NATIONAL INSTITUTE OF STANDARDS &

TECENOLOGY

Research Infonnation Center

Gaithersburg, MD 2089©

[46

U.S. DEPARTMENT OF COMMERCE / National Bureau of Standards

The Theory of the Optical

Wedge Beam Splitter
NATIONAL BUREAU OF STANDARDS

The National Bureau of Standards' was established by an act of Congress March 3, 1901.

The Bureau's overall goal is to strengthen and advance the Nation's science and technology

and facilitate their effective application for public benefit. To this end, the Bureau conducts

research and provides: (I) a basis for the Nation's physical measurement system, (2) scientific and technological services for industry and government, (3) a technical basis for equity in trade,

and (4) technical services to promote public safety. The Bureau consists of the Institute for

Basic Standards, the Institute for Materials Research, the Institute for Applied Technology,

the Institute for Computer Sciences and Technology, and the Office for Information Programs.

THE INSTiTUTE FOR BASIC STANDARDS provides the central basis within the United

States of a complete and consistent system of physical measurement; coordinates that system

with measurement systems of other nations; and furnishes essential services leading to accurate and uniform physical measurements throughout the Nation's scientific community, industry,

and commerce. The Institute consists of a Center for Radiation Research, an Office of Measurement Services and the following divisions: Applied Mathematics — Electricity — Mechanics — Heat — Optical Physics — Nuclear

Sciences - — Applied Radiation ^ — Quantum Electronics " — Electromagnetics ' — Time

and Frequency ' — Laboratory Astrophysics " — Cryogenics

THE INSTITUTE FOR MATERIALS RESEARCH conducts materials research leading to improved methods of measurement, standards, and data on the properties of well-characterized

materials needed by industry, commerce, educational institutions, and Government; provides

advisory and research services to other Government agencies; and develops, produces, and

distributes standard reference materials. The Institute consists of the Office of Standard

Reference Materials and the following divisions:

Analytical Chemistry — Polymers — Metallurgy — Inorganic Materials — Reactor

Radiation — Physical Chemistry.

THE INSTITUTE FOR APPLIED TECHNOLOGY provides technical services to promote

the use of available technology and to facilitate technological innovation in industry and

Government; cooperates with public and private organizations leading to the development of

technological standards (including mandatory safety standards), codes and methods of test; and provides technical advice and services to Government agencies upon request. The Institute consists of a Center for Building Technology and the following divisions and offices: Engineering and Product Standards — Weights and Measures — Invention and Innovation — Product Evaluation Technology — Electronic Technology — Technical Analysis — Measurement Engineering — Structures, Materials, and Life Safety* — Building

Environment * — Technical Evaluation and Application * — Fire Technology.

THE INSTITUTE FOR COMPUTER SCIENCES AND TECHNOLOGY conducts research

and provides technical services designed to aid Government agencies in improving cost effec- tiveness in the conduct of their programs through the selection, acquisition, and effective

utilization of automatic data processing equipment; and serves as the principal focus within

the executive branch for the development of Federal standards for automatic data processing

equipment, techniques, and computer languages. The Institute consists of the following

divisions: Computer Services — Systems and Software — Computer Systems Engineering — Information Technology.

THE OFFICE FOR INFORMATION PROGRAMS promotes optimum dissemination and

accessibility of scientific information generated within NBS and other agencies of the Federal

Government; promotes the development of the National Standard Reference Data System and

a system of information analysis centers dealing with the broader aspects of the National

Measurement System; provides appropriate services to ensure that the NBS staff has optimum

accessibility to the scientific information of the world. The Office consists of the following

organizational units:

Office of Standard Reference Data — Office of Information Activities — Office of Technical

Publications — Library — Office of International Relations.

1 Headquarters and Laboratories at Gaithersburg, Maryland, unless otherwise noted; mailing address Washington, D.C. 20234.

' Part of the Center for Radiation Research.

' Located at Boulder, Colorado 80302. Part of the Center for Building Technology.
V 1 1974

y^e^ The Theory of the Optical

Wedge Beam Splitter

,2-

Yardley Beers

Institute for Basic Standards

tl , 5 . National Bureau of Standards

Boulder, Colorado 80302

U.S. DEPARTMENT OF COMMERCE, Frederick B. Dent, Secreiary

NATIONAL BUREAU OF STANDARDS, Richard W. Roberts, Director

Issued October 1974
Library of Congress Cataloging in Publication Data

Beers, Yar.lley, 1913- The theory of the optical wedge beam splitter.

(National Bureau of Standards Monograph 146)

Supt. of Docs. No.: C13.44:146

Includes bibliographical references.

1. Beam splitters. I. Title. II. Series: United States. National Bureau of Stan<lards. Monograph 146.

QC100.U556 No. 146 [QC373.B3] 389'.08s [535.5'028] 74-19135

National Bureau of Standards Monograph 146

Nat. Bur. Stand. (U.S.), Monogr. 146, 31 pages (Oct. 1974)

CODEN: NBSMA6

U.S. GOVERNMENT PRINTING OFFICE

WASHINGTON: 1974

For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, D.C. 20402

(Order by SD Catalog No. C13.44:146). Price 80 cents.
COMEMTS

1. lETRODUCTION - 1

2. PROPERTIES OF THE PLAEE INTERFACE BETWEEN TWO MEDIA 5

3. STEPWISE DETERMIMTION OF ATTENUATION FACTOR - - ____ 8

k. CALCULATION OF ANGLES - 9

5. PRACTICAL MEASUREMENTS WITH BEAM SPLITTERS 12

6. NUMERICAL RESULTS AND CONCLUSIONS - ik

7. ACKNOWLEDGMENTS 26

8. REFERENCES ---------- ________ 26

THE THEORY OF THE OPTICAL WEDGE BEAM SPLITTER

Yardley Beers

An optical wedge beam splitter consists of a prism of transparent

material with a very small apex angle, usually about one degree. If

a pencil beam of radiation is incident upon it, a portion enters the

material and undergoes a series of reflections at the surfaces. At

each reflection a refracted beam emerges from the material. This paper

gives the basic theory for computing the ratio of the intensity of the

incident beam to the intensity of any selected emerging beam and also

for computing the direction of the emerging beam, assuming that the wedge

angle, index of refraction, angle of incidence, and number of reflections are known.

