Kitabı oku: «Optical Engineering Science», sayfa 9

4.6 Abbe Sine Condition

Long before the advent of powerful computer ray tracing models, there was a powerful incentive to develop simple rules of thumb to guide the optical design process. This was particularly true for the complex task of ameliorating system aberrations. Working in the nineteenth century, Ernst Abbe set out the Abbe sine condition, which directly relates the object and image space numerical apertures for a ‘perfect’, unaberrated system. Essentially, the Abbe sine condition articulates a specific requirement for a system to be free of spherical aberration and coma, i.e. aplanatic. The Abbe sine condition is expressed for an infinitesimal object and image height and its justification is illustrated in Figure 4.18.

In the representation in Figure 4.18 we trace a ray from the object to a point, P, located on a reference sphere whose centre lies on axis at the axial position of the object and whose vertex lies at the entrance pupil. At the same time, we also trace a marginal ray from the object location to the entrance pupil. The conjugate point to P, designated, P′, is located nominally at the exit pupil and on a sphere whose centre lies at the paraxial image location. For there to be perfect imaging, then the OPD associated with the passage of the marginal ray must be zero. Furthermore, the OPD of the ray from object to image must also be zero. It is also further assumed that the relative OPD of the object to image ray when compared to the marginal ray is zero on passage from points P to P′. This assumption is justified for an infinitesimal object height. Therefore, it is possible to compute the total object to image OPD by simply summing the path differences relative to the marginal ray between the object and point P and between the image and point P′. For there to be perfect imaging this difference must, of course be zero.

Figure 4.18 Abbe sine condition.

(4.46)

n is the refractive index in object space and n′ is the refractive index in image space.

Equation 4.46 is one formulation of the Abbe sine condition which, nominally, applies for all values of θ and θ′, including paraxial angles. If we represent the relevant paraxial angles in object and image space as θ_p and θ_p' then the Abbe sine condition may be rewritten as:

(4.47)

One specific scenario occurs where the object or image lies at the infinite conjugate. For example, one might imagine an object located on axis at the first focal point. In this case, the height of any ray within the collimated beam in image space is directly proportional to the numerical aperture associated with the input ray.

Figure 4.19 illustrates the application of the Abbe sine condition for a specific example. As highlighted previously, the sine condition effectively seeks out the aplanatic condition in an optical system. In this example, a meniscus lens is to be designed to fulfil the aplanatic condition. However, its conjugate parameter is adjusted around the ideal value and the spherical aberration and coma plotted as a function of the conjugate parameter. In addition, the departure from the Abbe sine condition is also plotted in the same way. All data is derived from detailed ray tracing and values thus derived are presented as relative values to fit reasonably into the graphical presentation. It is clear that elimination of spherical aberration and coma corresponds closely to the fulfilment of the Abbe sine condition.

The form of the Abbe sine condition set out in Eq. (4.46) is interesting. It may be compared directly to the Helmholtz equation which has a similar form. However, instead of a relationship based on the sine of the angle, the Helmholtz equation is defined by a relationship based on the tangent of the angle:

It is quite apparent that the two equations present something of a contradiction. The Helmholtz equation sets the condition for perfect imaging in an ideal system for all pairs of conjugates. However, the Abbe sine condition relates to aberration free imaging for a specific conjugate pair. This presents us with an important conclusion. It is clear that aberration free imaging for a specific conjugate (Abbe) fundamentally denies the possibility for perfect imaging across all conjugates (Helmholtz). Therefore, an optical system can only be designed to deliver aberration free imaging for one specific conjugate pair.

Figure 4.19 Fulfilment of Abbe sine condition for aplanatic meniscus lens.

