Abstract

We demonstrate phase locking between two pairs of nanosecond laser pulses generated from independent sources. We achieve phase locking experimentally by separately mixing two uncorrelated dye lasers of frequencies ω1(a) and ω1(b), with a common beam of frequency ω0, thereby generating two additional frequencies ω2(b)ω1(a)+ω0 and ω2(a)ω1(b)+ω0. We demonstrate that there are well-defined phase relationships between any two-photon process using the ω1(a) and the ω2(a) pair of frequencies versus any two-photon process that uses the ω1(b) and the ω2(b) pair. In particular, interference between the two identical sum frequencies ωtotal=ω1(a)+ω2(a) and ωtotal=ω1(b)+ω2(b), which we generate in a separate pair of mixing crystals, yields stable interference fringes with measured modulation depths of ±40%. Well-defined phase relationships are especially useful for two-photon versus two-photon coherent control experiments. In addition, the system can be used to transport, with a high degree of stability, the phase of a given input laser frequency ω0 to higher frequencies ωtotal by use of carrier lasers that need not be correlated.

© 2003 Optical Society of America

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  1. M. Shapiro and P. Brumer, Principles of the Quantum Control of Molecular Processes (Wiley, Hoboken, N.J., 2003).
  2. S. A. Rice, “Interfering for the good of a chemical reaction,” Nature 409, 422–426 (2001).
    [CrossRef] [PubMed]
  3. R. N. Zare, “Laser control of chemical reactions,” Science 279, 1875–1879 (1998).
    [CrossRef] [PubMed]
  4. R. J. Gordon and S. A. Rice, “Active control of the dynamics of atoms and molecules,” Annu. Rev. Phys. Chem. 48, 601–641 (1997).
    [CrossRef] [PubMed]
  5. M. Shapiro and P. Brumer, “Coherent control of atomic, molecular, and electronic processes,” Adv. At., Mol., Opt. Phys. 42, 287–345 (2000).
    [CrossRef]
  6. R. J. Gordon, L. Zhu, and T. Seideman, “Coherent control of chemical reactions,” Acc. Chem. Res. 32, 1007–1016 (1999).
    [CrossRef]
  7. R. Bersohn, “Coherent control of photoexcitation processes,” J. Mol. Struct. 480–481, 231–235 (1999).
    [CrossRef]
  8. M. Shapiro and P. Brumer, “Quantum control of chemical reactions,” J. Chem. Soc., Faraday Trans. 93, 1263–1277 (1997).
    [CrossRef]
  9. P. Brumer and M. Shapiro, “Coherence chemistry: controlling chemical reactions with lasers,” Acc. Chem. Res. 22, 407–413 (1989).
    [CrossRef]
  10. H. Rabitz, R. de Vivie-Riedle, M. Motzkus, and K. Kompa, “Whither the future of controlling quantum phenomena?” Science 288, 824–828 (2000).
    [CrossRef] [PubMed]
  11. A. Assion, T. Baumert, M. Bergt, T. Brixner, B. Kiefer, V. Seyfried, M. Strehle, and G. Gerber, “Control of chemical reactions by feedback-optimized phase-shaped femtosecond laser pulses,” Science 282, 919–922 (1998).
    [CrossRef] [PubMed]
  12. H. Rabitz, T. Kobayashi, R. W. Field, and D. J. Tannor, “Ramifications of feedback for control of quantum dynamics,” Adv. Chem. Phys. 101, 315–325 (1997).
  13. W. S. Warren, H. Rabitz, and M. Dahleh, “Coherent control of quantum dynamics: the dream is alive,” Science 259, 1581–1589 (1993).
    [CrossRef] [PubMed]
  14. D. J. Tannor and S. A. Rice, “Coherent pulse sequence control of product formation in chemical reactions,” Adv. Chem. Phys. 70, 441–523 (1988).
  15. P. Brumer and M. Shapiro, “Control of unimolecular reactions using coherent light,” Chem. Phys. Lett. 126, 541–546 (1986).
    [CrossRef]
  16. M. Shapiro and P. Brumer, “Laser control of product quantum state populations in unimolecular reactions,” J. Chem. Phys. 84, 4103–4104 (1986).
    [CrossRef]
  17. M. Shapiro, J. W. Hepburn, and P. Brumer, “Simplified laser control of unimolecular reactions: simultaneous (ω1, ω3) excitation,” Chem. Phys. Lett. 149, 451–454 (1988).
    [CrossRef]
  18. C. Chen, Y.-Y. Yin, and D. S. Elliott, “Interference between optical transitions,” Phys. Rev. Lett. 64, 507–510 (1990).
    [CrossRef] [PubMed]
  19. S. M. Park, S.-P. Lu, and R. J. Gordon, “Coherent laser control of the resonance-enhanced multiphoton ionization of HCl,” J. Chem. Phys. 94, 8622–8624 (1991).
    [CrossRef]
  20. S.-P. Lu, S. M. Park, Y. Xie, and R. J. Gordon, “Coherent laser control of bound-to-bound transitions of HCl and CO,” J. Chem. Phys. 96, 6613–6620 (1992).
    [CrossRef]
  21. V. D. Kleiman, L. Zhu, X. Li, and R. J. Gordon, “Coherent phase control of the photoionization of H2S,” J. Chem. Phys. 102, 5863–5866 (1995).
    [CrossRef]
  22. L. Zhu, V. D. Kleiman, X. Li, S. P. Lu, K. Trentelman, and R. J. Gordon, “Coherent laser control of the product distribution obtained in the photoexcitation of HI,” Science 270, 77–80 (1995).
    [CrossRef]
  23. X. Wang, R. Bersohn, K. Takahashi, M. Kawasaki, and H. L. Kim, “Phase control of absorption in large polyatomic molecules,” J. Chem. Phys. 105, 2992–2997 (1996).
    [CrossRef]
  24. Little modulation of the phase-sensitive signal will be ap-parent if the experimental signal from one pathway greatly exceeds that of the other.
  25. R. J. Gordon, S.-P. Lu, S. M. Park, K. Trentelman, Y. Xie, L. Zhu, A. Kumar, and W. J. Meath, “The use of coherent phase control of multiphoton ionization to measure the refractive indices of H2 and Ar between 1100 and 1150 Å,” J. Chem. Phys. 98, 9481–9486 (1993).
    [CrossRef]
  26. E. Papastathopoulos, D. Xenakis, and D. Charalambidis, “Phase-sensitive ionization through multiphoton-excitation schemes involving even numbers of photons,” Phys. Rev. A 59, 4840–4842 (1999).
    [CrossRef]
  27. Z. Chen, P. Brumer, and M. Shapiro, “Coherent radiative control of molecular photodissociation via resonant two-photon versus two-photon interference,” Chem. Phys. Lett. 198, 498–504 (1992).
    [CrossRef]
  28. Z. Chen, P. Brumer, and M. Shapiro, “Multiproduct coherent control of photodissociation via two-photon versus two-photon interference,” J. Chem. Phys. 98, 6843–6852 (1993).
    [CrossRef]
  29. S. T. Pratt, “Interference effects in the two-photon ionization of nitric oxide,” J. Chem. Phys. 104, 5776–5783 (1996).
    [CrossRef]
  30. F. Wang, C. Chen, and D. S. Elliott, “Product state control through interfering excitation routes,” Phys. Rev. Lett. 77, 2416–2419 (1996).
    [CrossRef] [PubMed]
  31. N. Ph. Georgiades, E. S. Polzik, and H. J. Kimble, “Frequency metrology by use of quantum interference,” Opt. Lett. 21, 1688–1690 (1996).
    [CrossRef]
  32. J. J. McFerran and A. N. Luiten, “Coherent bisection of 141 THz using sum frequency generation of 1064 nm and 709 nm radiation,” Opt. Quantum Electron. 34, 841–858 (2002).
    [CrossRef]
  33. N. F. Scherer, A. J. Ruggiero, M. Du, and G. R. Fleming, “Time resolved dynamics of isolated molecular systems studied with phase-locked femtosecond pulse pairs,” J. Chem. Phys. 93, 856–857 (1990).
    [CrossRef]
  34. J. H. Yi, S. H. Kim, and Y. K. Kwak, “A nanometric displacement measurement method using the detection of fringe peak movement,” Meas. Sci. Technol. 11, 1352–1358 (2000).
    [CrossRef]
  35. These red wavelengths are resonant with the transitions X 1Σg+(v=0,  J=15)→A 1Σu+(v=7,  J=14) and X 1Σg+(v=0,  J=15)→A 1Σu+(v=6,  J=14) in Na2, which are being used for coherent control photodissociation experiments currently under way in our laboratory.