The paper also gives the results of numerical calculations based

upon this theory for sample situations which are of interest. It is

shown that polarization effects can be minimized by the use of a small

wedge angle and by the proper selection of the angle of incidence. In

particular, it is shown that it is possible by the use of four reflections

and a wedge angle of one degree to obtain attenuation factors of about

400,000 (56db) , and that the effect of changes in polarization on the

attenuation factor can be held down to about one percent.

Key Words: Optical attenuation, optical beam splitter.

1. INTRODUCTION

A wedge beam splitter is a prism of transparent material such as glass with a very

small apex angle. If a narrow pencil beam is approximately at normal incidence to one face,

it pentrates the prism and undergoes multiple reflections. At each reflection a portion of

the beam is refracted and gives rise to an external beam, as illustrated in figure 1. The

various beams are identified by a number, which we shall call the "order" of the beam,

which is the number of reflections the radiation has encountered between incidence upon the

prism and emergence from it. It is to be noted that there are two first order beams, denoted by -1 and +1, respectively. The + and - signs are arbitrary labels used to distinguish between these beams using a notation which is in the vocabulary of any computer:
0

Figure 1. Propagation of a pencil beam in a wedge of small angle.

otherwise they have no significance. The -1 beam is reflected from the first surface.

Therefore its properties are independent of the wedge angle, and they are an even function

of the angle of incidence B: that is, they are the same for +B as for -B. The properties

of all the other beams depend upon the wedge angle, and, in general they do not have the

same values at -B as at +B.

There are two principal uses for such a device. One is to permit the monitoring of a

source of radiation which varies in the emitted power. In such a case, one beam is incident

upon a power meter, while another is used for some other purpose. The other is to provide

a fixed amount of attenuation, which can be determined by both experiment and theory, and

it is our purpose to indicate how the attenuation can be determined from theory.

In practice, the experimental value of the attenuation factor is more reliable in most

cases, since the theory, for simplicity, must make some assumptions such as the neglect of

surface scattering and imperfections in the glass. Also we are assuming that the incident

beam is large enough to be considered a portion of a plane wave but small enough such that

the various beams do not overlap and produce interference effects. Laser beams can not

always be accurately described as plane waves. Thus our assumption implies an approximation.

In fact it has been [1,2] shown that the mode structure of a laser beam can change on reflection. On the other hand, the theory gives guidance as to which beam to employ for any given

purpose, and what conditions to use to make the attenuation as nearly independent of variations in angle of incidence or the state of polarization as possible. Furthermore the theory

serves as a check upon experiment: any large discrepancy may be the result of scattered light.

2
It is desirable to make the attenuation factor as nearly independent of the state of

polarization as possible, as often the state of polarization is unknown, or we may wish to

compare sources of different polarizations. Also, there has come to our attention the case

of a laser whose power output was reasonably constant but whose state of polarization was

subject to large fluctuations. We shall show that the attenuation ratio depends very little

upon the state of polarization if the wedge angle is very small and if approximately normal

incidence is used at the first surface, and under these conditions errors due to polarization effects usually are negligible.

In principle, polarization effects can be eliminated by using a symmetrical arrangement

of two identical beam splitters in cascade with their apex lines perpendicular to one

another. Such an arrangement in a particular configuration is described in a U.S. Patent [3],

and is incorporated in a commercial device. However, such an arrangement involves a number

of obvious complications. We, therefore, are focusing our attention on the use of a single

beam splitter and in finding under what conditions the errors are negligible.

We shall refer to the points where the various beams emerge, (such as b, c, d, e

in figure 1) as "ports". If we assume as an approximation, that the energy left in the

internal beam after the m'th reflection is negligible, the splitter can be considered as

the analog of a 2m port microwave junction. The significance of the factor of 2 will become

apparent later when we consider in detail the properties of the plane interface between two

transparent media.

In developing the theory it is convenient to keep in mind the limiting situation when

the wedge angle goes to zero: i.e., when the sides or the glass are parallel, as Illustrated

in figure 2. All of the beams emerging from the far side, the even orders, are all parallel

INC. -1 •! 3 5

Figure 2. Propagation of a pencil beam in

transparent material with parallel plane

sides.

3
to the incident beam INC but displaced from it by an amount which depends upon the angle of

incidence, the thickness, and the index of refraction. Under practical conditions this displacement is small, and the effect of changing the angle of incidence is minor. The odd

order beams are parallel to each other but deviate from the incident beam by twice the angle

of incidence. •

The reflectivity R is defined as the ratio of the power per unit area of the interface in the reflected beam to the power per unit area of the interface of the incident beam.

For beams of finite cross section this is the same as the ratio of the total power in the

reflected beam to the total power in the incident beam. Similiarly, the transmissivity T is

the ratio of the total power in the refracted beam to that in the incident beam. In general,

these are functions of the angle of incidence B, the index of refraction N, and the state

of polarization. Since the angle of refraction C is given by Snell's law in terms of B and

N, it may be used as an independent variable instead of N.

