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Imaging of Optical Modes - Resonators with Internal Lenses
By HERWIG KOGELNIK
(Manuscript received November 10, 1964)
This paper discusses the modes of optical resonators, and optical modes of propagation or Gaussian beams of light. The passage of Gaussian beams through lenses, telescopes, sequences of lenses, and lenslike media is studied. Mode matching formulae are derived. A complex beam parameter is introduced for which the law of transformation by any given optical structure can bewrittenin the simpleform of a bilinear transformation (ABOD law). Resonators with internal optical elements and their transmission line duals are also investigated. Effective Fresnel numbers and curvature parameters are determined which allow one to infer the diffraction losses, the resonant conditions, and the mode patterns of the various systems. Results are obtained for 1'esonatol'S with internal. lenses, sequences of lenses with irises inserted between the lenses, 1'esonators with intemal lenslike media, transmission lines consisting of a lenslike medium with periodically spaced irises, and resonators with one very large mirror.
I. INTRODUCTION
The theory of Fresnel diffraction is the basis for an understanding of optical resonators':" and of optical modes of propagation.t-t-! Fresnel diffraction explains the mode patterns and diffraction losses of optical resonators, and the beam waist and spreading of the modes of propagation or "Gaussian beams." In this paper we will discuss how these Gaussian beams of light are transformed on their passage through lenses, telescopes, various lens combinations, and lenslike (guiding) media, and how these optical systems affect the properties of optical resonators when inserted between the resonator mirrors.
We will assume that no additional aperture diffraction effects are introduced by these optical systems, i.e., that the apertures of the internal lenses can be regarded as infinitely large. The imaging laws of geometrical optics are therefore expected to apply, and we will use them
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THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1965

wherever possible, as they generally simplify the algebraic derivations and at the same time provide some physical insight.
Some of the problems to be investigated here in greater detail have already been treated in the literature. Goubau" has given some mathematical relations between the parameters of Gaussian beams transformed by a thin lens. The recently published mode matching formulae? are the result of a computation which will now be presented. Resonators with internal lenses have also been discussed in the literature,8-11 and we have used the concept of an effective distance9,l o in a previous publication.! In several cases an alternative to our algebraic approach is the graphical method of Collins.t! who introduced the circle diagramll ,12 for Gaussian beams.
In the following we will first establish the rules of imaging for Fresnel diffraction with attention to the imaging of the phase fronts which are of particular importance for optical modes. Then we will list expressions for the focal length and the principal planes of various optical systems of interest, because these parameters are needed later for application of the imaging rules. This listing includes the parameters of the telescope, of sequences of lenses, and of sections of lenslike medium. Armed with these tools we will study the passage of Gaussian beams through lenses and various optical systems. The paper is concluded by an investigation of optical resonators with internal optical elements and their transmission line duals. Effective Fresnel numbers and curvature parameters are determined which allow one to infer the diffraction losses, the resonant conditions, and the mode patterns of the various systems. Results are obtained for resonators with internal lenses, sequences of lenses with irises inserted between the lenses, resonators with internal lenslike media, transmission lines consisting of a guiding medium with periodically spaced irises, and resonators with one very large mirror.

II. IMAGING RULES
While geometrical optics deals with rays, the theory of Fresnel diffraction deals with (scalar) fields. To describe the field distribution, we use complex amplitudes E(x,y,z) and a Cartesian (x,y,z) coordinate system. We consider a wave that propagates in the direction of the optic axis (z axis). Within the assumptions of Fresnel diffraction an
ideal thin lens of focal length f transforms the incoming wave with a
field Eloft(X,y,z = const) immediately to the left of the lens into a wave with the field
t2
Erighj(:r;,y,z = const) = E1ort(x,y,z = const) exp ( -jk x y) (1)

OPTICAL MODES

457

immediately to the right of the lens. Here lc is the propagation constant. The thin lens produces a phase shift which is proportional to the square of the distance to the optic axis, while the intensity distribution is the same on both sides of the lens.
Consideration of spherical waves provides a link between (1) and the laws of geometrical optics according to which a spherical wave with a radius of curvature R1 at the left of the lens is transformed into a wave with curvature radius R2 as shown in Fig. 1. The radii R1 and R2 are related by
(2)

For Fresnel diffraction the transverse field distribution of a wave with

a

spherical

phase

front

of

radius

R

is

given2

3 •

by

E(x,y,z = const) = exp (-jkr2j 2R )

(3)

where
(4)
and R is counted positive for a phase front that is concave if observed from the left. For spherical phase fronts of radius R1 on the left and - R2 on the right of the lens (where the phase front curvature is negative, as shown in Fig. 1) we can express E 1eft and Eright with the help of (3), compare the exponents in (1), and find the same relation (2) between
R1 , R2 , and f as for the spherical waves of geometrical optics.
To discuss imaging consider an object; i.e., the field E1(Xl, Yl) in an object plane, and its image E2(X2 , Y2) in the corresponding image plane (see Fig. 2). The distances d1 and d2 between the lens and the two planes are related by
(5)

(T I_--PHASE FRONTS

I _R1-

-

-

-

-

-

-

-""
\\

II' -........- .........R__z

---- --------

.....

I
f
Fig. 1 - Lens transforming phase front of spherical wave.

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THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1965

OBJECT PLANE

IMAGE PLANE

X.

X2

t.------
I

dt

-

-

-

-

-

+
I

-

-

-

-

-

-

-

d2

---------4
I

I --- -----

I
f

Fig. 2 - Imaging of field distribution by a thin lens.

We know from geometrical optics that the intensity distributions of the object and the image are similar. This is, of course, still true for Fresnel diffraction by any field aperture in the object plane. Assuming that no aperture diffraction effects are introduced by the thin lens, one can use the Fresnel diffraction formula to relate E1 and E2 (see Appendix A) and arrive at

-~Y2) ~~ ~ E2( X2,Y2) = - E1 ( - X2 , (6) 2 + + . exp -:J•k ( dl d2 2r2/ d~l )

+ with rl:.-= X22 Y22• The factor dl/d2 in this equation follows from

conservation of energy; the arguments - (dI/d2)X2 and - (dI/d2)Y2

+ indicate that the image is inverted and magnified by d2/ dl . The first
two terms in the exponent are simply due to the phase shift k(d l d2)

which the light wave suffers in propagating from the object to the image

plane, while the third and last term is of particular importance for our

considerations. It describes an additional phase shift proportional to

2 r2

which

appears

in

the

field distribution

of

the

image.

Apart

from

this additional phase shift the amplitude and phase distribution of the

image and the object are scales of each other.

The expression for the additional phase shift follows also from geo-

metrical optics (see e.g. Appendix B), and it is related to the thick-

mirror formulae," as we shall see later. it is also obtained by studying

OPTICAL MODES

459

the passage of Gaussian beams through a lens," The additional phase
shift does not appear in Abbe's theory of imaging; he finds that the
image is strictly similar to the object, both as regards the amplitude and phase distribution.'! But Abbe used the Fraunhofer diffraction theory, where phase terms proportional to r 2 are neglected.
For Fresnel diffraction the r2 dependence of the additional phase shift suggests that one should use spherical reference surfaces instead of plane ones, as shown in Fig. 3. By proper choice of the curvature of
these surfaces tangential to the image and object planes, one can achieve
an image field on one surface that strictly reproduces the object on the
other surface in amplitude and phase. For an object reference surface of radius R1 and an image reference surface of radius R2 one gets for the fieldsadditional phase factors of exp (_jkrI2j 2R1 ) and exp (_jkr22j 2R2) , respectively. These phase factors cancel the additional phase shift in (6) if

(7)

After some algebraic manipulations involving (5) this relation can be rewritten as

+1 + 1

1

dl RI d2 - R2 f .