2002 (1)

J. J. McFerran and A. N. Luiten, “Coherent bisection of 141 THz using sum frequency generation of 1064 nm and 709 nm radiation,” Opt. Quantum Electron. 34, 841–858 (2002).
[CrossRef]

2001 (1)

S. A. Rice, “Interfering for the good of a chemical reaction,” Nature 409, 422–426 (2001).
[CrossRef] [PubMed]

2000 (3)

M. Shapiro and P. Brumer, “Coherent control of atomic, molecular, and electronic processes,” Adv. At., Mol., Opt. Phys. 42, 287–345 (2000).
[CrossRef]

H. Rabitz, R. de Vivie-Riedle, M. Motzkus, and K. Kompa, “Whither the future of controlling quantum phenomena?” Science 288, 824–828 (2000).
[CrossRef] [PubMed]

J. H. Yi, S. H. Kim, and Y. K. Kwak, “A nanometric displacement measurement method using the detection of fringe peak movement,” Meas. Sci. Technol. 11, 1352–1358 (2000).
[CrossRef]

1999 (3)

E. Papastathopoulos, D. Xenakis, and D. Charalambidis, “Phase-sensitive ionization through multiphoton-excitation schemes involving even numbers of photons,” Phys. Rev. A 59, 4840–4842 (1999).
[CrossRef]

R. J. Gordon, L. Zhu, and T. Seideman, “Coherent control of chemical reactions,” Acc. Chem. Res. 32, 1007–1016 (1999).
[CrossRef]

R. Bersohn, “Coherent control of photoexcitation processes,” J. Mol. Struct. 480–481, 231–235 (1999).
[CrossRef]

1998 (2)

A. Assion, T. Baumert, M. Bergt, T. Brixner, B. Kiefer, V. Seyfried, M. Strehle, and G. Gerber, “Control of chemical reactions by feedback-optimized phase-shaped femtosecond laser pulses,” Science 282, 919–922 (1998).
[CrossRef] [PubMed]

R. N. Zare, “Laser control of chemical reactions,” Science 279, 1875–1879 (1998).
[CrossRef] [PubMed]

1997 (3)

R. J. Gordon and S. A. Rice, “Active control of the dynamics of atoms and molecules,” Annu. Rev. Phys. Chem. 48, 601–641 (1997).
[CrossRef] [PubMed]

H. Rabitz, T. Kobayashi, R. W. Field, and D. J. Tannor, “Ramifications of feedback for control of quantum dynamics,” Adv. Chem. Phys. 101, 315–325 (1997).

M. Shapiro and P. Brumer, “Quantum control of chemical reactions,” J. Chem. Soc., Faraday Trans. 93, 1263–1277 (1997).
[CrossRef]

1996 (4)

X. Wang, R. Bersohn, K. Takahashi, M. Kawasaki, and H. L. Kim, “Phase control of absorption in large polyatomic molecules,” J. Chem. Phys. 105, 2992–2997 (1996).
[CrossRef]

S. T. Pratt, “Interference effects in the two-photon ionization of nitric oxide,” J. Chem. Phys. 104, 5776–5783 (1996).
[CrossRef]

F. Wang, C. Chen, and D. S. Elliott, “Product state control through interfering excitation routes,” Phys. Rev. Lett. 77, 2416–2419 (1996).
[CrossRef] [PubMed]

N. Ph. Georgiades, E. S. Polzik, and H. J. Kimble, “Frequency metrology by use of quantum interference,” Opt. Lett. 21, 1688–1690 (1996).
[CrossRef]

1995 (2)

V. D. Kleiman, L. Zhu, X. Li, and R. J. Gordon, “Coherent phase control of the photoionization of H2S,” J. Chem. Phys. 102, 5863–5866 (1995).
[CrossRef]

L. Zhu, V. D. Kleiman, X. Li, S. P. Lu, K. Trentelman, and R. J. Gordon, “Coherent laser control of the product distribution obtained in the photoexcitation of HI,” Science 270, 77–80 (1995).
[CrossRef]

1993 (3)

W. S. Warren, H. Rabitz, and M. Dahleh, “Coherent control of quantum dynamics: the dream is alive,” Science 259, 1581–1589 (1993).
[CrossRef] [PubMed]