Let us define the attenuation factor F as the ratio of the power in the incident beam

to the power in the m'th order beam. Assuming that the sides are parallel, that the glass

is homogeneous, and that scattering and absorption are negligible, for the -1 beam,

F = 1/R, (1)

and for any other order m

F = 1/(tV). (2)

In eq (2) it is assumed, as will be shown later, that T has the same value for the incident

beam as for the emerging beam. For glass of low index of refraction (e.g., 1.5) R is

about 1/25 while for high index (e.g. , 1.75) it is about 1/13. For rough calculations T may be taken to be approximately unity. In terms of decibels, the attenuation

factor is about 14 db per reflection for low index glass and about 11 db per reflection for

high index glass. These figures can serve as a rough guide in preliminary design.

Because of the parallelism of the beams, a piece of glass with parallel sides is not

generally useful as a beam splitter, since the beams are insufficiently separated to be distinguished by most detectors. Therefore, a practical splitter requires a small but finite

angle between the sides allowing the adjacent beams to diverge as shown in figure 1. Later

it will be shown that successive angles of incidence for beams inside the glass on the same

4
side (going from left to right in figure 1) increase by 2A, where A is the prism angle.

For angles small enough such that the sine of an angle can be replaced by the angle in

radians, it follows that successive emerging beams in air deviate by an angle 2NA, approximately. However, because of the addition of successive increments of 2A, the angle of

incidence ultimately gets so large that this approximation does not hold. In fact, even with

modest values of A, it may get so large that it exceeds the critical angle, and then the

internal beams are totally reflected, R and T are independent of the state of polarization for normal incidence, and their variation with the polarization increases with the angle

of incidence. Therefore, in general, polarization effects are more serious with the higher

order beams.

As far as the forward (even order) beams are concerned, rotation of the beam splitter

has little effect on their directions if the wedge angle is small: the beam splitter, other

than giving rise to this divergence of the beams acts approximately like a piece of glass

with parallel sides., However, the position of any backward (odd order) beam may be shifted

to any arbitrary direction by rotating the prism. In general, as suggested by figure 1, the

higher order beams are deviated towards the base of the prism and away from the apex line,

under conditions which normally prevail in practice.

2. PR0PE;RTIES of the plane INTERFACE BETWEEN TWO MEDIA

The calculation of the attenuation factor F for any arbitrary beam requires knowledge

of the coefficients R and T at each encounter between the radiation and the interface

between glass and air. And to evaluate R and T we must determine, using the laws of

geometrical optics, the angle of incidence at each encounter.

The calculation of R and T for the plane interface between two homogeneous isotropic

media is discussed in most books in optics and is well known. For convenience we summarize

the results here. In spite of all that has been written on the subject, there are also

some details which we shall supply that do not seem to be well known.

In figure 3 ft and fs denote the boundary rays of a pencil beam incident on the interface in medium 1 (which for the moment we shall assume to be air) at an angle of incidence B.

For brevity we shall denote this beam as f, and we shall use an analogous notation for the

5
9'

Figure 3. Reflection and refraction at

plane interface.

y h

Other beams. This gives rise to a reflected beam g at an angle of reflection B and a

refracted beam h at an angle of refraction C, where, of course, from Snell's Law,

where N is the index of refraction of the second medium (glass) relative to the first (air).

If we send the light on to the interface backwards along the direction of g , we

produce a beam backwards along f. The beam h is not produced, but we produce a different

refracted beam j at an angle C measured in the other direction. Similarly if the incident light is sent backwards along the beam direction h, the beam j is excited by reflection and f by refraction, while incident light along j excites beams h and g.

Thus we have a situation analogous to a hybrid circuit in electrical circuit theory, the

best known special case of which is the microwave "Magic Tee" junction.

In general for calculating the transmissivity T and the reflectivity R, we must

distinguish between the case where the electric vector of the incident radiation is perpindicular to the plane of incidence (denoted by the subscript e) and when it is parallel (denoted by the subscript a). As can be found in many textbooks, [4]

sin B = N sin C, (3)

R sin (B-C)/sin (B+C)

,

tan^ (B-C) /tan^ (B+C)

(4) e

R (5) a

6
= (sin 2B)(sin 2C)/sin^ (B+C) , (6)

and

T = (sin 2B)(sin 2C)/(sin^ (B+C) cos^ (B-C) , (7)

Si

In principle, by the use of Snell's Law, eq (3), these coefficients can be given as

functions of B and N rather than of B and C, but the form of these equations is not

convenient.

When B and C are small, the sines and tangents can be approximated by the appropriate angles in radians, and we note, approximately, R =R , T = T , and polarization

e a e a

effects disappear.

However, at normal incidence itself, B = 0, C = 0, and eqs (4-7) become indeterminant

.

In such a case it can be shown that [4]

R = (N-1)^/(N+1)^ (8)

and

T = 4N/(N+1)^ . (9)

Also, for all cases, in accordance with conservation of energy,

R + T = 1. (10)

It is to be noted, that in eqs (4-7), R and T are functions only of B+C and the

magnitude (not the sign) of B-C. Also in eqs (8) and (9), R and T are invariant with

the replacement of N by 1/N, Therefore, for a beam reflected in medium 1 at an angle B,

R has the same value as for a beam reflected in medium 2 with an angle C, where B and C

are related by eq (3). Similarly, for a beam incident at an angle B in medium 1 and

transmitted into medium 2, T has the same value for a wave incident at an angle C in

medium 2 and transmitted into medium 1.

Therefore, in principle, all of the R and T coefficients involved in a high order

beam attenuation factor F can be determined experimentally. For example, if we want to

know R and T at the point f in figure 1, we send in a calibrated beam backwards

along the direction that the third order beam had emerged. This generates a reflected beam

at f (which is not shown in the diagram). If the power in this beam is measured by a

suitable calibrated detector, and if this power is divided by that in the incident beam, the
ratio is R, and this is the same for the internally reflected beams shown in the diagram.

