Abstract

The dependence of the population of the target state, and the excitation cross section and rate, on pulse duration Q, laser intensity I, and a molecular matrix element A is discussed for two-photon molecular excitation. Perturbative and rotating wave approximation (RWA) expressions for the observables are obtained; the latter are used to discuss the validity of the former. For example, the perturbative cross section and rate increasingly underestimate the RWA results as I increases for given A and Q. Two- and ten-level model dipolar molecules are employed for illustrative purposes. The results are relevant for understanding two-photon excitation processes and their enhancement and include discussions of the roles the permanent dipole- and virtual-state mechanisms in such processes and of the validity of using intensity-independent cross sections to gauge the strength of such excitations.

© 2008 Optical Society of America

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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef] [PubMed]
  46. I. Schek, J. Jortner, and M. L. Sage, “Application of the Magnus expansion for higher-order multiphoton excitation,” Chem. Phys. 59, 11-27 (1981).
    [CrossRef]
  47. M. Quack, “Reaction dynamics and statistical mechanics of the preparation of highly excited states by intense infrared radiation,” Adv. Chem. Phys. 50, 395-473 (1982).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]

2007 (2)

K. S. Samkoe, A. A. Clancy, A. Karotki, B. C. Wilson, and D. T. Cramb, “Complete blood vessel occlusion in the chick chorioallantoic membrane using two-photon excitation photodynamic therapy: implications for treatment of wet age-related macular degeneration,” J. Biomed. Opt. 12, 034025 (2007).
[CrossRef] [PubMed]

J. Fu, L. A. Padilha, D. J. Hagan, E. W. Van Stryland, O. V. Przhonska, M. V. Bondar, Y. L. Slominsky, and A. D. Kachkovski, “Molecular structure-two-photon absorption property relations for polymethine dyes,” J. Opt. Soc. Am. B 24, 56-66 (2007).
[CrossRef]

2006 (4)

S. R. Marder, “Organic nonlinear optical materials: where we have been and where we are going,” Chem. Commun. (Cambridge) 2006, 131-134 (2006).
[CrossRef]

M. Khurana, A. Karotki, H. Collins, H. L. Anderson, and B. C. Wilson, “In vitro studies of the efficiency of two-photon activation of photodynamic therapy agents,” Proc. SPIE 6343, 634306 (2006).
[CrossRef]

A. Karotki, M. Khurana, J. P. Lepock, and B. C. Wilson, “Simultaneous two-photon excitation of photofrin in relation to photodynamic therapy,” Photochem. Photobiol. 82, 443-452 (2006).
[CrossRef] [PubMed]

W. J. Meath, B. N. Jagatap, and A. E. Kondo, “The mechanisms for, and the enhancement of, the simultaneous absorption of two photons by molecules,” J. Phys. B 39, S605-S620 (2006).
[CrossRef]

2005 (1)

C. W. Spangler, J. R. Starkey, F. Meng, A. Gong, M. Drobizhev, A. Rabane, and B. Moss, “Targeted two-photon photodynamic therapy for the treatment of subcutaneous tumors,” Proc. SPIE 5689, 141-148 (2005).
[CrossRef]

2004 (3)

G. S. He, T. C. Lin. J. Dai, P. N. Prasad, R. Kannan, A. G. Dombroskie, R. A. Vaia, and L. S. Tan, “Degenerate two-photon absorption spectral studies of highly two-photon active organic chromophores,” J. Chem. Phys. 120, 5275-5284 (2004).
[CrossRef] [PubMed]

E. Zojer, D. Beljonne, P. Pacher, and J. L. Brédas, “Two-photon absorption in quadrupolar π-conjugated molecules: influence of the nature of the conjugated bridge and the donor-acceptor separation,” Chem.-Eur. J. 10, 2668-2680 (2004).
[CrossRef] [PubMed]

J. M. Dela Cruz, I. Pastirk, M. Comstock, V. V. Lozovoy, and M. Dantus, “Use of coherent control methods through scattering biological tissue to achieve functional imaging,” Proc. Natl. Acad. Sci. U.S.A. 101, 16996-17001 (2004).
[CrossRef] [PubMed]

2002 (1)

