Influence of Interference Fringes of Equal Inclination on the Reflection of Laser Beams from Plane Parallel Plates Franz Hillenkamp The coefficient of reflection of plane parallel plate beam splitters is very often taken as twice that of a single surface reflection. It is shown that this introduces appreciable errors in the majority of cases, be- cause of the interference fringes of equal inclination. Formulas are derived for the angular dependence of the coefficient of reflection as well as for its average value. The angular distance between adjacent fringes has been evaluated and represented graphically. It is finally demonstrated that the influence of incomplete interference due to the lateral displacement of the reflected beam can be neglected if the beam splitter is followed by an integrating sphere or a similar element. Introduction In laser applications, plane parallel glass or quartz flats are commonly used as beam splitters. The re- flected portion of the light is then taken as a measure of the intensity as well as the energy of the main beam. These flats should be made of very homogeneous mate- rial and polished to a high degree of flatness, in order to avoid disturbance of the transmitted wavefront and to prevent surface damage at the high intensities of Q- switched lasers. Small angles of incidence are usually preferable for two reasons. Firstly, the reflection co- efficient for a single reflection is small. This keeps the usable energy high and helps to keep the photodetector from being saturated. Secondly, the coefficient of re- flection is almost independent of the plane of polariza- tion of the incident light (the relative difference be- tween the reflection coefficient for a single reflection and waves polarized parallel and perpendicular to the plane of incidence is 2% for a = 5 and 8% for a = 10°). The fact that such plane parallel plates, particularly when of good quality, exhibit interference fringes of equal inclination was well known for a long time. Before the advent of lasers, this was, however, of more or less academic interest. It is demonstrated in the following that this interference must be carefully ob- served in many applications of laser beam splitters. Interference of a Uniform Plane Wave The reflection of a uniform plane wave of infinite extent from a plane parallel plate is first considered. The author is with the Gesellschaft fur Strahlenforschung mbH, Physikalisch-Technische Abteilung, 8042 Neuherberg bei Mfinchen, Germany. Received 25 September 1968. The influence of nonuniformity and finite cross section of the laser beam is then discussed. From geometrical considerations, the phase difference 8 between the two waves, reflected at the front and the back surface is' (see Fig. 1): = 47D (n2 -sin2a)l . X (1) The sign of the additional phase r depends on the plane of polarization of the incident light, as well as on the angle of incidence a. For 8 equal to even multiples of 7r, the interference will be additive, for 8 equal to odd multiples of r it will be subtractive. Because of the variation of with the angle of incidence, successive reflection minima and maxima will occur as a function of a. A straightforward method of finding an expression for the coefficient of reflection is to use the impedance for- mulation. The characteristic impedance of region 3 is taken as the load at the 2-3 interface, and, with complex wave equation, transformed to the 1-2 boundary, where it then acts as the true load for the incoming wave. The two components of this wave, polarized parallel (denoted by a subscript p) and perpendicular (denoted by a subscript s) to the plane of incidence must be treated separately. The plane containing the E vector is taken as the plane of polarization. The characteristic impedances and the phase constants are given ex- plicitly: region 1 and 3 1 Up= - Cosa no 1 -s =-seca no region 2 1 VP- -cos3 n 1 Is =- sect n (2) February 1969 / Vol. 8, No. 2 / APPLIED OPTICS 351
Transcript
Influence of Interference Fringes of Equal Inclination on theReflection of Laser Beams from Plane Parallel Plates
Franz Hillenkamp
The coefficient of reflection of plane parallel plate beam splitters is very often taken as twice that of asingle surface reflection. It is shown that this introduces appreciable errors in the majority of cases, be-cause of the interference fringes of equal inclination. Formulas are derived for the angular dependenceof the coefficient of reflection as well as for its average value. The angular distance between adjacentfringes has been evaluated and represented graphically. It is finally demonstrated that the influence ofincomplete interference due to the lateral displacement of the reflected beam can be neglected if the beamsplitter is followed by an integrating sphere or a similar element.
