Abstract

An oblique reflection of a laser pulse from a fully reflective mirror is treated using the fundamental nonrelativistic conservation principles of energy and momentum. Since the mirror is considered as an elastic object, the reflection of light gives rise to an elastic wave with measurable amplitude that propagates within the mirror. Our results predict a larger Doppler shift in the reflected pulse for the most common setting, when the mirror is initially at rest, compared to the results obtained when the mirror is treated as rigid.

© 2013 Optical Society of America

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References

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  1. C. B. Scruby and L. E. Drain, Laser Ultrasonics: Techniques and Applications (Adam Hilger, 1990).
  2. R. M. White, J. Appl. Phys. 34, 3559 (1963).
    [CrossRef]
  3. J. Možina and M. Dovč, Mod. Phys. Lett. B 8, 1791 (1994).
    [CrossRef]
  4. T. Požar and J. Možina, Phys. Rev. Lett. 111, 185501 (2013).
    [CrossRef]
  5. J. Možina and J. Diaci, Appl. Phys. B 105, 557 (2011).
    [CrossRef]
  6. J. D. Achenbach, Int. J. Solids Struct. 37, 13 (2000).
    [CrossRef]
  7. M. Mansuripur, Opt. Commun. 283, 1997 (2010).
    [CrossRef]
  8. T. J. Kippenberg and K. J. Vahala, Science 321, 1172 (2008).
    [CrossRef]
  9. O. Arcizet, P. F. Cohadon, T. Briant, M. Pinard, and A. Heidmann, Nature (London) 444, 71 (2006).
    [CrossRef]
  10. G. Berden, R. Peeters, and G. Meijer, Int. Rev. Phys. Chem. 19, 565 (2000).
    [CrossRef]
  11. R. Y. Kezerashvili, Adv. Space Res. 48, 1683 (2011).
    [CrossRef]
  12. A. Yariv, IEEE J. Quantum Electron. 1, 28 (1965).
    [CrossRef]
  13. R. A. Beth, Phys. Rev. 50, 115 (1936).
    [CrossRef]
  14. G. Molina-Terriza, J. P. Torres, and L. Torner, Nat. Phys. 3, 305 (2007).
    [CrossRef]
  15. T. Požar and J. Možina, Opt. Lett. 38, 1754 (2013).
    [CrossRef]
  16. M. Mansuripur, Proc. SPIE 8122, 81220D (2011).
    [CrossRef]
  17. A. Gjurchinovski, Eur. J. Phys. 34, L1 (2013).
  18. J. R. Galli and F. Amiri, Am. J. Phys. 80, 680 (2012).
    [CrossRef]
  19. G. Goedecke, V. Toussaint, and C. Cooper, Am. J. Phys. 80, 684 (2012).
    [CrossRef]
  20. A. Einstein, Ann. Phys. (Berlin) 322, 891 (1905).
    [CrossRef]
  21. T. Požar and J. Možina, Appl. Phys. A 91, 315 (2008).
    [CrossRef]
  22. T. Požar, P. Gregorčič, and J. Možina, Opt. Express 17, 22906 (2009).
    [CrossRef]
  23. L. Bernal and L. Bilbao, Am. J. Phys. 75, 216 (2007).
    [CrossRef]
  24. R. L. Whitman and A. Korpel, Appl. Opt. 8, 1567 (1969).
    [CrossRef]

2013 (3)

T. Požar and J. Možina, Phys. Rev. Lett. 111, 185501 (2013).
[CrossRef]

T. Požar and J. Možina, Opt. Lett. 38, 1754 (2013).
[CrossRef]

A. Gjurchinovski, Eur. J. Phys. 34, L1 (2013).

2012 (2)

J. R. Galli and F. Amiri, Am. J. Phys. 80, 680 (2012).
[CrossRef]

G. Goedecke, V. Toussaint, and C. Cooper, Am. J. Phys. 80, 684 (2012).
[CrossRef]

2011 (3)

M. Mansuripur, Proc. SPIE 8122, 81220D (2011).
[CrossRef]

R. Y. Kezerashvili, Adv. Space Res. 48, 1683 (2011).
[CrossRef]

J. Možina and J. Diaci, Appl. Phys. B 105, 557 (2011).
[CrossRef]

2010 (1)

