Abstract

The paper introduces the complete model of the general astigmatic Gaussian beam as the most general case of the Gaussian beam in the fundamental mode. This includes the laws of propagation, reflection, and refraction as well as the equations for extracting from the complex-valued beam description its real-valued parameters, such as the beam spot radii and the radii of curvature of the wavefront. The suggested model is applicable to the case of an oblique incidence of the beam at any 3D surface that can be approximated by the second-order equation at the point of incidence. Thus it can be used in simulations of a large variety of 3D optical systems. The provided experimental validation of the model shows good agreement with simulations.

© 2013 Optical Society of America

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References

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  1. H. Kogelnik and T. Li, “Laser beams and resonators,” Appl. Opt. 5, 1550–1567 (1966).
    [CrossRef]
  2. J. Alda, “Laser and Gaussian beam propagation and transformation,” in Encyclopedia of Optical Engineering (Marcel Dekker, 2003), pp. 999–1013.
  3. G. A. Massey and A. E. Siegman, “Reflection and refraction of Gaussian light beams at tilted ellipsoidal surfaces,” Appl. Opt. 8, 975–978 (1969).
    [CrossRef]
  4. J. A. Arnaud and H. Kogelnik, “Gaussian light beams with general astigmatism,” Appl. Opt. 8, 1687–1694 (1969).
    [CrossRef]
  5. A. Rohani, A. A. Shishegar, and S. Safavi-Naeini, “A fast Gaussian beam tracing method for reflection and refraction of general vectorial astigmatic Gaussian beams from general curved surfaces,” Opt. Commun. 232, 1–10 (2004).
    [CrossRef]
  6. E. Kochkina, G. Heinzel, G. Wanner, V. Müller, C. Mahrdt, B. Sheard, S. Schuster, and K. Danzmann, “Simulating and optimizing laser interferometers,” in 9th LISA Symposium, Paris, ASP Conference Series (Astronomical Society of the Pacific, 2013), vol. 467, pp. 291–292.
  7. G. Wanner, G. Heinzel, E. Kochkina, C. Mahrdt, B. Sheard, S. Schuster, and K. Danzmann, “Methods for simulating the readout of lengths and angles in laser interferometers with Gaussian beams,” Opt. Commun. 285, 4831–4839 (2012).
    [CrossRef]
  8. J. Alda, S. Wang, and E. Bernabeu, “Analytical expression for the complex radius of curvature tensor Q for generalized Gaussian beams,” Opt. Commun. 80, 350–352 (1991).
    [CrossRef]
  9. A. E. Siegman, Lasers (University Science, 1986), pp. 626–697.
  10. A. B. Plachenov, V. N. Kudashov, and A. M. Radin, “Simple formula for a Gaussian beam with general astigmatism in a homogeneous medium,” Opt. Spectrosc. 106, 910–912 (2009).
    [CrossRef]
  11. J. Serna and G. Nemes, “Decoupling of coherent Gaussian beams with general astigmatism,” Opt. Lett. 18, 1774–1776 (1993).
    [CrossRef]
  12. G. A. Deschamps, “Ray techniques in electromagnetics,” Proc. IEEE 60, 1022–1035 (1972).
    [CrossRef]
  13. G. J. James, Geometrical Theory of Diffraction for Electromagnetic Waves, 3rd ed. (Peter Peregrinus, 1986), pp. 96–116.
  14. E. Hecht, Optics (Addison-Wesley, 1974), pp. 60–98.

2012 (1)

G. Wanner, G. Heinzel, E. Kochkina, C. Mahrdt, B. Sheard, S. Schuster, and K. Danzmann, “Methods for simulating the readout of lengths and angles in laser interferometers with Gaussian beams,” Opt. Commun. 285, 4831–4839 (2012).
[CrossRef]

2009 (1)

A. B. Plachenov, V. N. Kudashov, and A. M. Radin, “Simple formula for a Gaussian beam with general astigmatism in a homogeneous medium,” Opt. Spectrosc. 106, 910–912 (2009).
[CrossRef]

2004 (1)

A. Rohani, A. A. Shishegar, and S. Safavi-Naeini, “A fast Gaussian beam tracing method for reflection and refraction of general vectorial astigmatic Gaussian beams from general curved surfaces,” Opt. Commun. 232, 1–10 (2004).
[CrossRef]

1993 (1)

1991 (1)

J. Alda, S. Wang, and E. Bernabeu, “Analytical expression for the complex radius of curvature tensor Q for generalized Gaussian beams,” Opt. Commun. 80, 350–352 (1991).
[CrossRef]

1972 (1)

G. A. Deschamps, “Ray techniques in electromagnetics,” Proc. IEEE 60, 1022–1035 (1972).
[CrossRef]

1969 (2)

1966 (1)

Alda, J.