At the same time the transmission coefficient at f for the emerging third order beam is

given by 1 - R by eq (10)

3. STEPWISE DETERMINATION OF ATTENUATION FACTOR

For a finite wedge angle A, eq (2) must be replaced by

F = 1/(T^I^R^R2....R^_^TJ , (11)

where T. is the transmission coefficient at the point of entrance of the incident beam, T

1 ^ ' m

is the transmission coefficient at the point of exit of the m'th order beam, and R^ is

the reflection coefficient at the point of emergence of the k'th order beam. These coefficients, of course, are assumed to be evaluated at the appropriate respective angles of

incidence.

Now let us suppose that the incident beam had been backwards along the direction of the

original emergent beam of order m. From the discussion of the properties of a single interface, it can be inferred that a beam would emerge in the direction of the original incident

beam. Since the order of the factors in eq (11) is unimportant, the attenuation factor F

is the same as for the original configuration. In other words, in the language of circuit

theory, the beam splitter is a reciprocal device.

2 Next let us multiply numerator and denominator of eq (11) by T^i > where m' is intermediate between 0 and m, and group the factors as follows:

F = [ 1 it) m -1 m

In principle, each quantity in brackets can be determined experimentally. The first

quantity is just the attenuation factor of the m' order beam. The last factor is the

attenuation factor which applies when the incident light is sent backwards along the direction of the m order beam, and when the observation is made on a beam emerging from m'

port (but leaving the surface with an opposite angle to the normal). The middle quantitity

can be found by sending a beam backwards along the original m' beam direction and observing

its -1 reflection. This gives R , directly, and T , can be found from it by eq (10).

m

m

T |R i,i*.oR iT

m m +1 m-1 m

(12)

8
Therefore if the original factor is too large to be measured directly because of the

limited range of the instruments, it can be broken down into the product of quantities which

can be brought within range, and yet it is not necessary to measure individually every coefficient, as had been suggested earlier.

4. CALCULATION OF ANGLES

In applying the formulas that have been given for calculating the individual R and T

coefficients and thus for calculating the attenuation factor, it remains for us to develop

formulas for the angles of incidence. For this purpose we refer to figure 4, which duplicates the first few beams of figure 1 with the same notation but, for clarity, with an expanded scale and with the angles made much larger.

By applying the theorem that the sum of the angles of a triangle is 180° to the triangle

abc, we find that

Ej^ = A + C . (13)

By applying this theorem to the triangle bed , we find that

= 2E^ -C

= C + 2A.

Thus, exceeds C by 2A.

We can see by inspection that if we were to repeat this argument to the triangles cde

and def we can show that

G2 = Gj^ + 2A ,

and, in general,

G = C + 2LA, (14)

where L is any integer.

By a similar argument we can show that

Ej^ = 2(L - 1) A + Ej^ (15)

= C + (2L - 1)A . (16)

It is important to point out that there is an implied sign convention with regard to

the angles B, C, E^ , and G^. These angles are all positive as shown in figures 1 and 4. If

the incident beam associated with any of these beams should be on the opposite side of the

respective normal to the interface, it should be considered as a negative angle.

9
0

Figure 4. Details of propagation in a wedge.

For negative values of the angle of incidence of the entering beam B, some of these

angles may become negative, and in extreme cases the light may be propagated internally towards the apex of the prism. However, ultimately, when L becomes large enough, there are

enough increments of 2A to make E and G. positive, and the light is reversed in direc1j Li

tion. Then the higher order beams emerge directed towards the base. In the usual situation

prevalent under practical conditions, all even order beams emerge directed towards the plane

of the base, and, the odd order ones emerge in the sequence shown in figure 1.

Since E and G are smaller for negative values of B than for positive ones , the

Jul 1j minimum polarization effects occur at small negative values of B, as to be shown later by

the graphs which have been based upon calculations from the theory. Therefore, especially,

in working with higher order beams, one should avoid using a positive value of B.

It is convenient to record the relationship between the beam order m and the index L.

For odd orders

,

m = 2L - 1, (17)

and for even orders, m = 2(L - 1). (18)

The angles of refraction of the emerging beams, F^ and may be calculated by

Snell's law from E^ and G^ respectively. In principle, it is possible to derive expressions for explicit formulas for these in terms of A, B, and N, but if the exact form

10
of Snell's law is used, these formulas become very cumbersom^, and we shall not derive them.

Instead, for convenience, we shall assume that the angles are small enough so that their

sines may be approximated by the angles in radians. Under practical conditions, this approximation is excellent, at least for beams of low order. On the other hand, with numerical

computations carried out by a computer it is entirely practical to use the exact form of the

law throughout, and such has been done in the preparation of the graphs to be presented

later.

With this approximation, it can be shown that

= B + (2L -1)NA, (19)

and

H = B + 2LNA . (20)

Li

therefore „ „ „„.

-

- 1 =

and

H - H , = 2NA .

Li Jj " L

The deviation of the -1 beam from the incident beam is given by

D = 2B . (21)

For other odd orders it is given by

D = B + H

J-i

= 2B + 2LNA. (22)

For even orders it is given by

D = F - B - A

i-i

= (2L - 1)NA - A , (23)

which is independent of the angle of incidence B. Experimentally, these even order beams

are. found to shift in position slightly with changes in B. These shifts are due to the

breakdown of the approximation that the sines of angles be replaced by the angles.

