Abstract

We put forward a simple, scalable and robust technique for generating periodically structured light beams with intensity patterns, e.g. of the form cos2n(kxx) cos2m(kyy), where kx and ky are real numbers that can be tailored and n and m are integers. The technique combines the use of Gaussian beams with curved wavefronts, birefringent crystals (Savart plates) and linear polarizers. Applications range from photolithography to fabrication of micro-lens array and fiber Bragg gratings, 3D printing and tailoring of optical lattices for trapping atoms and molecules.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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2016 (2)

R. A. Terborg, J. Pello, I. Mannelli, J. P. Torres, and V. Pruneri, “Ultrasensitive interferometric on-chip microscopy of transparent objects,” Sci. Adv. 2, e1600077 (2016).
[Crossref] [PubMed]

B. G. Zimmerman and T. G. Brown, “Star test image-sampling polarimeter,” Opt. Express 24, 23154–23161 (2016).
[Crossref] [PubMed]

2013 (1)

2012 (1)

T. D. O’Sullivan, A. E. Cerussi, B. J. Tromberg, and D. J. Cuccia, “Diffuse optical imaging using spatially and temporally modulated light,” J. Biomed. Opt. 17, 071311 (2012).
[Crossref]

2011 (1)

2010 (3)

2006 (1)

C. López-Mariscal, M. A. Bandres, and J. Gutiérrez-Vega, “Observation of the experimental propagation properties of helmholtz-gauss beams,” Opt. Eng. 45, 068001 (2006).
[Crossref]

2005 (3)

D. McGloin and K. Dholakia, “Bessel beams: Diffraction in a new light,” Contemp. Phys. 46, 15–28 (2005).
[Crossref]

D. J. Cuccia, F. Bevilacqua, A. J. Durkin, and B. J. Tromberg, “Modulated imaging: quantitative analysis and tomography of turbid media in the spatial-frequency domain,” Opt. Lett. 30, 1354–1356 (2005).
[Crossref] [PubMed]

I. Bloch, “Ultracold quantum gases in optical lattices,” Nat. Phys. 1, 23–30 (2005).
[Crossref]

2004 (1)

E. G. Abramochkin and V. G. Volostnikov, “Spiral light beams,” Phys.-Uspekhi 47, 1177–1203 (2004).
[Crossref]

2000 (1)

M. G. L. Gustafsson, “Surpassing the lateral resolution limit by a factor of two using structured illumination microscopy,” J. Microsc. 198, 82–87 (2000).
[Crossref] [PubMed]

1993 (1)

E. Abramochkin and V. Volostnikov, “Spiral-type beams,” Opt. Commun. 102, 336–350 (1993).
[Crossref]

1987 (1)

J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[Crossref] [PubMed]

Abramochkin, E.

E. Abramochkin and V. Volostnikov, “Spiral-type beams,” Opt. Commun. 102, 336–350 (1993).
[Crossref]

Abramochkin, E. G.

E. G. Abramochkin and V. G. Volostnikov, “Spiral light beams,” Phys.-Uspekhi 47, 1177–1203 (2004).
[Crossref]

Alonso, M. A.

Bandres, M. A.

C. López-Mariscal, M. A. Bandres, and J. Gutiérrez-Vega, “Observation of the experimental propagation properties of helmholtz-gauss beams,” Opt. Eng. 45, 068001 (2006).
[Crossref]

Bevilacqua, F.

Bloch, I.

I. Bloch, “Ultracold quantum gases in optical lattices,” Nat. Phys. 1, 23–30 (2005).
[Crossref]

Brown, T. G.

Cerussi, A. E.

T. D. O’Sullivan, A. E. Cerussi, B. J. Tromberg, and D. J. Cuccia, “Diffuse optical imaging using spatially and temporally modulated light,” J. Biomed. Opt. 17, 071311 (2012).
[Crossref]

Cuccia, D. J.

T. D. O’Sullivan, A. E. Cerussi, B. J. Tromberg, and D. J. Cuccia, “Diffuse optical imaging using spatially and temporally modulated light,” J. Biomed. Opt. 17, 071311 (2012).
[Crossref]

D. J. Cuccia, F. Bevilacqua, A. J. Durkin, and B. J. Tromberg, “Modulated imaging: quantitative analysis and tomography of turbid media in the spatial-frequency domain,” Opt. Lett. 30, 1354–1356 (2005).
[Crossref] [PubMed]

Dholakia, K.

D. McGloin and K. Dholakia, “Bessel beams: Diffraction in a new light,” Contemp. Phys. 46, 15–28 (2005).
[Crossref]

Durkin, A. J.

Durnin, J.

J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[Crossref] [PubMed]

Eberly, J. H.

J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[Crossref] [PubMed]

Geng, J.

Gustafsson, M. G. L.

M. G. L. Gustafsson, “Surpassing the lateral resolution limit by a factor of two using structured illumination microscopy,” J. Microsc. 198, 82–87 (2000).
[Crossref] [PubMed]

Gutiérrez-Vega, J.