4.7 Chromatic Aberration

4.7.1 Chromatic Aberration and Optical Materials

Hitherto, we have only considered the classical monochromatic aberrations. At this point, we must introduce the phenomenon of chromatic aberration where imperfections in the imaging of an optical system are produced by significant variation in optical properties with wavelength. All optical materials are dispersive to some degree. That is to say, their refractive indices vary with wavelength. As a consequence, all first order properties of an optical system, such as the location of the cardinal points, vary with wavelength. Most particularly, the paraxial focal position of an optical system with dispersive components will vary with wavelength, as will its effective focal length. Therefore, for a given axial position in image space, only one wavelength can be in focus at any one time.

Dispersion is a property of transmissive optical materials, i.e. glasses. On the other hand, mirrors show no chromatic variation and their incorporation is favoured in systems where chromatic variation is particularly unwelcome. Such a system, where the optical properties do not vary with wavelength, is said to be achromatic. As argued previously, a mirror behaves as an optical material with a refractive index of minus one, a value that is, of course, independent of wavelength. In general, the tendency in most optical materials is for the refractive index to decrease with increasing wavelength. This behaviour is known as normal dispersion. In certain very specific situations, for certain materials at particular wavelengths, the refractive index actually decreases with wavelength; this phenomenon is known as anomalous dispersion.

Although dispersion is an issue of concern covering all wavelengths of interest from the ultraviolet to the infrared, for obvious reasons, historically, there has been particular focus on this issue within the visible portion of the spectrum. Across the visible spectrum, for typical glass materials, the refractive index variation might amount to 0.7–2.5%. This variation in the dispersive properties of different materials is significant, as it affords a means to reduce the impact of chromatic aberration as will be seen shortly. Figure 4.20 shows a typical dispersive plot, for the glass material, SCHOTT BK7®.

Figure 4.20 Refractive index variation with wavelength for SCHOTT BK7 glass material.

Because of the historical importance of the visible spectrum, glass materials are typically characterised by their refractive properties across this portion of the spectrum. More specifically, glasses are catalogued in terms of their refractive indices at three wavelengths, nominally ‘blue’, ‘yellow’, and ‘red’. In practice, there are a number of different conventions for choosing these reference wavelengths, but the most commonly applied uses two hydrogen spectral lines – the ‘Balmer-beta’ line at 486.1 nm and the ‘Balmer-alpha’ line at 656.3, plus the sodium ‘D’ line at 589.3 nm. The refractive indices at these three standard wavelengths are symbolised as nF, nC, and nD respectively. At this point, we introduce the Abbe number, VD, which expresses a glass's dispersion by the ratio of its optical power to its dispersion:

(4.48)

The numerator in Eq. (4.48) represents the effective optical or focusing power at the ‘yellow’ wavelength, whereas the denominator describes the dispersion of the glass as the difference between the ‘blue’ and the ‘red’ indices. It is important to recognise that the higher the Abbe number, then the less dispersive the glass, and vice versa. Abbe numbers vary, typically between about 20 and 80. Broadly speaking, these numbers express the ratio of the glass's focusing power to its dispersion. Hence, for a material with an Abbe number of 20, the focal length of a lens made from this material will differ by approximately 5% (1/20) between 486.1 and 656.3 nm.

4.7.2 Impact of Chromatic Aberration

The most obvious effect of chromatic aberration is that light is broad to a different focus for different wavelengths. This effect is known as longitudinal chromatic aberration and is illustrated in Figure 4.21.

As can be seen from Figure 4.21, light at the shorter, ‘blue’ wavelengths are focused closer to the lens, leading to an axial (longitudinal) shift in the paraxial focus for the different wavelengths. In summary, longitudinal chromatic aberration is associated with a shift in the paraxial focal position as a function of wavelength. Thus the effect of longitudinal chromatic aberration is to produce a blur spot or transverse aberration whose magnitude is directly proportional to the aperture size, but is independent of field angle. However, there are situations where, to all intents and purposes, all wavelengths share the same paraxial focal position, but the principal points are not co-located. That is to say, whilst all wavelengths are focused at a common point, the effective focal length corresponding to each wavelength is not identical. This scenario is illustrated in Figure 4.22.

Figure 4.21 Longitudinal chromatic aberration.

Figure 4.22 Transverse chromatic aberration.

