Abstract

A beam shaper for dynamic transversal shaping of broadband laser pulses that utilizes a blazed ruled grating and a blazed-type grating simulated on Spatial Light Modulator was demonstrated. The introduced shaper scheme is an extension of 2f-2f scheme [Mariyenko, et al., Opt. Express 13, 7599 (2005)] where the two thin holograms with matched grating constants performed light shaping. The new scheme utilizes the diffraction gratings with different grating constants. Dispersion-free light shaping is achieved by means of the intermediate transversal light beam magnification. The magnification balances the mismatch in the grating constants resulting in total residual angular dispersion compensation. In turn, the magnified beam covers a greater area on the modulator matrix thus reducing the incident light power density by a value equal to square of the magnification factor. It translates to the safe-operation threshold extension of the modulator allowing shaping pulses that are powerful enough to be used in the applications. With a proper components selection, the throughput efficiency of the shaper can be well above 40%. A proper shaper operation was demonstrated with the 140-fs Ti:Sapphire oscillator. Theoretical calculations support the conclusions.

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References

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  1. M. Vasnetsov, and K. Staliunas, eds., “Optical Vortices,” in Horizons in World Physics, (Nova Science Publishers, Commack, New York, 1999) Vol. 228.
  2. L. Allen, S. M. Barnett, and M. J. Padgett, Optical Angular Momentum, (Taylor & Francis, 2003).
  3. D. G. Grier, “A revolution in optical manipulation,” Nature 424(6950), 810–816 (2003).
    [CrossRef] [PubMed]
  4. A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature 412(6844), 313–316 (2001).
    [CrossRef] [PubMed]
  5. M. F. Andersen, C. Ryu, P. Cladé, V. Natarajan, A. Vaziri, K. Helmerson, and W. D. Phillips, “Quantized rotation of atoms from photons with orbital angular momentum,” Phys. Rev. Lett. 97(17), 170406 (2006).
    [CrossRef] [PubMed]
  6. L. Bergé, S. Skupin, R. Nuter, J. Kasparian, and J.-P. Wolf, “Ultrashort filaments of light in weakly ionized, optically transparent media,” Rep. Prog. Phys. 70(10), 1633–1713 (2007).
    [CrossRef]
  7. K. Cook, R. McGeorge, A. K. Kar, M. R. Taghizadeh, and R. A. Lamb, “Coherent array of white-light continuum filaments produced by diffractive microlenses,” Appl. Phys. Lett. 86(2), 021105 (2005).
    [CrossRef]
  8. Z. Jin, J. Zhang, M. H. Xu, X. Lu, Y. T. Li, Z. H. Wang, Z. Y. Wei, X. H. Yuan, and W. Yu, “Control of filamentation induced by femtosecond laser pulses propagating in air,” Opt. Express 13(25), 10424–10430 (2005).
    [CrossRef] [PubMed]
  9. V. P. Kandidov, N. Ak¨ozbek, M. Scalora, O. G. Kosareva, A. V. Nyakk, Q. Luo, S. A. Hosseini, and S. L. Chin, “Towards a control of multiple filamentation by spatial regularization of a high-power femtosecond laser pulse,” Appl. Phys. B 80(2), 267–275 (2005).
    [CrossRef]
  10. A. Vinçotte and L. Bergé, “Femtosecond optical vortices in air,” Phys. Rev. Lett. 95(19), 193901 (2005).
    [CrossRef] [PubMed]
  11. M. Reiser, Theory and Design of Charged Particle Beams, (Wiley, NY, 2005)
  12. C. Limborg-Deprey and P. Bolton, “Optimum electron distributions for space charge dominated beams in photoinjectors,” Nucl. Instrum. Methods. Phys. Res, Sect. A 557, 106 (2006).
    [CrossRef]
  13. K. Bezuhanov, A. Dreischuh, G. G. Paulus, M. G. Schätzel, and H. Walther, “Vortices in femtosecond laser fields,” Opt. Lett. 29(16), 1942–1944 (2004).
    [CrossRef] [PubMed]
  14. I. G. Mariyenko, J. Strohaber, and C. J. G. J. Uiterwaal, “Creation of optical vortices in femtosecond pulses,” Opt. Express 13(19), 7599–7608 (2005).
    [CrossRef] [PubMed]
  15. A. Schwarz and W. Rudolph, “Dispersion-compensating beam shaper for femtosecond optical vortex beams,” Opt. Lett. 33(24), 2970–2972 (2008).
    [CrossRef] [PubMed]
  16. Communications with HOLOEYE Photonics AG.
  17. V. Y. Bazhenov, M. S. Soskin, and M. V. Vasnetsov, “Screw Dislocations in Light Wavefronts,” J. Mod. Opt. 39(5), 985–990 (1992).
    [CrossRef]
  18. N. R. Heckenberg, R. McDuff, C. P. Smith, and A. G. White, “Generation of optical phase singularities by computer-generated holograms,” Opt. Lett. 17(3), 221–223 (1992).
    [CrossRef] [PubMed]
  19. J. F. Nye, Natural Focusing and Fine Structure of Light, (Institute of Physics, 1999).
  20. V. G. Denisenko, A. Minovich, A. S. Desyatnikov, W. Krolikowski, M. S. Soskin, and Y. S. Kivshar, “Mapping phases of singular scalar light fields,” Opt. Lett. 33(1), 89–91 (2008).
    [CrossRef]
  21. J. Strohaber, I. Mariyenko, and C. J. G. J. Uiterwaal, “Generation and diagnosis of focused ultrashort pulses of complex light,” Proc. SPIE Vol. 6483 (2007).
  22. J. Leach and M. J. Padgett, “Observation of chromatic effects near a white-light vortex,” N. J. Phys. 5, 1–7 (2003).
    [CrossRef]
  23. J. Leach, G. M. Gibson, M. J. Padgett, E. Esposito, G. McConnell, A. J. Wright, and J. M. Girkin, “Generation of achromatic Bessel beams using a compensated spatial light modulator,” Opt. Express 14(12), 5581–5587 (2006).
    [CrossRef] [PubMed]
  24. A. J. Wright, J. M. Girkin, G. M. Gibson, J. Leach, and M. J. Padgett, “Transfer of orbital angular momentum from a super-continuum, white-light beam,” Opt. Express 16(13), 9495–9500 (2008).
    [CrossRef] [PubMed]

