Abstract

Sinusoidal-Gaussian beams have recently been obtained as exact solutions of the paraxial wave equation for propagation in complex optical systems. Another useful set of beam solutions for Cartesian coordinate systems is based on Hermite–Gaussian functions. A generalization of these solution sets is developed here. The new solutions are referred to as Hermite–sinusoidal-Gaussian beams, because they are in the form of a product of Hermite-polynomial functions of either complex or real argument, sinusoidal functions of complex argument, and Gaussian functions of complex argument. These beams are valid for propagation through systems that can be represented in terms of complex beam matrices, and the previous beam solution sets are special cases of these more general results. Propagation characteristics and applications of these beams are discussed, including their use as a basis set for propagation of arbitrary electromagnetic beams.

© 1998 Optical Society of America

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References

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  1. N. G. Vakhimov, “Open resonators with mirrors having variable reflection coefficients,” Radio Eng. Electron. Phys. 10, 1439–1446 (1965).
  2. J. A. Arnaud, “Nonorthogonal optical waveguides and resonators,” Bell Syst. Tech. J. 49, 2311–2348 (1970).
    [CrossRef]
  3. J. A. Arnaud, “Mode coupling in first order optics,” J. Opt. Soc. Am. 61, 751–758 (1971).
    [CrossRef]
  4. A. E. Siegman, “Hermite–Gaussian functions of complex argument as optical-beam eigenfunctions,” J. Opt. Soc. Am. 63, 1093–1094 (1973).
    [CrossRef]
  5. J. A. Arnaud, Beam and Fiber Optics (Academic, New York, 1976), especially Subsecs. 2.16, 2.17, and 4.18–4.20.
  6. L. W. Casperson, “Beam modes in complex lenslike media and resonators,” J. Opt. Soc. Am. 66, 1373–1379 (1976).
    [CrossRef]
  7. R. Pratesi, L. Ronchi, “Generalized Gaussian beams in free space,” J. Opt. Soc. Am. 67, 1274–1276 (1977).
    [CrossRef]
  8. M. Nazarathy, A. Hardy, J. Shamir, “Generalized mode propagation in first-order optical systems with loss or gain,” J. Opt. Soc. Am. 72, 1409–1420 (1982).
    [CrossRef]
  9. A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986), especially Subsec. 20.5.
  10. A. A. Tovar, L. W. Casperson, “Off-axis complex-argument polynomial-Gaussian beams in optical systems,” J. Opt. Soc. Am. A 8, 60–68 (1991).
    [CrossRef]
  11. L. W. Casperson, “Gaussian light beams in inhomogeneous media,” Appl. Opt. 12, 2434–2441 (1973).
    [CrossRef] [PubMed]
  12. A. A. Tovar, L. W. Casperson, “Generalized beam matrices: Gaussian beam propagation in misaligned complex optical systems,” J. Opt. Soc. Am. A 12, 1522–1533 (1995).
    [CrossRef]
  13. A. A. Tovar, L. W. Casperson, “Generalized beam matrices. II. Mode selection in lasers and periodic misaligned complex optical systems,” J. Opt. Soc. Am. A 13, 90–96 (1996).
    [CrossRef]
  14. L. W. Casperson, D. G. Hall, A. A. Tovar, “Sinusoidal-Gaussian beams in complex optical systems,” J. Opt. Soc. Am. A 14, 3341–3348 (1997).
    [CrossRef]
  15. A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986), pp. 798–804.
  16. See, for example, J. Mathews, R. L. Walker, Mathematical Methods of Physics (Benjamin, New York, 1964), pp. 96–97.
  17. M. R. Spiegel, Mathematical Handbook of Formulas and Tables, Schaum’s Outline Series (McGraw-Hill, New York, 1991), pp. 131–135.
  18. L. W. Casperson, “Spatial modulation of Gaussian laser beams,” Opt. Quantum Electron. 10, 483–493 (1978).
    [CrossRef]

1997 (1)

1996 (1)

1995 (1)

1991 (1)