2001 (2)

C. K. Wang, P. Macak, Y. Luo, and H. Ågren, “Effects of π centres and symmetry on two-photon absorption cross sections of organic chromophores,” J. Chem. Phys. 114, 9813-9820 (2001).
[CrossRef]

D. T. Cramb and R. Goyan, “Dynamic behaviour of photosensitizers after multiphoton excitation,” Proc. SPIE 4262, 41-47 (2001).
[CrossRef]

2000 (3)

K. König, “Multiphoton microscopy in life science,” J. Microsc. 200, 83-104 (2000).
[CrossRef] [PubMed]

A. Brown, W. J. Meath and P. Tran, “Rotating-wave approximation for the interaction of a pulsed laser with a two-level system possessing permanent dipole moments,” Phys. Rev. A 63, 013403 (2000).
[CrossRef]

B. N. Jagatap and W. J. Meath, “On the control of the production of hydrogen atom 2s-2p resonance hybrids through the use of competitive one- and two-photon transitions from the ground state,” J. Chem. Phys. 113, 1501-1507 (2000).
[CrossRef]

1999 (1)

A. Jenei, A. K. Krisch, V. Subramaniam, D. J. Arndt-Jovin, and T. M. Jovin, “Picosecond multiphoton scanning near-field optical microscopy,” Biophys. J. 76, 1092-1100 (1999).
[CrossRef] [PubMed]

1998 (3)

T. J. Dougherty, C. J. Gomer, B. W. Henderson, G. Jori, D. Kessel, M. Korbelik, J. Moan, and Q. Peng, “Photodynamic Therapy,” J. Natl. Cancer Inst. 90, 889-905 (1998).
[CrossRef] [PubMed]

A. Salem and W. J. Meath, “On enantiomeric excesses obtained from racemic mixtures by using circularly polarized pulsed lasers of varying duration,” Chem. Phys. 288, 115-129 (1998).
[CrossRef]

M. Albota, D. Beljonne, J. L. Brédas, J. E. Ehrlich, J. Y. Fu, A. A. Heikal, S. E. Hess, T. Kogej, M. D. Levin, S. R. Marder, D. McCord-Maughon, J. W. Perry, H. Röckel, M. Rumi, G. Subramaniam, W. W. Webb, X. L. Wu, and C. Wu, “Design of organic molecules with large two-photon absorption cross sections,” Science 281, 1653-1656 (1998).
[CrossRef] [PubMed]

1996 (2)

A. Brown and W. J. Meath, “Role of permanent dipoles and orientational averaging in the phase control of two-colour, simultaneous one- and three-photon molecular excitations,” Phys. Rev. A 53, 2571-2586 (1996).
[CrossRef] [PubMed]

C. Xu, W. Zipfel, J. B. Shear, R. M. Williams, and W. W. Webb, “Multiphoton fluorescence excitation: new spectral window for biological nonlinear microscopy,” Proc. Natl. Acad. Sci. U.S.A. 93, 10763-10768 (1996).
[CrossRef] [PubMed]

1994 (1)

A. E. Kondo, W. J. Meath, S. H. Nilar, and A. J. Thakkar, “Pump-probe studies of the effects of permanent dipoles in one- and two-colour molecular excitations,” Chem. Phys. 186, 375-394 (1994).
[CrossRef]

1993 (1)

S. H. Nilar, A. J. Thakkar, A. E. Kondo, and W. J. Meath, “Electronic energies, dipole moment matrix elements, and static polarizabilities and hyperpolarizabilities for some diphenyl molecules,” Can. J. Chem. 71, 1663-1671 (1993).
[CrossRef]

1992 (1)

S. Nakai and W. J. Meath, “The rotating wave approximation including the incorporation and importance of diagonal dipole moment matrix elements, for infrared multiphoton excitations,” J. Chem. Phys. 96, 4991-5008 (1992).
[CrossRef]

1990 (2)

M. A. Kmetic and W. J. Meath, “Perturbative corrections to the rotating-wave approximation for two-level molecules and the effects of permanent dipoles on single-photon and multiphoton spectra,” Phys. Rev. A 41, 1556-1568 (1990).
[CrossRef] [PubMed]

