P. A. Ameen Yasir and J. Solomon Ivan, "Realization of first-order optical systems using thin convex lenses of fixed focal length," J. Opt. Soc. Am. A 31, 2011-2020 (2014)

A general axially symmetric first-order optical system is realized using free propagation and thin convex lenses of fixed focal length. It is shown that not more than five convex lenses of fixed focal length are required to realize the most general first-order optical system, with the required number of lenses depending on the situation. The free propagation distances are evaluated explicitly in each situation. The optimality of the decomposition obtained in each situation is brought out. Decompositions for some familiar subgroups of $S{L}_{2}(\mathbb{R})$ are also worked out. Convex or concave lenses of arbitrary focal length are realized using three or two convex lenses of fixed focal length, respectively. It is further shown that three convex lenses of arbitrary focal length are sufficient to realize the most general first-order optical system.

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Allowed Signatures for $a$, $b$, $c$, and $d$ so that Matrix $M$ Describes a First-Order Optical System^{a}

a

b

c

d

$S{L}_{2}(\mathbb{R})$ Family

G

G

G

G

Yes

G

G

G

L

No

G

G

L

G

Yes

G

G

L

L

Yes

G

L

G

G

Yes

G

L

G

L

Yes

G

L

L

G

Yes

G

L

L

L

No

L

G

G

G

No

L

G

G

L

Yes

L

G

L

G

Yes

L

G

L

L

Yes

L

L

G

G

Yes

L

L

G

L

Yes

L

L

L

G

No

L

L

L

L

Yes

Here we have assumed $a$, $b$, $c$, and $d$ to be nonsingular. Note that only for 12 of the signature possibilities, the constraint $ad-bc=1$ is satisfied. L and G stand for less than zero and greater than zero, respectively.

Table 2.

Summary of the Decompositions for Ray Transfer Matrices with Nonsingular Entries^{a}

Note: The first column enumerates the cases with increasing usage of lenses. Column 6 lists the decomposition, and column 7 lists the required number of lenses

Table 3.

Summary of Decompositions for Ray Transfer Matrices with Any One or Two of Its Entries To Be Singular^{a}

Two considerations go into the choice indicated in column 7 (refer to Table 2). If the singular entry does not occur in the denominator in any of the free propagation matrices $F$ composing the listed special cases [refer to Eqs. (32)–(43)], then the decomposition using the least number of lenses is chosen. If the singular entry does occur in the denominator in any of the free propagation matrices $F$ (in which case the interlens distance is unphysical) comprising the listed special cases, only the decomposition choices where this does not occur are taken into consideration, and then the decomposition using the minimum number of lenses is chosen.

Tables (3)

Table 1.

Allowed Signatures for $a$, $b$, $c$, and $d$ so that Matrix $M$ Describes a First-Order Optical System^{a}

a

b

c

d

$S{L}_{2}(\mathbb{R})$ Family

G

G

G

G

Yes

G

G

G

L

No

G

G

L

G

Yes

G

G

L

L

Yes

G

L

G

G

Yes

G

L

G

L

Yes

G

L

L

G

Yes

G

L

L

L

No

L

G

G

G

No

L

G

G

L

Yes

L

G

L

G

Yes

L

G

L

L

Yes

L

L

G

G

Yes

L

L

G

L

Yes

L

L

L

G

No

L

L

L

L

Yes

Here we have assumed $a$, $b$, $c$, and $d$ to be nonsingular. Note that only for 12 of the signature possibilities, the constraint $ad-bc=1$ is satisfied. L and G stand for less than zero and greater than zero, respectively.

Table 2.

Summary of the Decompositions for Ray Transfer Matrices with Nonsingular Entries^{a}

Note: The first column enumerates the cases with increasing usage of lenses. Column 6 lists the decomposition, and column 7 lists the required number of lenses

Table 3.

Summary of Decompositions for Ray Transfer Matrices with Any One or Two of Its Entries To Be Singular^{a}

Two considerations go into the choice indicated in column 7 (refer to Table 2). If the singular entry does not occur in the denominator in any of the free propagation matrices $F$ composing the listed special cases [refer to Eqs. (32)–(43)], then the decomposition using the least number of lenses is chosen. If the singular entry does occur in the denominator in any of the free propagation matrices $F$ (in which case the interlens distance is unphysical) comprising the listed special cases, only the decomposition choices where this does not occur are taken into consideration, and then the decomposition using the minimum number of lenses is chosen.