C. López-Mariscal, M. A. Bandres, and J. Gutiérrez-Vega, “Observation of the experimental propagation properties of helmholtz-gauss beams,” Opt. Eng. 45, 068001 (2006).
[Crossref]

Heintzmann, R.

L. Schermelleh, R. Heintzmann, and H. Leonhardt, “A guide to super-resolution fluorescence microscopy,” J. Cell Biol. 190, 165–175 (2010).
[Crossref] [PubMed]

R. Heintzmann, Structured Illumination Methods (SpringerUS, 2006), pp. 265–279.

Hernández-Hernández, R. J.

Kashyap, R.

R. Kashyap, Fiber bragg gratings (Academic, 2009).

Leonhardt, H.

L. Schermelleh, R. Heintzmann, and H. Leonhardt, “A guide to super-resolution fluorescence microscopy,” J. Cell Biol. 190, 165–175 (2010).
[Crossref] [PubMed]

López-Mariscal, C.

C. López-Mariscal, M. A. Bandres, and J. Gutiérrez-Vega, “Observation of the experimental propagation properties of helmholtz-gauss beams,” Opt. Eng. 45, 068001 (2006).
[Crossref]

Mannelli, I.

R. A. Terborg, J. Pello, I. Mannelli, J. P. Torres, and V. Pruneri, “Ultrasensitive interferometric on-chip microscopy of transparent objects,” Sci. Adv. 2, e1600077 (2016).
[Crossref] [PubMed]

McGloin, D.

D. McGloin and K. Dholakia, “Bessel beams: Diffraction in a new light,” Contemp. Phys. 46, 15–28 (2005).
[Crossref]

Miceli, J. J.

J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[Crossref] [PubMed]

O’Sullivan, T. D.

T. D. O’Sullivan, A. E. Cerussi, B. J. Tromberg, and D. J. Cuccia, “Diffuse optical imaging using spatially and temporally modulated light,” J. Biomed. Opt. 17, 071311 (2012).
[Crossref]

Pello, J.

R. A. Terborg, J. Pello, I. Mannelli, J. P. Torres, and V. Pruneri, “Ultrasensitive interferometric on-chip microscopy of transparent objects,” Sci. Adv. 2, e1600077 (2016).
[Crossref] [PubMed]

Pruneri, V.

R. A. Terborg, J. Pello, I. Mannelli, J. P. Torres, and V. Pruneri, “Ultrasensitive interferometric on-chip microscopy of transparent objects,” Sci. Adv. 2, e1600077 (2016).
[Crossref] [PubMed]

Ramkhalawon, R. D.

Ricardez-Vargas, I.

Schermelleh, L.

L. Schermelleh, R. Heintzmann, and H. Leonhardt, “A guide to super-resolution fluorescence microscopy,” J. Cell Biol. 190, 165–175 (2010).
[Crossref] [PubMed]

Terborg, R. A.

R. A. Terborg, J. Pello, I. Mannelli, J. P. Torres, and V. Pruneri, “Ultrasensitive interferometric on-chip microscopy of transparent objects,” Sci. Adv. 2, e1600077 (2016).
[Crossref] [PubMed]

R. J. Hernández-Hernández, R. A. Terborg, I. Ricardez-Vargas, and K. Volke-Sepúlveda, “Experimental generation of mathieu–gauss beams with a phase-only spatial light modulator,” Appl. Opt. 49, 6903–6909 (2010).
[Crossref]

Torres, J. P.

R. A. Terborg, J. Pello, I. Mannelli, J. P. Torres, and V. Pruneri, “Ultrasensitive interferometric on-chip microscopy of transparent objects,” Sci. Adv. 2, e1600077 (2016).
[Crossref] [PubMed]

Tromberg, B. J.

T. D. O’Sullivan, A. E. Cerussi, B. J. Tromberg, and D. J. Cuccia, “Diffuse optical imaging using spatially and temporally modulated light,” J. Biomed. Opt. 17, 071311 (2012).
[Crossref]

D. J. Cuccia, F. Bevilacqua, A. J. Durkin, and B. J. Tromberg, “Modulated imaging: quantitative analysis and tomography of turbid media in the spatial-frequency domain,” Opt. Lett. 30, 1354–1356 (2005).
[Crossref] [PubMed]

Volke-Sepúlveda, K.

Volostnikov, V.

E. Abramochkin and V. Volostnikov, “Spiral-type beams,” Opt. Commun. 102, 336–350 (1993).
[Crossref]

Volostnikov, V. G.

E. G. Abramochkin and V. G. Volostnikov, “Spiral light beams,” Phys.-Uspekhi 47, 1177–1203 (2004).
[Crossref]

Zimmerman, B. G.

