Now the photons are sorted according to n, and it is assumed that they leave the sphere after the discrete time tn = n·, where is the average time of flight ...
INTEGRATING SPHERE DIFFUSER FOR WIRELESS INFRARED COMMUNICATION V. Pohl, V. Jungnickel, R. Hentges and C. von Helmolt1
Abstract The application of an integrating sphere (IS) as a diffuser in a wireless infrared network is investigated. In order to obtain realistic data for the insertion loss, the transfer function and the far field light distribution, a random walk ray tracing model is used. As an example, a Spectralon® diffuser for a 155 Mbit/s 16-PPM IR transmitter is designed having a total efficiency of 86%, a cut-off frequency of 570 MHz and a nearly Lambertian intensity distribution. An additional target is used to create a more homogeneous far field without significant changes of the efficiency and the cut-off frequency. Introduction Wireless infrared (IR) networks are intended for indoors, especially when electromagnetic interference becomes a critical issue [1, 2]. The IR light carrying the information is usually broadcasted to the mobile users in the cell. However, most rooms are more wide than high, and a transmitter having a wide radiation cone is needed. Also, the eye safety condition for the transmitter is easier to be met with a large area emitter. A wide transmitter beam can be realised using a Lambertian diffuser. It’s cosine-law intensity distribution has a full-width-at-half-maximum (FWHM) angle of 120°. The diffuse reflection of many surfaces yields a nearly Lambertian intensity distribution [3]. But the source itself would then shadow a sector in the cell. Recently, holographic diffusers have been proposed for wireless IR networks [4]. However, their properties strongly depend on the wavelength and even at the optimum wavelength, a hot spot in the centre of the beam is present. Due to the eye safety limits, the hot spot reduces the maximum power that can be launched safely into the cell. In this paper, the application of an IS as a diffuser in a wireless IR network is studied. A small sphere is designed which creates a nearly Lambertian intensity distribution basically independent from the IR source characteristics. No hot spot is present, and high optical powers can be emitted into the cell. An internal target can be used to match the intensity distribution to the situation in a given room. Integrating Sphere Basics An ideal IS is a hollow spherical cavity having a diffusely reflecting internal surface. For the surface reflections, a Lambertian intensity distribution with the reflectivity ρ is assumed. The light is coupled into the sphere through a small entrance port with the area Ai. Due to the multiple diffuse reflections a constant radiance is created over the IS surface with the area As. Because of losses, the total radiant power after the exit port with the area Ae is reduced by the efficiency
η=ρ
Ae 1−α . Ae + A i 1 − αρ
(1)
The figure α = 1 – f is the complement of the fractional port area f = (Ae+Ai)/As [5]. Now the dynamic response of the IS is investigated. When the radiant power Φi at the input port is switched on, instantaneously, then the power at the exit port after n reflections builds up as
Φ n = η[1 − (αρ )n ]Φ i .
(2)
Now the photons are sorted according to n, and it is assumed that they leave the sphere after the discrete time tn = n·, where is the average time of flight between two reflections. In this way a discrete step response hσ(tn) is obtained from (2) 1
Heinrich-Hertz-Institut für Nachrichtentechnik Berlin GmbH, Einsteinufer 37, 10587 Berlin, Germany
−
hσ (t n ) = η[1 - e
tn
τ
]
(3)
where τ = - / ln(αρ) and = 2D/3c. The figures D and c are the IS diameter and the speed of light. When hσ(tn) is now interpolated on a continuous time scale the impulse response t
η − d hδ (t) = hσ (t) = e τ τ dt
(4)
can be obtained by deriving hσ(tn). Apparently, the temporal response is exponentially, and the decay time is increased when αρ is larger. The transfer function is then obtained by the Fourier transform. The IS behaves like a low-pass filter with the cut-off frequency f0.
H ( jf ) =
η
,
f 1+ j f0
where
j = -1
and
f0 =
1 2πτ
.
