Achromatic Doublets

This is a continuation from the previous tutorial - two lens systems.

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The singlet lens suffers from axial chromatic aberration, which is determined by the Abbe number \(V\) of the lens material and its \(\text{FN}\). A widely used lens form that corrects this aberration is the achromatic doublet as illustrated in Fig. 19.

 

Figure 19  Typical achromatic doublet lens.

 

An achromatic lens has equal focal lengths in \(c\) and \(f\) light. This lens comprises two lens elements where one element with a high \(V\)-number (crown glass) has the same power sign as the doublet and the other element has a low \(V\)-number (flint glass) with opposite power sign.

Three basic configurations are used. These are the cemented doublet, broken contact doublet, and the widely airspaced doublet (dialyte). The degrees of freedom are two lens powers, glasses, and shape of each lens.

The resultant power of two thin lenses in close proximity, \(s_2\rightarrow0\), is \(\phi_{ab}=\phi_a+\phi_b\) and the transverse primary chromatic aberration \(\text{TPAC}\) is

\[\tag{40}\text{TPAC}=-yf_{ab}\left[\frac{\phi_a}{V_a}+\frac{\phi_b}{V_b}\right]\]

where \(y\) is the marginal ray height.

Setting \(\text{TPAC}=0\) and solving for the powers of the lenses yields

\[\tag{41}\phi_a=\frac{V_a}{f_{ab}(V_a-V_b)}\]

\[\tag{42}\phi_b=\frac{-V_b\phi_a}{V_a}\]

The bending or shape of a lens is expressed by \(c=c_1-c_2\) and affects the aberrations of the lens. The bending of each lens is related to its power by \(c_a=\phi_a/(n_a-1)\) and \(c_b=\phi_b(n_b-1)\).

Since the two bendings can be used to correct the third-order spherical and coma, the equations for these aberrations can be combined to form a quadratic equation in terms of the curvature of the first surface \(c_1\). Solving for \(c_1\) will yield zero, one, or two solutions for the first lens. A linear equation relates \(c_1\) to \(c_2\) of the second lens.

While maintaining the achromatic correction of a doublet, the spherical aberration as a function of its shape (\(c_1\)) is described by a parabolic curve. Depending upon the choices of glasses, the peak of the curve may be above, below, or at the zero spherical aberration value.

When the peak lies in the positive spherical aberration region, two solutions with zero spherical aberration exist in which the solution with the smaller value of \(c_1\) is called the left-hand solution (Fraunhofer or Steinheil forms) and the other is called the right-hand solution (Gaussian form).

Two additional solutions are possible by reversal of the glasses. These two classes of designs are denoted as crown-in-front and flint-in-front designs. Depending upon the particular design requirements, one should examine all four configurations to select the most appropriate.

The spherical aberration curve can be raised or lowered by the selection of the \(V\) difference or the \(n\) difference. Specifically, the curve will be lowered as the \(V\) difference is increased or if the \(n\) difference is reduced. As for the thin singlet lens, the coma will be zero for the configuration corresponding to the peak of the spherical aberration curve.

Although the primary chromatic aberration may be corrected, a residual chromatic error often remains and is called the secondary spectrum, which is the difference between the ray intercepts in \(d\) and \(c\).

 

Figure 20   An F/5 airspaced doublet using conventional glasses is shown in (a) and exhibits residual secondary chromatic aberration. A similar lens is shown in (b) that uses a new glass to effectively eliminate the secondary color.

 

Figure 20(a) illustrates an F/5 airspaced doublet that exhibits well-corrected spherical light and primary chromatic aberrations and has notable secondary color. The angular secondary spectrum for an achromatic thin-lens doublet is given by

\[\tag{43}\text{SAC}=\frac{-(P_a-P_b)}{2(\text{FN})(V_a-V_b)}\]

where \(P=(n_\lambda-n_c)/(n_f-n_c)\) is the partial dispersion of a lens material.