2008

2007

L. Bergé, S. Skupin, R. Nuter, J. Kasparian, and J.-P. Wolf, “Ultrashort filaments of light in weakly ionized, optically transparent media,” Rep. Prog. Phys. 70(10), 1633–1713 (2007).
[CrossRef]

2006

C. Limborg-Deprey and P. Bolton, “Optimum electron distributions for space charge dominated beams in photoinjectors,” Nucl. Instrum. Methods. Phys. Res, Sect. A 557, 106 (2006).
[CrossRef]

J. Leach, G. M. Gibson, M. J. Padgett, E. Esposito, G. McConnell, A. J. Wright, and J. M. Girkin, “Generation of achromatic Bessel beams using a compensated spatial light modulator,” Opt. Express 14(12), 5581–5587 (2006).
[CrossRef] [PubMed]

M. F. Andersen, C. Ryu, P. Cladé, V. Natarajan, A. Vaziri, K. Helmerson, and W. D. Phillips, “Quantized rotation of atoms from photons with orbital angular momentum,” Phys. Rev. Lett. 97(17), 170406 (2006).
[CrossRef] [PubMed]

2005

I. G. Mariyenko, J. Strohaber, and C. J. G. J. Uiterwaal, “Creation of optical vortices in femtosecond pulses,” Opt. Express 13(19), 7599–7608 (2005).
[CrossRef] [PubMed]

K. Cook, R. McGeorge, A. K. Kar, M. R. Taghizadeh, and R. A. Lamb, “Coherent array of white-light continuum filaments produced by diffractive microlenses,” Appl. Phys. Lett. 86(2), 021105 (2005).
[CrossRef]

Z. Jin, J. Zhang, M. H. Xu, X. Lu, Y. T. Li, Z. H. Wang, Z. Y. Wei, X. H. Yuan, and W. Yu, “Control of filamentation induced by femtosecond laser pulses propagating in air,” Opt. Express 13(25), 10424–10430 (2005).
[CrossRef] [PubMed]