(8)

This simply means that the center of curvature 0 1 of the object surface

r---- ----+------- -------1 OBJECT PLANE dI

IMAGE PLANE
d2

\1
c. _----R,--

I

\1

_---R C2

2- - -

SPHERICAL

FPRHOANSTE---~I I

I

I

f

SPHERICAL --PHASE
FRONT

Fig. 3 - Imaging of fields with spherical wave fronts; centers of curvature are images of each other. The corresponding spherical reference surfaces are used when fields with nonsnherieal phase fronts are imaged.

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THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1965

is imaged onto the center of curvature C2 of the image surface. Thus, whenever the centers of curvature of the image and object surfaces are images of each other we have an image which is a strict (scaled) reproduction of the object as regards both the amplitude and phase distribution, with no additional phase shift.
The imaging rules discussed above can also be used to study imaging by a combination of lenses (or by any optical system that can be regarded as such). It is not necessary to apply the rules step by step to each individual thin lens of the combination. It is generally simpler to determine the parameters of the equiv:alent thick lens as usual in geo-
metrical optics. The place of f is then taken by the combined focal
length of the system, and object and image distances (dl and d2) are measured from the principal planes of the thick lens.

III. FOCAL LENGTHS OF VARIOUS OPTICAL SYSTEMS
3.1 The Ray Matrix
When one traces a paraxial ray through combinations of lenses and lenslike media, the quantities of interest are the position Xl and the slope Xl' of the ray in the input plane, and the corresponding quantities
X2 and x/ in the output plane (see Fig. 4). There is in general a linear
relation16.16.17 between the output and input quantities which can be written in matrix form as

,=

(9)

X2

CD

We will call this ABCD matrix the ray matrix of the system. Because of reciprocity the determinant of the ray matrix is generally unity:

AD - BC = 1.

(10)

It is easy to determine the focal length and the principal planes from the elements of the ray matrix of an optical system. By tracing a beam
that leaves the output plane parallel to the optic axis (x/ = 0) we find
the location of the focal point on the input side. Its distance 81 from
the input plane is obtained as

(11)

where we refer to Fig. 4. Similarly, we find for the distance 82 between

OPTICAL MODES
INPUT RAY_

OUTPUT RAY
I f

461

OPTICS

r----~----Sl--- h,j --~ f-----

L~--h-z---

---Sz---~
---f----1

FOCAL' PLANE

I

I I j

..

-

----I .1'RINCIPAL PLANES

I

I
I

INPUT PLANE

OUTPUT PLANE

FOCALlpLANE

Fig. 4 - Reference planes for optical system.

the output plane and the corresponding focal point

I 82 =

_

2, X

= - ~•

X2 %1'=0

C

(12)

To find the principal plane on the input side we follow an input ray from the focal point until its distance from the axis is equal to the position X2 of the corresponding output ray and have

(13)

where the distance hi between the principal plane and the input plane is measured positive as shown in Fig. 4. On the output side we find similarly

(14)

The focal length f of the system is obtained by calculating the distance
between a principal plane and the corresponding focal point

+ + X21 XII f = 81

hi = 82

xl ~ = -

%2'~O =

-

-

•

X2' %1'-0

(15)

Using the linear relations of (9) together with the last three expressions, one finally gets

f = - (1/C)

(16)

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THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1965

hi = (D - I)IC

(17)

112 = (A - O/C

(18)

where the thick-lens parameters are expressed in terms of the elements of the ray matrix. For later use we also write down the matrix elements as functions of the lens parameters which follow from the last expressions

A = 1 - (h21f)

(19)

B = hi + 112 - (hl1l2lf)

(20)

C = - (llf)

(21)

D = 1 - (hIlf).

(22)

3.2 The Two-Lens Combination-Telescope

The lens parameters of a combination of two lenses are well known

and are listed here for completeness and for later use. The combination is

shown in Fig. 5. For lenses of focal lengths 11 and 12 spaced at a distance

d we have

+ III = (1/iI) (1/h) - (dillh)

(23)

hI =/d2-l

(24)

~ = dl
II

(25)

where the lens planes are used as input and output planes.
r------d-----1

I

I

II I

Fig. 5 - The two-lens combination.

OPTICAL MODES

463

For a slightly misadjusted telescope the lens spacing is

d = 11 + 12 - t1d

(26)

where t1d measures the misadjustment. The lens parameters of the telescope can be written as

f =fd2

(27)

t1d

11.1

=

ltd t1d

(28)

11.2

=

I~
t1d'

(29)

3.3 Sequence of Lenses

A periodic sequence of lenses of equal focal length fo and lens spacing
d is shown in Fig. 6. The reference planes are chosen just to the right
of each lens. The elements of the ray matrix B of one section of the
sequence (i.e., one lens spaced at a distance d from the input plane) are well known":" and are given by

1

d

1 -fd-o

(30)

INPUT
L---d---i---d---l
I I-d/21 I :r,.0'

n Fig. 6 - Sequence of lenses of equal focal length.

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THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1965

They relate the position and slope (Xl and x/) of the ray just after the first lens to the ray position and slope just after the zeroth lens

, =S ,

(31)

Xl

Xu

The ray to the right of the nth lens is related to the input ray by the nth power of the ray matrix of one section

s: X n

Xu

,= ,

(32)

Xn

Xu

s: The matrix elements of can be computed with the help of Sylvester's

theorem" and are well known,":" One has*

.S"

1 =-.

smO

sin nO - sin (n - 1)0

d sin nO

(33)

1.
- -smnO
fu

(1 - fu)sinno - sin (n - 1)0

with

cos 0 = 1 - (d/2fo).

(34)

We can now employ (16) and obtain for the focal length f of n sec-
tions of a periodic sequence of lenses

f = fu(sin Olsin nO).

(35)

The distance of the two principal planes from the input and output

planes (zeroth and nth lens) follows also from (33) with the help of

(17) and (18). One finds

+ hi = (d/2) f(1 - cos nO),

(36)

and

+ h2 = - (dI2) f(1 - cos nO).

(37)

If we measure the distance h of the principal planes from the midplanes between the lenses as shown in Fig. 6 we have

h = f(1 - cos nO).

(38)

'" These matrix elements can be written in terms of Chebyshev polynomials of the second kind of the variable [1 - (df2fo)].

OPTICAL MODES

465

A more complicated sequence of lenses is shown in Fig. 7. Here a
lens of focal length fl is followed by a lens of focal length h and vice

versa. The lens spacings are dl and d2 in sequence as shown in the figure. This sequence of lenses can be reduced to the simpler type discussed

above. We can regard it as a sequence of thick lenses formed by lens
pairs of focal lengths /I and f2 . The focal length fo of the thick lens is,

according to (23), given by

+ lifo = (1IfI) 01/2) - (d1l f d 2),

(39)

and expressions for the principal planes are given in (24) and (25). The distance d between the output principal plane of a thick lens and the input principal plane of the consecutive thick lens is obtained as

j) . d = d2 + hi + h2 = d2 + fodl (~ +

(40)

With the principal planes as reference planes, rays passing through this

sequence of thick lenses behave the same way as rays passing through

a sequence of lenses of equal focal length that are equally spaced. We

can therefore use the expressions (33) and (34) obtained above. With

(39) and (40) the latter becomes

(!.. !) + + + cos (J = 1 _ dl d2

dld2 •

(41)

2 fl f2 2fd2

3.4 Lenslike Medium

A lenslike medium or "guiding medium" is one whose refractive index n varies near the optic axis as in

r-- --+ -- --+ -- d2

d,---t- d2

d,---t--d2--'

I I I II I

II

!I

It;
I
" ' - _ - - - ,_ _..J

If,
I
"------,------'/

Fig. 7 - Sequence of lenses of alternating focal length with alternating lens spacings.