In general, the ratio \((P_a-P_b)/(V_a-V_b)\) is nearly a constant which means little can be done to correct the \(\text{SAC}\). A few glasses exist that allow \(P_a-P_b\approx0\), but the \(V_a-V_b\) is often small, which results in lens element powers of rather excessive strength in order to achieve achromatism.

Figure 20(b) shows an F/5 airspaced doublet using a relatively new pair of glasses that have a small \(P_a-P_b\) and a more typical \(V_a-V_b\). Both the primary and secondary chromatic aberration are well corrected. Due to the relatively low refractive index of the crown glass, the higher power of the elements results in spherical aberration through the seventh order. Almost no spherochromatism (variation of spherical aberration with wavelength) is observed. The 80 percent blur diameter is almost the same for both lenses and is 0.007.

Table 3 contains the prescriptions for these lenses.

 

Table 3  Prescriptions for Achromatic Doublets Shown in Fig. 20

 

When the separation between the lens elements is made a finite value, the resultant lens is known as a dialyte and is illustrated in Fig. 21.

 

Figure 21  Widely separated achromatic doublet known as the dialyte lens.

 

As the lenses are separated by a distance \(s_d\), the power of the flint or negative lens increases rapidly. The distance \(s_d\) may be expressed as a fraction of the crown-lens focal length by \(p=s_d/f_a\). Requiring the chromatic aberration to be zero implies that

\[\tag{44}\frac{y_a^2}{f_aV_a}+\frac{y_b^2}{f_bV_b}=0\]

By inspection of the figure and the definition of \(p\), it is evident that \(y_b=y_a(1-p)\) from which it follows that

\[\tag{45}f_bV_b=-f_aV_a(1-p)^2\]

The total power of the dialyte is

\[\tag{46}\phi=\phi_a+\phi_b(1-p)\]

Solving for the focal lengths of the lenses yields

\[\tag{47}f_a=f_{ab}\left[1-\frac{V_b}{V_a(1-p)}\right]\]

and

\[\tag{48}f_b=f_{ab}(1-p)\left[1-\frac{V_a(1-p)}{V_b}\right]\]

The power of both lenses increases as \(p\) increases.

The typical dialyte lens suffers from residual secondary spectrum; however, it is possible to design an airspaced achromatic doublet with only one glass type that has significantly reduced secondary spectrum.

Letting \(V_a=V_b\) results in the former equations becoming

\[\tag{49}f_a=\frac{pf_{ab}}{p-1}\qquad{f_b}=-pf_{ab}(p-1)\qquad{s_d}=pf_a\qquad{bfl}=-f_{ab}(p-1)\]

When \(f_{ab}\gt0\), then \(p\) must be greater than unity, which means that the lens is quite long. The focal point lies between the two lenses, which reduces its general usefulness. This type of lens is known as the Schupmann lens, based upon his research in the late s. Several significant telescopes, as well as eyepieces, have employed this configuration.

For \(f_{ab}\lt0\), the lens can be made rather compact and is sometimes used as the rear component of some telephoto lenses.

 

The next tutorial discusses about triplet lenses. 

 

Lens that is designed to limit the effects of chromatic and spherical aberration

"Achromat" redirects here. For the form of color blindness, see achromatopsia

Chromatic aberration of a single lens causes different wavelengths of light to have differing focal lengths. An achromatic doublet brings red and blue light to the same focus, and is the earliest example of an achromatic lens. In an achromatic lens, two wavelengths are brought into the same focus, here red and blue.

An achromatic lens or achromat is a lens that is designed to limit the effects of chromatic and spherical aberration. Achromatic lenses are corrected to bring two wavelengths (typically red and blue) into focus on the same plane. Wavelengths in between these two then have better focus error than could be obtained with a simple lens.

The most common type of achromat is the achromatic doublet, which is composed of two individual lenses made from glasses with different amounts of dispersion. Typically, one element is a negative (concave) element made out of flint glass such as F2, which has relatively high dispersion, and the other is a positive (convex) element made of crown glass such as BK7, which has lower dispersion. The lens elements are mounted next to each other, often cemented together, and shaped so that the chromatic aberration of one is counterbalanced by that of the other.