1982 (1)

1978 (1)

L. W. Casperson, “Spatial modulation of Gaussian laser beams,” Opt. Quantum Electron. 10, 483–493 (1978).
[CrossRef]

1977 (1)

1976 (1)

1973 (2)

1971 (1)

1970 (1)

J. A. Arnaud, “Nonorthogonal optical waveguides and resonators,” Bell Syst. Tech. J. 49, 2311–2348 (1970).
[CrossRef]

1965 (1)

N. G. Vakhimov, “Open resonators with mirrors having variable reflection coefficients,” Radio Eng. Electron. Phys. 10, 1439–1446 (1965).

Arnaud, J. A.

J. A. Arnaud, “Mode coupling in first order optics,” J. Opt. Soc. Am. 61, 751–758 (1971).
[CrossRef]

J. A. Arnaud, “Nonorthogonal optical waveguides and resonators,” Bell Syst. Tech. J. 49, 2311–2348 (1970).
[CrossRef]

J. A. Arnaud, Beam and Fiber Optics (Academic, New York, 1976), especially Subsecs. 2.16, 2.17, and 4.18–4.20.

Casperson, L. W.

Hall, D. G.

Hardy, A.

Mathews, J.

See, for example, J. Mathews, R. L. Walker, Mathematical Methods of Physics (Benjamin, New York, 1964), pp. 96–97.

Nazarathy, M.

Pratesi, R.

Ronchi, L.

Shamir, J.

Siegman, A. E.

A. E. Siegman, “Hermite–Gaussian functions of complex argument as optical-beam eigenfunctions,” J. Opt. Soc. Am. 63, 1093–1094 (1973).
[CrossRef]

A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986), pp. 798–804.

A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986), especially Subsec. 20.5.

Spiegel, M. R.

M. R. Spiegel, Mathematical Handbook of Formulas and Tables, Schaum’s Outline Series (McGraw-Hill, New York, 1991), pp. 131–135.

Tovar, A. A.

Vakhimov, N. G.

N. G. Vakhimov, “Open resonators with mirrors having variable reflection coefficients,” Radio Eng. Electron. Phys. 10, 1439–1446 (1965).

Walker, R. L.

See, for example, J. Mathews, R. L. Walker, Mathematical Methods of Physics (Benjamin, New York, 1964), pp. 96–97.

Appl. Opt. (1)

Bell Syst. Tech. J. (1)

J. A. Arnaud, “Nonorthogonal optical waveguides and resonators,” Bell Syst. Tech. J. 49, 2311–2348 (1970).
[CrossRef]

J. Opt. Soc. Am. (5)

J. Opt. Soc. Am. A (4)

Opt. Quantum Electron. (1)

L. W. Casperson, “Spatial modulation of Gaussian laser beams,” Opt. Quantum Electron. 10, 483–493 (1978).
[CrossRef]

Radio Eng. Electron. Phys. (1)

N. G. Vakhimov, “Open resonators with mirrors having variable reflection coefficients,” Radio Eng. Electron. Phys. 10, 1439–1446 (1965).

Other (5)

J. A. Arnaud, Beam and Fiber Optics (Academic, New York, 1976), especially Subsecs. 2.16, 2.17, and 4.18–4.20.

A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986), especially Subsec. 20.5.

A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986), pp. 798–804.

See, for example, J. Mathews, R. L. Walker, Mathematical Methods of Physics (Benjamin, New York, 1964), pp. 96–97.

M. R. Spiegel, Mathematical Handbook of Formulas and Tables, Schaum’s Outline Series (McGraw-Hill, New York, 1991), pp. 131–135.

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Figures (3)

Fig. 1
Fig. 1

Intensity plots of m=4 Hermite–sinusoidal-Gaussian beam profiles from Eq. (45). The parameter a represents approximately the ratio of the width associated with the Hermite–Gaussian factors to the width associated with the sinusoidal portion of the beam, and in the plots a takes on the values 0.0, 0.5, 1.0, and 2.0.