M. Mansuripur, Opt. Commun. 283, 1997 (2010).
[CrossRef]

2009 (1)

2008 (2)

T. Požar and J. Možina, Appl. Phys. A 91, 315 (2008).
[CrossRef]

T. J. Kippenberg and K. J. Vahala, Science 321, 1172 (2008).
[CrossRef]

2007 (2)

G. Molina-Terriza, J. P. Torres, and L. Torner, Nat. Phys. 3, 305 (2007).
[CrossRef]

L. Bernal and L. Bilbao, Am. J. Phys. 75, 216 (2007).
[CrossRef]

2006 (1)

O. Arcizet, P. F. Cohadon, T. Briant, M. Pinard, and A. Heidmann, Nature (London) 444, 71 (2006).
[CrossRef]

2000 (2)

G. Berden, R. Peeters, and G. Meijer, Int. Rev. Phys. Chem. 19, 565 (2000).
[CrossRef]

J. D. Achenbach, Int. J. Solids Struct. 37, 13 (2000).
[CrossRef]

1994 (1)

J. Možina and M. Dovč, Mod. Phys. Lett. B 8, 1791 (1994).
[CrossRef]

1969 (1)

1965 (1)

A. Yariv, IEEE J. Quantum Electron. 1, 28 (1965).
[CrossRef]

1963 (1)

R. M. White, J. Appl. Phys. 34, 3559 (1963).
[CrossRef]

1936 (1)

R. A. Beth, Phys. Rev. 50, 115 (1936).
[CrossRef]

1905 (1)

A. Einstein, Ann. Phys. (Berlin) 322, 891 (1905).
[CrossRef]

Achenbach, J. D.

J. D. Achenbach, Int. J. Solids Struct. 37, 13 (2000).
[CrossRef]

Amiri, F.

J. R. Galli and F. Amiri, Am. J. Phys. 80, 680 (2012).
[CrossRef]

Arcizet, O.

O. Arcizet, P. F. Cohadon, T. Briant, M. Pinard, and A. Heidmann, Nature (London) 444, 71 (2006).
[CrossRef]

Berden, G.

G. Berden, R. Peeters, and G. Meijer, Int. Rev. Phys. Chem. 19, 565 (2000).
[CrossRef]

Bernal, L.

L. Bernal and L. Bilbao, Am. J. Phys. 75, 216 (2007).
[CrossRef]

Beth, R. A.

R. A. Beth, Phys. Rev. 50, 115 (1936).
[CrossRef]

Bilbao, L.

L. Bernal and L. Bilbao, Am. J. Phys. 75, 216 (2007).
[CrossRef]

Briant, T.

O. Arcizet, P. F. Cohadon, T. Briant, M. Pinard, and A. Heidmann, Nature (London) 444, 71 (2006).
[CrossRef]

Cohadon, P. F.

O. Arcizet, P. F. Cohadon, T. Briant, M. Pinard, and A. Heidmann, Nature (London) 444, 71 (2006).
[CrossRef]

Cooper, C.

G. Goedecke, V. Toussaint, and C. Cooper, Am. J. Phys. 80, 684 (2012).
[CrossRef]

Diaci, J.

J. Možina and J. Diaci, Appl. Phys. B 105, 557 (2011).
[CrossRef]

Dovc, M.

J. Možina and M. Dovč, Mod. Phys. Lett. B 8, 1791 (1994).
[CrossRef]

Drain, L. E.

C. B. Scruby and L. E. Drain, Laser Ultrasonics: Techniques and Applications (Adam Hilger, 1990).

Einstein, A.

A. Einstein, Ann. Phys. (Berlin) 322, 891 (1905).
[CrossRef]

Galli, J. R.

J. R. Galli and F. Amiri, Am. J. Phys. 80, 680 (2012).
[CrossRef]

Gjurchinovski, A.

A. Gjurchinovski, Eur. J. Phys. 34, L1 (2013).

Goedecke, G.

G. Goedecke, V. Toussaint, and C. Cooper, Am. J. Phys. 80, 684 (2012).
[CrossRef]

Gregorcic, P.

Heidmann, A.

O. Arcizet, P. F. Cohadon, T. Briant, M. Pinard, and A. Heidmann, Nature (London) 444, 71 (2006).
[CrossRef]

Kezerashvili, R. Y.

