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An **achromatic doublet** brings red and blue light to the same focus, and is the earliest example of an achromatic lens.

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

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 1750s, 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 1758, 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 1763.^[2]

Types

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, $R 1$ denotes the first lens surface counted from the object. A doublet lens has four surfaces with radii $R 1$ through $R 2$ . Surfaces with positive radii curve away from the object ( $R 1$ positive is a convex first surface); negative radii curve toward the object ( $R 1$ 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

Uses an equiconvex crown glass lens (i.e. $R 1 > 0$ with $- R 1 = R 2$ ) and a complementary-curved second flint glass lens (with $R 3 = R 2$ ). The back of the flint glass lens is flat ( $R 4 = \infty$ ). A Littrow doublet can produce a ghost image between $R 2$ and $R 3$ because the lens surfaces of the two lenses have the same radii.

Fraunhofer doublet (Fraunhofer objective)

The first lens has positive refractive power, the second negative. $R 1 > 0$ is set greater than $- R 2$ , and $R 3$ is set close to, but not quite equal to, $- R 2$ . $R 4$ is usually greater than $- R 3$ . In a Fraunhofer doublet, the dissimilar curvatures of $- R 2$ and $R 3$ 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

Early Clark lenses follow the Fraunhofer design. After the late 1860s, they changed to the Littrow design, approximately equiconvex crown, $R 1 = R 2$ , and a flint with $R 3 ≃ R 2$ and $R 4 ≫ R 3$ . By about 1880, Clark lenses had $R 3$ set slightly shorter than $R 2$ to create a focus mismatch between $R 2$ and $R 3$ , thereby avoiding ghosting caused by reflections within the airspace.^[8]

Oil-spaced doublet

The use of oil between the crown and flint eliminates the effect of ghosting, particularly where $R 2 \approx R 3$ . It can also increase light transmission slightly and reduce the impact of errors in $R 2$ and $R 3$ .

Steinheil doublet

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

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 $R 2$ and $R 3$ have different absolute values.^[11]

Design

The first-order design of an achromat involves choosing the overall power $\ {\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 $n_{d}$ (for the refractive index at the Fraunhofer "d" spectral line wavelength), and the Abbe number $V$ (for the reciprocal of the glass dispersion). To make the linear dispersion of the system zero, the system must satisfy the equations

{\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 $\ {\frac {1}{\ f\ }}\$ for a lens with focal length $f$ . Solving these two equations for $\ f_{1}\$ and $\ f_{2}\$ gives

{\frac {f_{1}}{\ f_{\mathsf {dblt}}\ }}={\frac {+V_{1}-V_{2}\;}{V_{1}}}\

and

\ {\frac {f_{2}}{\ f_{\mathsf {dblt}}\ }}={\frac {-V_{1}+V_{2}\;}{V_{2}}}~.

Since $\ f_{1}=-f_{2}\ {\frac {\ V_{2}\ }{V_{1}}}\ ,$ 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

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}\$ and separately $\ f_{2}~.$ This leaves a continuum of different combinations of front and back lens curvatures for design tweaks ( $\ R_{1}\$ and $\ R_{2}\$ for lens 1; and $\ R_{3}\$ and $\ R_{4}\$ for lens 2) that will all produce the same $\ f_{1}\$ and $\ 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

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.

References

^ Daumas, Maurice, Scientific Instruments of the Seventeenth and Eighteenth Centuries and Their Makers, Portman Books, London 1989 ISBN 978-0-7134-0727-3
^ ^a ^b ^c Watson, Fred (2007). Stargazer: the life and times of the telescope. Allen & Unwin. pp. 140–55. ISBN 978-1-74175-383-7.
^ Fred Hoyle, Astronomy; A history of man's investigation of the universe, Rathbone Books, 1962, LCCN 62-14108
^ J. A. B. "Peter Dollond answers Jesse Ramsden". Sphaera 8. Museum of the History of Science, Oxford. Retrieved November 27, 2017. – A review of the events of the invention of the achromatic doublet with emphasis on the roles of Hall, Bass, John Dollond and others.
^ Dokland, Terje; Ng, Mary Mah-Lee (2006). Techniques in microscopy for biomedical applications. p. 23. ISBN 981-256-434-9.
^ "Chester Moor Hall". Encyclopædia Britannica. Retrieved 16 February 2019.
^ Wolfe, William L. (2007). Optics Made Clear: The nature of light and how We Use it. Press monograph. Vol. 163 (illustrated ed.). SPIE. p. 38. ISBN 9780819463074.
^ Warner, Deborah Jean; Ariail, Robert B. (1995). Alvan Clark & Sons, Artists In Optics (2nd ed.). Willmann-Bell. p. 174.
^ Kidger, M.J. (2002). Fundamental Optical Design. Bellingham, WA: SPIE Press. p. 174 ff.
^ Manly, Peter L. (1995). Unusual Telescopes. Cambridge University Press. p. 55. ISBN 978-0-521-48393-3.
^ Carson, Fred A. Basic Optics and Optical Instruments. p. AJ-4.