The well known case of the minimum deviation of the zero order beam occurs when C = Ej^,

giving

B = -NA/2 . (24)

11
5 . PRACTICAL MEASUREMENTS WITH BEAM SPLITTERS

The measurement of the attenuation factor for the beam of order which now we shall

denote by F^, requires putting a detector first in the incident beam and then in the M'th

order beam. In general, the power of the beam can not be expected to remain constant,

although, if it is reasonably constant, the effects of instabilities can be eliminated to a

large extent by alternating the observations and taking the average. However, the effect

can be eliminated by observing two beams simultaneously, but neither of these obviously can

be the incident beam, as a detector placed in it prevents the radiation in reaching the beam

splitter. Therefore, in many applications which are too obvious to enumerate, one interested

in the ratio of two other beams m and m'. Therefore, it is convenient to define the beam

splitter ratio S , as such a ratio, and mm '

S , = F /F , , (25) mm m m

which can be calculated theoretically by application of eq (11).

In order to apply the theory, it is necessary of course, to have values of N and A.

The measurement of these quantities is contained in a standard course in elementary optics.

However, when the wedge angle A becomes very small the standard methods are difficult or

impossible to apply.

'The standard method of measuring the angle A is to mount the prism on a spectrometer

with an angular scale and to use with it a telescope with a Gauss eyepiece. With this,

one face is made perpendicular to the axis of the telescope, and then the prism is rotated

through an angle until the other face is perpendicular. Then A is just 180 degrees minus

the angle through which the prism is rotated. The modern version of this method is to use

a laser beam instead of a telescope and to adjust the positions of the prism to produce beams

that are reflected along the incident beam. When the angle A is large, there is no difficulty in recognizing when the beam is reflected off the front face (i.e., the -1 beam).

However, there are also two positions where the +1 beam is also reflected along the incident

beam, and when A is small it may be difficult to identify which beam is which. Here the

laser method has a distinct advantage in identifying the beams, since even with a low powered

laser it is usually possible to observe the second and third order beams, and, since under

practical conditions, these are in the sequence shown in figure 1, they may be used to dis-
tinguish between the two first order beams. At any rate, this method of determining A is

likely to have low precision, since it is necessary to take the small difference between

two large angles. In practice, it may be better to obtain A indirectly as discussed below and to use the direct value as a check.

The standard method for measuring the index of refraction of a prism of large angle A

which has previously been determined, is to measure the angle of minimum deviation D and

with the formula for it to calculate N. However, as we have seen, with small values of A

the angle of deviation is essentially independent of the angle of incidence, and the formul

can not be applied.

As a more practical procedure, it is suggested that the index of refraction N be determined by measuring the beam splitter ratio ^, which is independent of the angle A.

To reduce the effect of inhomogenieties , measurements should be made on both faces and the

average computed. By application of eqs(25), (1 ), and (11) it can be seen that

However, if the wedge angle A is small and if nearly normal incidence is employed, we may

drop the subscripts on the right and by use of eq (10 ) see that

Sq_^ = (1-R)^/R . (27)

The reflection coefficient R is found by inserting the measured value of ^ and

solving, and then the index of refraction is found by substituting this value of R into

eq ( 8 ) and solving.

The quantity NA can be determined by measuring the angular separation between two

adjacent beams (which, in radians, is 2NA) . For this purpose the most convenient pair are

-1 and +1. With a laser source, these can make bright spots on a screen, which should be

sufficiently distant to allow an accurate measurement of the separation. Finally the value

of A can be obtained by dividing the previously determined value of N into this result.

In determining a high order attenuation factor or beam splitter' ratio experimentally,

it is desirable to make observations of the weak beam with the detector at various distances

from the beam splitter because of possible scattering due to dirt and imperfections of the

13
splitter. The intensity of the true beam is independent of the distance while scattered

light varies approximately with the inverse square of the distance „ Therefore correction

for effects of scattering can be accomplished by extrapolating the observed ratio quadratically to infinite distance.

60 NUMERICAL RESULTS AND CONCLUSIONS

The equations that have been presented in the earlier sections have been incorporated

in a computer program which, given the wedge angle A, the index of refraction N, the angle

of incidence B, and the order of the reflection m, calculates the angular position and the

attenuation factor of the emerging beam for both polarization of the incident beam.

Illustrations of attenuation factor calculations are contained in graphical form in

figures 5 through 14, In the upper left portion of the first of these, figure 5, is shown

the attenuation factor for the perpendicular polarization as a function of the angle of incidence for a range of ± 5° from the maximum for the -1 order, for A = 1"*

, and for N = 1.75

the value of the attenuation factor is normalized to the maximum value. The actual value at

the maximum is given on the left axis of the graph, and the actual value at any other angle

can be found by multiplying it by the ordinate of the graph for that angle.

The graph in the lower left corner gives the ratio of the attenuation factor for the

parallel polarization to that for the perpendicular polarization for the same parameters as

the graph above. It appears that this has a minimum value at the same angle as the maximum

in the graph above. Such a correspondence appears to hold with all of the other pairs of

graphs which follow, but the equations are too complicated to make it practical to prove

whether these stationary values occur at exactly the same angle or not, but within the accuracy of the graphs they appear to occur at the same angle.

The graph of the ratio of the two attenuation factors is useful for evaluating the

limit of error which results if the attenuation factor is measured or calculated for one

polarization and then the splitter used with a beam of unknown polarization. (Incidentally,

one laser used by our colleagues has a nearly constant power output, but its state of

polarization is subject to serious fluctuations.) This error is given by taking the ordinate from the graph and subtracting unity from it. The perpendicular polarization is of

greater interest than the parallel one since some of the better engineered lasers have beams

14
RATIO PARALLEL TO PERPENDICULAR ATTENUATION PERPENDICULAR SCALE = 1.34507.001

S S 3

zo

tUUtT H:>t:l<

RATIO PARALLEL TO PER?ENDICLTj\R ATTENUATION PERPENDICULAR SCALE = 2.S0145»00I

m

^ -

5

Xz

«/«l/T< M:|l:||

«/«I/T< H II H

Figure 5,

15
that are vertically polarized and, if they are used with beam splitters that are oriented to

spread the beams out in a horizontal distribution, this is the relevant case. Therefore

the attenuation factor for this polarization was chosen for the ordinate in the left hand

graph.