V. P. Kandidov, N. Ak¨ozbek, M. Scalora, O. G. Kosareva, A. V. Nyakk, Q. Luo, S. A. Hosseini, and S. L. Chin, “Towards a control of multiple filamentation by spatial regularization of a high-power femtosecond laser pulse,” Appl. Phys. B 80(2), 267–275 (2005).
[CrossRef]

A. Vinçotte and L. Bergé, “Femtosecond optical vortices in air,” Phys. Rev. Lett. 95(19), 193901 (2005).
[CrossRef] [PubMed]

2004

2003

D. G. Grier, “A revolution in optical manipulation,” Nature 424(6950), 810–816 (2003).
[CrossRef] [PubMed]

J. Leach and M. J. Padgett, “Observation of chromatic effects near a white-light vortex,” N. J. Phys. 5, 1–7 (2003).
[CrossRef]

2001

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature 412(6844), 313–316 (2001).
[CrossRef] [PubMed]

1992

Ak¨ozbek, N.

V. P. Kandidov, N. Ak¨ozbek, M. Scalora, O. G. Kosareva, A. V. Nyakk, Q. Luo, S. A. Hosseini, and S. L. Chin, “Towards a control of multiple filamentation by spatial regularization of a high-power femtosecond laser pulse,” Appl. Phys. B 80(2), 267–275 (2005).
[CrossRef]

Andersen, M. F.

M. F. Andersen, C. Ryu, P. Cladé, V. Natarajan, A. Vaziri, K. Helmerson, and W. D. Phillips, “Quantized rotation of atoms from photons with orbital angular momentum,” Phys. Rev. Lett. 97(17), 170406 (2006).
[CrossRef] [PubMed]

Bazhenov, V. Y.

V. Y. Bazhenov, M. S. Soskin, and M. V. Vasnetsov, “Screw Dislocations in Light Wavefronts,” J. Mod. Opt. 39(5), 985–990 (1992).
[CrossRef]

Bergé, L.

L. Bergé, S. Skupin, R. Nuter, J. Kasparian, and J.-P. Wolf, “Ultrashort filaments of light in weakly ionized, optically transparent media,” Rep. Prog. Phys. 70(10), 1633–1713 (2007).
[CrossRef]

A. Vinçotte and L. Bergé, “Femtosecond optical vortices in air,” Phys. Rev. Lett. 95(19), 193901 (2005).
[CrossRef] [PubMed]

Bezuhanov, K.

Bolton, P.

C. Limborg-Deprey and P. Bolton, “Optimum electron distributions for space charge dominated beams in photoinjectors,” Nucl. Instrum. Methods. Phys. Res, Sect. A 557, 106 (2006).
[CrossRef]

Chin, S. L.

V. P. Kandidov, N. Ak¨ozbek, M. Scalora, O. G. Kosareva, A. V. Nyakk, Q. Luo, S. A. Hosseini, and S. L. Chin, “Towards a control of multiple filamentation by spatial regularization of a high-power femtosecond laser pulse,” Appl. Phys. B 80(2), 267–275 (2005).
[CrossRef]

Cladé, P.

M. F. Andersen, C. Ryu, P. Cladé, V. Natarajan, A. Vaziri, K. Helmerson, and W. D. Phillips, “Quantized rotation of atoms from photons with orbital angular momentum,” Phys. Rev. Lett. 97(17), 170406 (2006).
[CrossRef] [PubMed]

Cook, K.

K. Cook, R. McGeorge, A. K. Kar, M. R. Taghizadeh, and R. A. Lamb, “Coherent array of white-light continuum filaments produced by diffractive microlenses,” Appl. Phys. Lett. 86(2), 021105 (2005).
[CrossRef]

Denisenko, V. G.

Desyatnikov, A. S.

Dreischuh, A.

Esposito, E.

Gibson, G. M.

Girkin, J. M.

Grier, D. G.

D. G. Grier, “A revolution in optical manipulation,” Nature 424(6950), 810–816 (2003).
[CrossRef] [PubMed]

Heckenberg, N. R.

Helmerson, K.

M. F. Andersen, C. Ryu, P. Cladé, V. Natarajan, A. Vaziri, K. Helmerson, and W. D. Phillips, “Quantized rotation of atoms from photons with orbital angular momentum,” Phys. Rev. Lett. 97(17), 170406 (2006).
[CrossRef] [PubMed]

Hosseini, S. A.