466

THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1965

n = no (1 - 2~)

(42)

where no is a constant, r is the distance from the optic axis, and b meas-
ures the degree of the variation of n. A medium of this kind can be produced by inhomogeneities in laser crystals19,20 or by a radial variation
of the gain in high-gain gaseous lasers." Another important example is the medium of the recently reported gas lens.22,23,24
To trace rays in a lenslike medium one uses the differential equation for light rays.25 For paraxial rays this ray equation has the form

d2x
no -dz2

' =

a -axn

=

x
-4no -b2

(43)

for the distance x (z) of the ray from the z axis. A corresponding relation holds for y (z). The solution is, again, a linear relation between the ray position and slope in the output plane (x and x') and the corresponding input quantities xo and xo'

x

cos 2 bz -2b si.n 2 -bz xo

(44)

,
x

- -b2 S.In 2 bz- cos 2 bz

,
xo

A typical ray path is shown in Fig. 8. To calculate the optical parameters for a section of lenslike medium immersed in a medium with a refractive index of unity (vacuum), we invoke Snell's law to relate the ray slopes

LIGHT BEAM

n=1

n=1

Fig. 8 - Hay path in lenslike medium.

OPTICAL MODES

467

at the section boundaries. For paraxial rays we have approximately

,

,

X vac = no.'l: j xOvac = noXo .

(45)

Now we use (16) and find for the focal length of a section of length 1

f= b

2no sin 2 ~.

(46)

(for no = 1 this formula has been given in Refs. 23 and 26, for example). The distance h of the principal planes from the input and output planes respectively (see Fig. 8) is computed with the help of (17) and (18). One obtains

h = -2bntoa n -1b.

(47)

The above expressions have been derived for a focusing medium where b2 ~ O. For a defocusing medium we have b2 ~ O,·and the .expression
for the focal length becomes

(48)

The location of the corresponding principal planes is described by

h

_
-

Ib I
2nc tanh

fbl i'

(49)

IV. OPTICAL TRANSFORMATION OF GAUSSIAN BEAMS

4.1 Light Propagation in Free Space

Near the optic axis an optical mode of propagation or Gaussian beam is regarded as a TEM wave with a spherical phase-front and a transverse field distribution that is described by Laguerre-Gaussians or Hermite-Gaussian" functions. The two beam parameters of interest are the "spot size" or beam radius w(z) and the radius of the phase front R(z). In any beam cross section of a fundamental mode the field varies as

exp ( -

w1-'22

-

j

k1-'2) 2R

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THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1965

and is specified by wand R. The light beam expands as it propagates through space as shoowwn imFI'g.9. Th e I aw 0 f expa' nsion .182'3"6 26,27

(50)

Here the z is measured from the beam waist where the phase front is plane and the beam reaches its minimum radius wo. For R(z) we have 2,3,6,2o,27

R=Z[l+(~:)].

(51)

Dividing (50) by (51) we find

~2 >..z

>..R - ~o2

(52)

which we can use to rewrite the terms in the round brackets, and express Wo and z in terms of wand R

+ ( 2
Wo

=

1

~2) 2
>..R

(53)

R

1+

(

->. R )
~W2

2.

(54)

~-----------z ---------1

II

\\ I

I

f-

Wo I
~-- ------

/
I ,""--PHASE FRONT
Fig. 9 - Contour of Gaussian beam of light.

OPTICAL MODES

469

4~2 Beam Transformation by a Lens

When a Gaussian beam passes through a lens a new beam waist is formed, and the parameters in the expansion laws are changed. Assume the light beam is propagating to the right. Before passing through the lens it has a beam waist a distance d1 away from the lens with a beam radius WI as shown in Fig. 10. The lens produces another beam waist a distance d2 away with a beam radius W2. The distances d1 and d2 are measured positive as shown in the figure (for a negative diane has a virtual waist). In the following we will establish some relationships between beam parameters of the incoming beam (identified by the subscript 1) and the parameters of the transformed beam (subscript 2).
The far field angles" 81 and 82 of the two beams are computed from (50) as

81 = >"/7rWl; 82 = >"/7rW~.

(55)

From these two angles follow immediately the beam radii WI/ and W2/ in the two focal planes of the lens where the image of the far field appears

WI! = j82 = Xj/ 7rW2

(5&)-

W2/ = j81 = Xj/7rWI.

(57)

The beam radius in ono of tho focal planes is, of course, independent of the spacing between the lens and the beam waist of the other beam. It follows from (51) that the center of curvature of the far field phase front is in the beam waist. According to the imaging rules of Section II, corresponding centers of curvature are images of each other (where we take the phase fronts as reference surfaces). We therefore have to determine the image of a beam waist to find the curvature center of the phase front in the focal plane on the other side of the lens. The distance d2' between the lens and the image of the waist W2 follows from

Fig. 10 - Gaussian beam transformed by a lens.

470

THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1965

(58)

and the radius of curvature Rl/ in the left focal plane is equal to the spacing between that image and the focal plane

Rl/ = d2' - f.

(59)

Combining (58) and (59) we have

l Rlf = d _ f

(60)

2

and correspondingly

l R21 =dl---f

(61)

for the radius of curvature in the right focal plane. Rlf and R2! are independent of the beam radii W2 and WI, respectively, a fact that can be used for mode matching into confocal resonators.
To relate the beam waists we use (56) and (60) to write

'1rW1/ _ X(d2 - f)

(62)

XRl/ -

1rW2 2

and similarly

'1rW2/ _ X(d1 - f)

(63)

XR2! -

1rW1 2

To express W2 in terms of WI and d1 we insert (57) and (63) into (53) and find

(64)

This relation, first given by Goubau," relates the beam radius of the waist of the transformed beam to the parameters of the incoming beam. A corresponding relation for the spacing ~ between the lens and the beam waist W2 is found by inserting (61) and (63) in (54). The result is

J2

(65)

The above expressions were derived with the help of the imaging rules of Section II. As mentioned before, these rules apply not only to thin lenses but also to thick lenses and lens combinations. Therefore,

OPTICAL MODES

471

if dl and rh are measured from the principal planes the results given
above are valid for the transformation of Gaussian beams by thick lenses.

4.3 M ode Matching

In experiments with optical modes one often wants to transform a beam with a given beam radius WI at the waist into another beam of waist radius W2 • One wants to "match" the modes of one optical system (like a laser resonator) to the modes of another one (an optical transmission line for example). This can be done by selecting a suitable lens and by properly adjusting the waist spacings dl and d2 , where we refer to Fig. 10. The proper spacings are given by the mode matching formulae" derived below.
We combine (62) or (63) with (52) and obtain

dl
d2

-
-

11 2 WI = W22 '

(
66)

This is used to rewrite (64) in the form

«, - + 12 1

1(

W22 = W22f2

1) (d2 - 1) T 1 (1rWI)2

(67)

l Multiplying

(57)

by

2 W2

we arrive at

l - (di - f) (d2 - f) =

102

(68)

where we have defined

= fo 1r(WIW2/"1I.).