The two graphs on the right side of this page (figure 5) are analogous respectively to

the two on the left except that the index of refraction had been changed from 1.75 to 1.5.

The graphs on the next page, figure 6, pertain to the same set of conditions (order -1)

as the ones in corresponding positions in figure 5 except that the wedge angle A has been

changed from 1° to 2° .

The remaining pairs of figures are arranged in the same manner except that succeeding

pairs pertain respectively to orders m = + 1, 2, 3, and 4.

In view of the fact the reflection for the -1 order is independent of A, figures 5

and 5 are identical except for the labeling.

From the examination of these graphs, it is possible to draw several useful conclusions:

(1) As we have said before, the maximum attenuation factor and minimum polarization

effect angle occur at essentially the same angle of incidence. In order to minimize the

polarization error and the error due to uncertainties in the angle of incidence, attempt

should be made to use this angle of incidence in experimental situations.

(2) As suggested by the previous theory, this optimum angle of incidence is negative in

sign and becomes larger as the beam order increases.

(3) The principal effects of increasing the index of refraction N is to lower the

attenuation factor for the beam of any given order and to widen the angular separation between

adjacent beams.

(4) Increasing the wedge angle A has little effect upon the attenuation factor of any

given beam but it makes the polarization effects more serious. Also the optimum artgle of

incidence is shifted to larger negative value.

(5) By restricting the wedge angle A to 1° or less and by the use of the optimum

angle of incidence it is possible to produce attenuation factors of the order of 400,000

(56 db) with polarization errors of just slightly more than one percent. It is doubtful

whether such attenuation factors can be measured or calculated to such an accuracy. Therefore, with attenuation factors of this size it seems unnecessary to go to the complication

of two symmetrically place beam splitters to cancel out polarization effects if proper care

is taken.

16
RATIO PARALLEL TO PERPENDICULAR ATTENUATION PERPENDICULAR SCALE < l.34S0T<001

a s a s a

RATIO PARALLEL TO PERPENDICULAR

ATTENUATION PERPENDICULAR SCALE ' Z.50I4S«001

a s a : ix

O m

mntm m.u-.u mntm M:ii:ii

Figure 6.

17
RATIO PARAU-EL TO PERPENDICULAR V

ATTENUATION PERPENDICULAR SCALE = I. 57022.001

a s » s s

K/tl/n N:ll.'lt

1 A

H/ mntfit M:|l:l<

RATIO PARALLEL TO PERPENDICULAR

ATTENUATION PERPENDICULAR SCALE = 2.71449.001

s s s s bs

Figure 7,

18
ATTENUATION PERPENDICULAR SCALE = I 5710G.0OI RATIO PARALLEL TO PERTENDICULAR

o

/

—

\

\

»/tuU II II II

M/ll/n M:llrn

RATIO PARALLEL TO PERPENDICULAR

ATTENUATION PERPENDICULAR SCALE = 2.715I2»001

1

II/II/T4 II It II

mnt/u Mil:

Figure 8.

19
RATIO PARALLEL TO PCRPENDIOTLAR

mnt/H Miiiiif

ATTENUATION PERPCNDICULAR SCALE ' 2. llUOtOtZ

s s !« : s

•(/•l/n WrlliW

X

7

RATIO PARALLEL TO PERPENDICl/LAR

s = is S 3

ATTENUATION PERPENDICULAR SCALE : «. 7B7J9<002

S it » : 3

Figure 9.

20
RATIO PARALLEL TO PERFENDIOn^R

9 = is at 3

ATTCNUATtON WIPCNOICULAR SCALE < 2. Ilt45«0lt

.9 it ii : a

St

I

Ml/M MiMilt

RATIO PARALLEL TO PERPENDICULAR

* 2 5 » 3 ix

/

ATTENUATION PERPENDICULAR SCALE < t. T(IS5<e02

s it ii s ix

S5

mnt/n w:ii:ii mm/u (({11 III

Figure 10.

21
«ATIO PARAUEX TO PERPENDICULAR

m

. 3

ATTENUATION PERPENDICULAR SCALE = 2.BJ654.003

z o

*/>l/n N:ll:ll U/H/ft M il If

RATIO PARALLEL TO PERPENDICULAR

m

ATTENUATION PERPENDICULAR SCALE = I.6958W004

o

lt/tl/T4 M:|I:K

Figure 11.

22
RATIO PARALLEL TO PERPENDICULAR

o r .- .- r r ATTENUATION PERPENDICULAR SCALE • Z.SZ4l2<a05

* lt/tl/T4 N:llin

RATIO PARALLEL TO PERPENDICULAR ATTENUATION PERPENDICULAR SCALE I.C(t09.«t4

^ii = s 3 3 is j.a < 'a : s

Figure 12.

23
RATIO PARAUEL TO PERPENDICULAR

ATTENUATION PERPENDICULAR SCALE = 3.60429«004

8 ill 's is

o

M/lt/T4 It fi 15

RATIO PARALLEL TO PERPENDICULAR

m

tt/ll/T4 M,||:|T

ATTENUATION PERPENDICULAR SCALE = 4.25069.005

is a is a

U/t\/U II It It

Figure 13.