External links

Media related to Achromatic lenses at Wikimedia Commons

[daumas-1] Daumas, Maurice, Scientific Instruments of the Seventeenth and Eighteenth Centuries and Their Makers, Portman Books, London 1989 ISBN 978-0-7134-0727-3

[Stargazer-2] Watson, Fred (2007). Stargazer: the life and times of the telescope. Allen & Unwin. pp. 140–55. ISBN 978-1-74175-383-7.

[3] Fred Hoyle, Astronomy; A history of man's investigation of the universe, Rathbone Books, 1962, LCCN 62-14108

[Dollond-4] J. A. B. "Peter Dollond answers Jesse Ramsden". Sphaera 8. Museum of the History of Science, Oxford. Retrieved November 27, 2017. – A review of the events of the invention of the achromatic doublet with emphasis on the roles of Hall, Bass, John Dollond and others.

[5] Dokland, Terje; Ng, Mary Mah-Lee (2006). Techniques in microscopy for biomedical applications. p. 23. ISBN 981-256-434-9.

[6] "Chester Moor Hall". Encyclopædia Britannica. Retrieved 16 February 2019.

[7] Wolfe, William L. (2007). Optics Made Clear: The nature of light and how We Use it. Press monograph. Vol. 163 (illustrated ed.). SPIE. p. 38. ISBN 9780819463074.

[8] Warner, Deborah Jean; Ariail, Robert B. (1995). Alvan Clark & Sons, Artists In Optics (2nd ed.). Willmann-Bell. p. 174.

[9] Kidger, M.J. (2002). Fundamental Optical Design. Bellingham, WA: SPIE Press. p. 174 ff.

[10] Manly, Peter L. (1995). Unusual Telescopes. Cambridge University Press. p. 55. ISBN 978-0-521-48393-3.

[11] Carson, Fred A. Basic Optics and Optical Instruments. p. AJ-4.

