J. Eur. Opt. Society-Rapid Publ. 22, 30( 2026) 303
P k ¼ðn k u k � n kþ1 u kþ1 Þ = h �
P 2m ¼ nl 2m = q � l 2mþ1 = h ¼ ð nl2m = q � l 2m þ hKz 2m Þ = h
¼ Kz 2m þ ðn = q � 1Þl 2m = h:
P 2m�1 ¼ ðl 2m�1 � nl 2m = qÞ = h ¼ ðl 2m þ hKz 2m�1 � nl 2m = qÞ = h ¼ Kz 2m�1 � ðn = q � 1Þl 2m = h
equation( 12) gives n q � 1 ¼ 2ðn2 � 1Þ n þ 2
and from equations( 21) and( 19) we obtain! l 2m = h ¼�K X2m�1 z i � a i¼1 ð23Þ
ð24Þ
ð25Þ
The surface powers are then
P 2m�1 ¼ K z 2m�1 þ 2 ð n2 �1
2m�1
Þ P z ðnþ2Þ i � a i ð26Þ
P 2m ¼ K z 2m � 2 ð n2 �1
2m�1
Þ P z i � a ðnþ2Þ
and the corresponding surface curvatures result then from equation( 3) as c k ¼ P k = ðn kþ1 � n k Þ: ð27Þ It follows from equation( 26) that the power of lens m, P m ¼ P 2m�1 þ P 2m is simply
P m ¼ Kz ð 2m�1 þ z 2m Þ: ð28Þ
Note from equations( 26) and( 28) that, for each lens surface, z k has a term proportional to the surface power, plus a correction term that is exactly compensated by a correction term of equal magnitude and opposite sign coming from the other surface of the same lens. The power of each lens is then proportional to the sum of the z-values of its two surfaces. Because they can be viewed intuitively as power-like quantities, we refer to the variables z k as“ quasi-powers”.
When the group of thin lenses forms the entire system, the position of the object s o and that of the image s i with respect to the lens group and the transverse magnification M T are determined by the angles u 1 ¼ a and u 2Lþ1 ¼ b,
s o ¼�h = a; s i ¼�h = b; M T ¼ s i = s o ¼ a = b: ð29Þ
Using equations( 29), equation( 17) becomes after division by h the well-known Lensmaker’ s Formula( 1) = s i � 1 = s o ¼ K.
4 Other aberrations
4.1 Axial colour, astigmatism and Petzval sum
The simple relation( 28) between the power P m of a lens and the two quasi-powers leads immediately to the expression for the total axial colour of the thin lens group expressed in terms of quasi-powers. As well known, the axial
i colour contribution of each lens in the group is proportional to its lens power [ 1 ]. The total axial colour of the thin lens group is then the sum of the contributions of the individual lenses
A ¼�h 2 K X L ð z 2m�1 þ z 2m Þ; ð30Þ
V �1 m¼1 m
where V m is the Abbe number for lens m. When equation( 30) is used, the Abbe numbers can be different, but the refractive index n needs to be the same for all lenses in the thin group.
For the thin lens group, several primary aberrations do not depend on the quasi-powers and have well-known expressions [ 1 ]. If the aperture stop is placed at the thin lens group, the 3rd-order distortion and lateral colour vanish.
0
The total astigmatism T 1 and Petzval sum T 1 of the lens group are
and k¼1
T 1 ¼ H 2 K
T 0 1 ¼ T 1 = n;
k¼1 ð31Þ
ð32Þ
where H is the Lagrange invariant of the entire system.
The last Seidel aberration that remains to be expressed in terms of the quasi-powers is coma. The same approach as for spherical aberration can be used to obtain a simple formula for the coma contribution of thin lenses in contact. The Seidel sum for the 3rd-order coma is [ 1 ]
C ¼ X2L
C k ¼� X2L u kþ1 A k A k h k � u k ð33Þ n kþ1 n k
When the aperture stop is placed at the group of thin lenses, the chief-ray height at the group is zero. The paraxial refraction invariant A k for the chief ray( which has a formula similar to equation( 2), but using the chief-ray height and angle) is then given by the chief ray angle u before and after the group, A k ¼ u 1 ¼ u 2Lþ1 ¼ u.
4.2 Derivation of the simple coma expression
Readers primarily interested in the results may skip directly to Section 4.3.
For odd and even surfaces we have the surface contributions
C 2m�1 ¼ t 0 ðu 2m�1 � u 2m Þðnu 2m�1 � u 2m Þ C 2m ¼ t 0 ðu 2mþ1 � u 2m Þðu 2m � nu 2mþ1 Þ
t 0 ¼ hu ð34Þ n � 1 or, using equation( 7) C k ¼ ð�1Þ k�1 t 0 ðu~ k � u 2m Þðn u~ k � u 2m Þ: ð35Þ
When expanded, the coma surface coefficient contains three quadratic terms in the angles u. We look for new forms of equation( 34) as a perfect square plus two correction terms. For odd surfaces we look for a form
C 2m�1 ¼ c C1 ðl 2m � l 2m�1 Þ 2 þ c C2 l 2
2m�1 þ c C3l 2 2m: ð36Þ