(5)
0 1 f0x D [GHz cm]
10 log η [dB]
-3
f = 0.1 f = 0.05
-6
f = 0.02 f = 0.01
ρ = 0.94 (Infragold)
ρ = 0.96 (Duraflect)
-9 Infragold Duraflect Spectralon 0,92
0,94
0,96
internal reflectivity ρ
0,98
1
ρ = 0.99 (Spectralon) 0,1
0,01
0,1
fractional port area f
Fig. 1 Left: Sphere efficiency versus internal reflectivity for various fractional port areas f. Right: Cut-off frequency f0 times sphere diameter as a function of the fractional port area for typical diffuser materials. The two filled circles refer to the sphere described in the text.
Usually ISs are relatively large (2 to 40 inches diameter). They are made of moderate reflectivity materials like Duraflect or Infragold [6], and they have fractional port areas smaller than 0.01 in order to maintain good Lambertian properties. In Fig. 1 the sphere efficiency losses and the cut-off frequency according to (1) and (5) are investigated in detail. From the figure it follows that usual ISs have a rather poor efficiency. However, the efficiency increases when the internal reflectivity and the fractional port area are larger. The best values are obtained for Spectralon® [6] with fractional port areas up to 0.1. Spectralon® is a highly reflective synthetic diffuser material which can be processed using machine tools for large numbers of units. For example, an efficiency of -0.65 dB is expected with an exit port area of f = 0,06. Because usual spheres are so large, they have a relatively low cut-off frequency. But when the diameter is 1 cm only, the cut-off frequency is increased to 515 MHz. Random Walk Ray Tracing Due to the large port area which is necessary for a high efficiency, deviations from the Lambertian intensity distribution will occur which are not described by the theory. Recently, Ziegler et al. [7] proposed a computer model for such IS.
6
target position z
~ 10 photons diffuse reflection x0 x2 x1
0 target x3
- D/2 Θ statistics on Θ and time of flight
Fig. 2 Principle of random walk ray tracing after Ziegler et al. [6] used for the simulation. The random path of each photon is tracked. After a large number of trials statistical data for the diffuser properties are obtained.
In Fig. 2 the principle of the simulation model is shown. Starting from the entrance port (top), the random path of each photon is tracked. At the reflection points on the inner surface a random number generator (see Appendix) creates a new direction as well as a decision whether the photon is being reflected or absorbed. When the photon arrives at the exit (bottom), the final direction of it’s flight as well as the total path length in the sphere are noticed. After a large number of trials, the insertion loss is obtained by counting the fraction of photons leaving the sphere. The angular intensity distribution is obtained by sorting the photons according to their final direction of flight. The impulse response comes from sorting the photons according to their total time of flight. Note, that a separate impulse response for each direction can be obtained. The transfer function is obtained from the discrete Fourier transform of the impulse response. Because of the Poisson distribution, a population of 106 photons must be investigated in order to get an error of 10-3 for the statistical data. The computation time for the C++ computer program is then about 15 minutes on a Pentium II PC at 266 MHz.
normalised intensity [a.u.]
Simulation Results In the following, an IS with a diameter of 1 cm is studied. For the inner surface a Lambertian diffuser with ρ = 0.99 is assumed. In order to obtain axial symmetry, the entrance port and the exit port are placed opposite to each other, and a small target is placed on the axis. After passing the entrance port, at first the light ray falls onto the horizontal target where it is diffusely reflected (see Fig. 2). 100
80
120
60
140
160
180
40 Lambertian f = 0.01 f = 0.06
20
0
Fig. 3 Intensity distribution for different port areas normalised to the radiant power in the hemisphere.
In Fig. 3 the angular intensity distributions are shown when the port area is increased. For f = 0.01 the simulation yields a nearly Lambertian intensity distribution, and the data for the efficiency (η = -3.3 dB) and the exponential decay time (τ = 1.093 ns) are close to the theoretical results (η = -3.0 dB, τ = 1.1 ns). When f is increased from 0.01 to 0.06, the cone is getting slightly more narrow. The FWHM angle of the intensity distribution is reduced from 114° down to 104°. But the efficiency is improved to –0.66 dB and the cut-off frequency increases from 145 MHz to 570 MHz. For all figures the small differences compared to the
relative efficiency 10 log η [dB] η0
theoretical values are due to the target (1 mm diameter, thickness is neglected). When the target is removed and a perpendicular port geometry is used, simulation and theory agree very well with each other. 0
-3 -6
number of photons [a.u.]