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@@ Line 1: / Line 1: @@
+{{Short description|Lens that is designed to limit the effects of chromatic and spherical aberration}}
+{{Redirect|Achromat|the form of color blindness|achromatopsia}}
 [[File:Chromatic aberration lens diagram.svg|thumb|right|Chromatic aberration of a single lens causes different wavelengths of light to have differing focal lengths.]]
 [[File:Lens6b-en.svg|thumb|right|An '''achromatic doublet''' brings red and blue light to the same focus, and is the earliest example of an achromatic lens.]]
 [[File:Achromatic focal curve.svg|thumb|right|In an achromatic lens, two wavelengths are brought into the same focus, here red and blue.]]
-An '''achromatic lens''' or '''achromat''' is a [[lens (optics)|lens]] that is designed to limit the effects of [[chromatic aberration|chromatic]] and [[spherical aberration]]. Achromatic lenses are corrected to bring two wavelengths (typically red and blue) into focus on the same plane.
+An '''achromatic lens''' or '''achromat''' is a [[lens (optics)|lens]] that is designed to limit the effects of [[chromatic aberration|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 (lens)|doublet]]''', which is composed of two individual lenses made from [[glass]]es with different amounts of [[Dispersion (optics)|dispersion]]. Typically, one element is a negative ([[Lens (optics)#Types of simple lenses|concave]]) element made out of [[flint glass]] such as F2, which has relatively high dispersion, and the other is a positive ([[Lens (optics)#Types of simple lenses|convex]]) element made of [[Crown glass (optics)|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.
@@ Line 10: / Line 13: @@
 ==History==
-Theoretical considerations of the feasibility of correcting chromatic aberration were debated in the 18th century following [[Isaac Newton|Newton]]'s statement that such a correction was impossible (see [[History of the telescope#Achromatic refracting telescopes|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]].<ref name="daumas">Daumas, Maurice,  ''Scientific Instruments of the Seventeenth and Eighteenth Centuries and Their Makers'', Portman Books, London 1989  {{ISBN|978-0-7134-0727-3}}</ref><ref name=Stargazer>{{cite book |url=https://books.google.com/books?id=2LZZginzib4C&pg=PA140 |title=Stargazer: the life and times of the telescope |first=Fred |last=Watson |publisher=Allen & Unwin |year=2007 |pages=140–55 |isbn=978-1-74175-383-7}}</ref> Hall wished to keep his work on the achromatic lenses a secret and contracted the manufacture of the [[Crown glass (optics)|crown]] and [[flint glass|flint]] lenses to two different opticians, Edward Scarlett and James Mann.<ref>[[Fred Hoyle]], ''Astronomy; A history of man's investigation of the universe'', Rathbone Books, 1962, {{LCCN|6214108}}</ref><ref name="Dollond">{{cite web | url=http://www.mhs.ox.ac.uk/sphaera/index.htm?issue8/articl5 | title=Peter Dollond answers Jesse Ramsden | publisher=[[Museum of the History of Science, Oxford]] | work=Sphaera 8 |author=J. A. B.| issue=Autumn, 1998 | accessdate=November 27, 2017}}- A review of the events of the invention of the achromatic doublet with emphasis on the roles of Hall, Bass, [[John Dollond]] and others.</ref><ref>{{cite book |title=Techniques in microscopy for biomedical applications |first1=Terje |last1=Dokland |first2=Mary Mah-Lee |last2=Ng |page=23 |year=2006 |isbn=981-256-434-9 |url=https://books.google.com/books?id=Ix3G9_Rr0EAC&pg=PA23&lpg=PA23&dq=achromatic+lens+subcontract#v=onepage&q=&f=false |accessdate=July 31, 2009}}</ref>  They in turn sub-contracted the work to the same person, [[George Bass (optician)|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.<ref>{{cite web |url=http://www.britannica.com/biography/Chester-Moor-Hall |title=Chester Moor Hall |website=Encyclopædia Britannica |accessdate=16 February 2019}}</ref>