In the most common type (shown), the positive power of the crown lens element is not quite equalled by the negative power of the flint lens element. Together they form a weak positive lens that will bring two different wavelengths of light to a common focus. Negative doublets, in which the negative-power element predominates, are also made.

History

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Theoretical considerations of the feasibility of correcting chromatic aberration were debated in the 18th century following Newton's statement that such a correction was impossible (see History of the telescope). Credit for the invention of the first achromatic doublet is often given to an English barrister and amateur optician named Chester Moore Hall.[1][2] Hall wished to keep his work on the achromatic lenses a secret and contracted the manufacture of the crown and flint lenses to two different opticians, Edward Scarlett and James Mann.[3][4][5] They in turn sub-contracted the work to the same person, George Bass. He realized the two components were for the same client and, after fitting the two parts together, noted the achromatic properties. Hall used the achromatic lens to build the first achromatic telescope, but his invention did not become widely known at the time.[6]

In the late s, Bass mentioned Hall's lenses to John Dollond, who understood their potential and was able to reproduce their design.[2] Dollond applied for and was granted a patent on the technology in , which led to bitter fights with other opticians over the right to make and sell achromatic doublets.

Dollond's son Peter invented the apochromat, an improvement on the achromat, in .[2]

Types

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Several different types of achromat have been devised. They differ in the shape of the included lens elements as well as in the optical properties of their glass (most notably in their optical dispersion or Abbe number).

In the following, R denotes the radius of the spheres that define the optically relevant refracting lens surfaces. By convention, R1 denotes the first lens surface counted from the object. A doublet lens has four surfaces with radii R1 through R2 . Surfaces with positive radii curve away from the object (R1 positive is a convex first surface); negative radii curve toward the object (R1 negative is a concave first surface).

The descriptions of the achromat lens designs mention advantages of designs that do not produce "ghost" images. Historically, this was indeed a driving concern for lens makers up to the 19th century and a primary criterion for early optical designs. However, in the mid 20th century, the development of advanced optical coatings for the most part has eliminated the issue of ghost images, and modern optical designs are preferred for other merits.

Littrow doublet

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Uses an equiconvex crown glass lens (i.e. R1 > 0 with &#;R1 = R2 ) and a complementary-curved second flint glass lens (with R3 = R2 ). The back of the flint glass lens is flat ( R4 = &#; ). A Littrow doublet can produce a ghost image between R2 and R3 because the lens surfaces of the two lenses have the same radii.

Fraunhofer doublet (Fraunhofer objective)

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The first lens has positive refractive power, the second negative. R1 > 0 is set greater than &#;R2 , and R3 is set close to, but not quite equal to, &#;R2 . R4 is usually greater than &#;R3 . In a Fraunhofer doublet, the dissimilar curvatures of &#;R2 and R3 are mounted close, but not quite in contact.[7] This design yields more degrees of freedom (one more free radius, length of the air space) to correct for optical aberrations.

Clark doublet

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Early Clark lenses follow the Fraunhofer design. After the late s, they changed to the Littrow design, approximately equiconvex crown, R1 = R2 , and a flint with R3 &#; R2 and R4 &#; R3 . By about , Clark lenses had R3 set slightly shorter than R2 to create a focus mismatch between R2 and R3, thereby avoiding ghosting caused by reflections within the airspace.[8]

Oil-spaced doublet

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The use of oil between the crown and flint eliminates the effect of ghosting, particularly where R2 &#; R3 . It can also increase light transmission slightly and reduce the impact of errors in R2 and R3 .