IntroductionIn laser applications, plane parallel glass or quartz
flats are commonly used as beam splitters. The re-flected portion of the light is then taken as a measure ofthe intensity as well as the energy of the main beam.These flats should be made of very homogeneous mate-rial and polished to a high degree of flatness, in order toavoid disturbance of the transmitted wavefront and toprevent surface damage at the high intensities of Q-switched lasers. Small angles of incidence are usuallypreferable for two reasons. Firstly, the reflection co-efficient for a single reflection is small. This keeps theusable energy high and helps to keep the photodetectorfrom being saturated. Secondly, the coefficient of re-flection is almost independent of the plane of polariza-tion of the incident light (the relative difference be-tween the reflection coefficient for a single reflection andwaves polarized parallel and perpendicular to the planeof incidence is 2% for a = 5 and 8% for a = 10°).The fact that such plane parallel plates, particularlywhen of good quality, exhibit interference fringes ofequal inclination was well known for a long time.Before the advent of lasers, this was, however, of moreor less academic interest. It is demonstrated in thefollowing that this interference must be carefully ob-served in many applications of laser beam splitters.
Interference of a Uniform Plane WaveThe reflection of a uniform plane wave of infinite
extent from a plane parallel plate is first considered.
The author is with the Gesellschaft fur StrahlenforschungmbH, Physikalisch-Technische Abteilung, 8042 Neuherberg beiMfinchen, Germany.
Received 25 September 1968.
The influence of nonuniformity and finite cross sectionof the laser beam is then discussed. From geometricalconsiderations, the phase difference 8 between the twowaves, reflected at the front and the back surface is'(see Fig. 1):
= 47D (n2 -sin2a)l .
X (1)
The sign of the additional phase r depends on the planeof polarization of the incident light, as well as on theangle of incidence a. For 8 equal to even multiples of7r, the interference will be additive, for 8 equal to oddmultiples of r it will be subtractive. Because of thevariation of with the angle of incidence, successivereflection minima and maxima will occur as a function ofa.
A straightforward method of finding an expression forthe coefficient of reflection is to use the impedance for-mulation. The characteristic impedance of region 3 istaken as the load at the 2-3 interface, and, with complexwave equation, transformed to the 1-2 boundary, whereit then acts as the true load for the incoming wave.The two components of this wave, polarized parallel(denoted by a subscript p) and perpendicular (denotedby a subscript s) to the plane of incidence must betreated separately. The plane containing the E vectoris taken as the plane of polarization. The characteristicimpedances and the phase constants are given ex-plicitly:
In agreement with Eq. (1), both coefficients* equal zerofor every a given by
sina, = [n2- 2( X/2D)2JJ; (5)
any particular n, D, and X given, Aa can easily be deter-mined with the help of these curves. If the example ofFig. 3 is taken up again, it turns out that the smallest
(2D/X) (n2- 1) < < 2nD/X; is the integer inter-
ference order.The fraction in the denominator of R, contains the
Brewster condition. It will become infinite for tana =n. In Fig. 2, the coefficient of reflection is plotted for acase which might well occur in practice. The flat is 1mm thick (size of a microscope slide), the wavelength isthat of a Nd laser. As said before, there is practicallyno difference between R, and R, in this region. It isobvious that, for these small angles of incidence, theangular distance between adjacent interference fringesis well above the divergence of laser beams. With acontinuous He-Ne laser, these fringes can easily bedemonstrated experimentally. Owing to the finitecross section of the beam, a dark strip will show up in thecenter of the spot instead of total extinction.
Figure 3 shows the coefficient of reflection for therather extreme case of the reflection from a pelliclebeam splitter. These pellicles are collodion membranesstretched over a suitable frame. They will in all casesexhibit a very strong dependence of the coefficient ofreflection on the angle of incidence.
