Abstract

We present two methods for the precise independent focusing of orthogonal linear polarizations of light at arbitrary relative locations. Our first scheme uses a displaced lens in a polarization Sagnac interferometer to provide adjustable longitudinal and lateral focal displacements via simple geometry; the second uses uniaxial crystals to achieve the same effect in a compact collinear setup. We develop the theoretical applications and limitations of our schemes, and provide experimental confirmation of our calculations.

© 2013 OSA

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  1. E. Higurashi, R. Sawada, and T. Ito, “Optically induced angular alignment of birefringent micro-objects by linear polarization,” Appl. Phys. Lett.73(21), 3034 (1998).
    [CrossRef]
  2. J. B. Lassiter, M. W. Knight, N. A. Mirin, and N. J. Halas, “Reshaping the plasmonic properties of an individual nanoparticle,” Nano Lett.9(12), 4326–4332 (2009).
    [CrossRef] [PubMed]
  3. M. O. El-Shenawee, “Polarization dependence of plasmonic nanotoroid dimer antenna,” Antennas and Wireless Propagation Letters, IEEE9, 463–466 (2010).
    [CrossRef]
  4. C. M. Dutta, T. A. Ali, D. W. Brandl, T. H. Park, and P. Nordlander, “Plasmonic properties of a metallic torus,” J. Chem. Phys.129(8), 084706 (2008).
    [CrossRef] [PubMed]
  5. K. Bonin, B. Kourmanov, and T. Walker, “Light torque nanocontrol, nanomotors and nanorockers,” Opt. Express10(19), 984–989 (2002).
    [CrossRef] [PubMed]
  6. A. A. Yanik, R. Adato, S. Erramilli, and H. Altug, “Hybridized nanocavities as single-polarized plasmonic antennas,” Opt. Express17(23), 20900–20910 (2009).
    [CrossRef] [PubMed]
  7. R. Rangarajan, L. E. Vicent, A. B. U’Ren, and P. G. Kwiat, “A. B. U'Ren, and P. G. Kwiat, “Engineering an ideal indistinguishable photon-pair source for optical quantum information processing,” J. Mod. Opt.58(3-4), 318–327 (2011).
    [CrossRef]
  8. L. E. Vicent, A. B. U’Ren, R. Rangarajan, C. I. Osorio, J. P. Torres, L. Zhang, and I. A. Walmsley, “Design of bright, fiber-coupled and fully factorable photon pair sources,” New J. Phys.12(9), 093027 (2010).
    [CrossRef]
  9. W. S. Bakr, A. Peng, M. E. Tai, R. Ma, J. Simon, J. I. Gillen, S. Fölling, L. Pollet, and M. Greiner, “Probing the superfluid-to-Mott insulator transition at the single-atom level,” Science329(5991), 547–550 (2010).
    [CrossRef] [PubMed]
  10. F. Kenny, D. Lara, O. G. Rodríguez-Herrera, and C. Dainty, “Complete polarization and phase control for focus-shaping in high-NA microscopy,” Opt. Express20(13), 14015–14029 (2012).
    [CrossRef] [PubMed]
  11. S. Sanyal and A. Ghosh, “High tolerance to spherical aberrations and defects of focus with a birefringent lens,” Appl. Opt.41(22), 4611–4619 (2002).
    [CrossRef] [PubMed]
  12. S. Sanyal, Y. Kawata, S. Mandal, and A. Ghosh, “High tolerance to off-axis aberrations with a birefringent lens,” Opt. Eng.43(6), 1381–1386 (2004).