Z. Chen, P. Brumer, and M. Shapiro, “Multiproduct coherent control of photodissociation via two-photon versus two-photon interference,” J. Chem. Phys. 98, 6843–6852 (1993).
[CrossRef]

R. J. Gordon, S.-P. Lu, S. M. Park, K. Trentelman, Y. Xie, L. Zhu, A. Kumar, and W. J. Meath, “The use of coherent phase control of multiphoton ionization to measure the refractive indices of H2 and Ar between 1100 and 1150 Å,” J. Chem. Phys. 98, 9481–9486 (1993).
[CrossRef]

1992 (2)

Z. Chen, P. Brumer, and M. Shapiro, “Coherent radiative control of molecular photodissociation via resonant two-photon versus two-photon interference,” Chem. Phys. Lett. 198, 498–504 (1992).
[CrossRef]

S.-P. Lu, S. M. Park, Y. Xie, and R. J. Gordon, “Coherent laser control of bound-to-bound transitions of HCl and CO,” J. Chem. Phys. 96, 6613–6620 (1992).
[CrossRef]

1991 (1)

S. M. Park, S.-P. Lu, and R. J. Gordon, “Coherent laser control of the resonance-enhanced multiphoton ionization of HCl,” J. Chem. Phys. 94, 8622–8624 (1991).
[CrossRef]

1990 (2)

C. Chen, Y.-Y. Yin, and D. S. Elliott, “Interference between optical transitions,” Phys. Rev. Lett. 64, 507–510 (1990).
[CrossRef] [PubMed]

N. F. Scherer, A. J. Ruggiero, M. Du, and G. R. Fleming, “Time resolved dynamics of isolated molecular systems studied with phase-locked femtosecond pulse pairs,” J. Chem. Phys. 93, 856–857 (1990).
[CrossRef]

1989 (1)

P. Brumer and M. Shapiro, “Coherence chemistry: controlling chemical reactions with lasers,” Acc. Chem. Res. 22, 407–413 (1989).
[CrossRef]

1988 (2)

D. J. Tannor and S. A. Rice, “Coherent pulse sequence control of product formation in chemical reactions,” Adv. Chem. Phys. 70, 441–523 (1988).

M. Shapiro, J. W. Hepburn, and P. Brumer, “Simplified laser control of unimolecular reactions: simultaneous (ω1, ω3) excitation,” Chem. Phys. Lett. 149, 451–454 (1988).
[CrossRef]

1986 (2)

P. Brumer and M. Shapiro, “Control of unimolecular reactions using coherent light,” Chem. Phys. Lett. 126, 541–546 (1986).
[CrossRef]

M. Shapiro and P. Brumer, “Laser control of product quantum state populations in unimolecular reactions,” J. Chem. Phys. 84, 4103–4104 (1986).
[CrossRef]

Assion, A.

A. Assion, T. Baumert, M. Bergt, T. Brixner, B. Kiefer, V. Seyfried, M. Strehle, and G. Gerber, “Control of chemical reactions by feedback-optimized phase-shaped femtosecond laser pulses,” Science 282, 919–922 (1998).
[CrossRef] [PubMed]

Baumert, T.

A. Assion, T. Baumert, M. Bergt, T. Brixner, B. Kiefer, V. Seyfried, M. Strehle, and G. Gerber, “Control of chemical reactions by feedback-optimized phase-shaped femtosecond laser pulses,” Science 282, 919–922 (1998).
[CrossRef] [PubMed]

Bergt, M.

A. Assion, T. Baumert, M. Bergt, T. Brixner, B. Kiefer, V. Seyfried, M. Strehle, and G. Gerber, “Control of chemical reactions by feedback-optimized phase-shaped femtosecond laser pulses,” Science 282, 919–922 (1998).
[CrossRef] [PubMed]

Bersohn, R.

R. Bersohn, “Coherent control of photoexcitation processes,” J. Mol. Struct. 480–481, 231–235 (1999).
[CrossRef]

X. Wang, R. Bersohn, K. Takahashi, M. Kawasaki, and H. L. Kim, “Phase control of absorption in large polyatomic molecules,” J. Chem. Phys. 105, 2992–2997 (1996).
[CrossRef]

Brixner, T.

A. Assion, T. Baumert, M. Bergt, T. Brixner, B. Kiefer, V. Seyfried, M. Strehle, and G. Gerber, “Control of chemical reactions by feedback-optimized phase-shaped femtosecond laser pulses,” Science 282, 919–922 (1998).
[CrossRef] [PubMed]

Brumer, P.

M. Shapiro and P. Brumer, “Coherent control of atomic, molecular, and electronic processes,” Adv. At., Mol., Opt. Phys. 42, 287–345 (2000).
[CrossRef]

M. Shapiro and P. Brumer, “Quantum control of chemical reactions,” J. Chem. Soc., Faraday Trans. 93, 1263–1277 (1997).
[CrossRef]

Z. Chen, P. Brumer, and M. Shapiro, “Multiproduct coherent control of photodissociation via two-photon versus two-photon interference,” J. Chem. Phys. 98, 6843–6852 (1993).
[CrossRef]

Z. Chen, P. Brumer, and M. Shapiro, “Coherent radiative control of molecular photodissociation via resonant two-photon versus two-photon interference,” Chem. Phys. Lett. 198, 498–504 (1992).
[CrossRef]

P. Brumer and M. Shapiro, “Coherence chemistry: controlling chemical reactions with lasers,” Acc. Chem. Res. 22, 407–413 (1989).
[CrossRef]

M. Shapiro, J. W. Hepburn, and P. Brumer, “Simplified laser control of unimolecular reactions: simultaneous (ω1, ω3) excitation,” Chem. Phys. Lett. 149, 451–454 (1988).
[CrossRef]

M. Shapiro and P. Brumer, “Laser control of product quantum state populations in unimolecular reactions,” J. Chem. Phys. 84, 4103–4104 (1986).
[CrossRef]

P. Brumer and M. Shapiro, “Control of unimolecular reactions using coherent light,” Chem. Phys. Lett. 126, 541–546 (1986).
[CrossRef]

Charalambidis, D.

E. Papastathopoulos, D. Xenakis, and D. Charalambidis, “Phase-sensitive ionization through multiphoton-excitation schemes involving even numbers of photons,” Phys. Rev. A 59, 4840–4842 (1999).
[CrossRef]

Chen, C.