-9
-12 -15 1M
10M
100M 1G frequency [Hz]
τ = 280
h2(t)
0
100
ps
h10(t)
200
10G
h20(t)
300
400 time [ps]
500
h30(t)
600
700
800
Fig. 4 Impulse response of an axial symmetric sphere (diameter = 1 cm, f = 0.06). Inset: Transfer function. For comparison, the ideal low-pass behaviour is plotted, too.
When the fractional port area becomes larger, deviations from the exponential time response gain in importance. At the bottom of Fig. 4 this is clearly resolved (f = 0.06). The impulse response has an initial delay of about 50 ps which is due to the shortest possible path in the sphere. During the next 100 ps, a number of distinct peaks occur until the predicted exponential decay takes place. In order to clarify the origin of these peaks, the distributions of the arrival time for all those photons is depicted separately, which leave the sphere after 2, 10, 20 and 30 reflections. During the first reflections, the temporal distributions are relatively sharp. But photons which have been reflected many times in the sphere do have a much wider temporal spread. From the data it turns out that the variance of the distributions increases linearly with the number of reflections n. This is well known from statistical theory. The time of flight between two reflections is a random variable which is distributed according to the geometrical distance to all other points on the surface and according to the actual back scattering characteristic. The time of flight after n reflections is then a sum of random variables. When n becomes large, the total distribution tends towards a Gaussian, and the variance increases linearly with n. Naturally, the photons do not arrive at the discrete times n· as it was assumed in the derivation of (4). But clearly, the sum of all distributions forms an exponential decay at later times. The increased magnitude of the initial peaks at larger port areas is explained by the larger fraction of photons which leave the sphere after each reflection. But when 1-αρ « 1, these peaks have no practical impact on the IS transfer function (see inset of Fig. 4). For relatively low frequencies (< 3 GHz) no significant deviations from the ideal low pass behaviour is observed. Significant modifications are found only at very high frequencies (> 10 GHz), where the initial peaks create a very broad base of the transfer function. As already mentioned, for large port areas the FWHM angle is slightly reduced compared to an ideal Lambertian source. Now the properties of the target in the sphere (see Fig. 2) are changed in order to create a wider intensity distribution. When a small target with a diameter of 1 mm is placed at the IS centre, the intensity distribution in Fig. 3 (f = 0.06) is obtained. At first the target diameter is increased to 5 mm, and the intensity distribution is deformed (see top of Fig. 5, z = 0 mm). The maximum intensity is then found 22,5° apart from the axis, and the FWHM angle is increased to 125°. The changes of the efficiency and of the cutoff frequency are negligible. Simply, the photons avoid the central paths in the sphere. A more significant change of the FWHM angle is obtained when the 5 mm target is moved towards the exit port. Now the target covers part of the hole and the photons leave the IS at even larger angles, preferably. The intensity in the
centre of the beam is further reduced by 44% (Fig. 5). The maximum intensity is shifted 40° apart from the axis, and a 156° FWHM angle significantly larger than that of a Lambertian source is obtained.
Fig. 5 Top: Intensity distribution as a function of the target position z (see Fig. 2). The curves are normalised to the same input power. The sphere centre refers to z = 0. Bottom: Efficiency and cut-off frequency versus target position.
eye safe transmitter power [mW]
At the bottom of Fig. 5 it is shown that at z = -3 mm an efficiency of -1.2 dB and a cut-off frequency of 330 MHz is found. While the decrease of the efficiency is often acceptable, the smaller f0 might be a problem for very high data rate applications. But of course, when all IS dimensions are reduced, proportionally, a higher cut-off frequency can be reached.
10
5
10
4
class 1 laser λ = 900 nm τ = 100 s
Lambertian source
103 10
2
10 collimated beam
1 0.01
0.1
10 1 source diameter [mm]
maximum permissible exposure x source area (skin)
100
Fig. 6 Class 1 eye safety limit according to IEC 825-1 [8] versus the source diameter.