    [CrossRef]
  13. Y. Unno, “Point-spread function for a rotationally symmetric birefringent lens,” J. Opt. Soc. Am. A19(4), 781–791 (2002).
    [CrossRef] [PubMed]
  14. H. Kikuta, K. Iwata, and H. Shimomura, “First-order aberration of a double-focus lens made of a uniaxial crystal,” J. Opt. Soc. Am. A9(5), 814–819 (1992).
    [CrossRef]
  15. J. P. Lesso, A. J. Duncan, W. Sibbett, and M. J. Padgett, “Aberrations introduced by a lens made from a birefringent material,” Appl. Opt.39(4), 592–598 (2000).
    [CrossRef] [PubMed]
  16. S. Sanyal, P. Bandyopadhayay, and A. Ghosh, “Vector wave imagery using a birefringent lens,” Opt. Eng.37(2), 592–599 (1998).
    [CrossRef]
  17. M. Avendaño-Alejo and M. Rosete-Aguilar, “Paraxial theory for birefringent lenses,” J. Opt. Soc. Am. A22(5), 881–891 (2005).
    [CrossRef] [PubMed]
  18. S. Sanyal and A. Ghosh, “High focal depth with a quasi-bifocus birefringent lens,” Appl. Opt.39(14), 2321–2325 (2000).
    [CrossRef] [PubMed]
  19. X. Liu, X. Cai, S. Chang, and C. Grover, “Cemented doublet lens with an extended focal depth,” Opt. Express13(2), 552–557 (2005).
    [CrossRef] [PubMed]
  20. R. Rangarajan . “Photonic sources and detectors for quantum information protocols: a trilogy in eight parts.” Diss. University of Illinois, Urbana-Champaign, 2010. Web. July 2010.
  21. We assume that each focal location could be anywhere within a Gaussian distribution with width equal to the error measured on the original data points (and centered about the measured value). We then calculate one million sets of simulated data, each set containing one value from the Gaussian distribution associated with each measured point. We then calculate the slope of each set, and the Gaussian width of the 1 million sets of slopes gives an accurate error value on our measured value of the slope.
  22. To see this, one can find the necessary lens displacement d for a given application by comparing the final beam waist wout,2 = √[λαf22/π(α2 + d2)] for two lenses with the final beam waist wout,1 = λf1/πw for one lens. For example, assume we want to achieve a focal size β times smaller than that possible with a single lens, so that β = wout,1/wout,2.Solving this equality for d gives d = √[(βf2)2 - α2]. If, then, we wish to focus five times more tightly (i.e., the desired beam waist is 5 times smaller than it was with only one lens), then d will be greater than 50, well within the range of validity of our equations.
  23. K. Yonezawa, Y. Kozawa, and S. Sato, “Focusing of radially and azimuthally polarized beams through a uniaxial crystal,” J. Opt. Soc. Am. A25(2), 469–472 (2008).
    [CrossRef] [PubMed]
  24. R. Goldstein, Electro-optic Devices in Review (Lasers and Applications, 1986).
  25. B. E. A. Saleh and M. C. Teich, Beam Optics, in Fundamentals of Photonics (John Wiley & Sons, 1991).

