Abstract

Gaussian beam propagation is well described by the q-parameter and the ABCD matrices. A variety of ABCD matrices are available that represent commonly occurring scenarios/components in optics. One important phenomenon that has not been studied in detail is the interference of two optical beams with different q-parameters undergoing interference. In this paper, we describe the effect of interference of two Gaussian beams. We derive an ABCD matrix for the addition of two beams that takes into account both the amplitude and phase difference between two beams. This ABCD matrix will help greatly in determining the propagation of beams inside complex interferometers and finding the solutions for the coupled cavity Eigenmodes.

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References

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  1. A. E. Siegman, Lasers, University Science books, Sausalito, CA (1984).
  2. R. P. Herloski, S. Marshall, and R. L. Antos, “Gaussian Beam Ray-Equivalent Modeling and Optical Design—Erratum,” Appl. Opt. 22(8), 1168–1174 (1983).
    [CrossRef] [PubMed]
  3. J. Arnaud, “Representation of Gaussian beams by complex rays,” Appl. Opt. 24(4), 538–543 (1985).
    [CrossRef] [PubMed]
  4. A. W. Greynolds, “Vector Formulation of the Ray-Equivalent Method for General Gaussian Beam Propagation,” Proceedings of SPIE, Current Developments in Optical Engineering and Diffractive Phenomena 679, 129–133 (1986).
  5. G. W. Forbes and M. A. Alonso, “Using rays better. I. Theory for smoothly varying media,” J. Opt. Soc. Am. A 18(5), 1132–1145 (2001).
    [CrossRef]
  6. M. A. Alonso and G. W. Forbes, “Using rays better. II. Ray families to match prescribed wave fields,” J. Opt. Soc. Am. A 18(5), 1146–1159 (2001).
    [CrossRef]
  7. M. Born, and E. Wolf, Principles of Optics, 7th (expanded)ed., Cambridge U. Press, Cambridge, UK, (1999).
  8. D. Z. Anderson, “Alignment of resonant optical cavities,” Appl. Opt. 23(17), 2944–2949 (1984).
    [CrossRef] [PubMed]
  9. H. Yamamoto, M. Barton, B. Bhawal, M. Evans, and S. Yoshida, “Simulation tools for future interferometers,” J. Phys.: Conf. Ser. 32, 398–403 (2006).
    [CrossRef]
  10. S. A. Collins., “Lens-system diffraction integral written in terms of matrix optics,” J. Opt. Soc. Am. 60(9), 1168–1177 (1970).
    [CrossRef]
  11. J. Arnaud, “Nonorthogonal optical Waveguides and Resonators,” Bell Syst. Tech. J. (November), 2311–2348 (1970).
  12. M. J. Bastiaans, “The Expansion of an Optical Signal into a Discrete Set of Gaussian Beams,” Optik (Stuttg.) 57, 95–101 (1980).
  13. M. A. Arain and G. Mueller, “Design of the Advanced LIGO recycling cavities,” Opt. Express 16(14), 10018–10032 (2008), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-14-10018 .
    [CrossRef] [PubMed]

2008

2006

H. Yamamoto, M. Barton, B. Bhawal, M. Evans, and S. Yoshida, “Simulation tools for future interferometers,” J. Phys.: Conf. Ser. 32, 398–403 (2006).
[CrossRef]

2001

1986

A. W. Greynolds, “Vector Formulation of the Ray-Equivalent Method for General Gaussian Beam Propagation,” Proceedings of SPIE, Current Developments in Optical Engineering and Diffractive Phenomena 679, 129–133 (1986).

1985

1984

1983

1980

M. J. Bastiaans, “The Expansion of an Optical Signal into a Discrete Set of Gaussian Beams,” Optik (Stuttg.) 57, 95–101 (1980).

1970

J. Arnaud, “Nonorthogonal optical Waveguides and Resonators,” Bell Syst. Tech. J. (November), 2311–2348 (1970).

S. A. Collins., “Lens-system diffraction integral written in terms of matrix optics,” J. Opt. Soc. Am. 60(9), 1168–1177 (1970).
[CrossRef]

Alonso, M. A.

Anderson, D. Z.

Antos, R. L.

Arain, M. A.

Arnaud, J.

J. Arnaud, “Representation of Gaussian beams by complex rays,” Appl. Opt. 24(4), 538–543 (1985).
[CrossRef] [PubMed]

J. Arnaud, “Nonorthogonal optical Waveguides and Resonators,” Bell Syst. Tech. J. (November), 2311–2348 (1970).

Barton, M.

H. Yamamoto, M. Barton, B. Bhawal, M. Evans, and S. Yoshida, “Simulation tools for future interferometers,” J. Phys.: Conf. Ser. 32, 398–403 (2006).
[CrossRef]

Bastiaans, M. J.

M. J. Bastiaans, “The Expansion of an Optical Signal into a Discrete Set of Gaussian Beams,” Optik (Stuttg.) 57, 95–101 (1980).

Bhawal, B.

H. Yamamoto, M. Barton, B. Bhawal, M. Evans, and S. Yoshida, “Simulation tools for future interferometers,” J. Phys.: Conf. Ser. 32, 398–403 (2006).
[CrossRef]

Collins, S. A.

Evans, M.

H. Yamamoto, M. Barton, B. Bhawal, M. Evans, and S. Yoshida, “Simulation tools for future interferometers,” J. Phys.: Conf. Ser. 32, 398–403 (2006).
[CrossRef]

Forbes, G. W.

Greynolds, A. W.

A. W. Greynolds, “Vector Formulation of the Ray-Equivalent Method for General Gaussian Beam Propagation,” Proceedings of SPIE, Current Developments in Optical Engineering and Diffractive Phenomena 679, 129–133 (1986).

Herloski, R. P.

