Abstract

Type I and type II second harmonic generation (SHG) of a beam transformed by the conical refraction phenomenon are presented. We show that, for type I, the second harmonic intensity pattern is a light ring with a point of null intensity while, for type II, the light ring possesses two dark regions. Taking into account the different two-photon processes involved in SHG, we have derived analytical expressions for the resulting transverse intensity patterns that are in good agreement with the experimental data. Finally, we have investigated the spatial evolution of the second harmonic signals, showing that they behave as conically refracted beams.

© 2013 Optical Society of America

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  1. T. K. Kalkandjiev and M. Bursukova, Proc. SPIE 6994, 69940B (2008).
    [CrossRef]
  2. A. M. Belskii and A. P. Khapalyuk, Opt. Spectrosc. 44, 436 (1978).
  3. C. V. Raman, Curr. Sci. 11, 44 (1942).
  4. M. V. Berry and M. R. Jeffrey, Prog. Opt. 50, 13 (2007).
    [CrossRef]
  5. A. M. Belsky and M. A. Stepanov, Opt. Commun. 167, 1 (1999).
    [CrossRef]
  6. H. Shih and N. Bloembergen, Phys. Rev. 184, 895 (1969).
    [CrossRef]
  7. A. J. Schell and N. Bloembergen, Phys. Rev. A 18, 2592 (1978).
    [CrossRef]
  8. V. I. Stroganov, A. I. Illarionov, and B. I. Kidyarov, J. Appl. Spectrosc. 32, 341 (1980).
    [CrossRef]
  9. T. S. Velichkina, O. I. Vasileva, A. I. Israilenko, and I. A. Yakovlev, Phys. Usp. 23, 176 (1980).
    [CrossRef]
  10. J. Kroupa, J. Opt. 12, 045706 (2010).
    [CrossRef]
  11. S. A. Zolotovskaya, A. Abdolvand, T. K. Kalkandjiev, and E. U. Rafailov, Appl. Phys. B 103, 9 (2011).
    [CrossRef]
  12. V. Peet and S. Shchemelyov, J. Opt. 13, 055205 (2011).
    [CrossRef]
  13. W. Boyd, Nonlinear Optics, 3rd ed. (Academic, 2008).

2011

S. A. Zolotovskaya, A. Abdolvand, T. K. Kalkandjiev, and E. U. Rafailov, Appl. Phys. B 103, 9 (2011).
[CrossRef]

V. Peet and S. Shchemelyov, J. Opt. 13, 055205 (2011).
[CrossRef]

2010

J. Kroupa, J. Opt. 12, 045706 (2010).
[CrossRef]

2008

T. K. Kalkandjiev and M. Bursukova, Proc. SPIE 6994, 69940B (2008).
[CrossRef]

2007

M. V. Berry and M. R. Jeffrey, Prog. Opt. 50, 13 (2007).
[CrossRef]

1999

A. M. Belsky and M. A. Stepanov, Opt. Commun. 167, 1 (1999).
[CrossRef]

1980

V. I. Stroganov, A. I. Illarionov, and B. I. Kidyarov, J. Appl. Spectrosc. 32, 341 (1980).
[CrossRef]

T. S. Velichkina, O. I. Vasileva, A. I. Israilenko, and I. A. Yakovlev, Phys. Usp. 23, 176 (1980).
[CrossRef]

1978

A. J. Schell and N. Bloembergen, Phys. Rev. A 18, 2592 (1978).
[CrossRef]

A. M. Belskii and A. P. Khapalyuk, Opt. Spectrosc. 44, 436 (1978).

1969

H. Shih and N. Bloembergen, Phys. Rev. 184, 895 (1969).
[CrossRef]

1942

C. V. Raman, Curr. Sci. 11, 44 (1942).

Abdolvand, A.

S. A. Zolotovskaya, A. Abdolvand, T. K. Kalkandjiev, and E. U. Rafailov, Appl. Phys. B 103, 9 (2011).
[CrossRef]

Belskii, A. M.

A. M. Belskii and A. P. Khapalyuk, Opt. Spectrosc. 44, 436 (1978).

Belsky, A. M.

A. M. Belsky and M. A. Stepanov, Opt. Commun. 167, 1 (1999).
[CrossRef]

Berry, M. V.

M. V. Berry and M. R. Jeffrey, Prog. Opt. 50, 13 (2007).
[CrossRef]

Bloembergen, N.