2012

2011

R. Rangarajan, L. E. Vicent, A. B. U’Ren, and P. G. Kwiat, “A. B. U'Ren, and P. G. Kwiat, “Engineering an ideal indistinguishable photon-pair source for optical quantum information processing,” J. Mod. Opt.58(3-4), 318–327 (2011).
[CrossRef]

2010

L. E. Vicent, A. B. U’Ren, R. Rangarajan, C. I. Osorio, J. P. Torres, L. Zhang, and I. A. Walmsley, “Design of bright, fiber-coupled and fully factorable photon pair sources,” New J. Phys.12(9), 093027 (2010).
[CrossRef]

W. S. Bakr, A. Peng, M. E. Tai, R. Ma, J. Simon, J. I. Gillen, S. Fölling, L. Pollet, and M. Greiner, “Probing the superfluid-to-Mott insulator transition at the single-atom level,” Science329(5991), 547–550 (2010).
[CrossRef] [PubMed]

M. O. El-Shenawee, “Polarization dependence of plasmonic nanotoroid dimer antenna,” Antennas and Wireless Propagation Letters, IEEE9, 463–466 (2010).
[CrossRef]

2009

J. B. Lassiter, M. W. Knight, N. A. Mirin, and N. J. Halas, “Reshaping the plasmonic properties of an individual nanoparticle,” Nano Lett.9(12), 4326–4332 (2009).
[CrossRef] [PubMed]

A. A. Yanik, R. Adato, S. Erramilli, and H. Altug, “Hybridized nanocavities as single-polarized plasmonic antennas,” Opt. Express17(23), 20900–20910 (2009).
[CrossRef] [PubMed]

2008

C. M. Dutta, T. A. Ali, D. W. Brandl, T. H. Park, and P. Nordlander, “Plasmonic properties of a metallic torus,” J. Chem. Phys.129(8), 084706 (2008).
[CrossRef] [PubMed]

K. Yonezawa, Y. Kozawa, and S. Sato, “Focusing of radially and azimuthally polarized beams through a uniaxial crystal,” J. Opt. Soc. Am. A25(2), 469–472 (2008).
[CrossRef] [PubMed]

2005

2004

S. Sanyal, Y. Kawata, S. Mandal, and A. Ghosh, “High tolerance to off-axis aberrations with a birefringent lens,” Opt. Eng.43(6), 1381–1386 (2004).
[CrossRef]

2002

2000

1998

S. Sanyal, P. Bandyopadhayay, and A. Ghosh, “Vector wave imagery using a birefringent lens,” Opt. Eng.37(2), 592–599 (1998).
[CrossRef]

E. Higurashi, R. Sawada, and T. Ito, “Optically induced angular alignment of birefringent micro-objects by linear polarization,” Appl. Phys. Lett.73(21), 3034 (1998).
[CrossRef]

1992

Adato, R.

Ali, T. A.

C. M. Dutta, T. A. Ali, D. W. Brandl, T. H. Park, and P. Nordlander, “Plasmonic properties of a metallic torus,” J. Chem. Phys.129(8), 084706 (2008).
[CrossRef] [PubMed]

Altug, H.

Avendaño-Alejo, M.

Bakr, W. S.

W. S. Bakr, A. Peng, M. E. Tai, R. Ma, J. Simon, J. I. Gillen, S. Fölling, L. Pollet, and M. Greiner, “Probing the superfluid-to-Mott insulator transition at the single-atom level,” Science329(5991), 547–550 (2010).
[CrossRef] [PubMed]

Bandyopadhayay, P.

S. Sanyal, P. Bandyopadhayay, and A. Ghosh, “Vector wave imagery using a birefringent lens,” Opt. Eng.37(2), 592–599 (1998).
[CrossRef]

Bonin, K.

Brandl, D. W.

C. M. Dutta, T. A. Ali, D. W. Brandl, T. H. Park, and P. Nordlander, “Plasmonic properties of a metallic torus,” J. Chem. Phys.129(8), 084706 (2008).
[CrossRef] [PubMed]

Cai, X.

Chang, S.

Dainty, C.

Duncan, A. J.

Dutta, C. M.

C. M. Dutta, T. A. Ali, D. W. Brandl, T. H. Park, and P. Nordlander, “Plasmonic properties of a metallic torus,” J. Chem. Phys.129(8), 084706 (2008).
[CrossRef] [PubMed]

El-Shenawee, M. O.

M. O. El-Shenawee, “Polarization dependence of plasmonic nanotoroid dimer antenna,” Antennas and Wireless Propagation Letters, IEEE9, 463–466 (2010).
[CrossRef]

Erramilli, S.

Fölling, S.

W. S. Bakr, A. Peng, M. E. Tai, R. Ma, J. Simon, J. I. Gillen, S. Fölling, L. Pollet, and M. Greiner, “Probing the superfluid-to-Mott insulator transition at the single-atom level,” Science329(5991), 547–550 (2010).
[CrossRef] [PubMed]

Ghosh, A.