V. P. Kandidov, N. Ak¨ozbek, M. Scalora, O. G. Kosareva, A. V. Nyakk, Q. Luo, S. A. Hosseini, and S. L. Chin, “Towards a control of multiple filamentation by spatial regularization of a high-power femtosecond laser pulse,” Appl. Phys. B 80(2), 267–275 (2005).
[CrossRef]

Jin, Z.

Kandidov, V. P.

V. P. Kandidov, N. Ak¨ozbek, M. Scalora, O. G. Kosareva, A. V. Nyakk, Q. Luo, S. A. Hosseini, and S. L. Chin, “Towards a control of multiple filamentation by spatial regularization of a high-power femtosecond laser pulse,” Appl. Phys. B 80(2), 267–275 (2005).
[CrossRef]

Kar, A. K.

K. Cook, R. McGeorge, A. K. Kar, M. R. Taghizadeh, and R. A. Lamb, “Coherent array of white-light continuum filaments produced by diffractive microlenses,” Appl. Phys. Lett. 86(2), 021105 (2005).
[CrossRef]

Kasparian, J.

L. Bergé, S. Skupin, R. Nuter, J. Kasparian, and J.-P. Wolf, “Ultrashort filaments of light in weakly ionized, optically transparent media,” Rep. Prog. Phys. 70(10), 1633–1713 (2007).
[CrossRef]

Kivshar, Y. S.

Kosareva, O. G.

V. P. Kandidov, N. Ak¨ozbek, M. Scalora, O. G. Kosareva, A. V. Nyakk, Q. Luo, S. A. Hosseini, and S. L. Chin, “Towards a control of multiple filamentation by spatial regularization of a high-power femtosecond laser pulse,” Appl. Phys. B 80(2), 267–275 (2005).
[CrossRef]

Krolikowski, W.

Lamb, R. A.

K. Cook, R. McGeorge, A. K. Kar, M. R. Taghizadeh, and R. A. Lamb, “Coherent array of white-light continuum filaments produced by diffractive microlenses,” Appl. Phys. Lett. 86(2), 021105 (2005).
[CrossRef]

Leach, J.

Li, Y. T.

Limborg-Deprey, C.

C. Limborg-Deprey and P. Bolton, “Optimum electron distributions for space charge dominated beams in photoinjectors,” Nucl. Instrum. Methods. Phys. Res, Sect. A 557, 106 (2006).
[CrossRef]

Lu, X.

Luo, Q.

V. P. Kandidov, N. Ak¨ozbek, M. Scalora, O. G. Kosareva, A. V. Nyakk, Q. Luo, S. A. Hosseini, and S. L. Chin, “Towards a control of multiple filamentation by spatial regularization of a high-power femtosecond laser pulse,” Appl. Phys. B 80(2), 267–275 (2005).
[CrossRef]

Mair, A.

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature 412(6844), 313–316 (2001).
[CrossRef] [PubMed]

Mariyenko, I. G.

McConnell, G.

McDuff, R.

McGeorge, R.

K. Cook, R. McGeorge, A. K. Kar, M. R. Taghizadeh, and R. A. Lamb, “Coherent array of white-light continuum filaments produced by diffractive microlenses,” Appl. Phys. Lett. 86(2), 021105 (2005).
[CrossRef]

Minovich, A.

Natarajan, V.

M. F. Andersen, C. Ryu, P. Cladé, V. Natarajan, A. Vaziri, K. Helmerson, and W. D. Phillips, “Quantized rotation of atoms from photons with orbital angular momentum,” Phys. Rev. Lett. 97(17), 170406 (2006).
[CrossRef] [PubMed]

Nuter, R.

L. Bergé, S. Skupin, R. Nuter, J. Kasparian, and J.-P. Wolf, “Ultrashort filaments of light in weakly ionized, optically transparent media,” Rep. Prog. Phys. 70(10), 1633–1713 (2007).
[CrossRef]

Nyakk, A. V.