J. Alda, S. Wang, and E. Bernabeu, “Analytical expression for the complex radius of curvature tensor Q for generalized Gaussian beams,” Opt. Commun. 80, 350–352 (1991).
[CrossRef]

J. Alda, “Laser and Gaussian beam propagation and transformation,” in Encyclopedia of Optical Engineering (Marcel Dekker, 2003), pp. 999–1013.

Arnaud, J. A.

Bernabeu, E.

J. Alda, S. Wang, and E. Bernabeu, “Analytical expression for the complex radius of curvature tensor Q for generalized Gaussian beams,” Opt. Commun. 80, 350–352 (1991).
[CrossRef]

Danzmann, K.

G. Wanner, G. Heinzel, E. Kochkina, C. Mahrdt, B. Sheard, S. Schuster, and K. Danzmann, “Methods for simulating the readout of lengths and angles in laser interferometers with Gaussian beams,” Opt. Commun. 285, 4831–4839 (2012).
[CrossRef]

E. Kochkina, G. Heinzel, G. Wanner, V. Müller, C. Mahrdt, B. Sheard, S. Schuster, and K. Danzmann, “Simulating and optimizing laser interferometers,” in 9th LISA Symposium, Paris, ASP Conference Series (Astronomical Society of the Pacific, 2013), vol. 467, pp. 291–292.

Deschamps, G. A.

G. A. Deschamps, “Ray techniques in electromagnetics,” Proc. IEEE 60, 1022–1035 (1972).
[CrossRef]

Hecht, E.

E. Hecht, Optics (Addison-Wesley, 1974), pp. 60–98.

Heinzel, G.

G. Wanner, G. Heinzel, E. Kochkina, C. Mahrdt, B. Sheard, S. Schuster, and K. Danzmann, “Methods for simulating the readout of lengths and angles in laser interferometers with Gaussian beams,” Opt. Commun. 285, 4831–4839 (2012).
[CrossRef]

E. Kochkina, G. Heinzel, G. Wanner, V. Müller, C. Mahrdt, B. Sheard, S. Schuster, and K. Danzmann, “Simulating and optimizing laser interferometers,” in 9th LISA Symposium, Paris, ASP Conference Series (Astronomical Society of the Pacific, 2013), vol. 467, pp. 291–292.

James, G. J.

G. J. James, Geometrical Theory of Diffraction for Electromagnetic Waves, 3rd ed. (Peter Peregrinus, 1986), pp. 96–116.

Kochkina, E.

G. Wanner, G. Heinzel, E. Kochkina, C. Mahrdt, B. Sheard, S. Schuster, and K. Danzmann, “Methods for simulating the readout of lengths and angles in laser interferometers with Gaussian beams,” Opt. Commun. 285, 4831–4839 (2012).
[CrossRef]

E. Kochkina, G. Heinzel, G. Wanner, V. Müller, C. Mahrdt, B. Sheard, S. Schuster, and K. Danzmann, “Simulating and optimizing laser interferometers,” in 9th LISA Symposium, Paris, ASP Conference Series (Astronomical Society of the Pacific, 2013), vol. 467, pp. 291–292.

Kogelnik, H.

Kudashov, V. N.

A. B. Plachenov, V. N. Kudashov, and A. M. Radin, “Simple formula for a Gaussian beam with general astigmatism in a homogeneous medium,” Opt. Spectrosc. 106, 910–912 (2009).
[CrossRef]

Li, T.

Mahrdt, C.