24
RATIO PARALLEL TO PERPENDICULAR ATTENUATION PERPENDICULAR SCALE - 5. 7S597>004

ATTENUATION PERPENDICULAR SCALE = 4.179494005 RATIO PARALLEL TO PERPENDICULAR „ _ - 5 s a

Figure 14,

25
7. ACKNOWLEDGMElttS

The author wishes to express his thanks to Eric Johnson for aid in computer programming,

especially modifying the original program written by the author to print out the graphs

which are contained in figures 5-14.

The author wishes to thank his colleagues, Bruce L. Danielson, E. D. West, T. W. Russell,

G. E. Charaberlin, W. E. Case. L. B. Schmidt, D. A. Jennings, and Carla Selby for helpful discussions and for correcting errors in portions of the manuscript.

This work was supported in part by the Department of Defense, Calibration Coordination

Group 74-92, U. S. Army Missile Command, Redstone Arsenal, Alabama 35809.

' 8 . REFERENCES

1. Yu A. Kurchatov and I. A. Malinov, Opt. Spektrosk, 31. 283 (1971); [Opt. Spectroc. 31., 151

(1971)].

2. A. M. Belskii and A. P. Khapalyuk, Opt. Spektrosk. 35, 117 (1973) [Opt. Spectrosc. 35,

67, 1973)],

3. P. T. Gates, U.S. Patent 3,463,575, "Light Beam Sampling Apparatus", August 26, 1969.

4. See for example, M. Born and E. Wolf, "Principles of Optics", Sec. 1.5.3 3d. Ed.,

Pergamon Press, N.Y. (1965).

26
NBS-1 14A (REV. 7-73)

U.S. DEPT. OF COMM.

RIRI in(^RAPHir° DATA

SHEET

1. I'UHI.K A l lON OK KI' I'OK T NO.

NBS Monograph 146

2, Ciov't Acfcs.sion

i\ { I

.

3. Kcc ipicni 's Ai cession No.

4. 11 TI.I-; AND SUH TI ri.i':

The Theory of the Optical Wedge Beam Splitter

5. I'libl icat it>n Date- October 1974

6. Performing Organization < ode

271.00

7. AU rilOK(S)

Yardley Beers

8» l*(.'rft>r m in^ Or^tWi. Kcpt^rl No.

9. PKKI- ORMINC ORCIAN l/.A I ION NAMl- AND AHnKlvSS

NATIONAL BUREAU OF STANDARDS Boulder Labs

DEPARTMENT OF COMMERCE ' WASHINGTON, D.C. 20234

10. Project/Task/Work Unit No. 2712122

11. Contract /Cirant No.

12. Sponsoring Organization Name and C omplctc- Addres.s (Street, City, State, ZIP)

Same as §^ -

13. Type of Keport & Period

( ovcred

Final

14. Sponsoring Agency C,c)de

15. SUPPl.l-MlCN'IAKY NOII-S

Library of Congress Catalog Card Number: 74-19135

16. All.STKA(; r (A 200-word or less factual summary ol most signHicant information. If document includes a significant

bibliof^rapfiy or literature survey, mention it here.)

An optical wedge beam splitter consists of a prism of transparent material

with a very small apex angle, usually about one degree. If a pencil beam of

radiation is incident upon it, a portion enters the material and undergoes a

series of reflections at the surfaces. At each reflection a refracted beam

emerges from the material. This paper gives the basic theory for computing

the ratio of the intensity of the incident beam to the intensity of any selected

emerging beam and also for computing the direction of the emerging beam, assuming

that the wedge angle, index of refraction, angle of incidence, and number of

reflections are known.

The paper also gives the results of numerical calculations based upon this

theory for sample situations which are of interest. It is shown that polarization

effects can be minimized by the use of a small wedge angle and be the proper

selection of the angle of incidence. In particular, it is shown that it is

possible by the? use of four reflections and a wedge angle of one degree to

obtain attenuation factors of about 400,000 (56 db), and that the effect of

changes in polarization on the attenuation factor can be held down to about one

percent.

17. KI^Y WOUnS (six to twelve entries; alphabetical order; capitalize only the first letter of the first key word unless a proper name; separated by semicolons

)

Optical attenuation; optical beam splitter.

18. AVAlI.AHIl.l TY [X UnUmitcd

I

' l-'or Official nistrilmt ion. Ho Not Kelease to NTIS

J^' Oriler 1' roni Sup. ol Hoc, U.S. dovernmcnt I'riiitiiii; Office

Washington, O.C . 2()U)2, SI1 ( at. No. (13 . 44 /14b

Order 1' rom National Tecliiiical Inform.itioii, Service (N Tl.S) Springfield, Virginia J21'S1

19. SIX.IIR! TY CLASS

(THIS RKPORD

UNCI. Assn-ii:n

20. si-;( ui<n Y ( I. ASS

( riiis PACK)

lINCl.ASSllll-:!')

21. NO. OP PAGHS

31

22. Price

80 cents

USCOMM-DC 29042-P74

U.S. GOVERNMENT PRINTING OFFICE: 1974—583-012:19

NBS TECHNICAL PUBLICATIONS

PERIODICALS JOURNAL OF RESEARCH reports National Bureau

of Standards research and development in physics,

mathematics, and chemistry. Comprehensive scientific papers pive complete details of the work, includinp:

laboratory data, experimental procedures, and theoretical and mathematical analyses. Illustrated with photographs, drawings, and charts. Includes listings of other

NBS papers as issued.

Published in two sections, available separately:

• Physics and Chemistry (Section A)

Papers of interest primarily to scientists working in these fields. This section covers a broad range of physical and chemical research, with major emphasis on standards of physical measurement, fundamental constants, and properties of matter. Issued six times a

year. Annual subscription: Domestic, $17.00; Foreign,

$21.25.