The effect illustrated is known as transverse chromatic aberration or lateral colour. Whilst no distinct blurring is produced by this effect, the fact that different wavelengths have different focal lengths inevitably means that system magnification varies with wavelength. As a result, the final image size or height of a common object depends upon the wavelength. This produces distinct coloured fringing around an object and the size of the effect is proportional to the field angle, but independent of aperture size.

Hitherto, we have cast the effects of chromatic aberration in terms of transverse aberration. However, to understand the effect on the same basis as the Gauss-Seidel aberrations, it is useful to express chromatic aberration in terms of the OPD. When applied to a single lens, longitudinal chromatic aberration simply produces defocus that is equal to the focal length divided by the Abbe number. Therefore, the longitudinal chromatic aberration is given by:

(4.49a)

f is the focal length of the lens and r the pupil position.

Figure 4.23 Huygens eyepiece.

Similarly, the transverse chromatic aberration can be expressed as an OPD:

(4.49b)

Examining Eqs. (4.49a) and (4.49b) reveals that the ratio of transverse to longitudinal aberration is given by the ratio of the field angle to the numerical aperture. In practice, for optical elements, such as microscope and telescope objectives, the field angle is very much smaller than the numerical aperture and thus longitudinal chromatic aberration may be expected to predominate. For eyepieces, the opposite is often the case, so the imperative here is to correct lateral chromatic aberration.

Worked Example 4.5 Lateral Chromatic Aberration and the Huygens Eyepiece

A practical example of the correction of lateral chromatic aberration is in the Huygens eyepiece. This very simple, early, eyepiece uses two plano-convex lenses separated by a distance equivalent to half the sum of their focal lengths. This is illustrated in Figure 4.23.

Since we are determining the impact of lateral chromatic aberration, we are only interested in the effective focal length of the system comprising the two lenses. Using simple matrix analysis as described in Chapter 1, the system focal length is given by:

If we assume that both lenses are made of the same material, then their focal power will change as a function of wavelength by a common proportion, α. In that case, the system focal power at the new wavelength would be given by:

For small values of α, we can ignore terms of second order in α, so the change in system power may be approximated by:

The change in system power should be zero and this condition unambiguously sets the lens separation, d, for no lateral chromatic aberration:

(4.50)

If this condition is fulfilled, then the Huygens eyepiece will have no transverse chromatic aberration. However, it must be emphasised that this condition does not provide immunity from longitudinal chromatic aberration.

Figure 4.24 Abbe diagram.

4.7.3 The Abbe Diagram for Glass Materials

For visible applications, the Abbe number for a glass is of equal practical importance as the refractive index itself. The Abbe diagram is a simple graphic tool that captures the basic refractive properties of a wide range of optical glasses. It comprises a simple 2D map with the horizontal axis corresponding to the Abbe number and the vertical axis to the glass index. A representative diagram is shown in Figure 4.24.

By referring to this diagram, the optical designer can make appropriate choices for specific applications in the visible. In particular, it helps select combinations of glasses leading to a substantially achromatic design. One special and key application is the achromatic doublet. This lens is composed of two elements, one positive and one negative. The positive lens is a high power (short focal length) element with low dispersion and the negative lens is a low power element with high dispersion. Materials are chosen in such a way that the net dispersion of the two elements cancel, but the powers do not. This will be considered in more detail in the next section.

The different zones highlighted in the Abbe diagram replicated in Figure 4.24 refer to the elemental composition of the glass. For example, ‘Ba’ refers to the presence of barium and ‘La’ to the presence of lanthanum. Originally, many of the dense, high index glasses used to contain lead, but these are being phased out due to environmental concerns. The Abbe diagram reveals a distinct geometrical profile with a tendency for high dispersion to correlate strongly with refractive index. In fact, it is the presence of absorption features within the glass (at very much shorter wavelengths) that give rise to the phenomenon of refraction and these features also contribute to dispersion.