Steinheil doublet

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The Steinheil doublet, devised by Carl August von Steinheil, is a flint-first doublet. In contrast to the Fraunhofer doublet, it has a negative lens first followed by a positive lens. It needs stronger curvature than the Fraunhofer doublet.[9]

Dialyte

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Dialyte lenses have a wide air space between the two elements. They were originally devised in the 19th century to allow much smaller flint glass elements down stream since flint glass was hard to produce and expensive.[10] They are also lenses where the elements can not be cemented because R2 and R3 have different absolute values.[11]

Design

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The first-order design of an achromat involves choosing the overall power   1   f d b l t     {\displaystyle \ {\frac {1}{\ f_{\mathsf {dblt}}\ }}\ } of the doublet and the two glasses to use. The choice of glass gives the mean refractive index, often written as n d {\displaystyle n_{d}} (for the refractive index at the Fraunhofer "d" spectral line wavelength), and the Abbe number V {\displaystyle V} (for the reciprocal of the glass dispersion). To make the linear dispersion of the system zero, the system must satisfy the equations

1   f 1   + 1   f 2   = 1   f d b l t     , 1   f 1   V 1   + 1   f 2   V 2   = 0   ; {\displaystyle {\begin{aligned}{\frac {1}{\ f_{1}\ }}+{\frac {1}{\ f_{2}\ }}&={\frac {1}{\ f_{\mathsf {dblt}}\ }}\ ,\\{\frac {1}{\ f_{1}\ V_{1}\ }}+{\frac {1}{\ f_{2}\ V_{2}\ }}&=0\ ;\end{aligned}}}

where the lens power is   1   f     {\displaystyle \ {\frac {1}{\ f\ }}\ } for a lens with focal length f {\displaystyle f} . Solving these two equations for   f 1   {\displaystyle \ f_{1}\ } and   f 2   {\displaystyle \ f_{2}\ } gives

f 1   f d b l t   = + V 1 &#; V 2 V 1   {\displaystyle {\frac {f_{1}}{\ f_{\mathsf {dblt}}\ }}={\frac {+V_{1}-V_{2}\;}{V_{1}}}\ }

  f 2   f d b l t   = &#; V 1 + V 2 V 2   . {\displaystyle \ {\frac {f_{2}}{\ f_{\mathsf {dblt}}\ }}={\frac {-V_{1}+V_{2}\;}{V_{2}}}~.}

Since   f 1 = &#; f 2     V 2   V 1   , {\displaystyle \ f_{1}=-f_{2}\ {\frac {\ V_{2}\ }{V_{1}}}\ ,} and the Abbe numbers are positive-valued, the power of the second element in the doublet is negative when the first element is positive, and vice-versa.

Removing other aberrations

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Optical aberrations other than just color are present in all lenses. For example, coma remains after spherical and chromatic aberrations are corrected. In order to correct other aberrations, the front and back curvatures of each of the two lenses remain free parameters, since the color correction design only prescribes the net focal length of each lens,   f 1   {\displaystyle \ f_{1}\ } and separately   f 2   . {\displaystyle \ f_{2}~.} This leaves a continuum of different combinations of front and back lens curvatures for design tweaks (   R 1   {\displaystyle \ R_{1}\ } and   R 2   {\displaystyle \ R_{2}\ } for lens 1; and   R 3   {\displaystyle \ R_{3}\ } and   R 4   {\displaystyle \ R_{4}\ } for lens 2) that will all produce the same   f 1   {\displaystyle \ f_{1}\ } and   f 2   {\displaystyle \ f_{2}\ } required by the achromat design. Other adjustable lens parameters include the thickness of each lens and the space between the two, all constrained only by the two required focal lengths. Normally, the free parameters are adjusted to minimize non-color-related optical aberrations.

Further color correction

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Focus error for four types of lens, over the visible and near infrared spectrum.

Lens designs more complex than achromatic can improve the precision of color images by bringing more wavelengths into exact focus, but require more expensive types of glass, and more careful shaping and spacing of the combination of simple lenses:

apochromatic lenses

bring three wavelengths into a common focus and requires costly materials

superachromatic lenses

bring four wavelengths into focus and must be manufactured with even more expensive fluoride glass and to considerably tighter tolerances

In theory, the process can continue indefinitely: Compound lenses used in cameras typically have six or more simple lenses (e.g. double-Gauss lens); several of those lenses can be made with different types of glass, with slightly altered curvatures, in order to bring more colors into focus. The constraint is extra manufacturing cost, and diminishing returns of improved image for the effort.

See also

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References

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• Achromatic lenses at Wikimedia Commons