S. Sanyal, Y. Kawata, S. Mandal, and A. Ghosh, “High tolerance to off-axis aberrations with a birefringent lens,” Opt. Eng.43(6), 1381–1386 (2004).
[CrossRef]

S. Sanyal and A. Ghosh, “High tolerance to spherical aberrations and defects of focus with a birefringent lens,” Appl. Opt.41(22), 4611–4619 (2002).
[CrossRef] [PubMed]

S. Sanyal and A. Ghosh, “High focal depth with a quasi-bifocus birefringent lens,” Appl. Opt.39(14), 2321–2325 (2000).
[CrossRef] [PubMed]

S. Sanyal, P. Bandyopadhayay, and A. Ghosh, “Vector wave imagery using a birefringent lens,” Opt. Eng.37(2), 592–599 (1998).
[CrossRef]

Gillen, J. I.

W. S. Bakr, A. Peng, M. E. Tai, R. Ma, J. Simon, J. I. Gillen, S. Fölling, L. Pollet, and M. Greiner, “Probing the superfluid-to-Mott insulator transition at the single-atom level,” Science329(5991), 547–550 (2010).
[CrossRef] [PubMed]

Greiner, M.

W. S. Bakr, A. Peng, M. E. Tai, R. Ma, J. Simon, J. I. Gillen, S. Fölling, L. Pollet, and M. Greiner, “Probing the superfluid-to-Mott insulator transition at the single-atom level,” Science329(5991), 547–550 (2010).
[CrossRef] [PubMed]

Grover, C.

Halas, N. J.

J. B. Lassiter, M. W. Knight, N. A. Mirin, and N. J. Halas, “Reshaping the plasmonic properties of an individual nanoparticle,” Nano Lett.9(12), 4326–4332 (2009).
[CrossRef] [PubMed]

Higurashi, E.

E. Higurashi, R. Sawada, and T. Ito, “Optically induced angular alignment of birefringent micro-objects by linear polarization,” Appl. Phys. Lett.73(21), 3034 (1998).
[CrossRef]

Ito, T.

E. Higurashi, R. Sawada, and T. Ito, “Optically induced angular alignment of birefringent micro-objects by linear polarization,” Appl. Phys. Lett.73(21), 3034 (1998).
[CrossRef]

Iwata, K.

Kawata, Y.

S. Sanyal, Y. Kawata, S. Mandal, and A. Ghosh, “High tolerance to off-axis aberrations with a birefringent lens,” Opt. Eng.43(6), 1381–1386 (2004).
[CrossRef]

Kenny, F.

Kikuta, H.

Knight, M. W.

J. B. Lassiter, M. W. Knight, N. A. Mirin, and N. J. Halas, “Reshaping the plasmonic properties of an individual nanoparticle,” Nano Lett.9(12), 4326–4332 (2009).
[CrossRef] [PubMed]

Kourmanov, B.

Kozawa, Y.

Kwiat, P. G.

R. Rangarajan, L. E. Vicent, A. B. U’Ren, and P. G. Kwiat, “A. B. U'Ren, and P. G. Kwiat, “Engineering an ideal indistinguishable photon-pair source for optical quantum information processing,” J. Mod. Opt.58(3-4), 318–327 (2011).
[CrossRef]

Lara, D.

Lassiter, J. B.

J. B. Lassiter, M. W. Knight, N. A. Mirin, and N. J. Halas, “Reshaping the plasmonic properties of an individual nanoparticle,” Nano Lett.9(12), 4326–4332 (2009).
[CrossRef] [PubMed]

Lesso, J. P.

Liu, X.

Ma, R.

W. S. Bakr, A. Peng, M. E. Tai, R. Ma, J. Simon, J. I. Gillen, S. Fölling, L. Pollet, and M. Greiner, “Probing the superfluid-to-Mott insulator transition at the single-atom level,” Science329(5991), 547–550 (2010).
[CrossRef] [PubMed]

Mandal, S.

S. Sanyal, Y. Kawata, S. Mandal, and A. Ghosh, “High tolerance to off-axis aberrations with a birefringent lens,” Opt. Eng.43(6), 1381–1386 (2004).
[CrossRef]

Mirin, N. A.