In many applications, the knowledge of just howlarge the angular distance between adjacent reflectionminima is will suffice. Because the quotient X/D will besmall in most cases of practical interest, the angulardistance Aa will also be small compared with the angleof incidence and small enough to replace sin ( Aa) by Aa.With these approximations, the following formula canbe derived:
Aa ( (n - sin2a) (D sin2a (6)
In Fig. 4, the function (n' - sina) /sin2a isplotted for two different indices of refraction. With
* In his book Optik (Springer Verlag, 1932; second unrevisededition, 1965) M. Born states on p. 119 that complete cancellationof the reflected beam due to interference cannot take place, be-cause the two beams, reflected from the front and back surface,respectively, are of unequal intensity. This statement, however,holds for noncoherent superposition of the beams only. Thoughof minor importance, readers should be aware of this error.
regionin=1l
Fig. 1. Reflection of a
0,14,, Rs
0.12A A `A
0.10
0.08
0,06
0,04
0.02
10
region 2n
collimated beamparallel plate.
20 .3 - 40
region3nd I
of light from a plane
.5
Fig. 2. Dependence of the coefficient of reflection of a planeparallel plate on the angle of incidence. n = 1.46; X = 1.06 X
angular distance occurs for a - 500. For this angleof incidence, Aa is equal to 1.35 mrad. This is stilllarger than the divergence of diffraction-limited laserbeams and approximately equal to that of mode con-trolled lasers. It may be worthwhile to note that thedistance between adjacent reflection minima increasesagain for larger angles of incidence. Figure 3 clearlydemonstrates this. The A's, however, decrease mono-tonically as one would expect. Equation (6) fails forangles of incidence near 00 and 90°.
Three different cases should now be distinguished:(1) beam divergence small compared with Aa; (2) beamdivergence comparable with Aa; and (3) beam diver-gence large compared with Aa. The first two casesshould be avoided if possible. In the first case, Eqs.(3) and (4) will give the right value for R. However,small variations in due to vibrations of the beamsplitter, thermal influences, or changes in the divergenceof the beam for different pump energies may cause largevariations of R. This happens because of the largederivative of R with respect to a at certain angles. To
find the coefficient of reflection in the second case, theconvolution integral of the laser intensity and R as afunction of a must be solved, which is quite impractical.Again, changes in the divergence of the beam may causeappreciable variations of the average coefficient of re-flection.
Average Coefficient of ReflectionIf at all possible, one should aim for the third case by
choosing a suitable plate thickness. The coefficient ofreflection can then be obtained by averaging R, and R,over one or more interference orders. The dependenceof these average values (, and R,) on a will, in almostall cases, be small enough so that the variation of thereflected light intensity with a can be neglected for smallvariations of a. The averaging process is straight-forward if the interference order m is chosen as thevariable:
g A 01A + +1 sin2(m - g)-r + 4[(mX/2D)/(1 - n2)2 + (mX/2D)/( - n 2)](
In accordance with Eq. (5), IL is that integer value of mwhich corresponds to the center angle of incidence ofthe beam. a, and a±+ may be obtained from Eq. (5).
To the author's knowledge, these integrals cannot besolved in closed form. They have therefore been eval-uated by a computer program for three different indicesof refraction and for all so's corresponding to any integermultiple of 5. The results are plotted in Fig. 5. Itis of course not surprising that these curves qualitativelyfollow that of single reflections. Though it seems diffi-cult to prove rigorously that the magnitude of R, andR, does not depend on X and D, the computer resultsare the same for different X's and D's within their limitsof accuracy, as one would expect.
Reflection of Nonuniform WavefrontsThe above results have been obtained for a uniform
wavefront of infinite extent. The laser wavefront is,however, nonuniform and finite. The influence of thiswill be twofold. The interference will be incomplete,because the two interfering beams will generally be ofunequal intensity. Moreover, the second beam will bedisplaced laterally by the amount:
a = D sin(2a)/[n' - sinal 4l.
