Codex @codex  0

An achromatic doublet is an achromatic lens with two optical lens elements. In the contact thin lens model, their optical powers add and their primary colour changes cancel according to achromatic doublet power balance. The conventional positive achromatic doublet uses a converging crown glass element and a weaker diverging flint glass element; these names describe a common choice, not a universal ordering of every modern optical glass composition.

Achromatic doublet power balance chooses two signed optical powers to meet two goals: obtain the required focal length and cancel the primary F–C colour difference. Model two thin lens elements in contact in air, using the paraxial approximation. Let $i=c,f$ label the crown glass and flint glass elements, and let ${\Phi }_{i,d}$ denote each element's optical power at the reference d line. The Abbe number for each material isThe subscripts label standard spectral lines: blue F near $486.1\mathrm{nm}$, red C near $656.3\mathrm{nm}$, and reference d near $587.6\mathrm{nm}$. These are three specified wavelengths, not colour bands. A larger Abbe number means less relative optical dispersion between F and C. ^[ref].

For a fixed thin lens shape in air, write the lensmaker's equation as ${\Phi }_i(\lambda )=K_i[n_i(\lambda )-1]$, with signed shape factor $K_i=1/R_{i,a}-1/R_{i,b}$. Subtracting its C value from its F value gives the primary optical power differenceThe negative element has negative $\Delta {\Phi }_i$: its blue optical power is more negative, not less negative. The contact rule for optical power therefore givesThis zero is equality of the total F and C optical powers, hence equality of their focal lengths when the total optical power is nonzero. The relation follows directly from the lensmaker's equation and the Abbe number; it is exact within the fixed-shape contact thin lens model. ^[ref]. Tatum uses a yellow D-line reference; here the Abbe number convention uses the d line consistently.

For a toy design, choose $V_c=60$, $V_f=30$, ${\Phi }_{c,d}=10\mathrm{D}$ and ${\Phi }_{f,d}=-5\mathrm{D}$. ThenThe positive element contributes $+1/6\mathrm{D}$ to the F–C difference and the negative element contributes $-1/6\mathrm{D}$. This retains the same reference focal length as the single $5\mathrm{D}$, $V_d=60$ thin lens in chromatic aberration, while cancelling its primary colour split. The values are illustrative design parameters, not a catalogue optical glass pair.

Use the same synthetic colour coordinate $q=-1/2,0,+1/2$ and toy optical dispersion as in chromatic aberration. The two element optical powers and their sum becomeAt the red endpoint, the element optical powers are $9.916667\mathrm{D}$ and $-4.916667\mathrm{D}$; at the blue endpoint, they are $10.083333\mathrm{D}$ and $-5.083333\mathrm{D}$. Both sums are $5\mathrm{D}$. In this first-order toy model alone, the sum is constant for every $q$ and the focal length remains $0.2\mathrm{m}$ throughout. Real material optical dispersion is not captured by these two straight slopes, so the flat line is not a claim that a real achromatic doublet focuses all colours perfectly.

For specified nonzero reference optical power ${\Phi }_d$ and distinct Abbe numbers, solving the two balance equations givesThus ${\Phi }_d>0$ and $V_c>V_f>0$ require positive crown glass optical power and negative flint glass optical power. If the two finite Abbe numbers were equal, cancellation would force zero total optical power. If they are close, the required element optical powers become large compared with the total: cancelling primary colour can demand much stronger individual elements.

The contact assumption matters. For two thin lens elements separated by distance $s$ in air, the equivalent optical power is ${\Phi }_c+{\Phi }_f-s{\Phi }_c{\Phi }_f$, evaluated at each wavelength; its colour change need not satisfy the contact balance. Finite thickness, departures from the paraxial approximation, and the secondary spectrum also limit the simple prediction.^[ref]

An achromatic lens combines optical lens elements with different optical dispersion so that two specified wavelengths have the same focal length in the paraxial approximation. This brings their parallel incoming light rays to a common optical focus, helping an optical camera form a sharper image under mixed-colour illumination. A common implementation is an achromatic doublet: a positive crown glass element paired with a negative flint glass element. ^[ref].

The achromatic doublet does not require either material to have zero optical dispersion. Instead, the positive element gains positive optical power toward blue, while the negative element becomes more negative. Suitable Abbe numbers and signed optical powers let these changes cancel even though their reference optical powers do not cancel. For example, the achromatic doublet power balance calculation below combines $+10\mathrm{D}$ with $-5\mathrm{D}$ to obtain $+5\mathrm{D}$, or focal length $0.2\mathrm{m}$, with zero primary F–C optical power difference. The smaller Abbe number of the negative element makes its weaker reference optical power sufficient to oppose the positive element's colour change.