+Theoretical considerations of the feasibility of correcting chromatic aberration were debated in the 18th century following [[Isaac Newton|Newton]]'s statement that such a correction was impossible (see [[History of the telescope#Achromatic refracting telescopes|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]].<ref name="daumas">Daumas, Maurice,  ''Scientific Instruments of the Seventeenth and Eighteenth Centuries and Their Makers'', Portman Books, London 1989  {{ISBN|978-0-7134-0727-3}}</ref><ref name=Stargazer>{{cite book |url=https://books.google.com/books?id=2LZZginzib4C&pg=PA140 |title=Stargazer: the life and times of the telescope |first=Fred |last=Watson |publisher=Allen & Unwin |year=2007 |pages=140–55 |isbn=978-1-74175-383-7}}</ref> Hall wished to keep his work on the achromatic lenses a secret and contracted the manufacture of the [[Crown glass (optics)|crown]] and [[flint glass|flint]] lenses to two different opticians, Edward Scarlett and James Mann.<ref>[[Fred Hoyle]], ''Astronomy; A history of man's investigation of the universe'', Rathbone Books, 1962, {{LCCN|6214108}}</ref><ref name="Dollond">{{cite web | url=http://www.mhs.ox.ac.uk/sphaera/index.htm?issue8/articl5 | title=Peter Dollond answers Jesse Ramsden | publisher=[[Museum of the History of Science, Oxford]] | work=Sphaera 8 |author=J. A. B.| issue=Autumn, 1998 | access-date=November 27, 2017}} – A review of the events of the invention of the achromatic doublet with emphasis on the roles of Hall, Bass, [[John Dollond]] and others.</ref><ref>{{cite book |title=Techniques in microscopy for biomedical applications |first1=Terje |last1=Dokland |first2=Mary Mah-Lee |last2=Ng |page=23 |year=2006 |isbn=981-256-434-9 |url=https://books.google.com/books?id=Ix3G9_Rr0EAC&pg=PA23}}</ref>  They in turn sub-contracted the work to the same person, [[George Bass (optician)|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.<ref>{{cite web |url=http://www.britannica.com/biography/Chester-Moor-Hall |title=Chester Moor Hall |website=Encyclopædia Britannica |access-date=16 February 2019}}</ref>
 In the late 1750s, Bass mentioned Hall's lenses to [[John Dollond]], who understood their potential and was able to reproduce their design.<ref name=Stargazer/> Dollond applied for and was granted a patent on the technology in 1758, which led to bitter fights with other opticians over the right to make and sell achromatic doublets.
@@ Line 19: / Line 22: @@
 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 [[sphere]]s that define the optically relevant [[refraction|refracting]] lens surfaces. By convention, R<sub>1</sub> denotes the first lens surface counted from the object. A doublet lens has four surfaces  with radii R<sub>1</sub> to R<sub>4</sub>.
+In the following, {{mvar|R}} denotes the [[radius]] of the [[sphere]]s that define the optically relevant [[refraction|refracting]] lens surfaces. By convention, {{math|''R''{{sub|1}} }} denotes the first lens surface counted from the object. A doublet lens has four surfaces with radii {{math|''R''{{sub|1}} }} through {{math|''R''{{sub|2}} }}. Surfaces with positive radii curve away from the object ({{math|''R''{{sub|1}} }} positive is a convex first surface); negative radii curve toward the object ({{math|''R''{{sub|1}} }} 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 coating]]s for the most part has eliminated the issue of ghost images, and modern optical designs are preferred for other merits.
 ===Littrow doublet===
-Uses an equiconvex crown glass lens with R<sub>1</sub> = R<sub>2</sub>, and a second flint glass lens with R<sub>3</sub> = -R<sub>2</sub>. The back of the flint glass lens is flat. A Littrow doublet can produce a ghost image between R<sub>2</sub> and R<sub>3</sub> because the lens surfaces of the two lenses have the same radii.
+Uses an equiconvex crown glass lens (i.e. {{nobr|{{math|''R''{{sub|1}} > 0}} }} with {{nobr|{{math|−''R''{{sub|1}} {{=}} ''R''{{sub|2}} }} )}} and a complementary-curved second flint glass lens (with {{nobr|{{math|''R''{{sub|3}} {{=}} ''R''{{sub|2}} }} ).}} The back of the flint glass lens is flat {{nobr|( {{math|''R''{{sub|4}} {{=}} ∞}} ).}} A Littrow doublet can produce a ghost image between {{math|''R''{{sub|2}} }} and {{math|''R''{{sub|3}} }} because the lens surfaces of the two lenses have the same radii.
 ===Fraunhofer doublet (Fraunhofer objective)===