R. Y. Kezerashvili, Adv. Space Res. 48, 1683 (2011).
[CrossRef]

Kippenberg, T. J.

T. J. Kippenberg and K. J. Vahala, Science 321, 1172 (2008).
[CrossRef]

Korpel, A.

Mansuripur, M.

M. Mansuripur, Proc. SPIE 8122, 81220D (2011).
[CrossRef]

M. Mansuripur, Opt. Commun. 283, 1997 (2010).
[CrossRef]

Meijer, G.

G. Berden, R. Peeters, and G. Meijer, Int. Rev. Phys. Chem. 19, 565 (2000).
[CrossRef]

Molina-Terriza, G.

G. Molina-Terriza, J. P. Torres, and L. Torner, Nat. Phys. 3, 305 (2007).
[CrossRef]

Možina, J.

T. Požar and J. Možina, Opt. Lett. 38, 1754 (2013).
[CrossRef]

T. Požar and J. Možina, Phys. Rev. Lett. 111, 185501 (2013).
[CrossRef]

J. Možina and J. Diaci, Appl. Phys. B 105, 557 (2011).
[CrossRef]

T. Požar, P. Gregorčič, and J. Možina, Opt. Express 17, 22906 (2009).
[CrossRef]

T. Požar and J. Možina, Appl. Phys. A 91, 315 (2008).
[CrossRef]

J. Možina and M. Dovč, Mod. Phys. Lett. B 8, 1791 (1994).
[CrossRef]

Peeters, R.

G. Berden, R. Peeters, and G. Meijer, Int. Rev. Phys. Chem. 19, 565 (2000).
[CrossRef]

Pinard, M.

O. Arcizet, P. F. Cohadon, T. Briant, M. Pinard, and A. Heidmann, Nature (London) 444, 71 (2006).
[CrossRef]

Požar, T.

T. Požar and J. Možina, Phys. Rev. Lett. 111, 185501 (2013).
[CrossRef]

T. Požar and J. Možina, Opt. Lett. 38, 1754 (2013).
[CrossRef]

T. Požar, P. Gregorčič, and J. Možina, Opt. Express 17, 22906 (2009).
[CrossRef]

T. Požar and J. Možina, Appl. Phys. A 91, 315 (2008).
[CrossRef]

Scruby, C. B.

C. B. Scruby and L. E. Drain, Laser Ultrasonics: Techniques and Applications (Adam Hilger, 1990).

Torner, L.

G. Molina-Terriza, J. P. Torres, and L. Torner, Nat. Phys. 3, 305 (2007).
[CrossRef]

Torres, J. P.

G. Molina-Terriza, J. P. Torres, and L. Torner, Nat. Phys. 3, 305 (2007).
[CrossRef]

Toussaint, V.

G. Goedecke, V. Toussaint, and C. Cooper, Am. J. Phys. 80, 684 (2012).
[CrossRef]

Vahala, K. J.

T. J. Kippenberg and K. J. Vahala, Science 321, 1172 (2008).
[CrossRef]

White, R. M.

R. M. White, J. Appl. Phys. 34, 3559 (1963).
[CrossRef]

Whitman, R. L.

Yariv, A.

A. Yariv, IEEE J. Quantum Electron. 1, 28 (1965).
[CrossRef]

Adv. Space Res. (1)

R. Y. Kezerashvili, Adv. Space Res. 48, 1683 (2011).
[CrossRef]

Am. J. Phys. (3)

J. R. Galli and F. Amiri, Am. J. Phys. 80, 680 (2012).
[CrossRef]

G. Goedecke, V. Toussaint, and C. Cooper, Am. J. Phys. 80, 684 (2012).
[CrossRef]

L. Bernal and L. Bilbao, Am. J. Phys. 75, 216 (2007).
[CrossRef]

Ann. Phys. (Berlin) (1)

A. Einstein, Ann. Phys. (Berlin) 322, 891 (1905).
[CrossRef]

Appl. Opt. (1)

Appl. Phys. A (1)

T. Požar and J. Možina, Appl. Phys. A 91, 315 (2008).
[CrossRef]

Appl. Phys. B (1)

J. Možina and J. Diaci, Appl. Phys. B 105, 557 (2011).
[CrossRef]

Eur. J. Phys. (1)

A. Gjurchinovski, Eur. J. Phys. 34, L1 (2013).

IEEE J. Quantum Electron. (1)