W. Denk, J. H. Strickler, and W. W. Webb, “Two-photon laser scanning fluorescence microscopy,” Science 248, 73-76 (1990).
[CrossRef] [PubMed]

1989 (1)

W. J. Meath, R. A. Thuraisingham, and M. A. Kmetic, “Applications of the Riemann product integral method to spectroscopic problems,” Adv. Chem. Phys. 73, 307-349 (1989).
[CrossRef]

1987 (1)

T. Hattori and T. Kobayashi, “Bloch-Siegert shift in giant dipole molecules,” Phys. Rev. A 35, 2733-2736 (1987).
[CrossRef] [PubMed]

1986 (1)

B. C. Wilson and M. S. Patterson, “The physics of photodynamic therapy,” Phys. Med. Biol. 31, 327-360 (1986).
[CrossRef] [PubMed]

1984 (3)

W. J. Meath and E. A. Power, “On the importance of permanent moments in multiphoton absorption using perturbation theory,” J. Phys. B 17, 763-781 (1984).
[CrossRef]

A. Bambini and M. Lindberg, “Transition probability of a two-level atom interacting with a time-symmetric pulse,” Phys. Rev. A 30, 794-802 (1984).
[CrossRef]

K. B. Whaley and J. C. Light, “Rotating-frame transformations: a new approximation for multiphoton absorption and dissociation in laser fields,” Phys. Rev. A 29, 1188-1207 (1984).
[CrossRef]

1982 (2)

M. Quack, “Reaction dynamics and statistical mechanics of the preparation of highly excited states by intense infrared radiation,” Adv. Chem. Phys. 50, 395-473 (1982).
[CrossRef]

B. Dick and G. Hohlneicher, “Importance of initial and final states as intermediate states in two-photon spectroscopy of polar molecules,” J. Chem. Phys. 76, 5755-5760 (1982).
[CrossRef]

1981 (1)

I. Schek, J. Jortner, and M. L. Sage, “Application of the Magnus expansion for higher-order multiphoton excitation,” Chem. Phys. 59, 11-27 (1981).
[CrossRef]

1977 (1)

J. L. Oudar and D. S. Chemla, “Hyperpolarizabilities of the nitroanilines and their relation to the excited state dipole moment,” J. Chem. Phys. 66, 2664-2668 (1977).
[CrossRef]

1972 (1)

P. W. Langhoff, S. T. Epstein, and M. Karplus, “Aspects of time-dependent perturbation theory,” Rev. Mod. Phys. 44, 602-644 (1972).
[CrossRef]

1965 (1)

J. H. Shirley, “Solution of the Schrödinger equation with a Hamiltonian periodic in time,” Phys. Rev. B 138, 979-987 (1965).
[CrossRef]

1963 (1)

J. H. Shirley, “Some causes of resonant frequency shifts in atomic beam machines. I. Shifts due to other frequencies of excitation,” J. Appl. Phys. 34, 783-788 (1963).
[CrossRef]

1932 (1)

N. Rosen and C. Zerner, “Double Stern-Gerlach experiment and related collision phenomena,” Phys. Rev. 40, 502-507 (1932).
[CrossRef]

1931 (1)

M. Göppert-Mayer, “Uber elementarakte mit zwei quantensprüngen,” Ann. Phys. 9, 273-295 (1931).
[CrossRef]

1085 (1)

M. A. Kmetic and W. J. Meath, “Permanent dipole moments and multi-photon resonances,” Phys. Lett. 108A, 340-343 (1085).
[CrossRef]

Ågren, H.

C. K. Wang, P. Macak, Y. Luo, and H. Ågren, “Effects of π centres and symmetry on two-photon absorption cross sections of organic chromophores,” J. Chem. Phys. 114, 9813-9820 (2001).
[CrossRef]

Albota, M.

M. Albota, D. Beljonne, J. L. Brédas, J. E. Ehrlich, J. Y. Fu, A. A. Heikal, S. E. Hess, T. Kogej, M. D. Levin, S. R. Marder, D. McCord-Maughon, J. W. Perry, H. Röckel, M. Rumi, G. Subramaniam, W. W. Webb, X. L. Wu, and C. Wu, “Design of organic molecules with large two-photon absorption cross sections,” Science 281, 1653-1656 (1998).
[CrossRef] [PubMed]

Anderson, H. L.