(69)

To arrive at the mode-matching formulae we multiply or divide (68) by (66), extract the square root, and find

(70)

or (71)

As discussed in Ref. 7, one achieves mode matching by choosing a lens (or lens combination) with a focal length 1 that is larger than 10 or equal to it. For a given lens there are generally two ways open to match the modes. One can choose either the plus sign in both (70) and

472

THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1965

(71), or the minus sign. Fori = io there is only one set of proper spacings d1 = d2 = i = io .

4.4 Complex Beam Parameter - A BCD Law

In the foregoing we have used two parameters to characterize a Gaussian beam in a given beam cross section: the spot size or beam radius w, and the radius of phase front curvature R. We define now a more abstract complex beam parameter q

(72)

The propagation and transformation laws for this beam parameter are
particularly simple and allow one to trace Gaussian beams through
more complicated optical structures. The old parameters Rand w can,
of course, be recovered from the real and imaginary parts of llq. Note that we can regard the circle diagram of Collinsll as plotted in the complex plane of the variable jlq, and the circle diagram of Li12 as plotted
in the complex plane of jq*.
In terms of the complex beam parameter the laws of propagation
(50) and (51) have the simple and compact form t

q=qo+z

(73)

as one easily verifies by inserting (50) and (51) into (72). Here

qo = j(lIWo2/X)

(74)

is the complex beam parameter at the beam waist. Because of the linearity of (73) the parameters ql and q2 of two arbitrary beam cross sections are related by

(75)

where d is the distance between the two planes of interest measured positive in the direction of the optic axis.
The beam parameters ql and q2 to the left and to the right of a lens are related by

(76)

which simultaneously states the transformation of the phase fronts as in (1) or (2), and the fact that the beam radii (widths) are the same on both sides of the lens [compare (1)].
t Similar propagation laws for optical modes have been used independently by
D. A. Kleinman, A. Ashkin, and G. D. Boyd in an analysis of second-harmonic generation in crystals and by G. A. Deschamps and P. E. Mast in their recent paper in Proc. Symp. Quasi-Optics, Polytechnic lnst. Brooklyn, 1964, p. 379.

OPTICAL MODES

473

The imaging law (6) applied to Gaussian beams takes the form

~
q2

=

2
ddl22 •

!. + !
ql f

dl d2

(77)

if written in terms of the complex parameters ql and q2 of the beam in the object or image planes, respectively. Comparing with (7) and (8) one can also write this relation between the parameters of the object and the image as

dl

+1

+ _1_
ql d2 - q2

=!
f

.

(78)

Using (75) and (76) one can easily determine how an incoming beam with the parameter ql at a distance d: from a lens is transformed. The parameter q2 of the transformed beam at a distance d2 from the lens is obtained as

(79)

To establish a link to the transformation laws for the real parameters developed before, we multiply both sides of (79) with the denominator of the right side. Then we postulate that we have beam waists at dl and d2 by putting ql = j-TTWl2/X and q2 = j-TTW22/X. If we compare the real parts of the resulting expression, we obtain relation (68), and comparing the imaginary parts we find (66).
Let us now regard ql and q2 as the beam parameters in the input and output planes of an optical system described by its ray (ABCD) matrix as in Section III. This system is also described by its focal length and its principal planes as calculated from (16), (17), and (18); To relate
ql and q2 we use (79) and put d1 = hc , and d2 = ~. Comparing with
(19), (20), (21), and (22) we see that

q2

=

Aql Cql

++

B D

(80)

which we shall call the ABCD law. The q parameters of the input and the output are related by this bilinear transformation. The ABCD law says that the constants of this transformation are equal to the elements of the ray matrix. The ray matrices of several optical structures are given in Section III, and we shall use the ABCD law to study the passage of Gaussian beams through some of these structures.
There appears to be a very close connection between Gaussian light

474

THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1965

beams and the spherical waves of geometrical optics. In fact, all the

important laws of this chapter are formally the same for a spherical wave

with a radius of curvature q. One is therefore tempted to regard a Gaus-

sian beam as a spherical wave with a complex radius of curvature. For

the limit of infinitely small wavelengths the curvature radius becomes

real and one has a spherical wave of geometrical optics.

The ABCD law allows also a kind of "black box" approach to the

study of optical modes. One can, for example, inquire about the mode

parameters of a sequence of equal black boxes, i.e., optical structures

characterized by their ray matrix elements A, B, C, and D. For a mode

the beam parameter at the output of a black box is equal to the param-

eter at the input (ql = q2 = q). From (80) follows a quadratic equation

for the mode parameter q

cl + (D - A)q - B = O.

(81)

The solution of this equation can be written as

(82)

from which one can obtain the beam radius or spot size of the mode and the radius of curvature of its phase front.

4.5 Beam Tronsformaiioi: by a Telescope

In this chapter we shall study the passage of a Gaussian beam through a telescope consisting of two lenses of focal Iength jr andf2 , respectively,
+ spaced at a distance d = fl f2 - lid. The "misadjustment" lid is
assumed small. The focal length and the location of the principal planes of the telescope are given in (27), (28), and (29). We consider an incoming beam with a beam radius WI at its waist, and the waist spaced at a distance SI from the first lens as shown in Fig. 11. We want to determine the location S2 of the waist of the outgoing beam and its beam radius W2.
The distances of the waists to the corresponding principal planes are

(83)

From this we find with (24) and (27)

1). ~ _ 1 = .t! + lid (~ -

(84)

f

f2 f2fl

Inserting this expression together with (27) in (64) we get for the beam

OPTICAL MODES

475

Fig. 11 - Gaussian beam passing through a telescope.

waist Wz

W2 = WI ~ [1 + ~~ (1 - j~)J

(85)

which is correct to first order in Sd, We see that the ratio of the beam waists is more or less equal to the ratio of the focal lengths of the lenses. There is only a slight dependence on the position of the input beam waist for i1d ~ O.
To determine the location of the output beam waist we use (84) and the corresponding expression for (dz/f) - 1 to rearrange (68) as
- ~~2 [ (h - 1)(~ - 1) (86) +11~2 (1r W~2)].

Inserting (85) and expanding to first order in i1d this becomes

+ 81 1-12 II 82 1-22 /2

=

1i114d

[(
81

_

1)2 _
1

(1rWI 2)2J

A

•

(87)

For a well-adjusted telescope we have i1d = 0, and the distances
between the beam waists and the focal planes of the corresponding
lenses (i.e., 81 - II and 82 - iz) scale like the squares of the focal lengths
of the two lenses. The signs in (87) indicate that for an input waist
which lies to the left of the focal plane of the input lens one has an output beam waist to the left of the focal plane of the output lens, and conversely for an input waist to the right of the input focal plane.