V. P. Kandidov, N. Ak¨ozbek, M. Scalora, O. G. Kosareva, A. V. Nyakk, Q. Luo, S. A. Hosseini, and S. L. Chin, “Towards a control of multiple filamentation by spatial regularization of a high-power femtosecond laser pulse,” Appl. Phys. B 80(2), 267–275 (2005).
[CrossRef]

Padgett, M. J.

Paulus, G. G.

Phillips, W. D.

M. F. Andersen, C. Ryu, P. Cladé, V. Natarajan, A. Vaziri, K. Helmerson, and W. D. Phillips, “Quantized rotation of atoms from photons with orbital angular momentum,” Phys. Rev. Lett. 97(17), 170406 (2006).
[CrossRef] [PubMed]

Rudolph, W.

Ryu, C.

M. F. Andersen, C. Ryu, P. Cladé, V. Natarajan, A. Vaziri, K. Helmerson, and W. D. Phillips, “Quantized rotation of atoms from photons with orbital angular momentum,” Phys. Rev. Lett. 97(17), 170406 (2006).
[CrossRef] [PubMed]

Scalora, M.

V. P. Kandidov, N. Ak¨ozbek, M. Scalora, O. G. Kosareva, A. V. Nyakk, Q. Luo, S. A. Hosseini, and S. L. Chin, “Towards a control of multiple filamentation by spatial regularization of a high-power femtosecond laser pulse,” Appl. Phys. B 80(2), 267–275 (2005).
[CrossRef]

Schätzel, M. G.

Schwarz, A.

Skupin, S.

L. Bergé, S. Skupin, R. Nuter, J. Kasparian, and J.-P. Wolf, “Ultrashort filaments of light in weakly ionized, optically transparent media,” Rep. Prog. Phys. 70(10), 1633–1713 (2007).
[CrossRef]

Smith, C. P.

Soskin, M. S.

Strohaber, J.

Taghizadeh, M. R.

K. Cook, R. McGeorge, A. K. Kar, M. R. Taghizadeh, and R. A. Lamb, “Coherent array of white-light continuum filaments produced by diffractive microlenses,” Appl. Phys. Lett. 86(2), 021105 (2005).
[CrossRef]

Uiterwaal, C. J. G. J.

Vasnetsov, M. V.

V. Y. Bazhenov, M. S. Soskin, and M. V. Vasnetsov, “Screw Dislocations in Light Wavefronts,” J. Mod. Opt. 39(5), 985–990 (1992).
[CrossRef]

Vaziri, A.

M. F. Andersen, C. Ryu, P. Cladé, V. Natarajan, A. Vaziri, K. Helmerson, and W. D. Phillips, “Quantized rotation of atoms from photons with orbital angular momentum,” Phys. Rev. Lett. 97(17), 170406 (2006).
[CrossRef] [PubMed]

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature 412(6844), 313–316 (2001).
[CrossRef] [PubMed]

Vinçotte, A.

A. Vinçotte and L. Bergé, “Femtosecond optical vortices in air,” Phys. Rev. Lett. 95(19), 193901 (2005).
[CrossRef] [PubMed]

Walther, H.

Wang, Z. H.

Wei, Z. Y.

Weihs, G.

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature 412(6844), 313–316 (2001).
[CrossRef] [PubMed]

White, A. G.

Wolf, J.-P.

L. Bergé, S. Skupin, R. Nuter, J. Kasparian, and J.-P. Wolf, “Ultrashort filaments of light in weakly ionized, optically transparent media,” Rep. Prog. Phys. 70(10), 1633–1713 (2007).
[CrossRef]

Wright, A. J.

Xu, M. H.

Yu, W.

Yuan, X. H.

Zeilinger, A.

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature 412(6844), 313–316 (2001).
[CrossRef] [PubMed]

Zhang, J.

Appl. Phys. B

V. P. Kandidov, N. Ak¨ozbek, M. Scalora, O. G. Kosareva, A. V. Nyakk, Q. Luo, S. A. Hosseini, and S. L. Chin, “Towards a control of multiple filamentation by spatial regularization of a high-power femtosecond laser pulse,” Appl. Phys. B 80(2), 267–275 (2005).
[CrossRef]

Appl. Phys. Lett.