4.7.4 The Achromatic Doublet

As introduced previously, the achromatic doublet is an extremely important building block in a transmissive (non-mirror) optical design. The function of an achromatic doublet is illustrated in Figure 4.25.

Figure 4.25 The achromatic doublet.

The first element, often (on account of its shape) referred to as the ‘crown element’, is a high power positive lens with low dispersion. The second element is a low power negative lens with high dispersion. The focal lengths of the two elements are f₁ and f₂ respectively and their Abbe numbers V₁ and V₂. Since the intention is that the dispersions of the two elements should entirely cancel, this condition constrains the relative power of the two elements. Individually, the dispersion as measured by the difference in optical power between the red and blue wavelengths is proportional to the reciprocal of the focal power and the Abbe number for each element. Therefore:

(4.51)

From Eq. (4.51), it is clear that the ratio of the two focal lengths should be minus the inverse of the ratio of their respective Abbe numbers. In other words, the ratio of their powers should be minus the ratio of their Abbe numbers. The power of the system comprising the two lenses is, in the thin lens approximation, simply equal to the sum of their individual powers. Therefore, it is possible to calculate these individual focal lengths, f₁ and f₂, in terms of the desired system focal length of f:

Thus, the two focal lengths are simply given by:

(4.52)

In the thin lens approximation, therefore, light will be focused at the same point for the red and blue wavelengths. Consequentially, in this approximation, this system will be free from both longitudinal and transverse chromatic aberration. The simplicity of this approach may be illustrated in a straightforward worked example.

Worked Example 4.6 Simple Achromatic Doublet

We wish to construct and achromatic doublet with a focal length of 200 mm. The two glasses to be used are: SCHOTT N-BK7 for the positive crown lens and SCHOTT SF2 for the negative lens. Both these glasses feature on the Abbe diagram in Figure 4.24 and the Abbe number for these glasses are 64.17 and 33.85 respectively. The individual focal lengths may be calculated using Eq. (4.52):

Therefore, the focal length of the first ‘crown lens’ should be 94.5 mm and the focal length of the second diverging lens should be −179 mm.

Thus far, the analysis design of an achromatic doublet has been fairly elementary. In the previous worked example, we have constrained the focal lengths of the two lens elements to specific values. However, we are still free to choose the shape of each lens. That is to say, there are two further independent variables that can be adjusted. Achromatic doublets can either be cemented or air spaced. In the case of the cemented doublet, as presented in Figure 4.25, the second surface of the first lens must have the same radius as the first surface of the second lens. This provides an additional constraint; thus, for the cemented doublet, there is only one additional free variable to adjust. However, introduction of an air space between the two lenses removes this constraint and gives the designer an extra degree of freedom to play with. That said, the cemented doublet does offer greater robustness and reliability with respect to changes in alignment and finds very wide application as a standard optical component.

As a ‘stock component’ achromatic doublets are designed, generally, for the infinite conjugate. For cemented doublets, with the single additional degree of design freedom, these components are optimised to have zero spherical aberration at the central wavelength. This is an extremely important consideration, for not only are these doublets free of chromatic aberration, but they are also well optimised for other aberrations. Commercial doublets are thus extremely powerful optical components.

4.7.5 Optimisation of an Achromatic Doublet (Infinite Conjugate)

An air spaced achromatic doublet may be optimised to eliminate both spherical aberration and coma. The fundamental power of the wavefront approach in describing third order aberration is reflected in the ability to calculate the total system aberration as the sum of the aberration of the two lenses. In the thin lens approximation, we may simply use Eqs. (4.30a) and (4.30b) to express the spherical aberration and coma contribution for each lens element. We simply ascribe a variable shape parameter, s₁ and s₂ to each of the two lenses. The two conjugate parameters are fixed. In the particular case of a doublet designed for the infinite conjugate, the conjugate parameter for the first lens, t₁, is −1. In the case of the second lens, the conjugate parameter, t₂, is determined by the relative focal lengths of the two lenses and thus fixed by the ratio of the two Abbe numbers and, from Eq. (4.52), we get:

(4.53)

Without going through the algebra in detail, it is clear that having determined both t₁ and t₂, Eqs. (4.30a) and (4.30b) give us two expressions solely in terms of s₁ and s₂. These expressions for the spherical aberration and coma must be set to zero and can be solved for both s₁ and s₂. The important point to note about this procedure is that because Eq. 4.30a contains terms that are quadratic in shape factor, this is also reflected in the final solution. Therefore, in general, we might expect to find two solutions to the equation and this, in general, is true.