G. Wanner, G. Heinzel, E. Kochkina, C. Mahrdt, B. Sheard, S. Schuster, and K. Danzmann, “Methods for simulating the readout of lengths and angles in laser interferometers with Gaussian beams,” Opt. Commun. 285, 4831–4839 (2012).
[CrossRef]

E. Kochkina, G. Heinzel, G. Wanner, V. Müller, C. Mahrdt, B. Sheard, S. Schuster, and K. Danzmann, “Simulating and optimizing laser interferometers,” in 9th LISA Symposium, Paris, ASP Conference Series (Astronomical Society of the Pacific, 2013), vol. 467, pp. 291–292.

Massey, G. A.

Müller, V.

E. Kochkina, G. Heinzel, G. Wanner, V. Müller, C. Mahrdt, B. Sheard, S. Schuster, and K. Danzmann, “Simulating and optimizing laser interferometers,” in 9th LISA Symposium, Paris, ASP Conference Series (Astronomical Society of the Pacific, 2013), vol. 467, pp. 291–292.

Nemes, G.

Plachenov, A. B.

A. B. Plachenov, V. N. Kudashov, and A. M. Radin, “Simple formula for a Gaussian beam with general astigmatism in a homogeneous medium,” Opt. Spectrosc. 106, 910–912 (2009).
[CrossRef]

Radin, A. M.

A. B. Plachenov, V. N. Kudashov, and A. M. Radin, “Simple formula for a Gaussian beam with general astigmatism in a homogeneous medium,” Opt. Spectrosc. 106, 910–912 (2009).
[CrossRef]

Rohani, A.

A. Rohani, A. A. Shishegar, and S. Safavi-Naeini, “A fast Gaussian beam tracing method for reflection and refraction of general vectorial astigmatic Gaussian beams from general curved surfaces,” Opt. Commun. 232, 1–10 (2004).
[CrossRef]

Safavi-Naeini, S.

A. Rohani, A. A. Shishegar, and S. Safavi-Naeini, “A fast Gaussian beam tracing method for reflection and refraction of general vectorial astigmatic Gaussian beams from general curved surfaces,” Opt. Commun. 232, 1–10 (2004).
[CrossRef]

Schuster, S.

G. Wanner, G. Heinzel, E. Kochkina, C. Mahrdt, B. Sheard, S. Schuster, and K. Danzmann, “Methods for simulating the readout of lengths and angles in laser interferometers with Gaussian beams,” Opt. Commun. 285, 4831–4839 (2012).
[CrossRef]

E. Kochkina, G. Heinzel, G. Wanner, V. Müller, C. Mahrdt, B. Sheard, S. Schuster, and K. Danzmann, “Simulating and optimizing laser interferometers,” in 9th LISA Symposium, Paris, ASP Conference Series (Astronomical Society of the Pacific, 2013), vol. 467, pp. 291–292.

Serna, J.

Sheard, B.

G. Wanner, G. Heinzel, E. Kochkina, C. Mahrdt, B. Sheard, S. Schuster, and K. Danzmann, “Methods for simulating the readout of lengths and angles in laser interferometers with Gaussian beams,” Opt. Commun. 285, 4831–4839 (2012).
[CrossRef]

E. Kochkina, G. Heinzel, G. Wanner, V. Müller, C. Mahrdt, B. Sheard, S. Schuster, and K. Danzmann, “Simulating and optimizing laser interferometers,” in 9th LISA Symposium, Paris, ASP Conference Series (Astronomical Society of the Pacific, 2013), vol. 467, pp. 291–292.

Shishegar, A. A.

A. Rohani, A. A. Shishegar, and S. Safavi-Naeini, “A fast Gaussian beam tracing method for reflection and refraction of general vectorial astigmatic Gaussian beams from general curved surfaces,” Opt. Commun. 232, 1–10 (2004).
[CrossRef]

Siegman, A. E.

Wang, S.

J. Alda, S. Wang, and E. Bernabeu, “Analytical expression for the complex radius of curvature tensor Q for generalized Gaussian beams,” Opt. Commun. 80, 350–352 (1991).
[CrossRef]

Wanner, G.