F. Wang, C. Chen, and D. S. Elliott, “Product state control through interfering excitation routes,” Phys. Rev. Lett. 77, 2416–2419 (1996).
[CrossRef] [PubMed]

C. Chen, Y.-Y. Yin, and D. S. Elliott, “Interference between optical transitions,” Phys. Rev. Lett. 64, 507–510 (1990).
[CrossRef] [PubMed]

Chen, Z.

Z. Chen, P. Brumer, and M. Shapiro, “Multiproduct coherent control of photodissociation via two-photon versus two-photon interference,” J. Chem. Phys. 98, 6843–6852 (1993).
[CrossRef]

Z. Chen, P. Brumer, and M. Shapiro, “Coherent radiative control of molecular photodissociation via resonant two-photon versus two-photon interference,” Chem. Phys. Lett. 198, 498–504 (1992).
[CrossRef]

Dahleh, M.

W. S. Warren, H. Rabitz, and M. Dahleh, “Coherent control of quantum dynamics: the dream is alive,” Science 259, 1581–1589 (1993).
[CrossRef] [PubMed]

de Vivie-Riedle, R.

H. Rabitz, R. de Vivie-Riedle, M. Motzkus, and K. Kompa, “Whither the future of controlling quantum phenomena?” Science 288, 824–828 (2000).
[CrossRef] [PubMed]

Du, M.

N. F. Scherer, A. J. Ruggiero, M. Du, and G. R. Fleming, “Time resolved dynamics of isolated molecular systems studied with phase-locked femtosecond pulse pairs,” J. Chem. Phys. 93, 856–857 (1990).
[CrossRef]

Elliott, D. S.

F. Wang, C. Chen, and D. S. Elliott, “Product state control through interfering excitation routes,” Phys. Rev. Lett. 77, 2416–2419 (1996).
[CrossRef] [PubMed]

C. Chen, Y.-Y. Yin, and D. S. Elliott, “Interference between optical transitions,” Phys. Rev. Lett. 64, 507–510 (1990).
[CrossRef] [PubMed]

Field, R. W.

H. Rabitz, T. Kobayashi, R. W. Field, and D. J. Tannor, “Ramifications of feedback for control of quantum dynamics,” Adv. Chem. Phys. 101, 315–325 (1997).

Fleming, G. R.

N. F. Scherer, A. J. Ruggiero, M. Du, and G. R. Fleming, “Time resolved dynamics of isolated molecular systems studied with phase-locked femtosecond pulse pairs,” J. Chem. Phys. 93, 856–857 (1990).
[CrossRef]

Gerber, G.

A. Assion, T. Baumert, M. Bergt, T. Brixner, B. Kiefer, V. Seyfried, M. Strehle, and G. Gerber, “Control of chemical reactions by feedback-optimized phase-shaped femtosecond laser pulses,” Science 282, 919–922 (1998).
[CrossRef] [PubMed]

Gordon, R. J.

R. J. Gordon, L. Zhu, and T. Seideman, “Coherent control of chemical reactions,” Acc. Chem. Res. 32, 1007–1016 (1999).
[CrossRef]

R. J. Gordon and S. A. Rice, “Active control of the dynamics of atoms and molecules,” Annu. Rev. Phys. Chem. 48, 601–641 (1997).
[CrossRef] [PubMed]

L. Zhu, V. D. Kleiman, X. Li, S. P. Lu, K. Trentelman, and R. J. Gordon, “Coherent laser control of the product distribution obtained in the photoexcitation of HI,” Science 270, 77–80 (1995).
[CrossRef]

V. D. Kleiman, L. Zhu, X. Li, and R. J. Gordon, “Coherent phase control of the photoionization of H2S,” J. Chem. Phys. 102, 5863–5866 (1995).
[CrossRef]

R. J. Gordon, S.-P. Lu, S. M. Park, K. Trentelman, Y. Xie, L. Zhu, A. Kumar, and W. J. Meath, “The use of coherent phase control of multiphoton ionization to measure the refractive indices of H2 and Ar between 1100 and 1150 Å,” J. Chem. Phys. 98, 9481–9486 (1993).
[CrossRef]

S.-P. Lu, S. M. Park, Y. Xie, and R. J. Gordon, “Coherent laser control of bound-to-bound transitions of HCl and CO,” J. Chem. Phys. 96, 6613–6620 (1992).
[CrossRef]

S. M. Park, S.-P. Lu, and R. J. Gordon, “Coherent laser control of the resonance-enhanced multiphoton ionization of HCl,” J. Chem. Phys. 94, 8622–8624 (1991).
[CrossRef]

Hepburn, J. W.

M. Shapiro, J. W. Hepburn, and P. Brumer, “Simplified laser control of unimolecular reactions: simultaneous (ω1, ω3) excitation,” Chem. Phys. Lett. 149, 451–454 (1988).
[CrossRef]

Kawasaki, M.

X. Wang, R. Bersohn, K. Takahashi, M. Kawasaki, and H. L. Kim, “Phase control of absorption in large polyatomic molecules,” J. Chem. Phys. 105, 2992–2997 (1996).
[CrossRef]

Kiefer, B.

A. Assion, T. Baumert, M. Bergt, T. Brixner, B. Kiefer, V. Seyfried, M. Strehle, and G. Gerber, “Control of chemical reactions by feedback-optimized phase-shaped femtosecond laser pulses,” Science 282, 919–922 (1998).
[CrossRef] [PubMed]

Kim, H. L.

X. Wang, R. Bersohn, K. Takahashi, M. Kawasaki, and H. L. Kim, “Phase control of absorption in large polyatomic molecules,” J. Chem. Phys. 105, 2992–2997 (1996).
[CrossRef]

Kim, S. H.

J. H. Yi, S. H. Kim, and Y. K. Kwak, “A nanometric displacement measurement method using the detection of fringe peak movement,” Meas. Sci. Technol. 11, 1352–1358 (2000).
[CrossRef]

Kimble, H. J.

Kleiman, V. D.

V. D. Kleiman, L. Zhu, X. Li, and R. J. Gordon, “Coherent phase control of the photoionization of H2S,” J. Chem. Phys. 102, 5863–5866 (1995).
[CrossRef]

L. Zhu, V. D. Kleiman, X. Li, S. P. Lu, K. Trentelman, and R. J. Gordon, “Coherent laser control of the product distribution obtained in the photoexcitation of HI,” Science 270, 77–80 (1995).
[CrossRef]

Kobayashi, T.

