Systematic generation method of relay optical systems

Yukio Fukui, Tsunehiro Takeda, and Takeo lida

A relay optical system consists of a fixed, coaxially mounted pair of separated lenses. A general rule ispresented here which replaces either component of the relay system by a pair of lenses. This rule can berecursively applied and leads to many variations. A representative use illustrates the layout problem.Key words: Relay lens, relay optical system, optical design, Gaussian optics.

1. IntroductionA relay optical system, used in many optical instru-

ments, comprises a fixed, coaxially mounted pair oflenses.1 Assume that these are thin convex lenses offocal lengths f and f2, separated by the sum of theirfocal lengths; then the lateral, longitudinal (or axial),and angular magnifications of this system, denoted asMiat, M1on, and Mang, respectively, are represented asfollows (derived in Appendix A):

MIat =-f 2 /fl, (1)

Mlon = (f2/f1)2, (2)

Mang = f/f 2 , (3)

where the negative signs indicate image inversion.Total distance D from the object to the final image isobtained as follows:

D= (fi + f2)2/f1 + [f 2/f1 )2-1]sl, (4)

where s1 is the distance from the first lens of the systemto the object. Here the positive direction is defined asthe direction of lightwave travel through the system.So s is currently negative.

It is noteworthy that the above three magnificationsare independent of s. Furthermore, if both focallengths of the two lenses are the same, that is, fi = f2 =f, total distance D also becomes independent of s1 usingEq. (4). In this case, the absolute magnitudes of thethree magnifications all become unity and D becomes4f. This means that the object is imaged (or relayed),without distortion, at the 4f distant position along theaxis from the original place, regardless of the axialposition of the object. This system is therefore calleda typical relay optical system.

In some cases, however, a typical relay optical sys-tem is insufficient for meeting the requirements of theentire optical system at the conceptual design stage.

The authors are with Industrial Products Research Institute, 1-1-4 Higashi, Tsukuba, Ibaraki 305, Japan.

Received 26 July 1989.0003-6935/90/131947-05$02.00/0.© 1990 Optical Society of America.

The designer should then make another choice,2 3 re-taining the relaying nature as it is. This choice, how-ever, depends entirely on the designer's skill: whichvariation of the relay optical system would be bestsuited for the requirements of his particular case. Theproblem of laying out the optical components is impor-tant at the conceptual design stage, because once thecomponent layout is fixed, the detailed design wouldbe carried out in a more mathematical way, such as theray tracing method by a computer.

This paper presents a systematic generation methodof relay optical systems for the conceptual designstage. By this method, the designer should easily findthe best solution to his own problem.

II. Inverse Equivalent Relationship Between Convex andConcave Lenses

A convex lens converges the incident parallel beamsto the focal point, then diverges them, while a concavelens diverges the incident parallel beams from thebeginning. Both lenses, from a macroscopic point ofview, finally diverge the incident parallel beams, al-though the convex one inverts them. This relation-ship between the two lenses is illustrated in Fig. 1.Figure 1(a) shows a thin convex lens L1 of focal length fwith parallel beams incident on it. Figure 1(b) shows athin concave lens L2 of the same power located at theposition where the same divergent beams emerge as in(a). Figure 1(c) shows the positional relationship be-tween (a) and (b). As Fig. 1(c) suggests, thin convexlens L1 of focal lengthf can be thought to be an invertedconcave lens of the same power located at 2f distancefrom the convex lens. As this effect occurs on bothsides of the convex lens, principal planes H1 and H2 ofthis inverted concave lens are at 2f distance on bothsides of the convex lens. We call the relationshipbetween these convex and thick concave lenses theinverse equivalent. Figure 1(d) shows that a thickconvex lens L3 also has the same inverse equivalentproperty.

From Figs. 1(c) and (d) we find that 2f is the distancebetween two kinds of principal plane: one of the origi-nal convex lens and the other of its inverse equivalentconcave lens, whether the convex lens is thin or thick.