• Mathematical Sciences (Section B)

Studies and compilations designed mainly for the mathematician and theoretical physicist. Topics in mathematical statistics, theory of experiment design, numerical analysis, theoretical physics and chemistry, logical

design and programming of computers and computer

systems. Short numerical tables. Issued quarterly. Annual subscription: Domestic, $9.00; Foreign, $11.25.

DIMENSIONS/NBS (formerly Technical News Bulletin) —This monthly magazine is published to inform

scientists, engineers, businessmen, industry, teachers,

students, and consumers of the latest advances in science and technology, with primary emphasis on the work at NBS.

DIMENSIONS/NBS highlights and reviews such

issues as energy research, fire protection, building

technology, metric conversion, pollution abatement,

health and safety, and consumer product performance.

In addition, DIMENSIONS/NBS reports the results of Bureau programs in measurement standards and techniques, properties of matter and materials, engineeringstandards and services, instrumentation, and automatic

data processing.

Annual subscription: Domestic, $6.50; Foreign, $8.25.

NONPERIODICALS

Monographs—Major contributions to the technical liter- ature on various subjects related to the Bureau's scientific and technical activities. Handbooks—Recommended codes of engineering and

industrial practice (including safety codes) developed

in cooperation with interested industries, professional

organizations, and regulatory bodies. Special Publications—Include proceedings of high-level

national and international conferences sponsored by

NBS, precision measurement and calibration volumes,

NBS annual reports, and other special publications

appropriate to this grouping such as wall charts and

bibliographies.

Applied Mathematics Series—Mathematical tables,

manuals, and studies of special interest to physicists,

engineers, chemists, biologists, mathematicians, computer programmers, and others engaged in scientific and technical work.

National Standard Reference Data Series—Provides

quantitative data on the physical and chemical propei'-

ties of matei'ials, compiled from the world's literature and critically evaluated. Developed under a world-wide

program coordinated by NBS. Program under authority

of National Standard "Data Act (Public Law 90-,396). See also Section 1.2.3.

Building Science Series—Disseminates technical infor- mation developed at the Bureau on building materials,

components, systems, and whole structures. The series presents research results, test methods, and performance criteria related to the structural and environmental functions and the durability and safety characteristics of building elements and systems.

Technical Notes—Studies or reports which are complete

in themselves but restrictive in their treatment of a

subject. Analogous to monographs but not so comprehensive in scope or definitive in treatment of the subject area. Often serve as a vehicle for final reports of work performed at NBS under the sponsorship of other

government agencies.

Voluntary Product Standards—Developed under pro- cedures published by the Department of Commerce in Part 10, Title 15, of the Code of Federal Regulations.

The purpose of the standards is to establish nationally

recognized requii'ements for i)roducts, and to provide

all concei ned interests with a basis for common understanding of the characteristics of the products. The

National Bureau of Standards a<iministers the Voluntary Product Standards jirograin as a supplement to the activities of the private sector standardizing

organizations.

Federal Information Processing Standard.^ Publication.s (FIPS I'UBS)—Publications in this series collectively constitute the Federal Information Piocessing Standards Register. The purpose of the Register is to serve

as the official source of information in the Federal Government regarding standards issued by NBS luirsuant

to the Federal Propei-ty and Administrative Services

Act of 1949 as amended. Public Law 89-306 (79 Stat. 1127), and as implemented by F^xecutive Order 11717

(38 FR 12315, dated May 11, 1973) and Part (i of Title

15 CFR (Code of Federal Regulations). FIPS PUBS

will include approved Federal information processing

standards information of general interest, and a complete index of relevant standards publications.

Consumer Information Series—Practical information,

based on NBS research and experience, covering areas

of interest to the consumer. Easily understandable

language and illustrations ))rovide useful background

knowledge for shopping in today's technological

marketplace.

NBS Interagency Reports—A special series of interim

or final reports on work performed by NBS for outside sponsors (both government and non-government). In

general, initial distribution is handled by the sponsor;

public distribution is by the National Technical Infor- mation Service (Springfield, Va. 22151) in paper copy

or microfiche form.

Order NBS publications (except Bibliographic Subscription Services) from: Superintendent of Documents,

Government Printing Office, Washington, D.C. 20402.

BIBLIOGRAPHIC SUBSCRIPTION SERVICES

The following current-awareness and literature-survey

bibliographies are issued periodically by the Bureau:

Cryogenic Data Center Current Awareness Service

(Publications and Reports of Interest in Cryogenics).

A literature survey issued weekly. Annual subscription: Domestic, $20.00; foreign, $25.00.

Liquefied Natural Gas. A literature survey issued quarterly. Annual subscription: $20.00.

Superconducting Devices and Materials. A literature survey issued quarterly. Annual subscription: $20.00.

Send subscription orders and remittances for the preceding bibliographic services to the U.S. Department

of Commerce, National Technical Information Service, Springfield, Va. 22151. Electromagnetic Metrology Current Awareness Service

(Abstracts of Selected Articles on Measurement

Techniques and Standards of Electromagnetic Quantities from D-C to Millimeter-Wave Frequencies).

Issued monthly. Annual subscription: $100.00 (Special rates for multi-subscriptions). Send subscription

order and remittance to the Electromagnetic Metrology Information Center, Electromagnetics Division,

National Bureau of Standards, Boulder, Colo. 80302.
U.S. DEPARTMENT OF COMMERCE

National Bureau of Standards

Washington. D.C. 20234

OFFICIAL BUSINESS

Penalty for Private Use, S30a

POSTAGE AND FEES PAID

U.S. DEPARTMENT OF COMMERCE

COM-215