G. Wanner, G. Heinzel, E. Kochkina, C. Mahrdt, B. Sheard, S. Schuster, and K. Danzmann, “Methods for simulating the readout of lengths and angles in laser interferometers with Gaussian beams,” Opt. Commun. 285, 4831–4839 (2012).
[CrossRef]

E. Kochkina, G. Heinzel, G. Wanner, V. Müller, C. Mahrdt, B. Sheard, S. Schuster, and K. Danzmann, “Simulating and optimizing laser interferometers,” in 9th LISA Symposium, Paris, ASP Conference Series (Astronomical Society of the Pacific, 2013), vol. 467, pp. 291–292.

Appl. Opt. (3)

Opt. Commun. (3)

G. Wanner, G. Heinzel, E. Kochkina, C. Mahrdt, B. Sheard, S. Schuster, and K. Danzmann, “Methods for simulating the readout of lengths and angles in laser interferometers with Gaussian beams,” Opt. Commun. 285, 4831–4839 (2012).
[CrossRef]

J. Alda, S. Wang, and E. Bernabeu, “Analytical expression for the complex radius of curvature tensor Q for generalized Gaussian beams,” Opt. Commun. 80, 350–352 (1991).
[CrossRef]

A. Rohani, A. A. Shishegar, and S. Safavi-Naeini, “A fast Gaussian beam tracing method for reflection and refraction of general vectorial astigmatic Gaussian beams from general curved surfaces,” Opt. Commun. 232, 1–10 (2004).
[CrossRef]

Opt. Lett. (1)

Opt. Spectrosc. (1)

A. B. Plachenov, V. N. Kudashov, and A. M. Radin, “Simple formula for a Gaussian beam with general astigmatism in a homogeneous medium,” Opt. Spectrosc. 106, 910–912 (2009).
[CrossRef]

Proc. IEEE (1)

G. A. Deschamps, “Ray techniques in electromagnetics,” Proc. IEEE 60, 1022–1035 (1972).
[CrossRef]

Other (5)

G. J. James, Geometrical Theory of Diffraction for Electromagnetic Waves, 3rd ed. (Peter Peregrinus, 1986), pp. 96–116.

E. Hecht, Optics (Addison-Wesley, 1974), pp. 60–98.

E. Kochkina, G. Heinzel, G. Wanner, V. Müller, C. Mahrdt, B. Sheard, S. Schuster, and K. Danzmann, “Simulating and optimizing laser interferometers,” in 9th LISA Symposium, Paris, ASP Conference Series (Astronomical Society of the Pacific, 2013), vol. 467, pp. 291–292.

A. E. Siegman, Lasers (University Science, 1986), pp. 626–697.

J. Alda, “Laser and Gaussian beam propagation and transformation,” in Encyclopedia of Optical Engineering (Marcel Dekker, 2003), pp. 999–1013.

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Figures (8)

Fig. 1.
Fig. 1.

Nonorthogonal optical system consisting of the two spherical lenses. The plane of incidence for the beam transformation by the first lens and the plane of incidence for the beam transformation by the second lens are aligned at an oblique angle with respect to one other.

Fig. 2.
Fig. 2.

Gaussian beam (or Collins) chart. q1=i66mm, q2=500+i266mm at z=0. The axes are ω(z)=I(q)/|q|2 and ρ(z)=R(q)/|q|2. The line connecting two circles for the same value of z forms an angle, with tangent (ρ1ρ2)/(ω1ω2), with the ω axis.

Fig. 3.
Fig. 3.

Beam width evolution of (a) simple and (b) general astigmatic Gaussian beams with identical q-parameters and optical wavelengths (λ=1064nm; q1=i66mm, q2=500+i266mm at z=0; q1=500+i66mm, q2=i266mm at z=0.5m). The complex angle for the general astigmatic beam is θ=20°+i10°. In the simple astigmatic case one can observe two waists in two orthogonal planes (blue ring (left) corresponds to the waist in the XZ plane reached at z=0, green ring (right) corresponds to the waist in YZ plane reached at z=0.5m). In the general astigmatic case waists cannot be defined.

Fig. 4.
Fig. 4.