Simple examples of the inverse equivalent relation-ships are illustrated in Fig. 2. Figure 2(a) is a well-

1 May 1990 / Vol. 29, No. 13 / APPLIED OPTICS 1947

f_

I I- f (a)

f' (b)

(a)

as § .L2

_-~~ ~~ _n__

( b ) r

2f 2 H2

AL-------,}

(c)

{e' -_ l.(C)

L

H L3 H'

ll T _ -___EH 2

d 2 Fig. 1. Inverse equivalence of a convex lens to a concave lens: (a)convex lens and final divergent beams, (b) concave lens located toemerge as the same beams as in (a), (c) positional relationshipbetween (a) and (b), (d) thick convex lens and principal planes of its

equivalent concave lens.

known Galilean telescope. If the concave eyepiece isreplaced by its inverse equivalent convex one, the tele-scope becomes a Kepler as shown in (b). Both tele-scopes have the same function, with the difference ofimage inversion in (b). Figure 2(c) is a doublet com-posed of thin convex and concave lenses of the samepower. Parallel beams incident on it travel parallelafter refraction. If the concave lens part of this dou-blet is replaced by its inverse equivalent convex one asshown in (d), the doublet becomes a typical relay opti-cal system as shown in (e). Both (c) and (e) have thesame optical characteristics, with the difference of im-age inversion in (e).

Ill. Compound Pair and Its Inverse EquivalenceTwo thin lenses are compounded into a thick lens

when placed coaxially to each other with a finite dis-tance. The focal length and the principal planes of thecompound system are given by Eq. (5):

f = flf2 1(f 1+ f2 - d),

h, = fld(f, + f2- d), (5)

h2 = -f2 d/(f, + f 2 - d),

wheref is the focal length of the compound system,fi,f 2 are focal lengths of the component thin lenses,

d is the distance between the two component lenses,h, is the distance from the first lens to the primary

principal plane of the compound system, andh2 is the distance from the second lens to the sec-

ondary principal plane of the compound.

(e)

Fig. 2. Examples of inverse equivalence. The inverse equivalenceof the concave lens in (a) is the convex eyepiece in (b). The concavepart of the doublet (c), when replaced by its inverse equivalent

convex lens (d), becomes a typical relay system (e).

The compound focal length f is negative when fi + f2< d, while both fi and f2 are positive; this is illustratedin Fig. 3(a). Figure 3(b) shows the principal planes ofthe compound system (a) with its equivalent lenshalves on the planes in broken lines. This figure is alsoregarded as a general case of a thick concave lens. Byreversing the steps of the previous section, the inverseequivalent convex lens is obtained by shifting twoprincipal planes toward the inner side by 21fl distance;the primary principal plane is shifted by +21fI distance,and the secondary principal plane by -21fl. Note thatthe positive direction is always to the right.

Figure 3(c) illustrates the derived principal planesand lens halves of the inverse equivalents to (b). Fromthese three figures, the parameters of the inverseequivalent lens to the compound system are derived asfollows:

F = -f = -ff 2 1(f + f 2 -d),

H 1=h 1+2F

= fj(d - 2f2)/(f, + f2 - d), (6)

H2 = h2- 2F

= f2(2f, - d)/(f 1 + f2 -d),

where F is the focal length of the inverse equivalent lensto the compound system,

H 1 is the distance from the first lens to the primaryprincipal plane of the inverse equivalents, and

H2 is the distance from the second lens to the sec-ondary principal plane of the inverse equivalents.

From Figs. 3(a) and (c) it is clear that the thin convexlens pair is inverse equivalent to a thick convex one.

1948 APPLIED OPTICS / Vol. 29, No. 13 / 1 May 1990

H . Add I'

lZ Al. *l

F��_------ ___ it

4-

( a

(n

2 11 2111

Fig. 3. Compound system and its inverse equivalents: (a) com-pound system of two thin convex lenses, (b) principal planes of thecompound system (a), (c) principal planes of the inverse equivalent

to (a).

Equation (6) is one, and the only one, that is inverseequivalent to Eq. (5). The verification is shown inAppendix B.

Figure 4 shows a particular form of inverse equiva-lent lens used to generate a variation of an opticalsystem. Figure 4(a) is a typical relay optical system; itcomprises a pair of thin convex lenses L1,L2 of the samefocal length f, separated by 2f. Figure 4(b) shows acompound system composed of thin convex lensesL3, L4 of the same focal length f, separated by 3f. Thiscompound system is equivalent to a thick concave lensof focal length -f from Eq. (5), that is also inverseequivalent to a thick convex lens of focal length f fromEq. (6). Therefore the compound system (b) can beregarded as a thick inverse convex lens. This thickinverse convex lens is shown on its principal planes in(b).