476

THE B E L L SYSTEM TECHNICAL JOURNAL, MARCH 1965

4.6 Beam Transformation by a Sequence of Lenses
Consider now a sequence of lenses of equal focal length / spaced at a distance d as shown in Fig. 6. Immediately to the right of each lens an optical mode of this structure has a phase front with a radius of curvature of —2/ and a beam radius Wm given''*'^° by
λ 1 . /TT Τ sin θ

where θ is defined in (34). To the right of each lens the complex beam parameter (72) of a mode is therefore

1 = - 1 J?ÎΞ_^

qm

2f ·' d

(89)

Assume that a Gaussian beam is injected into the lens sequence, and call its complex beam parameter in the input plane q i . If gi = g„ , then we have launched a mode of the system, and the parameter of the beam to the right of every lens is qm • For qi ^ qm we use the ABCD law (80) to compute the beam parameter q2 to the right of the nth lens. The elements of the ray matrix of η sections of the lens sequence are given in (33), and we use them to apply the ABCD law. We have

[sin ηθ — sin (n — l)ö]ci + dsinnS qi = — J

, ^ . (90)

- J sin 710- gi + ^1 — jj sin ηθ — sin (η — 1)0

From (34) it follows that

sin ηθ — sin (n — 1)0 = (d/2/) sin ηθ + sin 0 cos ηθ (91)

which can be used together with (89) to rewrite (90) as

1 1

sm θ-β'"^

(92)

8ΐιιθ·β-ί"0 + d(- - —)smn0 \3i qmJ

After some further rearrangmg this can be written in the form

1

1

1 1 , 1 1 _ 1 2_ 1 exp ( - 2jne). (93)

32 qm qm f Lffl ffm Qm fj

For the case where the q parameter of the injected beam does not differ too much from the parameter qm of a mode we can put

Δ = (1/Si) - (l/qm)

(94)

OPTICAL MODES

477

and assume that a is small. Developing (92) in powers of a we obtain

!.. _ .!.. + 2 = a.e2j n8 _ ja2 1rWm (e2j n8 _ e4j n8 )

0(a3 ) .

(95)

q2 qm

2A

If we neglect all but the first-order term in (95) and compare the real

and imaginary parts, we arrive at approximate formulae* for the output

parameters R2 and W2 :

-+-= -+- 'I" 1
R2

1
2/

(1 R1

1) 2f

cos2nO+A-

(

- W1l2

-w- 1m2)

s.m2nO '

1 1 '11"(1 1). .(1 1) (96)

+ - + - - - -W22 -

w- m2 =

-

--
A R1

2/ sm 2nO

Wl2 wm2 cos 2 nO.

Comparing these expressions with (33) we see that the beam radius
W2 varies in z direction with a period that is half the period with which
a ray displacement varies. This fact has already been seen experimentally,' and has also been noted for other optical structures,"
The formulae (96) are valid for cases where the parameters WI and R1 of the input beam do not differ much from the parameters of the mode of the lens sequence (i.e. for small a). For cases where this condition is not fulfilled we have to go back to (90). Using (72) we re-express the q parameters in terms of WI , R1 , W2 , and R2 and compare the imaginary parts of 1/q2 as given by (90). After some algebra, where (91) is used to make simplifications, we obtain

::: =+~_[l1[+1:- ~w4~+4-(7(r1~_rWmm)_22)(2~(l-1++;/-1))]2J cos2n8 (97)

2

Wl4

A

R1 2/

+ ('II"~m) (kl + ;j) sin 2n 8.

In this exact expression for W2 we find the same periodicity in z direction
as in (96). As n is varied W2 goes through maximum values W m"" and minimum values Wmin. It is easy to show from (97) that

= Wmax Wmin

2 Wm •

Note that W max and Wmin are the extrema of the envelope curve obtained for continuously variable n. The extremal values of W2 actually occur at a lens only if the corresponding n is an integer.
* In a recent publication by J. Hirano and Y. Fukatsu in Proc, IEEE, 51, Nov.,
1964, p. 1284, similar expressions were derived by means of a perturbation technique in which the real beam parameters were used directly.

478

THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1965

An exact expression for R2 is obtained by comparing the real parts of l/q2 in (90) in a similar way.

4.7 Beam Transformation by a Lenslike Medium

The passage of Gaussian beams through a lenslike medium as described by (42) has been discussed by several investigators.19.28.29.30,31
We assume here for simplicity that no = 1, or a refractive index given by

(98) It is easy to ShOWI9.28,29.30,31 that for a Gaussian beam that is injected with a plane wave front and a beam radius Wo given by

wo2 = Xb/21r

(99)

the wave front remains plane, and the beam radius remains constant as the wave propagates. These light beams are called the modes of the lenslike medium. H the beam is injected with a beam radius WI ¢ Wo , the wave front and the beam radius will change as a function of z: This problem has been treated by Tien et aI.23with the help of a differential equation, and by Marcatili" who expanded the field distribution of the input beam in terms of the modes of the lenslike medium. We will show here that one can get the desired results in a rather simple fashion by employing the ABeD law (80).
The elements of the ray matrix of a medium section of length z are
given in (44). Using these together with (80) one computes for a beam with the complex parameter ql in the input plane a beam parameter

+ qlCOS

2

z b-

-b2 s.m2-zb

+ - ql 12i sm'2zb cos 2bz

(100)

in the output plane a distance z away from the input. Assuming an input beam with a plane wave front and a beam radius WI we have

(101)

Inserting this and (99) in (100) we obtain

+ _

~

sm.

2

z
b

2
J.W~ocos 2bz

1rWo2 cos 2-Zb -

J•

W w- ol22

•
sin

2Z-b

(102)

OPTICAL MODES

479

If we compare the imaginary parts in this expression we get

( + - 4 ) 2
W2

=

2 WI

cos22Zb

W Wl4O Sil.l22Zb-

(103)

which agrees with the results of Refs. 28 and 29. A comparison of the real parts yields an expression for the curvature of the wave front.

V. RESONATORS WITH INTERNAL OPTICAl, ELEMENTS

6.1 The Basic Resonator Parameters

A resonator consisting of two spherical mirrors spaced at a distance d is shown in Fig. 12. R1 and R2 are the radii of curvature of the two mirrors, measured positive as shown in the figure. The mirror diameters or widths are 2al and 2a2, respectively. The three basic parameters of such a resonator arelO,32,33

N = ala2
"Ad'

(104)

(1 - , G1 =

al a2

~)
R1

(105)

(1 - !£) . G2 =

a2 al

&

(106)