M. Khurana, A. Karotki, H. Collins, H. L. Anderson, and B. C. Wilson, “In vitro studies of the efficiency of two-photon activation of photodynamic therapy agents,” Proc. SPIE 6343, 634306 (2006).
[CrossRef]

Arndt-Jovin, D. J.

A. Jenei, A. K. Krisch, V. Subramaniam, D. J. Arndt-Jovin, and T. M. Jovin, “Picosecond multiphoton scanning near-field optical microscopy,” Biophys. J. 76, 1092-1100 (1999).
[CrossRef] [PubMed]

Bambini, A.

A. Bambini and M. Lindberg, “Transition probability of a two-level atom interacting with a time-symmetric pulse,” Phys. Rev. A 30, 794-802 (1984).
[CrossRef]

Beljonne, D.

E. Zojer, D. Beljonne, P. Pacher, and J. L. Brédas, “Two-photon absorption in quadrupolar π-conjugated molecules: influence of the nature of the conjugated bridge and the donor-acceptor separation,” Chem.-Eur. J. 10, 2668-2680 (2004).
[CrossRef] [PubMed]

M. Albota, D. Beljonne, J. L. Brédas, J. E. Ehrlich, J. Y. Fu, A. A. Heikal, S. E. Hess, T. Kogej, M. D. Levin, S. R. Marder, D. McCord-Maughon, J. W. Perry, H. Röckel, M. Rumi, G. Subramaniam, W. W. Webb, X. L. Wu, and C. Wu, “Design of organic molecules with large two-photon absorption cross sections,” Science 281, 1653-1656 (1998).
[CrossRef] [PubMed]

Birge, R. P.

R. P. Birge, “One-photon and two-photon excitation spectroscopy,” in Ultrasensitive Laser Spectroscopy, D.S.Kliger, ed. (Academic, 1983), pp. 109-174.

Bondar, M. V.

Brédas, J. L.

E. Zojer, D. Beljonne, P. Pacher, and J. L. Brédas, “Two-photon absorption in quadrupolar π-conjugated molecules: influence of the nature of the conjugated bridge and the donor-acceptor separation,” Chem.-Eur. J. 10, 2668-2680 (2004).
[CrossRef] [PubMed]

M. Albota, D. Beljonne, J. L. Brédas, J. E. Ehrlich, J. Y. Fu, A. A. Heikal, S. E. Hess, T. Kogej, M. D. Levin, S. R. Marder, D. McCord-Maughon, J. W. Perry, H. Röckel, M. Rumi, G. Subramaniam, W. W. Webb, X. L. Wu, and C. Wu, “Design of organic molecules with large two-photon absorption cross sections,” Science 281, 1653-1656 (1998).
[CrossRef] [PubMed]

Brown, A.

A. Brown, W. J. Meath and P. Tran, “Rotating-wave approximation for the interaction of a pulsed laser with a two-level system possessing permanent dipole moments,” Phys. Rev. A 63, 013403 (2000).
[CrossRef]

A. Brown and W. J. Meath, “Role of permanent dipoles and orientational averaging in the phase control of two-colour, simultaneous one- and three-photon molecular excitations,” Phys. Rev. A 53, 2571-2586 (1996).
[CrossRef] [PubMed]

Chemla, D. S.

J. L. Oudar and D. S. Chemla, “Hyperpolarizabilities of the nitroanilines and their relation to the excited state dipole moment,” J. Chem. Phys. 66, 2664-2668 (1977).
[CrossRef]

Clancy, A. A.

K. S. Samkoe, A. A. Clancy, A. Karotki, B. C. Wilson, and D. T. Cramb, “Complete blood vessel occlusion in the chick chorioallantoic membrane using two-photon excitation photodynamic therapy: implications for treatment of wet age-related macular degeneration,” J. Biomed. Opt. 12, 034025 (2007).
[CrossRef] [PubMed]

Collins, H.