H. Rabitz, T. Kobayashi, R. W. Field, and D. J. Tannor, “Ramifications of feedback for control of quantum dynamics,” Adv. Chem. Phys. 101, 315–325 (1997).

Kompa, K.

H. Rabitz, R. de Vivie-Riedle, M. Motzkus, and K. Kompa, “Whither the future of controlling quantum phenomena?” Science 288, 824–828 (2000).
[CrossRef] [PubMed]

Kumar, A.

R. J. Gordon, S.-P. Lu, S. M. Park, K. Trentelman, Y. Xie, L. Zhu, A. Kumar, and W. J. Meath, “The use of coherent phase control of multiphoton ionization to measure the refractive indices of H2 and Ar between 1100 and 1150 Å,” J. Chem. Phys. 98, 9481–9486 (1993).
[CrossRef]

Kwak, Y. K.

J. H. Yi, S. H. Kim, and Y. K. Kwak, “A nanometric displacement measurement method using the detection of fringe peak movement,” Meas. Sci. Technol. 11, 1352–1358 (2000).
[CrossRef]

Li, X.

V. D. Kleiman, L. Zhu, X. Li, and R. J. Gordon, “Coherent phase control of the photoionization of H2S,” J. Chem. Phys. 102, 5863–5866 (1995).
[CrossRef]

L. Zhu, V. D. Kleiman, X. Li, S. P. Lu, K. Trentelman, and R. J. Gordon, “Coherent laser control of the product distribution obtained in the photoexcitation of HI,” Science 270, 77–80 (1995).
[CrossRef]

Lu, S. P.

L. Zhu, V. D. Kleiman, X. Li, S. P. Lu, K. Trentelman, and R. J. Gordon, “Coherent laser control of the product distribution obtained in the photoexcitation of HI,” Science 270, 77–80 (1995).
[CrossRef]

Lu, S.-P.

R. J. Gordon, S.-P. Lu, S. M. Park, K. Trentelman, Y. Xie, L. Zhu, A. Kumar, and W. J. Meath, “The use of coherent phase control of multiphoton ionization to measure the refractive indices of H2 and Ar between 1100 and 1150 Å,” J. Chem. Phys. 98, 9481–9486 (1993).
[CrossRef]

S.-P. Lu, S. M. Park, Y. Xie, and R. J. Gordon, “Coherent laser control of bound-to-bound transitions of HCl and CO,” J. Chem. Phys. 96, 6613–6620 (1992).
[CrossRef]

S. M. Park, S.-P. Lu, and R. J. Gordon, “Coherent laser control of the resonance-enhanced multiphoton ionization of HCl,” J. Chem. Phys. 94, 8622–8624 (1991).
[CrossRef]

Luiten, A. N.

J. J. McFerran and A. N. Luiten, “Coherent bisection of 141 THz using sum frequency generation of 1064 nm and 709 nm radiation,” Opt. Quantum Electron. 34, 841–858 (2002).
[CrossRef]

McFerran, J. J.

J. J. McFerran and A. N. Luiten, “Coherent bisection of 141 THz using sum frequency generation of 1064 nm and 709 nm radiation,” Opt. Quantum Electron. 34, 841–858 (2002).
[CrossRef]

Meath, W. J.

R. J. Gordon, S.-P. Lu, S. M. Park, K. Trentelman, Y. Xie, L. Zhu, A. Kumar, and W. J. Meath, “The use of coherent phase control of multiphoton ionization to measure the refractive indices of H2 and Ar between 1100 and 1150 Å,” J. Chem. Phys. 98, 9481–9486 (1993).
[CrossRef]

Motzkus, M.

H. Rabitz, R. de Vivie-Riedle, M. Motzkus, and K. Kompa, “Whither the future of controlling quantum phenomena?” Science 288, 824–828 (2000).
[CrossRef] [PubMed]

Papastathopoulos, E.

E. Papastathopoulos, D. Xenakis, and D. Charalambidis, “Phase-sensitive ionization through multiphoton-excitation schemes involving even numbers of photons,” Phys. Rev. A 59, 4840–4842 (1999).
[CrossRef]

Park, S. M.

R. J. Gordon, S.-P. Lu, S. M. Park, K. Trentelman, Y. Xie, L. Zhu, A. Kumar, and W. J. Meath, “The use of coherent phase control of multiphoton ionization to measure the refractive indices of H2 and Ar between 1100 and 1150 Å,” J. Chem. Phys. 98, 9481–9486 (1993).
[CrossRef]

S.-P. Lu, S. M. Park, Y. Xie, and R. J. Gordon, “Coherent laser control of bound-to-bound transitions of HCl and CO,” J. Chem. Phys. 96, 6613–6620 (1992).
[CrossRef]

S. M. Park, S.-P. Lu, and R. J. Gordon, “Coherent laser control of the resonance-enhanced multiphoton ionization of HCl,” J. Chem. Phys. 94, 8622–8624 (1991).
[CrossRef]

Ph. Georgiades, N.

Polzik, E. S.

Pratt, S. T.

S. T. Pratt, “Interference effects in the two-photon ionization of nitric oxide,” J. Chem. Phys. 104, 5776–5783 (1996).
[CrossRef]

Rabitz, H.

H. Rabitz, R. de Vivie-Riedle, M. Motzkus, and K. Kompa, “Whither the future of controlling quantum phenomena?” Science 288, 824–828 (2000).
[CrossRef] [PubMed]

H. Rabitz, T. Kobayashi, R. W. Field, and D. J. Tannor, “Ramifications of feedback for control of quantum dynamics,” Adv. Chem. Phys. 101, 315–325 (1997).

W. S. Warren, H. Rabitz, and M. Dahleh, “Coherent control of quantum dynamics: the dream is alive,” Science 259, 1581–1589 (1993).
[CrossRef] [PubMed]

Rice, S. A.

S. A. Rice, “Interfering for the good of a chemical reaction,” Nature 409, 422–426 (2001).
[CrossRef] [PubMed]

R. J. Gordon and S. A. Rice, “Active control of the dynamics of atoms and molecules,” Annu. Rev. Phys. Chem. 48, 601–641 (1997).
[CrossRef] [PubMed]

D. J. Tannor and S. A. Rice, “Coherent pulse sequence control of product formation in chemical reactions,” Adv. Chem. Phys. 70, 441–523 (1988).

Ruggiero, A. J.

N. F. Scherer, A. J. Ruggiero, M. Du, and G. R. Fleming, “Time resolved dynamics of isolated molecular systems studied with phase-locked femtosecond pulse pairs,” J. Chem. Phys. 93, 856–857 (1990).
[CrossRef]

Scherer, N. F.