J. B. Lassiter, M. W. Knight, N. A. Mirin, and N. J. Halas, “Reshaping the plasmonic properties of an individual nanoparticle,” Nano Lett.9(12), 4326–4332 (2009).
[CrossRef] [PubMed]

Nordlander, P.

C. M. Dutta, T. A. Ali, D. W. Brandl, T. H. Park, and P. Nordlander, “Plasmonic properties of a metallic torus,” J. Chem. Phys.129(8), 084706 (2008).
[CrossRef] [PubMed]

Osorio, C. I.

L. E. Vicent, A. B. U’Ren, R. Rangarajan, C. I. Osorio, J. P. Torres, L. Zhang, and I. A. Walmsley, “Design of bright, fiber-coupled and fully factorable photon pair sources,” New J. Phys.12(9), 093027 (2010).
[CrossRef]

Padgett, M. J.

Park, T. H.

C. M. Dutta, T. A. Ali, D. W. Brandl, T. H. Park, and P. Nordlander, “Plasmonic properties of a metallic torus,” J. Chem. Phys.129(8), 084706 (2008).
[CrossRef] [PubMed]

Peng, A.

W. S. Bakr, A. Peng, M. E. Tai, R. Ma, J. Simon, J. I. Gillen, S. Fölling, L. Pollet, and M. Greiner, “Probing the superfluid-to-Mott insulator transition at the single-atom level,” Science329(5991), 547–550 (2010).
[CrossRef] [PubMed]

Pollet, L.

W. S. Bakr, A. Peng, M. E. Tai, R. Ma, J. Simon, J. I. Gillen, S. Fölling, L. Pollet, and M. Greiner, “Probing the superfluid-to-Mott insulator transition at the single-atom level,” Science329(5991), 547–550 (2010).
[CrossRef] [PubMed]

Rangarajan, R.

R. Rangarajan, L. E. Vicent, A. B. U’Ren, and P. G. Kwiat, “A. B. U'Ren, and P. G. Kwiat, “Engineering an ideal indistinguishable photon-pair source for optical quantum information processing,” J. Mod. Opt.58(3-4), 318–327 (2011).
[CrossRef]

L. E. Vicent, A. B. U’Ren, R. Rangarajan, C. I. Osorio, J. P. Torres, L. Zhang, and I. A. Walmsley, “Design of bright, fiber-coupled and fully factorable photon pair sources,” New J. Phys.12(9), 093027 (2010).
[CrossRef]

Rodríguez-Herrera, O. G.

Rosete-Aguilar, M.

Sanyal, S.

S. Sanyal, Y. Kawata, S. Mandal, and A. Ghosh, “High tolerance to off-axis aberrations with a birefringent lens,” Opt. Eng.43(6), 1381–1386 (2004).
[CrossRef]

S. Sanyal and A. Ghosh, “High tolerance to spherical aberrations and defects of focus with a birefringent lens,” Appl. Opt.41(22), 4611–4619 (2002).
[CrossRef] [PubMed]

S. Sanyal and A. Ghosh, “High focal depth with a quasi-bifocus birefringent lens,” Appl. Opt.39(14), 2321–2325 (2000).
[CrossRef] [PubMed]

S. Sanyal, P. Bandyopadhayay, and A. Ghosh, “Vector wave imagery using a birefringent lens,” Opt. Eng.37(2), 592–599 (1998).
[CrossRef]

Sato, S.

Sawada, R.

E. Higurashi, R. Sawada, and T. Ito, “Optically induced angular alignment of birefringent micro-objects by linear polarization,” Appl. Phys. Lett.73(21), 3034 (1998).
[CrossRef]

Shimomura, H.

Sibbett, W.

Simon, J.

W. S. Bakr, A. Peng, M. E. Tai, R. Ma, J. Simon, J. I. Gillen, S. Fölling, L. Pollet, and M. Greiner, “Probing the superfluid-to-Mott insulator transition at the single-atom level,” Science329(5991), 547–550 (2010).
[CrossRef] [PubMed]

Tai, M. E.

