Yardley Beers This paper An optical wedge beam splitter consists of a prism of transparent In The paper also gives the results of numerical calculations based Key Words: Optical attenuation, optical beam splitter. 1. INTRODUCTION The 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. 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 distin guish between these beams using a notation which is in the vocabulary of any computer: 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. 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 incidence, the thickness, and the index of refraction. Under practical conditions this dis placement 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 I 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, and for any other order m F = 1/(T2RTM). (2) about 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) 1/25 while for high index (e.g., 1.75) it is about 1/13. For rough calcula R is tions 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 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. PROPERTIES 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 f't 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 |