N. F. Scherer, A. J. Ruggiero, M. Du, and G. R. Fleming, “Time resolved dynamics of isolated molecular systems studied with phase-locked femtosecond pulse pairs,” J. Chem. Phys. 93, 856–857 (1990).
[CrossRef]

Seideman, T.

R. J. Gordon, L. Zhu, and T. Seideman, “Coherent control of chemical reactions,” Acc. Chem. Res. 32, 1007–1016 (1999).
[CrossRef]

Seyfried, V.

A. Assion, T. Baumert, M. Bergt, T. Brixner, B. Kiefer, V. Seyfried, M. Strehle, and G. Gerber, “Control of chemical reactions by feedback-optimized phase-shaped femtosecond laser pulses,” Science 282, 919–922 (1998).
[CrossRef] [PubMed]

Shapiro, M.

M. Shapiro and P. Brumer, “Coherent control of atomic, molecular, and electronic processes,” Adv. At., Mol., Opt. Phys. 42, 287–345 (2000).
[CrossRef]

M. Shapiro and P. Brumer, “Quantum control of chemical reactions,” J. Chem. Soc., Faraday Trans. 93, 1263–1277 (1997).
[CrossRef]

Z. Chen, P. Brumer, and M. Shapiro, “Multiproduct coherent control of photodissociation via two-photon versus two-photon interference,” J. Chem. Phys. 98, 6843–6852 (1993).
[CrossRef]

Z. Chen, P. Brumer, and M. Shapiro, “Coherent radiative control of molecular photodissociation via resonant two-photon versus two-photon interference,” Chem. Phys. Lett. 198, 498–504 (1992).
[CrossRef]

P. Brumer and M. Shapiro, “Coherence chemistry: controlling chemical reactions with lasers,” Acc. Chem. Res. 22, 407–413 (1989).
[CrossRef]

M. Shapiro, J. W. Hepburn, and P. Brumer, “Simplified laser control of unimolecular reactions: simultaneous (ω1, ω3) excitation,” Chem. Phys. Lett. 149, 451–454 (1988).
[CrossRef]

M. Shapiro and P. Brumer, “Laser control of product quantum state populations in unimolecular reactions,” J. Chem. Phys. 84, 4103–4104 (1986).
[CrossRef]

P. Brumer and M. Shapiro, “Control of unimolecular reactions using coherent light,” Chem. Phys. Lett. 126, 541–546 (1986).
[CrossRef]

Strehle, M.

A. Assion, T. Baumert, M. Bergt, T. Brixner, B. Kiefer, V. Seyfried, M. Strehle, and G. Gerber, “Control of chemical reactions by feedback-optimized phase-shaped femtosecond laser pulses,” Science 282, 919–922 (1998).
[CrossRef] [PubMed]

Takahashi, K.

X. Wang, R. Bersohn, K. Takahashi, M. Kawasaki, and H. L. Kim, “Phase control of absorption in large polyatomic molecules,” J. Chem. Phys. 105, 2992–2997 (1996).
[CrossRef]

Tannor, D. J.

H. Rabitz, T. Kobayashi, R. W. Field, and D. J. Tannor, “Ramifications of feedback for control of quantum dynamics,” Adv. Chem. Phys. 101, 315–325 (1997).

D. J. Tannor and S. A. Rice, “Coherent pulse sequence control of product formation in chemical reactions,” Adv. Chem. Phys. 70, 441–523 (1988).

Trentelman, K.

L. Zhu, V. D. Kleiman, X. Li, S. P. Lu, K. Trentelman, and R. J. Gordon, “Coherent laser control of the product distribution obtained in the photoexcitation of HI,” Science 270, 77–80 (1995).
[CrossRef]

R. J. Gordon, S.-P. Lu, S. M. Park, K. Trentelman, Y. Xie, L. Zhu, A. Kumar, and W. J. Meath, “The use of coherent phase control of multiphoton ionization to measure the refractive indices of H2 and Ar between 1100 and 1150 Å,” J. Chem. Phys. 98, 9481–9486 (1993).
[CrossRef]

Wang, F.

F. Wang, C. Chen, and D. S. Elliott, “Product state control through interfering excitation routes,” Phys. Rev. Lett. 77, 2416–2419 (1996).
[CrossRef] [PubMed]

Wang, X.

X. Wang, R. Bersohn, K. Takahashi, M. Kawasaki, and H. L. Kim, “Phase control of absorption in large polyatomic molecules,” J. Chem. Phys. 105, 2992–2997 (1996).
[CrossRef]

Warren, W. S.

W. S. Warren, H. Rabitz, and M. Dahleh, “Coherent control of quantum dynamics: the dream is alive,” Science 259, 1581–1589 (1993).
[CrossRef] [PubMed]

Xenakis, D.

E. Papastathopoulos, D. Xenakis, and D. Charalambidis, “Phase-sensitive ionization through multiphoton-excitation schemes involving even numbers of photons,” Phys. Rev. A 59, 4840–4842 (1999).
[CrossRef]

Xie, Y.

R. J. Gordon, S.-P. Lu, S. M. Park, K. Trentelman, Y. Xie, L. Zhu, A. Kumar, and W. J. Meath, “The use of coherent phase control of multiphoton ionization to measure the refractive indices of H2 and Ar between 1100 and 1150 Å,” J. Chem. Phys. 98, 9481–9486 (1993).
[CrossRef]

S.-P. Lu, S. M. Park, Y. Xie, and R. J. Gordon, “Coherent laser control of bound-to-bound transitions of HCl and CO,” J. Chem. Phys. 96, 6613–6620 (1992).
[CrossRef]

Yi, J. H.

J. H. Yi, S. H. Kim, and Y. K. Kwak, “A nanometric displacement measurement method using the detection of fringe peak movement,” Meas. Sci. Technol. 11, 1352–1358 (2000).
[CrossRef]

Yin, Y.-Y.

C. Chen, Y.-Y. Yin, and D. S. Elliott, “Interference between optical transitions,” Phys. Rev. Lett. 64, 507–510 (1990).
[CrossRef] [PubMed]

Zare, R. N.

R. N. Zare, “Laser control of chemical reactions,” Science 279, 1875–1879 (1998).
[CrossRef] [PubMed]

Zhu, L.