A. J. Schell and N. Bloembergen, Phys. Rev. A 18, 2592 (1978).
[CrossRef]

H. Shih and N. Bloembergen, Phys. Rev. 184, 895 (1969).
[CrossRef]

Boyd, W.

W. Boyd, Nonlinear Optics, 3rd ed. (Academic, 2008).

Bursukova, M.

T. K. Kalkandjiev and M. Bursukova, Proc. SPIE 6994, 69940B (2008).
[CrossRef]

Illarionov, A. I.

V. I. Stroganov, A. I. Illarionov, and B. I. Kidyarov, J. Appl. Spectrosc. 32, 341 (1980).
[CrossRef]

Israilenko, A. I.

T. S. Velichkina, O. I. Vasileva, A. I. Israilenko, and I. A. Yakovlev, Phys. Usp. 23, 176 (1980).
[CrossRef]

Jeffrey, M. R.

M. V. Berry and M. R. Jeffrey, Prog. Opt. 50, 13 (2007).
[CrossRef]

Kalkandjiev, T. K.

S. A. Zolotovskaya, A. Abdolvand, T. K. Kalkandjiev, and E. U. Rafailov, Appl. Phys. B 103, 9 (2011).
[CrossRef]

T. K. Kalkandjiev and M. Bursukova, Proc. SPIE 6994, 69940B (2008).
[CrossRef]

Khapalyuk, A. P.

A. M. Belskii and A. P. Khapalyuk, Opt. Spectrosc. 44, 436 (1978).

Kidyarov, B. I.

V. I. Stroganov, A. I. Illarionov, and B. I. Kidyarov, J. Appl. Spectrosc. 32, 341 (1980).
[CrossRef]

Kroupa, J.

J. Kroupa, J. Opt. 12, 045706 (2010).
[CrossRef]

Peet, V.

V. Peet and S. Shchemelyov, J. Opt. 13, 055205 (2011).
[CrossRef]

Rafailov, E. U.

S. A. Zolotovskaya, A. Abdolvand, T. K. Kalkandjiev, and E. U. Rafailov, Appl. Phys. B 103, 9 (2011).
[CrossRef]

Raman, C. V.

C. V. Raman, Curr. Sci. 11, 44 (1942).

Schell, A. J.

A. J. Schell and N. Bloembergen, Phys. Rev. A 18, 2592 (1978).
[CrossRef]

Shchemelyov, S.

V. Peet and S. Shchemelyov, J. Opt. 13, 055205 (2011).
[CrossRef]

Shih, H.

H. Shih and N. Bloembergen, Phys. Rev. 184, 895 (1969).
[CrossRef]

Stepanov, M. A.

A. M. Belsky and M. A. Stepanov, Opt. Commun. 167, 1 (1999).
[CrossRef]

Stroganov, V. I.

V. I. Stroganov, A. I. Illarionov, and B. I. Kidyarov, J. Appl. Spectrosc. 32, 341 (1980).
[CrossRef]

Vasileva, O. I.

T. S. Velichkina, O. I. Vasileva, A. I. Israilenko, and I. A. Yakovlev, Phys. Usp. 23, 176 (1980).
[CrossRef]

Velichkina, T. S.

T. S. Velichkina, O. I. Vasileva, A. I. Israilenko, and I. A. Yakovlev, Phys. Usp. 23, 176 (1980).
[CrossRef]

Yakovlev, I. A.

T. S. Velichkina, O. I. Vasileva, A. I. Israilenko, and I. A. Yakovlev, Phys. Usp. 23, 176 (1980).
[CrossRef]

Zolotovskaya, S. A.

S. A. Zolotovskaya, A. Abdolvand, T. K. Kalkandjiev, and E. U. Rafailov, Appl. Phys. B 103, 9 (2011).
[CrossRef]

Appl. Phys. B

S. A. Zolotovskaya, A. Abdolvand, T. K. Kalkandjiev, and E. U. Rafailov, Appl. Phys. B 103, 9 (2011).
[CrossRef]

Curr. Sci.

C. V. Raman, Curr. Sci. 11, 44 (1942).

J. Appl. Spectrosc.

V. I. Stroganov, A. I. Illarionov, and B. I. Kidyarov, J. Appl. Spectrosc. 32, 341 (1980).
[CrossRef]

J. Opt.

V. Peet and S. Shchemelyov, J. Opt. 13, 055205 (2011).
[CrossRef]

J. Kroupa, J. Opt. 12, 045706 (2010).
[CrossRef]

Opt. Commun.

