and chief ray height at the pupil. That is to say:

(4.41)

In this case, r₀ refers to the pupil radius at the stop and r₀′ to the effective pupil radius at the surface in question. As a consequence, we can re-cast all three equations in a more convenient form.

(4.42)

The angle, θ₀ is representative of the maximum system field angle and helps to define the eccentricity parameter and the Lagrange invariant. We already know the OPD when cast in terms of rx′, ry′, and θ, as this is as per the analysis for the case where the stop is at the optic itself. That is to say, the expression for the OPD is as given in Eqs. and these aberrations defined in terms of KSA′, KCO′, KAS′, KFC′, and KDI′. Therefore, the total OPD attributable to the five Gauss-Seidel aberrations is given by:

(4.43)

To determine the aberrations as expressed by the pupil co-ordinates for the new stop location, it is a simple matter of substituting Eq. (4.42) into Eq. (4.43). This results in the so-called stop shift equations:

(4.44a)

(4.44b)

(4.44c)

(4.44d)

(4.44e)

What this set of equations reveals is that there exists a ‘hierarchy’ of aberrations. Spherical aberration may be transmuted into coma, astigmatism, field curvature, and distortion by shifting the stop position. Similarly, coma may be transformed into astigmatism, field curvature, and distortion and both astigmatism and field curvature may produce distortion. However, coma can never produce spherical aberration and neither astigmatism nor field curvature is capable of generating spherical aberration or coma. Equation (4.44e) reveals, for the first time, that it is possible to generate distortion by shifting the stop. Our previous idealised analysis clearly suggested that distortion is not produced where the lens or optical surface is located at the stop.

Another important conclusion relating to Eqs. (4.44a)–(4.44e) is the impact of stop shift on the astigmatism and field curvature. Inspection of Eqs. (4.44c) and (4.44d) reveals that the change in field curvature produced by stop shift is precisely double that of the change in astigmatism in all cases. Therefore, the Petzval curvature, which is given by KFC−2KAS remains unchanged by stop shift. This further serves to demonstrate the fact that the Petzval curvature is a fundamental system attribute and is unaffected by changes in stop location and, indeed component location. Petzval curvature only depends upon the system power. Thus, it is important to recognise that the quantity KFC−2KAS is preserved in any manipulation of existing components within a system. If we express the Petzval curvature in terms of the tangential and sagittal curvature we find:

(4.45)

Since KPetz is not changed by any manipulation of component or stop positions, Eq. (4.45) implies that any change in the sagittal curvature is accompanied by a change three times as large in the tangential curvature. This is an important conclusion.

For small shifts in the position of the stop, the eccentricity parameter is proportional to that shift. Based on this and examining Eqs. (4.44a)–(4.44e), one can come to some general conclusions. For a system with pre-existing spherical aberration, additional coma will be produced in linear proportion to the stop shift. Similarly, the same spherical aberration will produce astigmatism and field curvature proportional to the square of the stop shift. The amount of distortion produced by pre-existing spherical aberration is proportional to the cube of the displacement. Naturally, for pre-existing coma, the additional astigmatism and field curvature produced is in proportion to the shift in the stop position. Additional distortion is produced according to the square of the stop shift. Finally, with pre-existing astigmatism and field curvature, only additional distortion may be produced in direct proportion to the stop shift.

As an example, a simple scenario is illustrated in Figure 4.16. This shows a symmetric system with a biconvex lens used to image an object in the 2f – 2f configuration. That is to say, the conjugate parameter is zero. In this situation, the coma may be expected, by virtue of symmetry, to be zero. For a simple lens, the distortion is also zero. The spherical aberration is, of course, non-zero, as are both the astigmatism and field curvature.

Using basic modelling software, it is possible to analyse the impact of small stop shifts on system aberration. The results are shown in Figure 4.17.

Clearly, according to Figure 4.17, the spherical aberration remains unchanged as predicted by Eq. (4.44a). For small shifts, the amount of coma produced is in proportion to the shift. Since there is no coma initially, the only aberration that can influence the astigmatism and field curvature is the pre-existing spherical aberration. As indicated in Eqs. (4.44c) and (4.44d), there should be a quadratic dependence of the astigmatism and field curvature on stop position. This is indeed borne out by the analysis in Figure 4.17. Similarly, the distortion shows a linear trend with stop position, mainly influenced by the initial astigmatism and field curvature that is present.

Although, in practice, these stop shift equations may not find direct use currently in optimising real designs, the underlying principles embodied are, nonetheless, important. Manipulation of the stop position is a key part in the optimisation of complex optical systems and, in particular, multi-element camera lenses. In these complex systems, the pupil is often situated between groups of lenses. In this case, the designer needs to be aware also of the potential for vignetting, should individual lens elements be incorrectly sized.

Figure 4.16 Simple symmetric lens system with stop shift.

Figure 4.17 Impact of stop shift for simple symmetric lens system.

The stop shift equations provide a general insight into the impact of stop position on aberration. Most significant is the hierarchy of aberrations. For example, no fundamental manipulation of spherical aberration may be accomplished by the manipulation of stop position. Otherwise, there some special circumstances it would be useful for the reader to be aware of. For example, in the case of a spherical mirror, with the object or image lying at the infinite conjugate, the placement of the stop at the mirror's centre of curvature altogether removes its contribution to coma and astigmatism; the reader may care to verify this.