One of the lens pairs in Fig. 4(a) can be replaced, ifimage inversion is accepted, by this inverse equivalentthick convex lens, namely, L3 and L4. The second lensL2 in (a), for example, is then replaced by L3 and L4 tocoincide the position of the first principal plane in (b)with the former position of lens L2 in (a). After thatthe replacement, inverse equivalent relay system hasthree components: L1, L3, and L4 as shown in Fig. 4(c).

IV. Replacement Dictionary of a Component Lens

The previous case is replacement of a convex lenswith lenses of the same power, but replacement withlenses of different power is more useful. We considerhow a general replacement rule can be formalized.Here we assume that the focal lengths of both replacingand being-replaced lenses are given as initial condi-tions. Then we find distance d between the two re-placing lenses using Eq. (7), which is derived from Eq.(6):

21 f 3 f f

I_ _ I

_ m-uv I-'( b L4

3 f 3 f

( c ) L, L3 14

Fig. 4. Particular form of inverse equivalents used: (a) originalrelay optical system, (b) replacing lens pair and its inverse equiva-

lent principal planes, (c) result of replacing L2 in (a) with (b).

d = fl + f2 + ff 2/F, (7)

where d is the distance between the replacing lens pair,flJ2 are focal lengths of the replacing lens pair, and

F is the focal length of the lens being replaced.The replacement position is obtained as the princi-

pal planes by Eq. (6). The replacement coincides theprimary principal plane Hi with the position of the lensto be replaced. When a lens of focal length f is to bereplaced by lenses of, for example, either f, 2f, or f/2focal length, all combinatorial cases of replacement areobtained and illustrated in Fig. 5. In the figure, thelens halves (in broken lines) show the position of thereplacement with the lens to be replaced. Figure 5 isthe only replacement dictionary using lenses of f, 2f,and f2 focal lengths.

V. Using the Method for Rotation and Translation of theOptical Axis by Mirrors

Among many variations of relay optical systems, atypical system composed of an equifocal lens pair is themost frequently used to produce an optically identicalimage of the object. In some cases, however, this two-lens system would be insufficient, if other featureswere to be added to it. Figure 6 illustrates one of thefeatures a relay optical system can have: the functionof rotation and translation of the optical axis usingrocking mirrors. Figure 6(a) shows the chief ray,emerging from origin 0 toward the z-axis, refractingtwo lenses which form a typical relay optical system,reflecting two rocking mirrors, going away through therectangular hole at the top. The final ray direction ischanged by rocking the lower mirror.

Figure 6(b) shows that the final ray (axis) position ischanged, without change of the direction, by rockingthe upper mirror. These effects of changing the opti-cal axis using rocking mirrors are independent of eachother and can be superimposed as shown in Fig. 6(c).

1 May 1990 / Vol. 29, No. 13 / APPLIED OPTICS 1949

I � -I.

L,

-1-

a �_ ��

f I

( as+ f 1 3t 1f

2 f 2

2f

Cc)

2ff

2

( e ; Stt2 ______ _____~ ~~1

U~~~~~~~~~~~~~~~~~~~~~~~~~( e t 7 t I~~~~~~~4.

Fig. 5. Inverse equivalent dictionary sample for replacement. Alens of focal length f is to be replaced with a pair of lenses of f, 2f, or

f/2 focal length.

This function can be added to a general relay opticalsystem by using the replacement dictionary as follows.If the optical axis has to be rotated slightly, a rockingmirror should be inserted where the incident beamsfocus. However, for the case shown in Fig. 7(a), focus-ing position B is occupied by the second componentlens. The second lens may then be replaced by anoth-er pair of lenses consulting the replacement dictionaryin Fig. 5. Figures 7(b) and (c) illustrate two such casesapplied by (a) and (b) in Fig. 5. If two rocking mirrorsare required for 2-D rotation around the axes perpen-dicular to the original optical axis, Fig. 7(c) turns out tobe unsatisfactory again, because another rocking mir-ror should be inserted at the second focal position C,while the small lens occupies it. The replacementprocedure should then be again carried out, resultingin (d), (e), or some other ones.

Figure 7(d) has successfully been used in a 3-D opto-meter, which changes the optical axis two-dimension-ally to follow the viewing direction of the human eyeautomatically.4 The optical system of the 3-D opto-meter is shown in Fig. 8, where two large lenses arereplaced by concave mirrors to obtain a wider rockingangle of the optical axis.

