Can one make a laser out of cardboard?

Krzysztof Burdzy; Tvrtko Tadić

doi:10.1214/16-AAP1180

August 2017 Can one make a laser out of cardboard?

Krzysztof Burdzy, Tvrtko Tadić

Ann. Appl. Probab. 27(4): 1951-1991 (August 2017). DOI: 10.1214/16-AAP1180

Abstract

We consider two-dimensional and three-dimensional semi-infinite tubes made of “Lambertian” material, so that the distribution of the direction of a reflected light ray has the density proportional to the cosine of the angle with the normal vector. If the light source is far away from the opening of the tube then the exiting rays are (approximately) collimated in two dimensions but are not collimated in three dimensions. An observer looking into the three-dimensional tube will see “infinitely bright” spot at the center of vision. In other words, in three dimensions, the light brightness grows to infinity near the center as the light source moves away.

References

1.

[1] Angel, O., Burdzy, K. and Sheffield, S. (2013). Deterministic approximations of random reflectors. Trans. Amer. Math. Soc. 365 6367–6383.[1] Angel, O., Burdzy, K. and Sheffield, S. (2013). Deterministic approximations of random reflectors. Trans. Amer. Math. Soc. 365 6367–6383.

2.

[2] Asmussen, S. (1998). A probabilistic look at the Wiener–Hopf equation. SIAM Rev. 40 189–201.[2] Asmussen, S. (1998). A probabilistic look at the Wiener–Hopf equation. SIAM Rev. 40 189–201.

3.

[3] Bingham, N. H., Goldie, C. M. and Teugels, J. L. (1987). Regular Variation. Encyclopedia of Mathematics and Its Applications 27. Cambridge Univ. Press, Cambridge.[3] Bingham, N. H., Goldie, C. M. and Teugels, J. L. (1987). Regular Variation. Encyclopedia of Mathematics and Its Applications 27. Cambridge Univ. Press, Cambridge.

4.

[4] Chow, Y. S. (1986). On moments of ladder height variables. Adv. in Appl. Math. 7 46–54.[4] Chow, Y. S. (1986). On moments of ladder height variables. Adv. in Appl. Math. 7 46–54.

5.

[5] Comets, F., Popov, S., Schütz, G. M. and Vachkovskaia, M. (2009). Billiards in a general domain with random reflections. Arch. Ration. Mech. Anal. 191 497–537.[5] Comets, F., Popov, S., Schütz, G. M. and Vachkovskaia, M. (2009). Billiards in a general domain with random reflections. Arch. Ration. Mech. Anal. 191 497–537.

6.

[6] Comets, F., Popov, S., Schütz, G. M. and Vachkovskaia, M. (2010). Knudsen gas in a finite random tube: Transport diffusion and first passage properties. J. Stat. Phys. 140 948–984.[6] Comets, F., Popov, S., Schütz, G. M. and Vachkovskaia, M. (2010). Knudsen gas in a finite random tube: Transport diffusion and first passage properties. J. Stat. Phys. 140 948–984.

7.

[7] Comets, F., Popov, S., Schütz, G. M. and Vachkovskaia, M. (2010). Quenched invariance principle for the Knudsen stochastic billiard in a random tube. Ann. Probab. 38 1019–1061. MR2674993 10.1214/09-AOP504 euclid.aop/1275486187 [7] Comets, F., Popov, S., Schütz, G. M. and Vachkovskaia, M. (2010). Quenched invariance principle for the Knudsen stochastic billiard in a random tube. Ann. Probab. 38 1019–1061. MR2674993 10.1214/09-AOP504 euclid.aop/1275486187

8.

[8] Doney, R. A. (1980). Moments of ladder heights in random walks. J. Appl. Probab. 17 248–252. MR557453 10.2307/3212942[8] Doney, R. A. (1980). Moments of ladder heights in random walks. J. Appl. Probab. 17 248–252. MR557453 10.2307/3212942

9.

[9] Durrett, R. (2010). Probability: Theory and Examples, 4th ed. Cambridge Univ. Press, Cambridge.[9] Durrett, R. (2010). Probability: Theory and Examples, 4th ed. Cambridge Univ. Press, Cambridge.