(a) Beam width evolution and (b) radius of curvature of the wavefront evolution for a general astigmatic Gaussian beam in comparison to the same quantities for a simple astigmatic Gaussian beam with identical q-parameters and optical wavelength. q1=i66mm, q2=500+i266mm at z=0. λ=1064nm. The complex angle for the general astigmatic beam is θ=20°+i10°. For any value of z: w1GA(z)<w1,2SA(z)<w2GA(z), w1GA(z)w2GA(z).

Fig. 5.
Fig. 5.

Geometry of the reflection and transmission of a general astigmatic Gaussian beam.

Fig. 6.
Fig. 6.

Simplified schematic of the experiment. The output from a laser was coupled into a fiber acting as a special mode filter. First, the beam exiting the fiber was characterized using the CCD camera, then two cylindrical lenses (focal length fi, distance to fiber collimator zi, angle αi to y axis) were inserted and the resulting beam intensity pattern was measured with the CCD camera at different positions behind the second lens.

Fig. 7.
Fig. 7.

(a) Intensity ellipse angle, and (b) semi-axes along the propagation behind the second lens. Measured points and simulated curves with nominal and fitted parameters (see Table 1). The threshold on the graph (b) is the maximum beam radius that can be measured by the CCD-camera.

Fig. 8.
Fig. 8.

Measured (left column) and simulated (right column) normalized intensity distributions at distances z behind the second lens, color-coded in a linear scale. The white lines represent the semi-axes. The parameters of the setup are listed in Table 1. In the simulation, fitted parameters were used. At z=11.5cm the measured intensity ellipse angle with respect to the global y axis φwm=69.3°, and the angle of simulated intensity ellipse is φws=69°. At z=14cm φwm=73.5°, and φws=73.6°. At z=16cm φwm=77.5°, and φws=77.4°.

Tables (1)

Tables Icon

Table 1. Parameters of the Setup, Including Measured Values, Tolerances, and Fitted Values

Equations (57)