K. Cook, R. McGeorge, A. K. Kar, M. R. Taghizadeh, and R. A. Lamb, “Coherent array of white-light continuum filaments produced by diffractive microlenses,” Appl. Phys. Lett. 86(2), 021105 (2005).
[CrossRef]

J. Mod. Opt.

V. Y. Bazhenov, M. S. Soskin, and M. V. Vasnetsov, “Screw Dislocations in Light Wavefronts,” J. Mod. Opt. 39(5), 985–990 (1992).
[CrossRef]

N. J. Phys.

J. Leach and M. J. Padgett, “Observation of chromatic effects near a white-light vortex,” N. J. Phys. 5, 1–7 (2003).
[CrossRef]

Nature

D. G. Grier, “A revolution in optical manipulation,” Nature 424(6950), 810–816 (2003).
[CrossRef] [PubMed]

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature 412(6844), 313–316 (2001).
[CrossRef] [PubMed]

Nucl. Instrum. Methods. Phys. Res, Sect. A

C. Limborg-Deprey and P. Bolton, “Optimum electron distributions for space charge dominated beams in photoinjectors,” Nucl. Instrum. Methods. Phys. Res, Sect. A 557, 106 (2006).
[CrossRef]

Opt. Express

Opt. Lett.

Phys. Rev. Lett.

A. Vinçotte and L. Bergé, “Femtosecond optical vortices in air,” Phys. Rev. Lett. 95(19), 193901 (2005).
[CrossRef] [PubMed]

M. F. Andersen, C. Ryu, P. Cladé, V. Natarajan, A. Vaziri, K. Helmerson, and W. D. Phillips, “Quantized rotation of atoms from photons with orbital angular momentum,” Phys. Rev. Lett. 97(17), 170406 (2006).
[CrossRef] [PubMed]

Rep. Prog. Phys.

L. Bergé, S. Skupin, R. Nuter, J. Kasparian, and J.-P. Wolf, “Ultrashort filaments of light in weakly ionized, optically transparent media,” Rep. Prog. Phys. 70(10), 1633–1713 (2007).
[CrossRef]

Other

M. Vasnetsov, and K. Staliunas, eds., “Optical Vortices,” in Horizons in World Physics, (Nova Science Publishers, Commack, New York, 1999) Vol. 228.

L. Allen, S. M. Barnett, and M. J. Padgett, Optical Angular Momentum, (Taylor & Francis, 2003).

M. Reiser, Theory and Design of Charged Particle Beams, (Wiley, NY, 2005)

J. F. Nye, Natural Focusing and Fine Structure of Light, (Institute of Physics, 1999).

Communications with HOLOEYE Photonics AG.

J. Strohaber, I. Mariyenko, and C. J. G. J. Uiterwaal, “Generation and diagnosis of focused ultrashort pulses of complex light,” Proc. SPIE Vol. 6483 (2007).

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Figures (3)

Fig. 1
Fig. 1

shows the optical scheme of 2f-2f shaper with magnification. The red line illustrates the space transformations of a broadband laser beam.

Fig. 2
Fig. 2

Reconstructed intensity “fingerprints” of OV of charge m = 3 in broadband laser field, (a), and CW laser field, (c); the angular dispersion is compensated for broadband field. (b) The cross-cut of the intensity distribution for both beams along the red lines: three zero points identify the individual single-charged vortex cores.

Fig. 3
Fig. 3

Intensity patterns of the Ti:S output laser beam focused by the simulated lens into a spot. Pattern (a) represents the focusing of pulsed laser output, (c) shows focused CW laser beam. The cross-cut of the intensity distribution for both beams along red lines is shown on the graph (b).

Equations (6)

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

E i n e x p { x 2 + y 2 w 0 2 }
E 1 e x p { x 2 w 0 2 + i ​   n 1 K 1 x }
E 2 e x p { i k 2 ( z 1 L R ) ( x i n 1 K 1 k L R ) 2 n 1 2 K 1 2 2 k L R i k x 2 2 f }
E 3 e x p { x 2 w 0 2 ( z 1 z 2 ) 2 + i k 2 f z 1 z 2 x 2 i z 1 z 2 n 1 K 1 x }
E o u t e x p { x 2 ( M w 0 ) 2 + i k 2 M f x 2 i n 1 K 1 M x + i n 2 K 2 x }
n 1 K 1 M + n 2 K 2 = 0

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