Eye Safety In wireless IR links, the transmitter power is limited due to eye safety regulations [8]. The limit depends on wavelength, beam characteristic and operation time of the source. For diffuse transmitters in the visible and the near IR spectral region the source diameter is an important figure since it is often used to transmit more power, safely. An image of the source is then formed on the retina, and the light is not focussed, like for a collimated beam. When the source diameter is increased, the image on the retina is growing, and, for the same critical intensity, a larger power can be emitted. In Fig. 6 it is shown that a collimated 900 nm class 1 laser should have a power of less than 0.5 mW. This figure is fully independent on the source diameter since the beam could be focussed by an optical instrument into a very fine spot. On the other hand, a Lambertian source is less dangerous because the eye catches only part of the light. But care must be taken that the skin comes not in close contact with a large area source. When the maximum permissible exposure (MPE) is multiplied with the emitter area, a smaller value than the eye safety limit is obtained for large source diameters. The diffuser must then be covered by a transparent shield to keep people outside the dangerous region. The 1 cm integrating sphere which is described in the text has an exit port diameter of 5 mm. It is eye safe up to 1 W assuming a Lambertian diffuser characteristic. Conclusions The use of an integrating sphere as a diffuser in a wireless infrared network has been investigated. A small sphere made of a highly reflective material with a relatively large exit port can be used for this application, advantageously. A random walk ray tracing model has been used to study the properties of a 1 cm diameter sphere. It has an efficiency of -0.66 dB, a cut-off frequency of 570 MHz and an eye safe transmitter power of up to 1 W at 900 nm. A target in the sphere allows a more homogeneous intensity distribution. FWHM angles of up to 156° significantly larger than that of a Lambertian source have been obtained. Appendix: A Random Direction Generator Let x be a continuous random variable having the probability density f(x) and the distribution function x
F( x ) = ∫ f ( t )dt . −∞
Let y = g(x) be a second random variable. When g(x) ≡ F(x) then it is known that y is equally distributed over [0, 1] (see proof in [9]). When F-1 exists, then the random variable x = F-1(y) can be obtained from y. Assuming a Lambertian diffuser, the probability density for the new direction is given by
f (θ ,φ ) =
1
π
sin(θ ) cos(θ ).
When a and b are two independent random variables equally distributed over [0, 1], the new direction is
1 2
φ (a ) = 2πa and θ ( b) = arccos( 2b − 1) . References [1] [2] [3] [4] [5] [6] [7] [8] [9]
F.R. Gfeller and U.H. Bapst “Wireless in-house data communication via diffuse infrared radiation”, Proc. IEE, 67, 11 (1979), pp. 1474-1486 J.M. Kahn and J.R. Barry “Wireless infrared communications”, Proc. IEEE, 85, 2 (1997), pp. 265-298 V. Jungnickel, C. v. Helmolt and U. Krüger “Broadband wireless IR LAN architecture compatible with the Ethernet protocol”, Electronics Letters, 34, 25 (1998), pp. 2371-2372 P.L. Eardley, D.R. Wiseley, D. Wood, P. McKee “Holograms for Optical Wireless LANs”, IEE Proc. Optoelectron., 143, 6 (1996), pp. 365-369 O.E. Miller, A.J. Sant “Incomplete Integrating Sphere”, J. Opt. Soc. Am., 48, 11 (1958), 828 Spectralon® is a registered trademark of Labsphere, Inc., see: http://www.labsphere.com A. Ziegler, H. Hess, H. Schimpl “Rechnersimulation von Ulbrichtkugeln”, Optik 101, 3 (1996), pp. 130-136 German version of IEC 825-1: DIN EN 60825-1:1994, “Sicherheit von Lasereinrichtungen, Teil 1: Klassifizierung von Anlagen, Anforderungen und Benutzer-Richtlinien”, VDE-Verlag, Berlin, 1997 M. Fisz: “Wahrscheinlichkeitsrechnung und mathematische Statistik”, Deuscher Verlag der Wissenschaften, Berlin 1980