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E(x,y,z)=E0(z)exp[iϕac+iη(z)ik2(x2qx(z)+y2qy(z))],
qx,y=1/Rx,yiλ/(πwx,y2),
ϕac=jkjlj,
η(z)=12[arctan(R(qx)I(qx))+arctan(R(qy)I(qy))].
E(x,y,z)=E0(z)exp{iϕac+iη(z)ik2[(cos2θq1(z)+sin2θq2(z))x2+(sin2θq1(z)+cos2θq2(z))y2+sin2θ(1q1(z)1q2(z))xy]}.
E(r,z)=E0(z)exp{iϕac+iη(z)ik2rTQ(z)r},
Q(z)=(cos2θq1(z)+sin2θq2(z)12sin2θ(1q1(z)1q2(z))12sin2θ(1q1(z)1q2(z))sin2θq1(z)+cos2θq2(z))
ϕ(r,z)=iϕac+iη(z)ik2rTQ(z)r.
Q=QI2+ΔzQ,
Q=(cos2θq1+Δz+sin2θq2+Δz12sin2θ(1q1+Δz1q2+Δz)12sin2θ(1q1+Δz1q2+Δz)sin2θq1+Δz+cos2θq2+Δz).
qj=qj+Δz,j=1,2.
P=XYI(x,y,z)dxdy.
I(x,y,z)=|E(x,y,z)|2=E0(z)2exp(g1(z)x2+g2(z)y2+g(z)xy),
g1(z)=kI(Q11),g2(z)=kI(Q22),g(z)=kI(Q12+Q21),
P=E02(z)2π4g1(z)g2(z)g2(z),
E0(z)=Pλ4I(Q11)I(Q22)I(Q12+Q21)2.
tan2θ=Q12+Q21Q11Q22.
qj=zz0j+izRj,j=1,2,
E(r,z)=E0(z)exp{k2rTI(Q)r}exp{ik2rTR(Q)riϕac+iη(z)}.
W(z)=(k/2)I(Q),C(z)=R(Q).
WSA(z)=(1wx2(z)001wy2(z)),CSA(z)=(1Rx(z)001Ry(z)),
E(r,z)=E0(z)exp{rTW(z)r}exp{ik2rTC(z)riϕac+iη(z)}.
WGA(z)=(cos2φww12(z)+sin2φww22(z)12sin2φw(1w12(z)1w22(z))12sin2φw(1w12(z)1w22(z))sin2φww12(z)+cos2φww22(z)),CGA(z)=(cos2φRR1(z)+sin2φRR2(z)12sin2φR(1R1(z)1R2(z))12sin2φR(1R1(z)1R2(z))sin2φRR1(z)+cos2φRR2(z)).
tan2φw=W12+W21W11W22,
tan2φR=C12+C21C11C22.
θ=α+iβ,1/qj=ρjiωj,j=1,2,ρj(z)=R(qj)/|qj|2,ωj(z)=I(qj)/|qj|2,
tan2φw0(z)=ρ1(z)ρ2(z)ω1(z)ω2(z)tanh2β,φw(z)=φw0(z)+α,
tan2φR0(z)=ω1(z)ω2(z)ρ1(z)ρ2(z)tanh2β,φR(z)=φR0(z)+α.
1w1,22(z)=k4{ω1(z)+ω2(z)±[ω1(z)ω2(z)]2cosh22β+[ρ1(z)ρ2(z)]2sinh22β},
1R1,2(z)=12{ρ1(z)+ρ2(z)±[ρ1(z)ρ2(z)]2cosh22β+[ω1(z)ω2(z)]2sinh22β}.
tan2(φw(z)φR(z))=sinh2βcosh2β(ρ1(z)ρ2(z)ω1(z)ω2(z)+ω1(z)ω2(z)ρ1(z)ρ2(z)).
tan2φw0(z)tan2φR0(z)=tanh22β
[ω1(z)ω2(z)]2cosh22β1+[ρ1(z)ρ2(z)]2sinh22β0[ω1(z)ω2(z)]2,
1w2GA2(z)<1w1,2SA2(z)<1w1GA2(z),
w1GA(z)<w1,2SA(z)<w2GA(z),
1R2GA(z)<1R1,2SA(z)<1R1GA(z),
ω1+ω2(ω1ω2)2cosh22β+(ρ1ρ2)2sinh22β,
cosh22β|q1q2*q1q2|.
s^(d)=d1d^1+d2d^212(dTCsd)n^,
f(x)=xTSx1=0,
S=(1/A20001/B20001/C2),
f(x)=(xt)TS(xt)1=0.
Cs=(2z/x22z/xy2z/yx2z/y2).
Cellipsoid=±1b(S11S33S132+gty2S12S33S13S23gtxtyS12S33S13S23gtxtyS22S33S232+gtx2),
b=[(S13tx+S23ty)2S33(S11tx2+2S12txty+S22ty21)]3/2,g=S11S232+S22S132+S33S1222S12S13S23S11S22S33,
Csphere=±(c00c),
z^r=z^i2(z^r·n^)n^,
z^t=nintz^i+(cosθtnintcosθi)n^,nisinθi=ntsinθt,
ϕ(rl,zl)=kl(zl+12rlTQl(zl)rl),l=i,r,t,
xl=s^(d)x^l=d1x^ld^1+d2x^ld^21/2(dCsdT)x^ln^,yl=s^(d)y^l=d1y^ld^1+d2y^ld^21/2(dCsdT)y^ln^,zl=s^(d)z^l=d1z^ld^1+d2z^ld^21/2(dCsdT)z^ln^.
Kl=(x^ld^1x^ld^2y^ld^1y^ld^2),l=i,r,t,
rl=Kld12(dCsdT)·(x^ln^y^ln^),zl=d1z^ld^1+d2z^ld^212(dCsdT)n^z^l.
ϕl=kl((z^ld^1)d1+(z^ld^2)d212(dCsdT)n^z^l+12(Kld)TQl(zl)(Kld))=kl((z^ld^1)d^1+(z^ld^2)d^2)pl(d1d^1+d2d^2)12kldT(KlTQl(zl)KlCn^z^l)Γld.
kiΓi=kiΓr=ktΓt,
Qr=(KrT)1(KiTQiKiC(n^z^in^z^r))Kr1,
Qt=n1n2(KtT)1(KiTQiKiC(n^z^in2n1n^z^t))Kt1.
wi(z)=w0i1+(λzz0iπw0i2)2,i=1,2,

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