Within the Fresnel diffraction theory of optical resonators these three

T-

I

-- I

I
2Ia,

---R ,~~

I I

~-~-

I

~~~

I I
I

--~-R2~ - - ~~

I

I

LI

1-1

~------------d ------------~

Fig. 12 - Empty spherical mirror resonator.

480

THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1965

parameters determine completely the diffraction losses, the resonant frequencies, and the mode patterns of the resonator."
In the following we will show that resonators in which lenses 01' similar optical structures are inserted between the resonator mirrors are equivalent to an empty resonator of the type shown in Fig. 12. By equivalent resonators we mean here resonators with the same diffraction losses, the same mode patterns except for a scaling factor, and the same resonant conditions. To specify an empty resonator equivalent to a resonator with internal optical elements we will compute its parameters N, G1 , and G2 •

5.2 Resonators with an Internal Lens
A resonator with an internal lens is shown in Fig. 13. A lens of focal
length f is spaced a distance d1 away from the left mirror and rh away
from the right mirror. As before we call the radii of curvature of the two mirrors R1 and R2, and their diameters 2al and 2a2 as shown. The internal lens is assumed to be so large that no additional aperture diffraction effects are introduced.
Suppose now that we know the modes of the resonator. We can apply the imaging rules of Section II and choose the mirror surface of the right mirror, say, as reference surface. The image of the mode pattern on this mirror appears a distance

-,-
a,

-,-
a,
Fig. 13 - Resonator with internal lens and equivalent empty resonator.

OPTICAL MODES

481

d 2'

=

~ d2 - f

(107)

away from the lens as shown in the figure. The field of the wave reflected from the mirror is zero outside the mirror aperture a2 . The field of the corresponding image is therefore zero outside an aperture a2' given by the magnification

a2 = ,_ d2 = 1 _ d2

~'

d2'

f.

(108)

The image is a scaled reproduction of the mode pattern on the mirror which is exact in amplitude and phase if a spherical reference surface is chosen. The correct curvature of this surface is found in accordance with (6) and (8) by imaging the center of curvature of the mirror on the right.
Consider now a mirror of diameter 2~' placed at the location of the image a distance

(109)

away from the original left mirror as shown in the lower part of Fig. 13. The mirror curvature is chosen to be the same as the curvature of the reference surface for the image. This mirror may be called _~ image mirror of the original mirror on the right. Apart from a phase difference
+ of 21e (d2 d2' ) it reflects a wave coming in from the left in exactly the
same way as the original mirror combined with the-Tens. The incoming wave produces the same (magnified) complex amplitude distribution or field pattern on the image mirror as on the original mirror on the right. The outgoing wave reflected by the image mirror has a field pattern at d2' that is identical to the field pattern at d2' of the outgoing wave reflected by the combination of the original mirror and the lens. The field patterns of the two outgoing waves are thus also identical in any other beam cross section, and in particular across the left mirror. Therefore the modes of the empty resonator formed by the image of the right mirror and the original left mirror as shown in the figure are equivalent to the original resonator with the internal lens. The mode patterns on the left mirror are identical in both cases, and the mode patterns of the corresponding mirrors on the right are scales of each other. The diffraction losses of the two systems are also the same, and there is only a small difference in the corresponding resonant conditions due to the difference
+ in phase shift of Ie (d2 d/) per transit.
The basic parameters of the equivalent empty resonator are easily

482

THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1965

obtained. According to (104) we have a Fresnel number of

and with (105)

N = alG.2'lAd

(110)

(111)

With (107), (108), and (109) these expressions can be written in terms of the dimensions of the resonator with the internal lens. One obtains
(112)

and By interchanging subscripts one gets

(113)

These three parameters determine the properties of the modes of the internal lens resonator. In the above expression one notes the appearance of the term
(115)
which one might call the effective distance between the mirrors. It is modified by the presence of the lens.
In Refs. 4 and 32 approximate expressions are given for the resonant condition and the beam radii (spot size) of the fundamental mode at the mirrors of an empty resonator that is stable. Recall that for a stable resonator there holds
(116)
We can apply these formulae to our equivalent resonator and obtain by imaging the corresponding expressions for the resonant wavelength A and the beam radii WI and W2 on the mirrors of our resonator with an internal lens. Using the parameters discussed above we get

OPTICAL MODES

483

+ + + + 2(d1 A d2) -_ q :1; (m

n

1)

cos-1

V-

I(f7lG U'lU2

(117)

where q is the longitudinal mode number, and m and n are the transverse mode numbers. The sign of the square root should be chosen equal to the sign of GJ (or G2 ) . For the beam radii we get

WIW2 = -Ado (1 - G1G2)-1, 1r

(118)

and

WI = ~ (G2)1.

(119)

W2 a2 G1

The image mirror discussed above can also be obtained from the concept of a "thick mirror." A thick mirror" is a combination of a spherical mirror and a lens. The optical characteristics of this combination are represented by a combined focal length and a principal plane." A mirror of this focal length located at the principal plane is equivalent to the thick mirror combination. This equivalent mirror is the same as
our image mirror. The equivalence of the empty resonator and the internal lens resonator
can also be shown by using the Fresnel diffraction formuhr in-the.manner of Appendix A. One obtains integral equations for the modes of an internal lens resonator. After performing the integration over the lens plane which involves infinite Fresnel integrals, the equivalence to the
empty resonator is easily seen.
For cases where the effective distance do as given by (115) is very
small, ray angles of interest become rather large and the theory of Fresnel diffraction is no longer expected to apply. We have to exclude these cases from our considerations.
Our discussion includes internal lens resonators with Bat mirrors as shown in Fig. 14. The basic parameters of this resonator type can be
obtained from (112), (113), and (114) by putting R1 = R2 = 00. Burch
and Toraldo di Francia" have discussed the confocal system of this
resonator family where G1 = G2 = O. The transmission line dual of an
internal lens resonator with Bat mirrors is also shown in Fig. 14. It is a sequence of lenses and irises spaced as shown. In this sequence the lenses
are large and the irises inserted between them control the modes of the
system. For a symmetric system of this kind where d1 = d2 = d and
al = a2 = a the above expressions simplify, and we have

G1 = G2 = 1 - (d/f) ,

(120)

484

THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1965

~ ~ I If F~--

·'-r- I ---.--+I----·'---TI--:'.,---+- ::--1

i
I

r
I

Fig. 14 - Internal lens resonator with Hat mirrors and its transmission line dual, a sequence of lenses with irises placed between the lenses.

and a Fresnel number of

(121)

5.3 Resonators with an Iniemol Optical System

As discussed before, the imaging rules of Section II apply not only
to thin lenses but to any optical system that can be characterized by a focal' length and by principal planes. The expressions derived above for internal lens resonators can therefore be applied also to spherical mirror resonators with an internal optical system. All one has to do is to inter-
pret f as the focal length of the system and put

(122)

where hI and 1'-2 measure the distances between the two principal planes and the corresponding mirrors.