W. S. Bakr, A. Peng, M. E. Tai, R. Ma, J. Simon, J. I. Gillen, S. Fölling, L. Pollet, and M. Greiner, “Probing the superfluid-to-Mott insulator transition at the single-atom level,” Science329(5991), 547–550 (2010).
[CrossRef] [PubMed]

Torres, J. P.

L. E. Vicent, A. B. U’Ren, R. Rangarajan, C. I. Osorio, J. P. Torres, L. Zhang, and I. A. Walmsley, “Design of bright, fiber-coupled and fully factorable photon pair sources,” New J. Phys.12(9), 093027 (2010).
[CrossRef]

U’Ren, A. B.

R. Rangarajan, L. E. Vicent, A. B. U’Ren, and P. G. Kwiat, “A. B. U'Ren, and P. G. Kwiat, “Engineering an ideal indistinguishable photon-pair source for optical quantum information processing,” J. Mod. Opt.58(3-4), 318–327 (2011).
[CrossRef]

L. E. Vicent, A. B. U’Ren, R. Rangarajan, C. I. Osorio, J. P. Torres, L. Zhang, and I. A. Walmsley, “Design of bright, fiber-coupled and fully factorable photon pair sources,” New J. Phys.12(9), 093027 (2010).
[CrossRef]

Unno, Y.

Vicent, L. E.

R. Rangarajan, L. E. Vicent, A. B. U’Ren, and P. G. Kwiat, “A. B. U'Ren, and P. G. Kwiat, “Engineering an ideal indistinguishable photon-pair source for optical quantum information processing,” J. Mod. Opt.58(3-4), 318–327 (2011).
[CrossRef]

L. E. Vicent, A. B. U’Ren, R. Rangarajan, C. I. Osorio, J. P. Torres, L. Zhang, and I. A. Walmsley, “Design of bright, fiber-coupled and fully factorable photon pair sources,” New J. Phys.12(9), 093027 (2010).
[CrossRef]

Walker, T.

Walmsley, I. A.

L. E. Vicent, A. B. U’Ren, R. Rangarajan, C. I. Osorio, J. P. Torres, L. Zhang, and I. A. Walmsley, “Design of bright, fiber-coupled and fully factorable photon pair sources,” New J. Phys.12(9), 093027 (2010).
[CrossRef]

Yanik, A. A.

Yonezawa, K.

Zhang, L.

L. E. Vicent, A. B. U’Ren, R. Rangarajan, C. I. Osorio, J. P. Torres, L. Zhang, and I. A. Walmsley, “Design of bright, fiber-coupled and fully factorable photon pair sources,” New J. Phys.12(9), 093027 (2010).
[CrossRef]

Antennas and Wireless Propagation Letters, IEEE

M. O. El-Shenawee, “Polarization dependence of plasmonic nanotoroid dimer antenna,” Antennas and Wireless Propagation Letters, IEEE9, 463–466 (2010).
[CrossRef]

Appl. Opt.

Appl. Phys. Lett.

E. Higurashi, R. Sawada, and T. Ito, “Optically induced angular alignment of birefringent micro-objects by linear polarization,” Appl. Phys. Lett.73(21), 3034 (1998).
[CrossRef]

J. Chem. Phys.

C. M. Dutta, T. A. Ali, D. W. Brandl, T. H. Park, and P. Nordlander, “Plasmonic properties of a metallic torus,” J. Chem. Phys.129(8), 084706 (2008).
[CrossRef] [PubMed]

J. Mod. Opt.

R. Rangarajan, L. E. Vicent, A. B. U’Ren, and P. G. Kwiat, “A. B. U'Ren, and P. G. Kwiat, “Engineering an ideal indistinguishable photon-pair source for optical quantum information processing,” J. Mod. Opt.58(3-4), 318–327 (2011).
[CrossRef]

J. Opt. Soc. Am. A

Nano Lett.

J. B. Lassiter, M. W. Knight, N. A. Mirin, and N. J. Halas, “Reshaping the plasmonic properties of an individual nanoparticle,” Nano Lett.9(12), 4326–4332 (2009).
[CrossRef] [PubMed]

New J. Phys.