R. J. Gordon, L. Zhu, and T. Seideman, “Coherent control of chemical reactions,” Acc. Chem. Res. 32, 1007–1016 (1999).
[CrossRef]

V. D. Kleiman, L. Zhu, X. Li, and R. J. Gordon, “Coherent phase control of the photoionization of H2S,” J. Chem. Phys. 102, 5863–5866 (1995).
[CrossRef]

L. Zhu, V. D. Kleiman, X. Li, S. P. Lu, K. Trentelman, and R. J. Gordon, “Coherent laser control of the product distribution obtained in the photoexcitation of HI,” Science 270, 77–80 (1995).
[CrossRef]

R. J. Gordon, S.-P. Lu, S. M. Park, K. Trentelman, Y. Xie, L. Zhu, A. Kumar, and W. J. Meath, “The use of coherent phase control of multiphoton ionization to measure the refractive indices of H2 and Ar between 1100 and 1150 Å,” J. Chem. Phys. 98, 9481–9486 (1993).
[CrossRef]

Acc. Chem. Res. (2)

R. J. Gordon, L. Zhu, and T. Seideman, “Coherent control of chemical reactions,” Acc. Chem. Res. 32, 1007–1016 (1999).
[CrossRef]

P. Brumer and M. Shapiro, “Coherence chemistry: controlling chemical reactions with lasers,” Acc. Chem. Res. 22, 407–413 (1989).
[CrossRef]

Adv. At., Mol., Opt. Phys. (1)

M. Shapiro and P. Brumer, “Coherent control of atomic, molecular, and electronic processes,” Adv. At., Mol., Opt. Phys. 42, 287–345 (2000).
[CrossRef]

Adv. Chem. Phys. (2)

H. Rabitz, T. Kobayashi, R. W. Field, and D. J. Tannor, “Ramifications of feedback for control of quantum dynamics,” Adv. Chem. Phys. 101, 315–325 (1997).

D. J. Tannor and S. A. Rice, “Coherent pulse sequence control of product formation in chemical reactions,” Adv. Chem. Phys. 70, 441–523 (1988).

Annu. Rev. Phys. Chem. (1)

R. J. Gordon and S. A. Rice, “Active control of the dynamics of atoms and molecules,” Annu. Rev. Phys. Chem. 48, 601–641 (1997).
[CrossRef] [PubMed]

Chem. Phys. Lett. (3)

P. Brumer and M. Shapiro, “Control of unimolecular reactions using coherent light,” Chem. Phys. Lett. 126, 541–546 (1986).
[CrossRef]

M. Shapiro, J. W. Hepburn, and P. Brumer, “Simplified laser control of unimolecular reactions: simultaneous (ω1, ω3) excitation,” Chem. Phys. Lett. 149, 451–454 (1988).
[CrossRef]

Z. Chen, P. Brumer, and M. Shapiro, “Coherent radiative control of molecular photodissociation via resonant two-photon versus two-photon interference,” Chem. Phys. Lett. 198, 498–504 (1992).
[CrossRef]

J. Chem. Phys. (9)

Z. Chen, P. Brumer, and M. Shapiro, “Multiproduct coherent control of photodissociation via two-photon versus two-photon interference,” J. Chem. Phys. 98, 6843–6852 (1993).
[CrossRef]

S. T. Pratt, “Interference effects in the two-photon ionization of nitric oxide,” J. Chem. Phys. 104, 5776–5783 (1996).
[CrossRef]

N. F. Scherer, A. J. Ruggiero, M. Du, and G. R. Fleming, “Time resolved dynamics of isolated molecular systems studied with phase-locked femtosecond pulse pairs,” J. Chem. Phys. 93, 856–857 (1990).
[CrossRef]

S. M. Park, S.-P. Lu, and R. J. Gordon, “Coherent laser control of the resonance-enhanced multiphoton ionization of HCl,” J. Chem. Phys. 94, 8622–8624 (1991).
[CrossRef]

S.-P. Lu, S. M. Park, Y. Xie, and R. J. Gordon, “Coherent laser control of bound-to-bound transitions of HCl and CO,” J. Chem. Phys. 96, 6613–6620 (1992).
[CrossRef]

V. D. Kleiman, L. Zhu, X. Li, and R. J. Gordon, “Coherent phase control of the photoionization of H2S,” J. Chem. Phys. 102, 5863–5866 (1995).
[CrossRef]

X. Wang, R. Bersohn, K. Takahashi, M. Kawasaki, and H. L. Kim, “Phase control of absorption in large polyatomic molecules,” J. Chem. Phys. 105, 2992–2997 (1996).
[CrossRef]

R. J. Gordon, S.-P. Lu, S. M. Park, K. Trentelman, Y. Xie, L. Zhu, A. Kumar, and W. J. Meath, “The use of coherent phase control of multiphoton ionization to measure the refractive indices of H2 and Ar between 1100 and 1150 Å,” J. Chem. Phys. 98, 9481–9486 (1993).
[CrossRef]

M. Shapiro and P. Brumer, “Laser control of product quantum state populations in unimolecular reactions,” J. Chem. Phys. 84, 4103–4104 (1986).
[CrossRef]

J. Chem. Soc., Faraday Trans. (1)

M. Shapiro and P. Brumer, “Quantum control of chemical reactions,” J. Chem. Soc., Faraday Trans. 93, 1263–1277 (1997).
[CrossRef]

J. Mol. Struct. (1)

R. Bersohn, “Coherent control of photoexcitation processes,” J. Mol. Struct. 480–481, 231–235 (1999).
[CrossRef]

Meas. Sci. Technol. (1)

J. H. Yi, S. H. Kim, and Y. K. Kwak, “A nanometric displacement measurement method using the detection of fringe peak movement,” Meas. Sci. Technol. 11, 1352–1358 (2000).
[CrossRef]

Nature (1)

S. A. Rice, “Interfering for the good of a chemical reaction,” Nature 409, 422–426 (2001).
[CrossRef] [PubMed]

Opt. Lett. (1)

Opt. Quantum Electron. (1)

J. J. McFerran and A. N. Luiten, “Coherent bisection of 141 THz using sum frequency generation of 1064 nm and 709 nm radiation,” Opt. Quantum Electron. 34, 841–858 (2002).
[CrossRef]

Phys. Rev. A (1)

E. Papastathopoulos, D. Xenakis, and D. Charalambidis, “Phase-sensitive ionization through multiphoton-excitation schemes involving even numbers of photons,” Phys. Rev. A 59, 4840–4842 (1999).
[CrossRef]

Phys. Rev. Lett. (2)