A. M. Belsky and M. A. Stepanov, Opt. Commun. 167, 1 (1999).
[CrossRef]

Opt. Spectrosc.

A. M. Belskii and A. P. Khapalyuk, Opt. Spectrosc. 44, 436 (1978).

Phys. Rev.

H. Shih and N. Bloembergen, Phys. Rev. 184, 895 (1969).
[CrossRef]

Phys. Rev. A

A. J. Schell and N. Bloembergen, Phys. Rev. A 18, 2592 (1978).
[CrossRef]

Phys. Usp.

T. S. Velichkina, O. I. Vasileva, A. I. Israilenko, and I. A. Yakovlev, Phys. Usp. 23, 176 (1980).
[CrossRef]

Proc. SPIE

T. K. Kalkandjiev and M. Bursukova, Proc. SPIE 6994, 69940B (2008).
[CrossRef]

Prog. Opt.

M. V. Berry and M. R. Jeffrey, Prog. Opt. 50, 13 (2007).
[CrossRef]

Other

W. Boyd, Nonlinear Optics, 3rd ed. (Academic, 2008).

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Figures (6)

Fig. 1.
Fig. 1.

(a) Focused randomly polarized Gaussian beam is transformed by a BC into a light ring at the ring plane of the system; (b) CR ring at the ring plane with the fine Poggendorff splitting. Double orange arrows show the polarization distribution along the ring. FL means focusing lens; o and e denote the points with ordinary and extraordinary polarizations, respectively. ΔBC=L(11/nBC) is a longitudinal shift of the ring plane’s position added by the BC, with intermediate refractive index nBC.

Fig. 2.
Fig. 2.

Evolution of the transverse intensity profile of the FH generated throughout the CR effect in a BC. The position of the ring plane, where the CR ring is most sharply resolved, is at Z=0. Experimental parameters: R0=476μm, w0=42μm, and zR=5.148mm.

Fig. 3.
Fig. 3.

Experimental setup. A randomly polarized input beam with a beam waist radius of w0=3.2mm is obtained from an Yb fiber laser generating light pulses at 1064   nm with pulse duration τ=(110±10)ns at a 20 kHz repetition rate and up to 10 W of nominal power. This beam is focused by a lens (FL) of 400 mm focal length to a KGd(WO4)2 BC of length L=28mm and conicity α=17mrad, yielding R0=476μm. At the ring plane, we place the NLCs: LBO (type I, deff=0.668pm/V, LLBO=10mm) and KTP (type II, deff=3.2598pm/V, LKTP=8mm). The imaging lens IL projects different planes of the SHG propagated beams onto the CCD camera. The infrared filter (IRF) eliminates the radiation at the FH. ΔNLC=LNLC(11/nNLC) is the longitudinal shift of the ring plane’s position added by the NLC.

Fig. 4.
Fig. 4.

Patterns of the (a) FH, (b) type I, and (c) type II SH generated with the NLCs placed at the ring plane. Patterns were captured by using the lens IL (see Fig. 3) to image the ring plane onto the CCD. Top and right insets are, respectively, the horizontal and vertical intensity profiles at the center of the images. Orange double arrows indicate the polarization plane.

Fig. 5.
Fig. 5.

Azimuthal intensity distribution of the final patterns for type I (LBO) and type II (KTP) SHG. Symbols represent the experimental data, while solid lines are the corresponding analytical solutions from Eqs. (2) and (3).

Fig. 6.
Fig. 6.

Evolution of the transverse intensity profile in type I (top row) and type II (bottom row) SHG when the NLCs are placed at the ring plane of the CR beam. The extraordinary polarization in the NLC was parallel to the plane of the optic axes of the BC, i.e., ϕ0=0°. We note that the Raman-like spots for the second harmonic, (d) and (h), have been observed on both sides from the ring plane.

Equations (3)

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|f(ξ,Z)|=eξ22wZ25/4wZ3/4D12(2ξwZ),
IType I=I2ω0|f(ξ,η)|4ρ02cos4(φ+ϕ02),
IType II=I2ω0|f(ξ,η)|4ρ02sin2(φ+ϕ0),

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