-The first lens has positive refractive power, the second negative. R<sub>1</sub> is set greater than R<sub>2</sub>, and R<sub>2</sub> is set close to, but not equal to, R<sub>3</sub>. R<sub>4</sub> is usually greater than R<sub>3</sub>. In a Fraunhofer doublet, the dissimilar curvatures of R<sub>2</sub> and R<sub>3</sub> are mounted close, but not in contact.<ref>{{cite book |url=https://books.google.com/books?id=0xxV3zbafeYC&pg=PA38&dq=Fraunhofer+dialyte&hl=en&ei=-JkHTYniEYH-8Aax9tjwDA&sa=X&oi=book_result&ct=result&resnum=8&ved=0CEQQ6AEwBw#v=onepage&q=Fraunhofer%20dialyte&f=false |title=Optics Made Clear: The Nature of Light and how We Use it |volume=163 |series=Press monograph |first=William L. |last=Wolfe |edition=illustrated |publisher=SPIE |date=2007 |isbn=9780819463074 |page=38}}</ref> This design yields more degrees of freedom (one more free radius, length of the air space) to correct for [[optical aberration]]s.
+The first lens has positive refractive power, the second negative. {{nobr|{{math|''R''{{sub|1}} > 0}} }} is set greater than {{math|−''R''{{sub|2}} }}, and {{math|''R''{{sub|3}} }} is set close to, but not quite equal to, {{math|−''R''{{sub|2}} }}. {{math|''R''{{sub|4}} }} is usually greater than {{math|−''R''{{sub|3}} }}. In a Fraunhofer doublet, the dissimilar curvatures of {{math|−''R''{{sub|2}} }} and {{math|''R''{{sub|3}} }} are mounted close, but not quite in contact.<ref>{{cite book |first=William L. |last=Wolfe |year=2007 |title=Optics Made Clear: The nature of light and how We Use it |volume=163 |series=Press monograph |edition=illustrated |publisher=SPIE |isbn=9780819463074 |url=https://books.google.com/books?id=0xxV3zbafeYC&pg=PA38 |page=38}}</ref> This design yields more degrees of freedom (one more free radius, length of the air space) to correct for [[optical aberration]]s.
 ===Clark doublet===
-Early Clark lenses follow the Fraunhofer design.  After the late 1860s, they changed to the Littrow design, approximately equiconvex crown, {{nowrap|1=''R''<sub>1</sub> = ''R''<sub>2</sub>}}, and a flint with {{nowrap|1=''R''<sub>3</sub> ≃ ''R''<sub>2</sub>}} and {{nowrap|1=''R''<sub>4</sub> ≫ ''R''<sub>3</sub>}}. By about 1880, Clark lenses had ''R''<sub>3</sub> set slightly shorter than ''R''<sub>2</sub> to create a focus mismatch between ''R''<sub>2</sub> and ''R''<sub>3</sub>, thereby reducing ghosting caused by reflections within the airspace.<ref>{{cite book |last1=Warner |first1=Deborah Jean |last2=Ariail |first2=Robert B. |title=Alvan Clark & Sons, Artists In Optics |edition=2nd |date=1995 |publisher=Willmann-Bell |page=174}}</ref>
+Early Clark lenses follow the Fraunhofer design.  After the late 1860s, they changed to the Littrow design, approximately equiconvex crown, {{nobr|{{math|''R''<sub>1</sub> {{=}} ''R''<sub>2</sub>}} ,}} and a flint with {{nobr|{{math|''R''<sub>3</sub> ≃ ''R''<sub>2</sub>}} }} and {{nobr|{{math|''R''<sub>4</sub> ≫ ''R''<sub>3</sub>}} .}} By about 1880, Clark lenses had {{math|''R''<sub>3</sub>}} set slightly shorter than {{math|''R''<sub>2</sub>}} to create a focus mismatch between {{math|''R''<sub>2</sub>}} and {{math|''R''<sub>3</sub>}}, thereby avoiding ghosting caused by reflections within the airspace.<ref>{{cite book |last1=Warner |first1=Deborah Jean |last2=Ariail |first2=Robert B. |title=Alvan Clark & Sons, Artists In Optics |edition=2nd |date=1995 |publisher=Willmann-Bell |page=174}}</ref>
 ===Oil-spaced doublet===
-The use of oil between the crown and flint eliminates the effect of ghosting, particularly where {{nowrap |1=''R''<sub>2</sub> = ''R''<sub>3</sub>}}. It can also increase light transmission slightly and reduce the impact of errors in ''R''<sub>2</sub> and ''R''<sub>3</sub>.
+The use of oil between the crown and flint eliminates the effect of ghosting, particularly where {{nobr|{{math| ''R''{{sub|2}} ≈ ''R''{{sub|3}}}} .}} It can also increase light transmission slightly and reduce the impact of errors in {{math|''R''{{sub|2}} }} and {{math|''R''{{sub|3}} }}.
 ===Steinheil doublet===
-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.<ref>Kidger, M.J. (2002) Fundamental Optical Design. SPIE Press, Bellingham, WA, pp. 174ff</ref>
+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.<ref>{{cite book |last=Kidger |first=M.J. |year=2002 |title=Fundamental Optical Design |publisher=SPIE Press |place=Bellingham, WA |page=174&nbsp;ff}}</ref>
 ===Dialyte===
-[[Dialyte lens]]es 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.<ref>{{cite book|author=Peter L. Manly|title=Unusual Telescopes|url=https://books.google.com/books?id=p282ijnF3C0C&pg=PA55|year=1995|publisher=Cambridge University Press|isbn=978-0-521-48393-3|page=55}}</ref> They are also lenses where the elements can not be cemented because ''R''<sub>2</sub> and ''R''<sub>3</sub> have different absolute values.<ref>Fred A. Carson, Basic optics and optical instruments, page AJ-4</ref>