F. Wang, C. Chen, and D. S. Elliott, “Product state control through interfering excitation routes,” Phys. Rev. Lett. 77, 2416–2419 (1996).
[CrossRef] [PubMed]

C. Chen, Y.-Y. Yin, and D. S. Elliott, “Interference between optical transitions,” Phys. Rev. Lett. 64, 507–510 (1990).
[CrossRef] [PubMed]

Science (5)

W. S. Warren, H. Rabitz, and M. Dahleh, “Coherent control of quantum dynamics: the dream is alive,” Science 259, 1581–1589 (1993).
[CrossRef] [PubMed]

H. Rabitz, R. de Vivie-Riedle, M. Motzkus, and K. Kompa, “Whither the future of controlling quantum phenomena?” Science 288, 824–828 (2000).
[CrossRef] [PubMed]

A. Assion, T. Baumert, M. Bergt, T. Brixner, B. Kiefer, V. Seyfried, M. Strehle, and G. Gerber, “Control of chemical reactions by feedback-optimized phase-shaped femtosecond laser pulses,” Science 282, 919–922 (1998).
[CrossRef] [PubMed]

R. N. Zare, “Laser control of chemical reactions,” Science 279, 1875–1879 (1998).
[CrossRef] [PubMed]

L. Zhu, V. D. Kleiman, X. Li, S. P. Lu, K. Trentelman, and R. J. Gordon, “Coherent laser control of the product distribution obtained in the photoexcitation of HI,” Science 270, 77–80 (1995).
[CrossRef]

Other (3)

Little modulation of the phase-sensitive signal will be ap-parent if the experimental signal from one pathway greatly exceeds that of the other.

These red wavelengths are resonant with the transitions X 1Σg+(v=0,  J=15)→A 1Σu+(v=7,  J=14) and X 1Σg+(v=0,  J=15)→A 1Σu+(v=6,  J=14) in Na2, which are being used for coherent control photodissociation experiments currently under way in our laboratory.

M. Shapiro and P. Brumer, Principles of the Quantum Control of Molecular Processes (Wiley, Hoboken, N.J., 2003).

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Figures (6)

Fig. 1
Fig. 1

Schematic diagram for a two-photon two-pathway optical system having distinct but degenerate optical pathways. For the frequency-mixing experiments described in this paper, the sequential steps correspond to mixing in nonlinear optical crystals. For molecular phase control experiments, the states excited by frequencies ω1(a) and ω1(b) could be resonant intermediate states, thereby enhancing and balancing the competing pathways. The ω2(a) and ω2(b) frequencies were chosen to satisfy the degeneracy condition ω1(a)+ω2(a)=ω1(b)+ω2(b).

Fig. 2
Fig. 2

Schematic diagram of the optical layout. Two dye laser frequencies [ω1(a) and ω1(b)] were generated independently and combined with the fundamental output of a standard Nd:YAG laser (ω0) in separate LBO crystals to produce additional frequencies labeled as ω2(b) and ω2(a), respectively (see text with regard to the labeling). The Nd:YAG fundamental for mixing in each LBO was produced at beam splitter BS. All four visible frequencies were combined by neutral-density beam recombiner BR1 and relayed over some distance to an optical system that consists of two separate BBO crystals. The actual layout uses two mirrors over a 0.7-m distance in this intermediate segment (not shown). Interference is produced at final beam recombiner BR2 and transmitted through two UV bandpass filters onto a linear array CCD whose output is digitized by an oscilloscope. Computer control of the overall optical phase difference was achieved by rotation of a glass phase plate (ΦP) inserted into one arm of the Nd:YAG fundamental laser beam ω0.

Fig. 3
Fig. 3

Spatial profiles of the UV lasers when generated separately (four left-hand traces) and when generated simultaneously (two right-hand traces). The lower plots were obtained after we set a small path-length difference for the Nd:YAG laser (ΔlYAG<2 mm), whereas the upper plots show no interference when the path-length difference was large (ΔlYAG=15 mm) compared with the coherence length of lcoh10 mm. All the profiles were measured on a 2048-pixel linear CCD array for single shots of the Nd:YAG laser.

Fig. 4
Fig. 4

Upper panel exhibits short-term stability by comparison of CCD fringes for a single shot of the Nd:YAG laser with those obtained by averaging 64 successive shots on a digital oscilloscope. The lower panel exhibits long-term stability with a measured phase drift of <1°/min that is due to the combined influence of all passive disturbances (i.e., the phase plate is fixed).

Fig. 5
Fig. 5

Calculated (continuous curves) and measured (points) phase differences for the He–Ne reference and four-color lasers as functions of the phase plate angle of incidence. Since the vertical scale is arbitrary for experimental measurements at each wavelength, agreement with each calculated curve is indicated by the parallelism of the respective data.

Fig. 6
Fig. 6

UV intensities measured by integration of the CCD signal when the two UV beams are aligned to be parallel. Since there are no fringes, each point is simply the total measured intensity for 16 laser shots at each setting of the phase plate from 1.28° to 3.38°. The average of three successive scans taken over 13 min is shown. The abscissa scale is taken from calculated curve (a) in Fig. 5.

Equations (11)

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

ω2(b)=ω1(a)+ω0,ω2(a)=ω1(b)+ω0.
ω1(a)+ω2(a)=ω1(b)+ω2(b).
ωtotal(a)=ω1(a)+ω2(a)=ω1(a)+[ω1(b)+ω0]
ωtotal(b)=ω1(b)+ω2(b)=ω1(b)+[ω1(a)+ω0]
ϕtotal(a)=ϕ1(a)+ϕ2(a)
ϕtotal(b)=ϕ1(b)+ϕ2(b)
ϕ2(a)=ϕ1(b)+ϕ0,ϕ2(b)=ϕ1(a)+ϕ0+Δϕ0.
Δϕtotalϕtotal(b)-ϕtotal(a)={ϕ1(b)+[ϕ1(a)+ϕ0+Δϕ0]}-{ϕ1(a)+[ϕ1(b)+ϕ0]}=Δϕ0.
ϕi(j)=ϕi(j)+2πΔl/λi(j)=ϕi(j)+ωi(j)Δl/c
(i=1, 2; j=a, b).
Δϕtotal=[ϕ1(b)+ϕ2(b)]-[ϕ1(a)+ϕ2(a)]=Δϕ0+(Δl/c)[ω1(b)+ω2(b)-ω1(a)-ω2(a)]=Δϕ0,

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