Fig. 2
Fig. 2

Example of an optical system containing a periodic transmission filter, as discussed in Section 4. In this system the filter is centered between two identical lenses.

Fig. 3
Fig. 3

Transmission characteristic |sin(xL/p)| for the amplitude filter in the system shown in Fig. 2.

Equations (53)

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2E(x, y, z)+k2(x, y, z)E(x, y, z)=0,
k2(x, y, z)=k0(z)[k0(z)-k1x(z)x-k1y(z)y-k2x(z)x2-k2y(z)y2].
Ex(x, y, z)=A(x, y, z)exp-i0zk0(z)dz,
2Ax2+2Ay2-2ik0 Az-i dk0dzA-k0(k1xx+k1yy+k2xx2+k2yy2)A=0,
Am,n(x, y, z)=A0Hm[ax(z)x+bx(z)]Hn[ay(z)y+by(z)]×exp-iQx(z)x22+Qy(z)y22+Sx(z)x+Sy(z)y+P(z),
Qx2+k0 dQxdz+k0k2x=0,
Qy2+k0 dQydz+k0k2y=0,
QxSx+k0 dSxdz+k0k1x2=0,
QySy+k0 dSydz+k0k1y2=0,
Qxax+k0 daxdz+iax3=0,
Qyay+k0 daydz+iay3=0,
Sxax+k0 dbxdz+iax2bx=0,
Syay+k0 dbydz+iay2by=0,
dPdz=-Sx2+Sy22k0-i Qx+Qy2k0-max2+nay2k0-i2k0dk0dz.
d2Hmdx2-2x dHmdx+2mHm=0.
Qxk0=1qx=1Rx-i λπwx2,
ux2(1/qx2)ux2Sx2ux2=AxBx0CxDx0GxHx1ux1(1/qx1)ux1Sx1ux1,
1qx2=Cx+Dx/qx1Ax+Bx/qx1,
1qy2=Cy+Dy/qy1Ay+By/qy1.
Sx2=Sx1+Gx+Hx/qx1Ax+Bx/qx1,
Sy2=Sy1+Gy+Hy/qy1Ay+By/qy1.
ax2=ax1Ax+Bx/qx11+2iax12k01BxAx+Bx/qx1-1/2,
ay2=ay1Ay+By/qy11+2iay12k01ByAy+By/qy1-1/2,
bx2=bx1-ax1Sx1k01BxAx+Bx/qx1×1+2iax12k01BxAx+Bx/qx1-1/2,
by2=by1-ay1Sy1k01ByAy+By/qy1×1+2iay12k01ByAy+By/qy1-1/2,
P2-P1=i2ln(AxDx-BxCx)-i2ln(Ax+Bx/qx1)-i2ln(Ay+By/qy1)+i2m ln1+2iax12k01BxAx+Bx/qx1+i2n ln1+2iay12k01ByAy+By/qy1-Bx2k01(Sx1+Gx+Hx/qx1)2Ax+Bx/qx1-By2k01(Sy1+Gy+Hy/qy1)2Ay+By/qy1+Hx2k01(2Sx1+Gx+Hx/qx1)+Hy2k01(2Sy1+Gy+Hy/qy1)+12k010zGx dHxdz-Hx dGxdzdz+12k010zGy dHydz-Hy dGydzdz.
P2=P1-Bx2k01(Sx1+Gx+Hx/qx1)2Ax+Bx/qx1+Hxk01Sx1+P0,
P0=i2ln(AxDx-BxCx)-i2ln(Ax+Bx/qx1)-i2ln(Ay+By/qy1)+i2m ln1+2iax12k01BxAx+Bx/qx1+i2n ln1+2iay12k01ByAy+By/qy1-By2k01(Sy1+Gy+Hy/qy1)2Ay+By/qy1+Hx2k01(Gx+Hx/qx1)+Hy2k01(2Sy1+Gy+Hy/qy1)+12k010zGx dHxdz-Hx dGxdzdz+12k010zGy dHydz-Hy dGydzdz.