M. Khurana, A. Karotki, H. Collins, H. L. Anderson, and B. C. Wilson, “In vitro studies of the efficiency of two-photon activation of photodynamic therapy agents,” Proc. SPIE 6343, 634306 (2006).
[CrossRef]

Comstock, M.

J. M. Dela Cruz, I. Pastirk, M. Comstock, V. V. Lozovoy, and M. Dantus, “Use of coherent control methods through scattering biological tissue to achieve functional imaging,” Proc. Natl. Acad. Sci. U.S.A. 101, 16996-17001 (2004).
[CrossRef] [PubMed]

Cramb, D. T.

K. S. Samkoe, A. A. Clancy, A. Karotki, B. C. Wilson, and D. T. Cramb, “Complete blood vessel occlusion in the chick chorioallantoic membrane using two-photon excitation photodynamic therapy: implications for treatment of wet age-related macular degeneration,” J. Biomed. Opt. 12, 034025 (2007).
[CrossRef] [PubMed]

D. T. Cramb and R. Goyan, “Dynamic behaviour of photosensitizers after multiphoton excitation,” Proc. SPIE 4262, 41-47 (2001).
[CrossRef]

Dai, T. C. Lin. J.

G. S. He, T. C. Lin. J. Dai, P. N. Prasad, R. Kannan, A. G. Dombroskie, R. A. Vaia, and L. S. Tan, “Degenerate two-photon absorption spectral studies of highly two-photon active organic chromophores,” J. Chem. Phys. 120, 5275-5284 (2004).
[CrossRef] [PubMed]

Dantus, M.

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M. Khurana, A. Karotki, H. Collins, H. L. Anderson, and B. C. Wilson, “In vitro studies of the efficiency of two-photon activation of photodynamic therapy agents,” Proc. SPIE 6343, 634306 (2006).
[CrossRef]

C. W. Spangler, J. R. Starkey, F. Meng, A. Gong, M. Drobizhev, A. Rabane, and B. Moss, “Targeted two-photon photodynamic therapy for the treatment of subcutaneous tumors,” Proc. SPIE 5689, 141-148 (2005).
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W. J. Meath, B. N Jagatap, and A. E. Kondo, are preparing a paper to be called “Effective two-level RWA for two-photon transitions in many-level molecules: the effects of permanent dipole moments.”

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Figures (2)

Fig. 1
Fig. 1

Steady-state populations of the excited states for the two-photon 1 2 and 1 5 excitations of the model molecule DANDY. The results are for a 100 fs pulse as a function of laser intensity I. The RWA results correspond to the total laser–molecule coupling ( d + v ) and the individual permanent dipole ( d ) and virtual-state ( v ) mechanisms. Also shown are the perturbative results for the total laser–molecule couplings (pert).

Fig. 2
Fig. 2

Laser intensity dependence of the correction function G corr that relates the RWA excitation cross sections and rates to the corresponding perturbative results for the two-photon 1 2 and 1 5 excitations of the model molecule DANDY. In each case results are given for the total laser–molecule coupling ( d + v ) and the individual permanent dipole ( d ) and virtual-state ( v ) mechanisms and for a pulse duration of 100 fs . Also included is G corr for the ( d + v ) 1 2 excitation for a pulse duration of 9.8332 × 100 fs ps (see main text).

Tables (1)

Tables Icon

Table 1 Universal Results for the RWA [ P f ( ) ] and Perturbative [ P f , X < ( ) ] Populations of the State f Formed by Two-Photon Excitation of the Ground State and for the Correction Function G corr Relating the RWA and Perturbative Results for the Two-Photon Excitation Cross Sections and Rates as a Function of the Variable Y = A I Q

Equations (29)