10.

[10] Erickson, K. B. (1970). Strong renewal theorems with infinite mean. Trans. Amer. Math. Soc. 151 263–291.[10] Erickson, K. B. (1970). Strong renewal theorems with infinite mean. Trans. Amer. Math. Soc. 151 263–291.

11.

[11] Evans, S. N. (2001). Stochastic billiards on general tables. Ann. Appl. Probab. 11 419–437.[11] Evans, S. N. (2001). Stochastic billiards on general tables. Ann. Appl. Probab. 11 419–437.

12.

[12] Kyprianou, A. E. (2006). Introductory Lectures on Fluctuations of Lévy Processes with Applications. Springer, Berlin.[12] Kyprianou, A. E. (2006). Introductory Lectures on Fluctuations of Lévy Processes with Applications. Springer, Berlin.

13.

[13] Lapidus, M. L. and Niemeyer, R. G. (2010). Towards the Koch snowflake fractal billiard: Computer experiments and mathematical conjectures. In Gems in Experimental Mathematics. Contemp. Math. 517 231–263. Amer. Math. Soc., Providence, RI.[13] Lapidus, M. L. and Niemeyer, R. G. (2010). Towards the Koch snowflake fractal billiard: Computer experiments and mathematical conjectures. In Gems in Experimental Mathematics. Contemp. Math. 517 231–263. Amer. Math. Soc., Providence, RI.

14.

[14] Lapidus, M. L. and Niemeyer, R. G. (2013). The current state of fractal billiards. In Fractal Geometry and Dynamical Systems in Pure and Applied Mathematics. II. Fractals in Applied Mathematics. Contemp. Math. 601 251–288. Amer. Math. Soc., Providence, RI.[14] Lapidus, M. L. and Niemeyer, R. G. (2013). The current state of fractal billiards. In Fractal Geometry and Dynamical Systems in Pure and Applied Mathematics. II. Fractals in Applied Mathematics. Contemp. Math. 601 251–288. Amer. Math. Soc., Providence, RI.

15.

[15] Lapidus, M. L. and Niemeyer, R. G. (2013). Sequences of compatible periodic hybrid orbits of prefractal Koch snowflake billiards. Discrete Contin. Dyn. Syst. 33 3719–3740.[15] Lapidus, M. L. and Niemeyer, R. G. (2013). Sequences of compatible periodic hybrid orbits of prefractal Koch snowflake billiards. Discrete Contin. Dyn. Syst. 33 3719–3740.

16.

[16] Mikosch, T. (1999). Regular variation, subexponentiality and their applications in probability theory. Lecture notes. [Online; accessed May 2015]. Available at http://www.math.ku.dk/~mikosch/Preprint/Eurandom/.[16] Mikosch, T. (1999). Regular variation, subexponentiality and their applications in probability theory. Lecture notes. [Online; accessed May 2015]. Available at http://www.math.ku.dk/~mikosch/Preprint/Eurandom/.

17.

[17] Obłój, J. (2004). The Skorokhod embedding problem and its offspring. Probab. Surv. 1 321–390.[17] Obłój, J. (2004). The Skorokhod embedding problem and its offspring. Probab. Surv. 1 321–390.

18.

[18] Rogozin, B. A. (1971). Distribution of the first laddar moment and height, and fluctuations of a random walk. Teor. Verojatnost. i Primenen. 16 539–613.[18] Rogozin, B. A. (1971). Distribution of the first laddar moment and height, and fluctuations of a random walk. Teor. Verojatnost. i Primenen. 16 539–613.

19.

[19] Spitzer, F. (1957). The Wiener–Hopf equation whose kernel is a probability density. Duke Math. J. 24 327–343.[19] Spitzer, F. (1957). The Wiener–Hopf equation whose kernel is a probability density. Duke Math. J. 24 327–343.

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Krzysztof Burdzy and Tvrtko Tadić "Can one make a laser out of cardboard?," The Annals of Applied Probability 27(4), 1951-1991, (August 2017). https://doi.org/10.1214/16-AAP1180

Received: 1 July 2015; Published: August 2017

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