A2,m,n(x, y)=A0Hm(ax2x+bx2)Hn(ay2y+by2)×exp-iQx2x22+Qy2y22+Sy2y×{12exp[-i(Sx2αx+P2α)]+12exp[-i(Sx2βx+P2β)]}.
Sx2α=Sx1α+Gx+Hx/qx1Ax+Bx/qx1,
Sx2β=Sx1β+Gx+Hx/qx1Ax+Bx/qx1,
P2α=P1α-Bx2k01(Sx1α+Gx+Hx/qx1)2Ax+Bx/qx1+Hxk01Sx1α+P0,
P2β=P1β-Bx2k01(Sx1β+Gx+Hx/qx1)2Ax+Bx/qx1+Hxk01Sx1β+P0.
Sx1α=Sx1-ax1,
Sx1β=Sx1+ax1,
P1α=P1-bx1,
P1β=P1+bx1.
A2,m,n(x, y)=A0Hm(ax2x+bx2)Hn(ay2y+by2)×exp-iQx2x22+Qy2y22+Sy2y×12exp-iSx1-ax1+Gx+Hx/qx1Ax+Bx/qx1x+P1-bx1-Bx2k01(Sx1-ax1+Gx+Hx/qx1)2Ax+Bx/qx1+Hxk01(Sx1-ax1)+P0+12exp-iSx1+ax1+Gx+Hx/qx1Ax+Bx/qx1x+P1+bx1-Bx2k01(Sx1+ax1+Gx+Hx/qx1)2Ax+Bx/qx1+Hxk01(Sx1+ax1)+P0.
A2,m,n(x, y)=A0Hm(ax2x+bx2)Hn(ay2y+by2)×exp-iQx2x22+Qy2y22+Sx2x+Sy2y+P2×expi ax122k01BxAx+Bx/qx1×12exp-i-ax1Ax+Bx/qx1x-bx1+Bxk01Sx1+Gx+Hx/qx1Ax+Bx/qx1ax1-Hxk01ax1+12exp+i-ax1Ax+Bx/qx1x-bx1+Bxk01Sx1+Gx+Hx/qx1Ax+Bx/qx1ax1-Hxk01ax1=A0Hm(ax2x+bx2)Hn(ay2y+by2)×exp-iQx2x22+Qy2y22+Sx2x+Sy2y+P2×expi ax122k01BxAx+Bx/qx1cos(ax2x+bx2),
ax2=ax1Ax+Bx/qx1,
bx2=bx1-Bxk01Sx1+Gx+Hx/qx1Ax+Bx/qx1ax1+Hxk01ax1.
A2,m,n(x, y)=A0Hm(ax2x+bx2)Hn(ay2y+by2)×exp-iQx2x22+Qy2y22+Sx2x+Sy2y+P2×expi ax122k01BxAx+Bx/qx1+i ay122k01ByAy+By/qy1×sin(ax2x+bx2)cos(ax2x+bx2)exp(iax2x+ibx2)sin(ay2y+by2)cos(ay2y+by2)exp(iay2y+iby2).
A1,m(x, y)=A0Hm(2x/w1)exp(-x2/w12)cosh(ax1x).
Am(x, y)=A1,m(x, y)A0=Hm(2x)exp(-x2)cosh(ax),
Im(x, y)=Am2(x, y)=Hm2(2x)exp(-2x2)cosh2(ax).
f(x)=nan exp(i2πnx/L),
an=-L/2L/2f(x)exp(-i2πnx/L)dx.
an=|an|exp(iϕn).
f(x)=n|an|exp[i(2πnx/L+ϕn)]=n|an|[cos(2πnx/L+ϕn)+i sin(2πnx/L+ϕn)].
T(x)=|sin(2πx/L)|,
T(x)=2π-4πcos(4πx/L)1×3+cos(8πx/L)3×5+cos(12πx/L)5×7+.
M1=1f0110-1/f1=0f-1/f1.
M2=10-1/f11f01=1f-1/f0.

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