Worked Example 4.7 Detailed Design of 200 mm Focal Length Achromatic Doublet

At this point we illustrate the design of an air spaced achromat by looking more closely at the previous example where we analysed a 200 mm achromat design. We are to design an achromat with a focal length of 200 mm working at the infinite conjugate, using SCHOTT N-BK7 and SCHOTT SF2 as the two glasses, with the less dispersive N-BK7 used as the positive ‘crown’ element. Again, the Abbe numbers for these glasses are 64.17 and 33.85 respectively and the n_d values (refractive index at 589.6 nm) 1.5168 and 1.647 69. From the previous example, we know that focal lengths of the two lenses are:

The two conjugate parameters are straightforward to determine. The first conjugate parameter, t₁, is naturally −1. Eq. (4.53) can be used to determine the second conjugate parameter, t₂. This gives:

We now substitute the conjugate parameter values together with the refractive index values (ND) into Eq. (4.30a). We sum the contributions of the two lenses giving the total spherical aberration which we set to zero. Calculating all coefficients we get a quadratic equation in terms of the two shape factors, s₁ and s₂.

(4.54)

We now repeat the same process for Eq. (4.30b), setting the total system coma to zero. This time we get a linear equation involving s₁ and s₂.

(4.55)

Substituting Eq. (4.55) into Eq. (4.54) gives the desired quadratic equation:

(4.56)

There are, of course, two sets of solutions to Eq. (4.56), with the following values:

Solution 1: s₁ = −0.194; s₂ = 1.823

Solution 2: s₁ = 3.198; s₂ = 2.929

There now remains the question as to which of these two solutions to select. Using Eq. (4.29) to calculate the individual radii of curvature from the lens shapes and focal length we get:

Solution 1: R₁ = 121.25 mm; R₂ = −81.78 mm; R₃₋81.29 mm; R₄ = −281.88 mm

Solution 2: R₁ = 23.26 mm; R₂ = 44.43 mm; R₃₋58.91 mm; R₄ = −119.68 mm

The radii R₁ and R₂ refer to the first and second surfaces of lens 1 and R₃ and R₄ to the first and second surfaces of lens 2. It is clear that the first solution contains less steeply curved surfaces and is likely to be the better solution, particularly for relatively large apertures. In the case of the second solution, whilst the solution to the third order equations eliminates third order spherical aberration and coma, higher order aberrations are likely to be enhanced.

The first solution to this problem comes under the generic label of the Fraunhofer doublet, whereas the second is referred to as a Gauss doublet. It should be noted that for the Fraunhofer solution, R₂ and R₃ are almost identical. This means that should we constrain the two surfaces to have the same curvature (in the case of a cemented doublet) and just optimise for spherical aberration, then the solution will be close to that of the ideal aplanatic lens. To do this, we would simply use Eq. 4.29, forcing R₂ and R₃ to be equal and to replace Eq. 4.55 constraining the total coma, providing an alternative relation between s₁ and s₂. However, the fact that the cemented doublet is close to fulfilling the zero spherical aberration and coma condition further illustrates the usefulness of this simple component.

The analysis presented applies only strictly in the thin lens approximation. In practice, optimisation of a doublet such as presented in the previous example would be accomplished with the aid of ray tracing software. However, the insights gained by this exercise are particularly important. For instance, in carrying out a computer-based optimisation, it is critically important to understand that two solutions exist. Furthermore, in setting up a computer-based optimisation, an exercise, such as this, provides a useful ‘starting point’.