We can also characterize the internal optical system by its ABeD or ray matrix as in (9). Inserting (122) in (112), (113), and (114) we compare the resulting expressions with (19) through (22). We note immediately that the three basic resonator parameters can be written in terms of the elements of the ray matrix in the form

N = ala2
>J3'

(123)

OPTICAL MODES

485 (124)

(125)

5.4 Internal Lenslike Medium - Guiding Medium with Apertures
In this section we consider a spherical mirror resonator with a lenslike medium inserted between the resonator mirrors. The optical properties of a lenslike medium have been discussed in Sections 3.4 and 4.7. The refractive index of this medium changes with the square of the distance from the optic axis and is described by (98) if we assume no = 1. The degree of this index variation is measured by the parameter b. As shown in Fig. 15, we assume that the medium fills the space between the resonator mirrors which are spaced at a distance l. The mirror diameters are 2al and 2CZ2 , respectively, and the corresponding radii of curvature are RI and R2 •
The three basic resonator parameters which describe the modal properties of this resonator with an internal lenslike medium are easily computed by using the results obtained before. The elements of the ray matrix for a medium section of length l are given in (44). Inserting these in (123), (124), and (125) we obtain for the Fresnel number of the system
(126)

and for the G parameters

R,

I

~--h--~-

I
~-- h--~

Fig. 15 - Resonator with an internal lenslike medium.

486

THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1965

G1 =

a-l ( cos 2 -l-b - .SID 2 -l)

a2

b 2R1

b'

(127)

G2

=

-a2 ( al

cos 2 l- -b - .SID 2 -l) .

b 2R 2

b

(128)

A special case of the above system is shown in Fig. 16, where the
mirrors are flat, i.e., R1 = R2 = 00. The transmission line dual of this
resonator is also shown in the figure. It is the interesting case of a lenslike
medium or a gas lens with periodically spaced irises as shown. For irises
of equal diameter (al = a2 = a) the above expressions simplify, and
we obtain for the Fresnel number of the system

2a2

N=

l'

Ab sin 2 lj

(129)

and

G1

=

G2

=

l cos2lj'

(130)

This system is confocal for l = (r/4)b. When the value of 2l/b approaehea.a multiple of r, N gets very large and we have a case where the effective distance between the mirrors is very small [compare (115)]. As discussed before, the theory of Fresnel diffraction is no longer expected to apply under these circumstances,

Fig. 16 - Resonator with flat mirrors and an internal lenslike medium, and its transmission line dual, a gas lens with periodically spaced irises.

OPTICAL MODES

487

In high-gain lasers the parameter lib can become rather large for certain frequencies." For frequencies where the laser medium is focusing
we have b2 > 0, while for frequencies where the medium is defocusing b2 < 0 and b is imaginary. It is interesting to study the stability" of a
resonator with an internal lenslike medium allowing for positive and
e. negative values of For simplicity we assume that the radii of curvature
of the two resonator mirrors are equal and put R1 = R2 = 21. With
(127) and (128) we obtain

G1G2 = G2 = ( cos 2 -lb -b-4f.sm 2 -lb)2
and write the stability condition (116) in the form -1 ~ G ~ 1.

(131) (132)

One can plot a stability diagram in which each resonator with given
parameters l, I, and b is represented by a point. Such a diagram is shown
in Fig. 17, where llj is plotted as ordinate and lib and jllb are plotted
as abscissae. Resonators with b2 > 0 are represented by points to the

Fig. 17 - Stability diagram for a resonator with an internallenslike medium.

488

THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1965

right of the llj axis, and resonators with b2 < 0 are represented by points
to the left. Points in the shaded regions correspond to unstable resonators, and resonators represented by points in the unshaded regions are stable. The boundaries between the stable and unstable regions follow from (131) and (132). They are described by the equations

l

1r

b = P 2'

P integer,

(133)

l j-

=

4 -lb cotbl- '

s l
f

=

-

4

l
b

t

an

l

(134) (135)

For b2 < 0, where b is imaginary, the trigonometric functions of (134) and (135) become hyperbolic functions as in (48) and (49). For b2 > 0
one gets periodically stable and unstable regions as lib is increased. We have not discussed in detail cases where the lenslike medium
occupies only a part of the space between the resonator mirrors. However, one can compute easily the basic parameters for resonators of this kind with the help of the matrix elements of (44), and the formulae (123), (124), and (125).

5.5 Resonators with One Very Large Mirror
Let us return to the case of an empty resonator. In some practical arrangements the diameter of one of the two mirrors, say 2a2, is so large that diffraction by its aperture can be neglected. The resonator modes are then more or less controlled by the aperture al of the other mirror. This statement is not true for resonators of the degenerate confocal type where the diffraction losses at each mirror are equal" for any aperture ratio a21al . We exclude resonators of this type from our present discussion.
The properties of the resonator modes are generally determined by the three basic parameters given in (104), (105), and (106). But for an infinitely large a2 the Fresnel number N and the parameter G2 be-
come infinitely large, and G1 = O. The resonator parameters are now
quite meaningless. It is, however, possible to construct an equivalent resonator with parameters of finite value, as we will show below.
Consider Fig. 18. An empty resonator with one mirror of large diameter is shown schematically at the top. Below it we have drawn its transmission line dual. It consists of a sequence of lenses where an apertured lens

OPTICAL MODES

489

~----d----~----d----~----d----~

I

I

I

I

r----- I

fz= RZ/ 2

I

d----+----d----,

Fig. 18 - Empty resonator with one very large mirror, its transmission line dual, and its equivalent internal lens resonator.

follows an unapertured lens of large diameter. But this transmission
line is also the dual of the resonator shown at the bottom of the figure.
This is a resonator formed by apertured mirrors of finite diameter 2al
with an internal lens of focal length f = R2/2. The lens is unapertured.
Internal lens resonators of this type have been considered before. We can compute the Fresnel number of this system from (112) and obtain

. N = 2Xd (1 - ~J

(136)

Equations (113) and (114) are used to calculate the G parameters with the result

490

THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1965

These parameters determine the properties of the modes of the internal lens resonator shown in Fig. 18. The mode patterns at the apertured mirrors of this resonator are, of course, equal to the mode pattern at the apertured mirror of the empty resonator. The one-trip diffraction loss of a mode of the internal lens resonator is equal to the return-trip diffraction loss of an empty resonator mode, as there are no diffraction losses at the infinitely large mirror.
For the special case where the large mirror is :fiat (R2 = 00) the above
discussed equivalences are well known. They follow from symmetry considerations.

VI. ACKNOWLEDGMENTS
Stimulating discussions with E. I. Gordon, J. P. Gordon, R. Kompfner, T. Li, and P. K. Tien in various phases of this work are gratefully acknowledged.

APPENDIX A

Imaging for Freenel Diffraction

The purpose of this appendix is to show how the imaging relation (6) of the main text is derived within the formalism of scalar Fresnel diffraction theory. Assume a light wave traveling in z direction and refer to Fig. 19. Call the object field E1(Xl, Yl) and the image field E2(X2, Y2). The distances d1 and d2 between the lens and the object and image planes, respectively, are related by
(138)

OBJECT PLANE

IMAGE PLANE

~r-------

d,

-------+~------

~
d,-------l

L--------p,-------- -------P2'------_~

I
f
Eo(X o; Yo) Eo (XoIYo)
Fig. 19- Dimensions of interest for Fresnel diffraction theory of imaging.

OPTICAL MODES

491

The field immediately to the left of the lens is denoted Eo(xo, Yo) and

the field to the right of the lens is Eo(xo, yo). According to (1) of the

main text we have for a large, ideal lens

t2