A. Yariv, IEEE J. Quantum Electron. 1, 28 (1965).
[CrossRef]

Int. J. Solids Struct. (1)

J. D. Achenbach, Int. J. Solids Struct. 37, 13 (2000).
[CrossRef]

Int. Rev. Phys. Chem. (1)

G. Berden, R. Peeters, and G. Meijer, Int. Rev. Phys. Chem. 19, 565 (2000).
[CrossRef]

J. Appl. Phys. (1)

R. M. White, J. Appl. Phys. 34, 3559 (1963).
[CrossRef]

Mod. Phys. Lett. B (1)

J. Možina and M. Dovč, Mod. Phys. Lett. B 8, 1791 (1994).
[CrossRef]

Nat. Phys. (1)

G. Molina-Terriza, J. P. Torres, and L. Torner, Nat. Phys. 3, 305 (2007).
[CrossRef]

Nature (London) (1)

O. Arcizet, P. F. Cohadon, T. Briant, M. Pinard, and A. Heidmann, Nature (London) 444, 71 (2006).
[CrossRef]

Opt. Commun. (1)

M. Mansuripur, Opt. Commun. 283, 1997 (2010).
[CrossRef]

Opt. Express (1)

Opt. Lett. (1)

Phys. Rev. (1)

R. A. Beth, Phys. Rev. 50, 115 (1936).
[CrossRef]

Phys. Rev. Lett. (1)

T. Požar and J. Možina, Phys. Rev. Lett. 111, 185501 (2013).
[CrossRef]

Proc. SPIE (1)

M. Mansuripur, Proc. SPIE 8122, 81220D (2011).
[CrossRef]

Science (1)

T. J. Kippenberg and K. J. Vahala, Science 321, 1172 (2008).
[CrossRef]

Other (1)

C. B. Scruby and L. E. Drain, Laser Ultrasonics: Techniques and Applications (Adam Hilger, 1990).

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Figures (2)

Fig. 1.
Fig. 1.

Schematic diagram of (a) the first and (b) the last contact during an oblique reflection of a laser pulse (magenta) from a 100% reflective elastic mirror (gray) that is initially at rest. The cutaway scheme shows only a limited cross-section area Ai of the whole incident pulse.

Fig. 2.
Fig. 2.

Surface cosαB=w/[ξ(1ξ)]>0 represents all the physically meaningful cases where the collision between the laser pulse and the mirror occurs. The height above the αB plane corresponds to the normalized amplitude of the elastic wave.

Tables (1)

Tables Icon

Table 1. Properties of the Incident and Reflected Laser Pulse

Equations (24)

Equations on this page are rendered with MathJax. Learn more.

tc=ti[1(β+B)/cosα]1,
u0=vtc=ctiβ[1(β+B)/cosα]1.
Ei+12MV2=Er+12(Mm)V2+12m(v+V)2+12mv2.
picosα+MV=prcosφ+(Mm)V+m(v+V),
pisinα=prsinφ.
amplitude of elastic wave:w=β[1(β+B)/cosα]1,
cons. of energy:η(1r)=β(β+B),
cons. ofx-momentum:η(cosα+rcosφ)=β,
cons. ofy-momentum:sinα=rsinφ,
η=Eimc2=EicosαρAic0c2tc=ξ2titc=ξ2(1β+Bcosα),
ξ=2qicosα/(ρc0c2ti)
β=[(i31)(H/F)(i3+1)F]/62B/3,
cosφ=2(β+B)+cosα[1+(β+B)2]12cosα(β+B)+(β+B)2,
r=12cosα(β+B)+(β+B)21(β+B)2,
β2η[(12η)cosαB],
r12Bcosα4ηcos2α,
cosφcosα(14ηsin2α)2Bsin2α,
β(η,B,α=0)={[(1+B)2+8η]1/2(1+B)}/2
r(β,B,α=0)=1(β+B)1+(β+B).
η(ξ,B,α)(ξ/2)(1ξB/cosα).
β(ξ,B,α)ξ[(12ξ)cosα2B],
r(ξ,B,α)12Bcosα2ξcos2α,
cosφ(ξ,B,α)cosα(12ξsin2α)2Bsin2α,
w(ξ,B,α)ξ(1ξ)(cosαB).

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