VI. ConclusionThis paper describes a new method of making all

variations of a relay optical system by recursively re-placing a component lens by a pair of lenses. First,inverse equivalent lenses of all kinds should be calcu-lated by a pair of all available lenses. The results arethen illustrated by their principal planes, which givethe replacing position, and by their component lenspairs, which give the replacing components. This listis the replacement dictionary. Once the replacementdictionary is obtained, the designer would easily buildup his desired optical system without further complexcalculation. Since this method is graphic, all the de-signer is required to do is to replace recursively acomponent lens by a pair of other lenses, by consultingthe dictionary. This method has been successfullyused for the conceptual design of a 3-D optometer.

(a) (b) (c)

Fig. 6. Rotation and translation of the optical axis: (a) lowerrocking mirror causes the change of ray direction, (b) upper rockingmirror causes the change of ray position, (c) both effects can be

superimposed.

I i

A

d)

(ed) - I e

Fig. 7. Replacements using the inverse equivalent dictionary: (a)original relay optical system, (b) system (a) replaced by Fig. 5(a), (c)system (a) replaced by Fig. 5(b), (d) system (c) again replaced by Fig.

5(a), (e) system (c) replaced by Fig. 5(b).

1950 APPLIED OPTICS / Vol. 29, No. 13 / 1 May 1990

rockingmirror convex

\ ,>, 1 lens

lens

Fig. 8. Optical system of the 3-D optometer.

Fig. 9. Relay optical system.

Therefore we have

Mlat =fhlfl

Mon = (f/fi)2,

Mang =-f1f2.

D = (fl + f2)2/fl + [(f21fl)2 - 1]s,.

Appendix AFigure 9 illustrates a pair of thin convex lenses of

focal lengths fi and f2, separated by distance (fi + f2).An object 0 at distance s (<0) from the first lens(positive direction is the light travel direction, that isto the right) forms an image I, at distance s' from thefirst lens, and forms another image 2 at distance s2from the second lens. Using the Gaussian formula andgeometric relationships, the following equations areobtained (symbols are defined in Fig. 9):

1/l- l/s = lf,

1/2 - /S2 = 1/2,

S2 = - (I + 2),

Mlat = (Si/sl) * (SOs 2),

D 4s2 +fl +f2 -s 1 ,

Mlon s2 /s 1,

tan 1 =hll(-sl),

tan' = tanG2 = hll(-sl),

tanO'2 =(-h)1

h2/hl = S2/SI,

Mang = tanG 2 /tanol.

From the above equations we obtain

Sl = fs1(f + S1),

S2 = (f2 + flf 2 + f2 sl)/(fl + S1),

S2 = f 2 + ff 2 + 2S )/.

(Al)

Appendix BBy definition the principal planes h and h2 in Eq.

(5) show that Eq. (Bi) holds:

1/(b - h2) - 1/(a - hl) = 1/f. (Bi)

Likewise, H 1 and H2 in Eq. (6) show that Eq. (B2)holds:

1/(b - H2) - 1/(a - H) = 1/F = -1/f. (B2)

What we have to prove is that Eq. (B2) is derived fromEq. (Bi) or vice versa. From Eq. (Bi) we have

b -h2 = f(a - h1 )/(f + a - h1)

[left-hand side term of Eq. (B2)]

= 1/(b - H2) - 1/(a - H,)

= 1/(b - h2 + 2F) - 1/(a - h- 2F)

= 1(b - h2- 2f) - 1/(a - h + 2f)

= 1/[f(a - h1 )/(f + a - h) - 2f] - 1/(a - h, + 2f)

[substituted by Eq. (B3)]

= -l/f

= [right-hand side term of Eq. (B2)].

(B3)

(B4)

As this process is mathematically equivalent, the prop-osition has been proved.

References1. L. T. Sachtleben, "Extending the Content and Expanding the

Usefulness of the Simple Gaussian Lens Equations," RCA Rev.39, 340-379 (1978).

2. M. Born and E. Wolf, Principles of Optics (Pergamon, Oxford,1975), p. 244.

3. H. D. Crane and M. R. Clark, "Three-Dimensional Visual Stimu-(A2) lus Deflector," Appl. Opt. 17, 706-714 (1978).

4. T. Takeda, Y. Fukui, and T. Iida, "Three-Dimensional Optome-ter," Appl. Opt. 27, 2595-2602 (1988).

1 May 1990 / Vol. 29, No. 13 / APPLIED OPTICS 1951

S phl

sphericalmirror

(A3)