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P f ( t ) = sin 2 ( X ( t ) ) = sin 2 ( 1 2 t 0 t C ( t ) d t ) ,
C pert pulse ( t ) = C pert CW f 2 ( t ) = C d , pert pulse ( t ) + C v , pert pulse ( t ) = [ C d , pert CW + C v , pert CW ] f 2 ( t ) .
X pulse ( t ) = C pert CW 2 t f 2 ( t ) d t = C pert CW 2 π 2 τ ( 1 0.5 erfc ( 2 t τ ) ) ,
d P g ( t ) d t = R ( t ) P g ( t ) + R ( t ) P f ( t ) ,
d P f ( t ) d t = R ( t ) P g ( t ) R ( t ) P f ( t ) .
R ( t ) = C pert pulse ( t ) 2 tan ( 2 X pulse ( t ) ) .
σ ( t ) = C pert pulse ( t ) 2 F 2 tan ( 2 X pulse ( t ) ) = ( ω ) 2 C pert pulse ( t ) 2 I 2 tan ( 2 X pulse ( t ) ) .
C pert CW = A I = C d , pert CW + C v , pert CW = ( A d + A v ) I ,
A d = [ 4 π ( 2 c ) ] 1 ω ( e ̂ μ ̱ g f ) ( e ̂ d ̱ f g ) ,
A v = [ 4 π ( 2 c ) ] j g , f ( e ̂ μ ̱ g j ) ( e ̂ μ ̱ j f ) ( ω j g ω ) .
P f ( ) = sin 2 ( C pert CW 2 π 2 τ ) = sin 2 ( A I 4 ln 2 π 2 Q ) = sin 2 ( 0.37635 A I Q ) ,
A I Q ( P f ( ) ) = ( 4 ln 2 π 2 ) sin 1 ( P f ( ) ) = 2.6571 sin 1 ( P f ( ) ) ,
P f , X < ( ) = [ X pulse ( ) ] 2 = π 32 ln 2 A 2 I 2 Q 2 = 0.14164 A 2 I 2 Q 2 .
R ¯ = 1 Q R ( t ) d t = C pert CW 2 Q e 2 t 2 τ 2 tan ( 2 X pulse ( t ) ) d t ,
R ¯ X < = C pert CW Q e 2 t 2 τ 2 X pulse ( t ) d t ,
R ¯ = A I 4 ln 2 e 2 s 2 tan [ A I π 2 Q 2 ln 2 ( 1 0.5 erfc ( 2 s ) ) ] d s = G corr R ¯ x < ,
R ¯ X < = A 2 I 2 16 ln 2 π Q [ 1 0.5 2 π e 2 s 2 erfc ( 2 s ) d s ] = A 2 I 2 32 ln 2 π Q .
G corr = R ¯ R ¯ X < = [ 1 + ( 1 24 ) ( π ( 2 ln 2 ) A I Q ) 2 + ( 1 360 ) ( π ( 2 ln 2 ) A I Q ) 4 + ] .
σ ¯ = R ¯ F 2 = G corr R ¯ X < F 2 = G corr σ ¯ X < ,
σ ¯ X < = R ¯ X < F 2 = A 2 ( ω ) 2 32 ln 2 π Q .
c ¯ = R ¯ I 2 = G corr R ¯ X < I 2 = G corr c ¯ X < ,
c ¯ X < = R ¯ X < I 2 = A 2 32 ln 2 π Q .
A = A d + A v = ( 71.61 + 6.60 ) cm 2 J ( 1 2 ) , = ( 71.24 + 174.01 ) cm 2 J ( 1 5 ) .
Y s = A I Q ( 0.5 ) = ( 4 ln 2 π 2 ) sin 1 ( 0.5 ) 2.0870 .
σ ¯ X < ( 1 2 ) = 7.969 × 10 35 Q cm 4 ( d + v ) , = 6.681 × 10 35 Q cm 4 ( d ) , = 5.675 × 10 37 Q cm 4 ( v ) ,
σ ¯ X < ( 1 5 ) = 1.680 × 10 33 Q cm 4 ( d + v ) , = 1.417 × 10 34 Q cm 4 ( d ) , = 8.456 × 10 34 Q cm 4 ( v ) ,
c ¯ X < ( 1 2 , i ) = 1.087 × 10 37 J 2 σ ¯ X < ( 1 2 , i ) ,
c ¯ X < ( 1 5 , i ) = 5.072 × 10 36 J 2 σ ¯ X < ( 1 5 , i ) ,
R ¯ X < ( A ) R ¯ X < ( B ) = c ¯ X < ( A ) c ¯ X < ( B ) = ( ω ( B ) ) 2 σ ¯ X < ( A ) ( ω ( A ) ) 2 σ ¯ X < ( B ) ,

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