Abstract

The propagation of Ince–Gaussian beams in media where the refractive index varies quadratically with the distance from the optical axis is studied. Explicit expressions for the complex beam parameter and the longitudinal phase shift are derived and discussed. Ince–Gaussian eigenmodes with constant width can be obtained by satisfying a relation between the beam width and the quadratic-medium coefficient. The derivation has included the possibility of propagation of Ince–Gaussian beams in complex lenslike media having quadratic transverse variations of the index of refraction and the gain or loss.

© 2005 Optical Society of America

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References

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  1. A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986).
  2. M. A. Bandres, J. C. Gutiérrez-Vega, “Ince–Gaussian beams,” Opt. Lett. 29, 144–146 (2004).
    [CrossRef] [PubMed]
  3. M. A. Bandres, J. C. Gutiérrez-Vega, “Ince–Gaussian modes of the paraxial wave equation and stable resonators,” J. Opt. Soc. Am. A 21, 873–880 (2004).
    [CrossRef]
  4. M. A. Bandres, “Elegant Ince–Gaussian beams,” Opt. Lett. 29, 1724–1726 (2004).
    [CrossRef] [PubMed]
  5. U. T. Schwarz, M. A. Bandres, J. C. Gutiérrez-Vega, “Observation of Ince–Gaussian modes in stable resonators,” Opt. Lett. 29, 1870–1872 (2004).
    [CrossRef] [PubMed]
  6. P. K. Tien, J. P. Gordon, J. R. Whinnery, “Focusing of light beam of Gaussian field distribution in continuous and periodic lenslike media,” Proc. IEEE 53, 129–136 (1965).
    [CrossRef]
  7. M. Newstein, K. Lin, “Laguerre–Gaussian periodically focusing beams in a quadratic index medium,” IEEE J. Quantum Electron. 23, 481–482 (1987).
    [CrossRef]
  8. F. M. Arscott, Periodic Differential Equations (Pergamon, Oxford, UK, 1964).
  9. H. Kogelnik, “On the propagation of Gaussian beams of light through lenslike media including those with a loss or gain variation,” Appl. Opt. 4, 1562–1569 (1965).
    [CrossRef]
  10. M. A. Bandres, J. C. Gutiérrez-Vega, “Ince–Gaussian series representation of the two-dimensional fractional Fourier transform,” Opt. Lett. (to be published).

2004 (4)

1987 (1)

M. Newstein, K. Lin, “Laguerre–Gaussian periodically focusing beams in a quadratic index medium,” IEEE J. Quantum Electron. 23, 481–482 (1987).
[CrossRef]

1965 (2)

H. Kogelnik, “On the propagation of Gaussian beams of light through lenslike media including those with a loss or gain variation,” Appl. Opt. 4, 1562–1569 (1965).
[CrossRef]

P. K. Tien, J. P. Gordon, J. R. Whinnery, “Focusing of light beam of Gaussian field distribution in continuous and periodic lenslike media,” Proc. IEEE 53, 129–136 (1965).
[CrossRef]

Arscott, F. M.

F. M. Arscott, Periodic Differential Equations (Pergamon, Oxford, UK, 1964).

Bandres, M. A.

Gordon, J. P.

P. K. Tien, J. P. Gordon, J. R. Whinnery, “Focusing of light beam of Gaussian field distribution in continuous and periodic lenslike media,” Proc. IEEE 53, 129–136 (1965).
[CrossRef]

Gutiérrez-Vega, J. C.

Kogelnik, H.

Lin, K.

M. Newstein, K. Lin, “Laguerre–Gaussian periodically focusing beams in a quadratic index medium,” IEEE J. Quantum Electron. 23, 481–482 (1987).
[CrossRef]

Newstein, M.

M. Newstein, K. Lin, “Laguerre–Gaussian periodically focusing beams in a quadratic index medium,” IEEE J. Quantum Electron. 23, 481–482 (1987).
[CrossRef]

Schwarz, U. T.

Siegman, A. E.

A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986).

Tien, P. K.

P. K. Tien, J. P. Gordon, J. R. Whinnery, “Focusing of light beam of Gaussian field distribution in continuous and periodic lenslike media,” Proc. IEEE 53, 129–136 (1965).
[CrossRef]

Whinnery, J. R.

P. K. Tien, J. P. Gordon, J. R. Whinnery, “Focusing of light beam of Gaussian field distribution in continuous and periodic lenslike media,” Proc. IEEE 53, 129–136 (1965).
[CrossRef]

Appl. Opt. (1)

IEEE J. Quantum Electron. (1)

M. Newstein, K. Lin, “Laguerre–Gaussian periodically focusing beams in a quadratic index medium,” IEEE J. Quantum Electron. 23, 481–482 (1987).
[CrossRef]

J. Opt. Soc. Am. A (1)

Opt. Lett. (3)

Proc. IEEE (1)

P. K. Tien, J. P. Gordon, J. R. Whinnery, “Focusing of light beam of Gaussian field distribution in continuous and periodic lenslike media,” Proc. IEEE 53, 129–136 (1965).
[CrossRef]

Other (3)

A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986).

F. M. Arscott, Periodic Differential Equations (Pergamon, Oxford, UK, 1964).

M. A. Bandres, J. C. Gutiérrez-Vega, “Ince–Gaussian series representation of the two-dimensional fractional Fourier transform,” Opt. Lett. (to be published).

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Figures (2)

Fig. 1
Fig. 1

Transverse field distributions of several even IG beams with =3. The IG beam (0, 0) reduces to the simple Gaussian beam. The even IG beam (1, 1) is equal to the HG beam (1, 0) and also to the even LG beam (0, 1).

Fig. 2
Fig. 2

Longitudinal phase-shift retardation ϕ(z)=arctan[tan(az)/azR] as a function of the normalized propagation distance z/zR for several values of azR.

Equations (18)

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t2+2ikz-k2a2r2Ψ(r)=0,
Ψ(r)=U(r)expikr22q(z),
dqdz-a2q2-1=0,
t2U+2ikUz+2ikq(U+tUrt)=0,
q(z2)=Aq(z1)+BCq(z1)+D,
A=cos[a(z2-z1)],B=sin[a(z2-z1)]/a,C=-a sin[a(z2-z1)],D=cos[a(z2-z1)].
q(z)=1asin(az)-iazR cos(az)cos(az)+iazR sin(az).
R(z)=tan(az)+a2zR2cot(az)a(1-a2zR2),
w(z)=w0[1+β sin2(az)]1/2,
U(r)=E(ξ)N(η)exp[iZ(z)],
d2Edξ2- sinh 2ξdEdξ-(μ-p cosh 2ξ)E=0,
d2Ndη2+ sin 2ηdNdη+(μ-p cos 2η)N=0,
dZdz-i1R(z)+i2(p+1)kw2(z)=0,
exp[iZ(z)]=w0w(z)exp-i(p+1)arctantan(az)azR.
IGp,me(r, t)=Cw0w(z)Cpm(iξ, )Cpm(η, )×expikr22q(z)expikz-i(p+1)×arctantan(az)azR-iωt,
w¯=w0a2π 02π/a[1+β sin2(az)]1/2dz
kz-(p+1)ϕ(z),
-IGp,mσIG¯p,mσdS=δσσδppδmm,

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