An achromatic lens is corrected at two chosen wavelengths, not necessarily at every wavelength between them. The remaining focal length variation is the secondary spectrum. Correcting axial chromatic aberration also does not automatically remove lateral chromatic aberration throughout an imaging system, spherical aberration, or diffraction. Actual designs therefore need ray tracing with the intended wavelengths, element thicknesses, separations and apertures. ^[ref]; ^[ref].

An optical camera can focus a white object sharply in one colour while leaving coloured fringes around its edges in another. Chromatic aberration is the dependence of image position or image scale on wavelength, caused by optical dispersion in the optical lens materials. Axial chromatic aberration separates the optical foci along the optical axis; lateral chromatic aberration changes the transverse magnification with wavelength. Moving the receiving plane can favour one colour without bringing all colours into focus. ^[ref]; ^[ref].

For ordinary optical glass in the visible range, the blue refractive index exceeds the red refractive index. For a fixed positive thin lens in air, the lensmaker's equation then givesHere $\lambda$ is wavelength, $n$ is refractive index, $\Phi$ is optical power, $f$ is focal length, and $R_a,R_b$ are the signed surface radius of curvature values in the lensmaker's equation. The blue light rays therefore bend more strongly and meet sooner. This ordering is a statement about ordinary visible optical dispersion, not a universal rule for every material and every spectral range. ^[ref].

For a worked example, take a single positive thin lens with reference optical power ${\Phi }_d=5\mathrm{D}$ and Abbe number $V_d=60$, hence reference focal length $f_d=0.2\mathrm{m}$. Its F–C optical power difference is $\Delta \Phi ={\Phi }_F-{\Phi }_C={\Phi }_d/V_d=5/60\mathrm{D}\approx 0.08333\mathrm{D}$, by the achromatic doublet power balance relation below. This difference alone does not locate the reference optical power within the F–C interval. To draw a transparent example, introduce a synthetic colour coordinate $q$, a dimensionless quantity, with red $q=-1/2$, reference $q=0$, and blue $q=+1/2$, and choose the toy optical dispersion modelThe diagram's red and blue endpoints represent the C and F endpoints of this toy model. Its reference point is placed symmetrically in optical power by construction; $q$ is not a measured wavelength scale, and the model is not measured optical glass data. The endpoint focal lengths areThus the axial chromatic aberration in this example separates the endpoint optical foci by about $3.334\mathrm{mm}$. A single receiving plane cannot coincide with both optical foci.

The useful design idea is to cancel changes of optical power while retaining a positive total optical power. An achromatic lens does this with materials having different Abbe numbers. It greatly reduces chromatic aberration, but correcting two chosen wavelengths still leaves the possibility of a secondary spectrum.

Set ${\Phi }_i=1/f_{id}$ and $\Phi =1/f_d$. The achromatic doublet power balance gives two linear equations:For distinct nonzero Abbe numbers, solving yields ${\Phi }_1=\Phi V_{1d}/(V_{1d}-V_{2d})$ and ${\Phi }_2=-\Phi V_{2d}/(V_{1d}-V_{2d})$, and thereforeEqual Abbe numbers force the total optical power to vanish under the achromatic condition; they cannot give a nonzero-power thin achromatic lens. Closely matched Abbe numbers require large opposing component powers and are generally unattractive because of the resulting sensitivity and other optical aberrations.

Write ${\Phi }_i=1/f_{id}$ and $\Phi =1/f_d$. The achromatic doublet power balance isSolving these two linear equations gives ${\Phi }_1=\Phi V_{1d}/(V_{1d}-V_{2d})$ and ${\Phi }_2=-\Phi V_{2d}/(V_{1d}-V_{2d})$. Taking reciprocals gives the required focal lengths:The design requires distinct Abbe numbers; identical relative dispersion cannot yield a nonzero total achromatic power from this two-element thin model.

At the intermediate $d$ reference wavelength, the lensmaker's equation gives $1/f_{id}={\rho }_i(n_{id}-1)$. Therefore the curvature-factor ratio from the previous part impliesand henceThe last equality uses the Abbe number definition and expresses the achromatic doublet power balance in terms of component focal lengths.

The secondary spectrum is residual chromatic aberration after an achromatic lens brings two selected wavelengths to the same optical focus. For example, equal red and blue focal lengths can coexist with a different green focal length. The Abbe number fixes one F–C relative difference, not the complete shape of optical dispersion between or beyond those wavelengths. The perfectly flat sum in the achromatic doublet power balance diagram deliberately omits this residual effect. ^[ref].