Marshall, S.

Mueller, G.

Yamamoto, H.

H. Yamamoto, M. Barton, B. Bhawal, M. Evans, and S. Yoshida, “Simulation tools for future interferometers,” J. Phys.: Conf. Ser. 32, 398–403 (2006).
[CrossRef]

Yoshida, S.

H. Yamamoto, M. Barton, B. Bhawal, M. Evans, and S. Yoshida, “Simulation tools for future interferometers,” J. Phys.: Conf. Ser. 32, 398–403 (2006).
[CrossRef]

Appl. Opt.

Bell Syst. Tech. J.

J. Arnaud, “Nonorthogonal optical Waveguides and Resonators,” Bell Syst. Tech. J. (November), 2311–2348 (1970).

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

J. Phys.: Conf. Ser.

H. Yamamoto, M. Barton, B. Bhawal, M. Evans, and S. Yoshida, “Simulation tools for future interferometers,” J. Phys.: Conf. Ser. 32, 398–403 (2006).
[CrossRef]

Opt. Express

Optik (Stuttg.)

M. J. Bastiaans, “The Expansion of an Optical Signal into a Discrete Set of Gaussian Beams,” Optik (Stuttg.) 57, 95–101 (1980).

Proceedings of SPIE, Current Developments in Optical Engineering and Diffractive Phenomena

A. W. Greynolds, “Vector Formulation of the Ray-Equivalent Method for General Gaussian Beam Propagation,” Proceedings of SPIE, Current Developments in Optical Engineering and Diffractive Phenomena 679, 129–133 (1986).

Other

M. Born, and E. Wolf, Principles of Optics, 7th (expanded)ed., Cambridge U. Press, Cambridge, UK, (1999).

A. E. Siegman, Lasers, University Science books, Sausalito, CA (1984).

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Figures (6)

Fig. 1
Fig. 1

FOM evaluated for various beam size mismatch as a function of phase angle between two Gaussian irradiance beams. Here w 1 = 5.2 cm, |α2|=2|α1|(cosθ+isinθ) and θ varies from zero to 360°. Although the plots shown are for 5.2 cm beam, the functional form remains same for widely different beam sizes.

Fig. 2
Fig. 2

FOM evaluated for various beam magnitude ratios. Negative values indicate that the two beams are 180° out of phase with each other. Various lines indicate a different beam size mismatch. Again w1=5 .2 cm .

Fig. 3
Fig. 3

Plot of two Gaussian irradiance beams with different magnitude and beam size ratio as indicated in the legend. Shown in red is the numerical sum of two beams while the approximation represented by Eq. (10) is shown via cyan color.

Fig. 4
Fig. 4

FOM evaluated for various ROC mismatches normalized to Rayleigh range Zr of about 8 km. The first beam is considered at the beam waist thus ΔR represents the ROC of the second beam.

Fig. 5
Fig. 5

FOM evaluated for same normalized beam size and ROC mismatch for various magnitude rations. The parameters of the first beam are kept constant where α1=1 , R1 = 1000 m, w1=5 .2 cm . The % change in the beam parameters of the second beam is plotted on x-axis.

Fig. 6
Fig. 6

Optical system to test the proposed ABCD formulism using example of Sec. 4.4. Here two beams (Beam 1 (red) and Beam 2 (blue)) are combined using a beam combiner and the resultant beam is calculated numerically as the sum of beam 1 and Beam 2. Beam 3 (green) is the beam obtained via the ABCD formulism proposed in this paper and propagated via the same optical system as Beam 1 and 2.

Equations (15)

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E1=α1er2[1w12+iπλR1],
E2=α2er2[1w22+iπλR2]
1q1,2=1R1,2iλπw1,22.
E1=α1er2[1w12+iπλR1]=α1[1r2w12iπr2λR1+...],
E2=α2er2[1w22+iπλR2]=α2[1r2w22iπr2λR2+...].
E=E1+E2=(α1+α2)[1r2(α1w12+α2w22)+iπλ(α1R1+α2R2)(α1+α2)+...].
1we2=(|a1|w12+|a2|w22)(|a1|+|a2|)=|a1|/w12(|a1|+|a2|)+|a2|/w22(|a1|+|a2|) , 
1Re=(|a1|R1+|a2|R2)(|a1|+|a2|)=|a1|R1(|a1|+|a2|)+|a2|R2(|a1|+|a2|)   .
1Reiλπwe2=|a1|(|a1|+|a2|)[1R1iλπw12]+|a2|(|a1|+|a2|)[1R2iλπw22].
1qe=|a1|(|a1|+|a2|)1q1+|a2|(|a1|+|a2|)1q2.
[ABCD] [11/qe]=[|a1|/(|a1|+|a2|)00|a1|/(|a1|+|a2|)] [11/q1]+[|a2|/(|a1|+|a2|)00|a2|/(|a1|+|a2|)] [11/q2]  . 
FOM=|(E1(r)+E2(r))E(r)dA|2|(E1(r)+E2(r))|2dx|E(r)|2dA.
FOM=|02πdφ0r[α1 er2[1w12+iπλR1]+α2 er2[1w22+iπλR2]][(α1+α2)er2[1we2iπλRe]]rdr|202πdφ0|(α1 er2[1w12+iπλR1]+α2 er2[1w22+iπλR2])|2rdr 02πdφ0|(α1+α2)er2[1we2+iπλRe]|2rdr.
0xea2x2dx=12a2.
FOM=4|{α11w12+1we2+iπλ(1R11Re)}+{α21w22+1we2+iπλ(1R21Re)}|2(we2)  ||a1|2w12+|a2|2w22+2[{α1×α2*1w12+1w22+iπλ(1R11R2)}+{α2×α1*1w12+1w22+iπλ(1R21R1)}]|.

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