L. E. Vicent, A. B. U’Ren, R. Rangarajan, C. I. Osorio, J. P. Torres, L. Zhang, and I. A. Walmsley, “Design of bright, fiber-coupled and fully factorable photon pair sources,” New J. Phys.12(9), 093027 (2010).
[CrossRef]

Opt. Eng.

S. Sanyal, Y. Kawata, S. Mandal, and A. Ghosh, “High tolerance to off-axis aberrations with a birefringent lens,” Opt. Eng.43(6), 1381–1386 (2004).
[CrossRef]

S. Sanyal, P. Bandyopadhayay, and A. Ghosh, “Vector wave imagery using a birefringent lens,” Opt. Eng.37(2), 592–599 (1998).
[CrossRef]

Opt. Express

Science

W. S. Bakr, A. Peng, M. E. Tai, R. Ma, J. Simon, J. I. Gillen, S. Fölling, L. Pollet, and M. Greiner, “Probing the superfluid-to-Mott insulator transition at the single-atom level,” Science329(5991), 547–550 (2010).
[CrossRef] [PubMed]

Other

R. Rangarajan . “Photonic sources and detectors for quantum information protocols: a trilogy in eight parts.” Diss. University of Illinois, Urbana-Champaign, 2010. Web. July 2010.

We assume that each focal location could be anywhere within a Gaussian distribution with width equal to the error measured on the original data points (and centered about the measured value). We then calculate one million sets of simulated data, each set containing one value from the Gaussian distribution associated with each measured point. We then calculate the slope of each set, and the Gaussian width of the 1 million sets of slopes gives an accurate error value on our measured value of the slope.

To see this, one can find the necessary lens displacement d for a given application by comparing the final beam waist wout,2 = √[λαf22/π(α2 + d2)] for two lenses with the final beam waist wout,1 = λf1/πw for one lens. For example, assume we want to achieve a focal size β times smaller than that possible with a single lens, so that β = wout,1/wout,2.Solving this equality for d gives d = √[(βf2)2 - α2]. If, then, we wish to focus five times more tightly (i.e., the desired beam waist is 5 times smaller than it was with only one lens), then d will be greater than 50, well within the range of validity of our equations.

R. Goldstein, Electro-optic Devices in Review (Lasers and Applications, 1986).

B. E. A. Saleh and M. C. Teich, Beam Optics, in Fundamentals of Photonics (John Wiley & Sons, 1991).

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Figures (14)

Fig. 1
Fig. 1

Schematic of Sagnac with offset lens for focal separation along propagation axis.

Fig. 2
Fig. 2

No-free-parameter theory and experimental data for focal separation versus lens displacement. Vertical and horizontal error are negligible.

Fig. 3
Fig. 3

Schematic of Sagnac setup for lateral focal separation induced by mirror offset. We show a large offset for illustration purposes. Note that a mirror translation induces both longitudinal and transverse separations; the ratio of these two effects can be tuned by changing the Sagnac geometry, specifically the deflection angle at the translating mirror.

Fig. 4
Fig. 4

Experimental data and theoretical slope of 1/√2 for focal separation versus mirror offset for small offsets (a) and larger offsets (b). Vertical error bars are small but included; horizontal error is negligible (< 10 μm).

Fig. 5
Fig. 5

Schematic of Sagnac for tighter focusing with two lenses.

Fig. 6
Fig. 6

(a). Data and fit for the focal separation Δ versus lens offset d; (b). data and fit for the beam waist wout versus lens offset d. Horizontal error is again negligible (< 10 μm).

Fig. 7
Fig. 7

(a). Schematic for one-block focusing; this model induces astigmatism. (b). Two-block setup for canceling aberrations; this model incurs additional lateral focal separations.