Those parts of the laser beam, which will not take partin the interference (either because the unequal intensityof the two interfering beams leaves a reflected residaulbeam, or because a second beam is not met at all),will undergo multiple reflections at the two interfaces.If-for the simplicity of the derivation-it is assumedthat no interference occurs ( > d), the intensity of thereflected parallel beams can be summed. If RI is the
(10)
Interference will therefore occur in the center part of thereflected beam only. The width of this region will be(- d). It may therefore seem that for this reason it isnot possible to determine the average coefficient ofreflection from the above formulas. There is, however,a solution to this problem.
Fig. 3. Dependence of the coefficient of reflection of a pelliclebeam splitter on the angle of incidence. n = 1.49; X = 6.943 X
Fig. 5. Average coefficient of reflection as a function of the angleof incidence.
intensity of the first beam reflected into region 1, theintensity of the kth reflected beam is
Rk = (1 - R1 )2R,1(2k-3), k > 2.
The intensities of the first n beams sum to:n 2(n-1)
Rn 1 R, = RI2(n-1) + 2 E (-.1)(k-l)Rkk=1 = 1
For an infinite number of reflections, this yields
R = R= 2R,/(1 + R).I,
(11)
(12)
(13)
intensity, if the size of the beam splitter allows at leasta few reflections. If now all the reflected light is gath-ered into an integrating sphere or onto a similar diffus-ing surface, the intensity measured by the detector willalways be proportional to the values given in Fig. 5,no matter what the degree of interference will be.
ConclusionIt has been shown that the influence of the inter-
ference fringes of equal inclination must be observedwhen plane parallel glass flats are used as beam splittersin laser applications. If the dielectric plates are not ofsufficient thickness, the reflectivity of the beam splitterwill strongly depend on small variations of the angle ofincidence due to vibration or thermal motion. A cal-ibration of such an arrangement will then be quite un-reliable. It might be suggested that the plane parallelflats could be replaced by interferometer flats with asmall tilt between the two surfaces. These flats will,however, exhibit interference fringes of equal thickness.For small tilt angles, the interference pattern, thoughfinite, will extend an unreasonable distance from theplate. If the angle between the two surfaces is large,the beam cross section will be distorted, as is the casewith Brewster angle components. Moreover, the sec-ond beam leaves the beam splitter at an appreciableangle to the first one. There is quite a chance that it, orone of the other reflected beams, will reach the detectorvia another reflection from the wall or some other sur-face. This will then disturb the measurement. It.will be particularly disturbing when Q-switched laserpulses are measured, or when arrays of beam splittersare used to attenuate the laser intensity to a value suit-able for photodetectors.
I would like to thank J. Kinder for stimulating dis-cussions, P. M\'ulser of the IPP, Garching, for doing thecomputer work, and M1. Poths for her help.
AppendixFor a = 0
R= R, =sin2(27rnD/X) (14>
sin'(27rnD/X) + 4n'/(n2 - 1)2
I E sin2 (2.7nD/X)1? J sin'(27rnD/X) + 4n2
/(n2
- d(2,rnD/X), (15)
1 + 4n2/(n 2 - )2' (16)
1 = (n - 1)2/(n2 + ), (17)
R= (n - 1)2/(n + 1)2, (18)
A comparison of the numerical values of R. and R revealsthat they are equal within their limits of computationalaccuracy. Unfortunately, this equality cannot beproved rigorously because of the complicated integralsof Eqs. (8) and (9). It can, however, be done for thevery similar case of normal incidence and variable Dor X (see Appendix). Because of the rapid decrease ofRk with k, R. will always be close to the actual reflected
R. = (n - 1)2/(n
2+ 1),
R = Ro. (20),
References1. M. Born and E. Wolf, Principles of Optics (Pergamon Press,.
Ltd., London, 1959), p. 281.2. See, e.g., S. Ramo and J. R. Whinnery, Fields and Waves
in Modern Radio (John Wiley & Sons, Inc., New York, 1953).