Eo = Eo exp ( -jk xo Yo)

(139)

where k = 27r/X is the propagation constant in the medium. With the help of the Fresnel diffraction formula the fields Eo and E2 can be expressed as

(140)

and where

1+'" - E2 = 2jkd

dxodyoEo exp (-jkp2)

7r 2 -'"

(14l)

and

The integrationin (140) is performed over the aperture area Al of the

object field, and the integration limits in (141) are extended to infinity

with the assumption that the lens is so large that no additional aperture

diffraction effects are introduced.

Combining (139), (140), and (141) we obtain by interchanging the

order of integration

f 1+'" 2
E2 = - 4 k2d d

dxIdylEI

dxodyo

7r I 2 Al

-'"

.exp [ - jk (PI + P2 + ;~)]

(144)

where

(145)

Now the expressions (142) and (143) for PI and P2 are inserted. One finds that in the exponential the terms proportional to ro2 cancel because
of (138). The integration with respect to Xo and yo can be performed by
noting that

492

'rHE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1965

1+ 00 -00

d:l:o

exp

[

J•k:l:o

(:al:;l

+

iXh.2)J

=

211"15 ([kX'ihl + Xd:i2J)

(146)

where 15 is the Dirac delta function." With this (144) becomes

E2= - d~:2 exp [ - jk (dl + d2+ ;~)]

.£J i ) dxldylEI exp ( -jk

(147)

This simplifies immediately with the help of the formalism of the delta function" to

Multiplying (138) by dl/d2 one finds that

(1!.
d2

+~) d2

=i!

~
d«

(149)

which is used to write (148) in the form of (6) of the main text.

APPENDIX B
Pl'incip~e of Equal Optical Path Leading to Additional Phase Shift in Image Plane
The process of imaging the field distribution in the object plane into the image plane can be understood in terms of the rays leaving each point (say PI) in the object plane at various angles as shown in Fig. 20. All rays originating from PI are collected at a corresponding point P 2 in the image plane. A form of the principle of equal optical path35 says that the optical path lengths from PI to P2 are the same for all rays regardless of initial slope.
To obtain an image which is an exact reproduction of the original amplitude and phase distribution it would be necessary for the various optical paths which connect corresponding points, say PI and P 2 or QI and Q2, to be equally long for all points regardless of their distance from the optic axis. That these path lengths are not the same for all

OBJECT PI. ...NE
I
I
P,

OPTICAL MODES

IM...GE PI. ...NE

II #:, W"'VE FRONT \
CORRESPONDING TO INCIDENT-~ \

PI....NE W"'VE

\

------_- \ Q2

493

Fig. 20 - Rays emerging from a point of the object collected at the image.
points but increase with increasing distance between the imaged point and the optic axis can be seen from the simple example of an ideal plane wave coming in from the left. This case furnishes an expression for the path length difference as a function of the distance between the imaged point and the axis. As the path length is independent of the field distribution imaged, this expression is valid for the general case. To derive it we recall that an ideal plane wave is transformed by an ideal lens into an ideal spherical wave with the focal point of the lens as its center. The rays connecting points which lie on corresponding wave fronts are equally long for all points on the wave front." Therefore all path lengths measured from the object plane to the spherical wave front which touches the image plane are equal. Paraxial rays (which are practically parallel to the optic axis) need an additional length equal to rN2f to reach a point (P2) in the image plane which is a distance r2 away from the axis. This additional ray length accounts for the additional phase shift given in (6) in the main text.
REFERENCES
1. Fox, A. G., and Li, Tingye, Resonant Modes in a Maser Interferometer, B.S.T.J., 40, March, 1961,p. 453; Proc. IRE, 48, 1960,p. 1904.
2. Goubau, G., and Schwering, F., On the Guided Propagation of Electromagnetic Wave Beams, Trans. IRE, AP-9, May, 1961,p. 248.
3. Boyd, G. D., and Gordon, J. P., Confocal Multimode Resonator for Milli-
meter Through Optical Wavelength Masers, B.S.T.J., 40, March, 1961, p.
489. 4. Boyd, G. D., and Kogelnik, H., Generalized Confocal Resonator Theory,
B.S.T.J., 41, July, 1962, p. 1347.
5. Fox, A. G., and Li, Tingye, Modes in a Maser Interferometer with Curved and Tilted Mirrors, Proc. IEEE, 51, Jan., 1963, p. SO.
6. Goubau, G., Optical Relations for Coherent Wave Beams, in Electromagnetic Theory and Antennas, ed. Jordan, E. C., Pergamon Press, 1963, p. 907.
7. Kogelnik, H., Matching of Optical Modes, B.S.T.J., 42, Jan., 1964,p. 334.

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THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1965

8. Burch, J. M., Design of Resonators, p. 1187 in Quantum Electronics III, ed. Grivet, P., and Bloembergen, N., Columbia University Press, New York, 1964; Toraldo di Francia, G., On the Theory of Optical Resonators, Proe. Symp. Optical Masers, New York, 1963, p. 157, Polytechnic Press, Brooklyn, New York.
9. Kogelnik, H., and Yariv, A., Considerations of Noise and Schemes for Its Reduction in Laser Amplifiers, Proc. IEEE, 52, Feb., 1964,p. 165.
10. Gloge, D., Analysis of Fabry-Perot Laser-Resonators by Means of Scattering Matrices, Arch. El. U., 18, March, 1964, p. 197.
11. Collins, S. A., Analysis of Optical Resonators Involving Focusing Elements, Appl. Opt., 9, Nov., 1964, p. 1263.
12. Li, T., Dual Forms of the Gaussian Beam Chart, Appl. Opt., 9, Nov., 1964, p.1315.
13. Jenkins, F. A., and White, H. E., Fundamentals of Optics, 3rd ed., McGrawHill Book Company, New York, 1957, p. 89.
14. Born, M., and Wolf, E., Principles of Optics, Pergamon Press, New York, 1959, p, 420.
15. Pierce, J. R., Theory and Design of Electron Beams, D. Van Nostrand Company, Ine., New York, 1954, p. 194.
16. Herriott, D. R., Kogelnik, H., and Kompfner, R., Off-Axis Paths in Spherical Mirror Interferometers, Appl. Opt., S, April, 1964, p. 523.
17. Bertolotti, M., Matrix Representation of Geometrical Properties of Laser Cavities, Nuovo Cimento, June, 1964, p. 1242; O'Neill, E. L., Introduction to Statistical Optics, Addison-Wesley Publ. Co., Reading, Mass., 1963; Brouwer, W., Matrix Methods in Optical Instrument Design, Benjamin, New York, 1964.
18. Richards, P. I., Manual of Mathematical Physics, Pergamon Press, London, 1959, p. 312.
19. Tonks, L.hFilamentary Standing-Wave Pattern in a Solid-State Maser, J. Appl. P ys., June, 1962, p. 1980.
20. Gudzenko, L. I., Concentration of a Light Wave in a Weakly Inhomogeneous Dielectric, J. Exp. Theor. Phys. (USSR), #, April, 1963, p. 1298; Sov. Phys. JETP, 17, Oct., 1963, p. 875.
21. Kompfner, R., Talk at the Electron Device Research Conference, Salt Lake City, Utah, 1963.
22. Berreman, D. W., A Lens or Light Guide Using Convectively Distorted
Thermal Gradients in Gases, B.S.T.J., 49, July, 1964, p. 1469; A Gas Lens Using Unlike, Counter-Flowing Gases, B.S.T.J., 49, July, 1964, p. 1476. 23. Marcuse, D., and Miller, S. E., Analysis of a Tubular Gas Lens, B.S.T.J., 49, July, 1964, p. 1759. 24. Beck, A. C., Thermal Gas Lens Measurements, B.S.T.J., 49, July, 1964, p. 1818; Gas Mixture Lens Measurements, B.S.T.J., 49, July, 1964, p. 1821.
25. Born and Wolf, op. cit., p. 121. 26. Kogelnik, H., Modes in Optical Resonators; in Advances in Lasers, ed. Levine,
A. K., Dekker Publishers, New York. 27. Yariv, A., and Gordon, J. P., The Laser, Proc. IEEE, 51, Jan., 1963, p, 4. 28. Tien, P. K., Gordon, J. P., and Whinnery, J. R., Focusing of a Light Beam
of Hermite-Gaussian Distribution in Continuous and Periodic Lenslike Media, to be published in Proc. IEEE. 29. Marcatili, E. A. J., Modes in a Sequence of Thick Astigmatic Lens-Like Focusers, B.S.T.J., 4$, November, 1964, p. 2887. 30. Gloge, D., Fpcusing of Coherent Light Rays in a Space Dependent Dielectric;
Arch. El. -0.,18, July, 1964, p. 451.
31. Pierce, J. R., unpublished work. 32. Gordon, J. P., and Kogelnik, H., Equivalence Relations among Spherical
Mirror Optical Resonators; B.S.T.J., 4$, November, 1964, p. 2873. 33. Streifer, W., and Garno, H., On the Schmidt Expansion for Optical Resonator
Modes, Proc. Symp. on Quasi-Optics, Polytechnic Inst. of Brooklyn, June, 1964. 34. Born and Wolf, op. cit., p, 752. 35. Born and Wolf, op. eit., p. 126.