Fig. 8
Fig. 8

Theoretical characterization of longitudinal focal separation (a) and lateral focal separation Δr = √[Δx2 + Δy2] of ordinary and extraordinary spots (b) per crystal thickness T versus the optic axis angle θ for the two-block scheme shown in Fig. 7(b). For simplicity, the focal separations were calculated assuming the two input polarizations were ordinary and extraordinary, not necessarily horizontal and vertical (see text for discussion).

Fig. 9
Fig. 9

Cross-sectional photographs of the extraordinary beam’s intensity every 0.6 mm along the propagation axis of the laser. a.) Elliptical beam distortions (with maximum aspect ratio 2:1) are quite evident in the cross-sectional intensity profile of the extraordinary beam after focusing through 79.4 mm of calcite with optic axis at angle 45°. b.) The circular, un-aberrated profile is restored if the same focusing beam instead traverses two 39.7-mm blocks of calcite, oriented as in our compensation scheme.

Fig. 10
Fig. 10

Collinear, four-block setup for purely longitudinal, aberration-free focal displacements.

Fig. 11
Fig. 11

Schematic of two Gaussian beams propagating through two lenses. The CW (a) and CCW (b) paths correspond to the collinear model of the same color displayed in (c). Our calculations for both of these paths are split into two parts, forward and backward, as shown with yellow boxes. Setting the forward and backward beams from the two calculations to have equal beam parameters at their intersection then provides our final result.

Fig. 12
Fig. 12

Propagation of the o-wave (a) and e-wave (b).

Fig. 13
Fig. 13

Schematic of k-surfaces [25].

Fig. 14
Fig. 14

Schematic for focal plane displacement of the top- and bottom-most e-rays in the x-z plane. The schematics for e-rays in the y-z plane, and for o-rays in both planes, are similar.

Equations (20)

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Δ= d f 2 2 α 2 + d 2
w out = λα f 2 2 π( α 2 + d 2 )
α ~ λ f 1 2 π w 2 ,
q a = z a +i z 0
q b = z b +i z 0 .
q'=z'+i z 0 '= A+ B q C+ D q = (Az+B)+i z 0 A (Cz+D)+i z 0 C
z'=Re[ (Az+B)+i z 0 A (Cz+D)+i z 0 C ]= (Az+B)(Cz+D)+( z 0 A)( z 0 C) (Cz+D) 2 + ( z 0 C) 2
z 0 '=Im[ (Az+B)+i z 0 A ( Cz+D )+i z 0 C ]= z 0 A(Cz+D)C(Az+B) (Cz+D) 2 + ( z 0 C) 2 .
z'= z( 1 z f 2 )  z 0 2 f 2 (1 z f 2 ) 2 + ( z 0 f 2 ) 2 ;       z o '= z 0 (1 z f 2 ) 2 + ( z 0 f 2 ) 2 .
z 0a '= z 0 ( 1 z a f 2 ) 2 + ( z 0 f 2 ) 2 = z 0 ( 1 z b f 2 ) 2 + ( z 0 f 2 ) 2 = z 0b '= z 0 '
( 1 z a f 2 )=±( 1 z b f 2 ) z a = z b  or  z a + z b =2 f 2 .
z ± '= f 2 f 2 2 δ δ 2 + z 0 2 ;  z o ± '= z 0 '= f 2 2 z 0 δ 2 + z 0 2 .
z i '= ( π w 2 ) 2 f 1 ( λ f 1 ) 2 + ( π w 2 ) 2 = π w 2 α f 1
z o i '= π w 2 λ f 1 2 ( λ f 1 ) 2 + ( π w 2 ) 2 α.
π w 2 α λ f 1 +L±d+i α= f 2 ± f 2 2 δ δ 2 + z 0 2 +i f 2 2 z 0 δ 2 + z 0 2
L= f 2 + π w 2 α λ f 1
w out = λα f 2 2 π( α 2 + d 2 ) = λ z 0 2 π
δ= d f 2 2 α 2 + d 2
α= π w 2 λ f 1 2 (π w 2 ) 2 + (λ f 1 ) 2 .
δz=T Abs[ δx + δx ] 2w/f ,

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