+[[Dialyte lens]]es 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.<ref>{{cite book |first=Peter L. |last=Manly |year=1995 |title=Unusual Telescopes |publisher=Cambridge University Press |isbn=978-0-521-48393-3 |page=55 |url=https://books.google.com/books?id=p282ijnF3C0C&pg=PA55 }}</ref> They are also lenses where the elements can not be cemented because {{math|''R''{{sub|2}} }} and {{math|''R''{{sub|3}} }} have different absolute values.<ref>{{cite book |first=Fred A. |last=Carson |title=Basic Optics and Optical Instruments |page=AJ-4}}</ref>
 ==Design==
-The first-order design of an achromat involves choosing the overall power <math>\phi_{\text{sys}}</math> of the doublet and the two glasses to use. The choice of glass gives the mean refractive index, often written as <math>n_d</math> (for the refractive index at the [[Fraunhofer diffraction|Fraunhofer]] [[Abbe number|"d" spectral line wavelength]]), and the [[Abbe number]] <math>V</math> (for the reciprocal of the glass [[dispersion (optics)|dispersion]]). To make the linear dispersion of the system zero, the system must satisfy the equations
+The first-order design of an achromat involves choosing the overall power <math>\ \frac{ 1 }{\ f_\mathsf{dblt}\ }\ </math> of the doublet and the two glasses to use. The choice of glass gives the mean refractive index, often written as <math>n_d</math> (for the refractive index at the [[Fraunhofer diffraction|Fraunhofer]] [[Abbe number|"d" spectral line wavelength]]), and the [[Abbe number]] <math>V</math> (for the reciprocal of the glass [[dispersion (optics)|dispersion]]). To make the linear dispersion of the system zero, the system must satisfy the equations
-:<math>\begin{align} \phi_1 + \phi_2 &= \phi_{\text{sys}} \\ \frac{\phi_1}{V_1} + \frac{\phi_2}{V_2} &= 0 \ ,\end{align}</math>
+:<math>\begin{align} \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{align}</math>
-where the [[optical power|lens power]] is <math>\phi = 1/f</math> for a lens with [[focal length]] <math>f</math>. Solving these two equations for <math>\phi_1</math> and <math>\phi_2</math> gives
+where the [[optical power|lens power]] is <math>\ \frac{1}{\ f\ }\ </math> for a lens with [[focal length]] <math>f</math>. Solving these two equations for <math>\ f_1\ </math> and <math>\ f_2\ </math> gives
-:<math>\frac{\phi_1}{\phi_{\text{sys}}} = \frac{V_1}{V_1 - V_2} \qquad \text{and} \qquad \frac{\phi_2}{\phi_{\text{sys}}} = \frac{-V_2}{V_1 - V_2} \ .</math>
+:<math>\frac{ f_1 }{\ f_\mathsf{dblt}\ } = \frac{ + V_1 - V_2\;}{ V_1 }\ </math> and <math>\ \frac{ f_2 }{\ f_\mathsf{dblt}\ } = \frac{ - V_1 + V_2\;}{ V_2 } ~.</math>
-Since <math>\phi_2 = -\phi_1 V_2 / V_1</math>, and the [[Abbe number]]s are positive-valued, the power of the second element in the doublet is negative when the first element is positive.
+Since <math>\ f_1 = - f_2\ \frac{\ V_2\ }{ V_1 }\ ,</math> and the [[Abbe number]]s 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 ===
-==See also==
+[[Optical aberration]]s other than just color are present in all lenses. For example, [[coma (optics)|''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, <math>\ f_1\ </math> and separately <math>\ f_2 ~.</math> This leaves a continuum of different combinations of front and back lens curvatures for design tweaks (<math>\ R_1\ </math> and <math>\ R_2\ </math> for lens&nbsp;1; and <math>\ R_3\ </math> and <math>\ R_4\ </math> for lens&nbsp;2) that will all produce the same <math>\ f_1\ </math> and <math>\ f_2\ </math> 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==
 {{Comparison_chromatic_focus_shift_plots.svg}}
+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:
+; [[apochromat|apochromatic lenses]]: bring ''three'' wavelengths into a common focus and requires costly materials
+; [[superachromat|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 lens]]es used in [[camera]]s 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==
 * [[Barlow lens]]
-* [[Superachromat]]
 ==References==
-{{reflist|30em}}
+{{clear}}
+{{reflist|25em}}
+==External links==
+*{{Commons category-inline}}
 [[Category:Lenses]]
+[[Category:Microscope components]]