Adv. Opt. Photon. (1)

Appl. Opt. (1)

Contemp. Phys. (1)

D. McGloin and K. Dholakia, “Bessel beams: Diffraction in a new light,” Contemp. Phys. 46, 15–28 (2005).
[Crossref]

J. Biomed. Opt. (1)

T. D. O’Sullivan, A. E. Cerussi, B. J. Tromberg, and D. J. Cuccia, “Diffuse optical imaging using spatially and temporally modulated light,” J. Biomed. Opt. 17, 071311 (2012).
[Crossref]

J. Cell Biol. (1)

L. Schermelleh, R. Heintzmann, and H. Leonhardt, “A guide to super-resolution fluorescence microscopy,” J. Cell Biol. 190, 165–175 (2010).
[Crossref] [PubMed]

J. Microsc. (1)

M. G. L. Gustafsson, “Surpassing the lateral resolution limit by a factor of two using structured illumination microscopy,” J. Microsc. 198, 82–87 (2000).
[Crossref] [PubMed]

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

Nat. Phys. (1)

I. Bloch, “Ultracold quantum gases in optical lattices,” Nat. Phys. 1, 23–30 (2005).
[Crossref]

Opt. Commun. (1)

E. Abramochkin and V. Volostnikov, “Spiral-type beams,” Opt. Commun. 102, 336–350 (1993).
[Crossref]

Opt. Eng. (1)

C. López-Mariscal, M. A. Bandres, and J. Gutiérrez-Vega, “Observation of the experimental propagation properties of helmholtz-gauss beams,” Opt. Eng. 45, 068001 (2006).
[Crossref]

Opt. Express (2)

Opt. Lett. (1)

Phys. Rev. Lett. (1)

J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[Crossref] [PubMed]

Phys.-Uspekhi (1)

E. G. Abramochkin and V. G. Volostnikov, “Spiral light beams,” Phys.-Uspekhi 47, 1177–1203 (2004).
[Crossref]

Sci. Adv. (1)

R. A. Terborg, J. Pello, I. Mannelli, J. P. Torres, and V. Pruneri, “Ultrasensitive interferometric on-chip microscopy of transparent objects,” Sci. Adv. 2, e1600077 (2016).
[Crossref] [PubMed]

Other (3)

R. Heintzmann, Structured Illumination Methods (SpringerUS, 2006), pp. 265–279.

R. Kashyap, Fiber bragg gratings (Academic, 2009).

Apple Inc., “Face ID,” https://www.apple.com/iphone-x/#face-id (2018 (accessed May, 2018)).

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Figures (5)

Fig. 1
Fig. 1 Scheme of the proposed technique to generate structured light patterns: (a) A SP divides an input beam into two parallel outgoing beams with orthogonal polarizations, displaced each other laterally a distance S. (b) and (c) Experimental setups that include a light source, Savart plates SP1 and SP2 and linear polarizers P1,P2 and P3. The outgoing curved beams share the same polarization and interfere to generate intensity patterns of the form cos2(kxx) and cos4(kxx) in (b) and (c), respectively. kx and ky are real numbers that can be tailored by the technique.
Fig. 2
Fig. 2 Patterns of light generated and description of the splitting of the optical beams in each SP with: (a) one, (b) two and (c) three SPs. The green dots depict the position of the beam centers after each SP. The displacement generated by each SP is indicated by arrows. In (c) the red ’x’ denotes where the centers of two beams overlap with an amplitude and phase relationship such that they interfere destructively. The dashed circles show that for these cases all the resulting points lie within a circle. In the text we show that these beams propagate preserving their structure. The bottom row shows the resulting output pattern for each case. Data: The input Gaussian beam (500 nm wavelength) shows a radius of curvature of R = 5 mm. The displacements introduced by the SPs are S1 = 60μm and S2 = S1/3).
Fig. 3
Fig. 3 Top row: Intensity patterns generated with three SPs after inducing a π-phase delay (or rotating the Ps by 90°) simultaneously to the three SPs (I1), only to the first (I2), only to the second (I3) or only to the third SP (I4). Center and bottom rows: Cumulative intensity patterns and close-ups from the resulting combinations: Ia = I1 + I2, Ib = I1 + I2 + I3, Ic = I1 + I2 + I3 + I4.
Fig. 4
Fig. 4 Examples of structured illumination patterns. The illumination is at λR = 617nm, the beam curvature is R = 45mm, and and the displacements introduced by the SPs are S = 50μm for SP1 and SP2. a) I = sin2(2γSx) for a configuration [P1][SP1][SP2][P2], with SP1 and SP2 aligned to double the displacement to 2S = 100μm ; b) I = sin2(γSx) with [P1][SP1][P2]: c) I ≈ sin4(γSx) with the configuration [P1][SP1][P2][SP2][P3]; d) I = sin2(γSx) cos2(γSy). e) Intensity profile along the dotted lines in a), b) and c) (red, blue and green lines, respectively). f) and g) Experimental patterns of the kind I = cos2(γSx) cos2(γSy) for an illumination at λB = 470nm with beam displacements S = 50μm and beam curvatures R = 45mm and R = 95mm, respectively. h) and i) Patterns at λG = 530nm of the kind I = cos2(γSx) cos2(2γSy) and I = (cos(γSx) + cos(2γSx))2 achieved with two different beam displacements S1 = 50μm and S2 = 100μm in orthogonal and in parallel directions.
Fig. 5
Fig. 5 Proof of concept surface structuring using the proposed technique: a) Illumination pattern of the form I = cos2(γSx) cos2(γSy) used for writing (central wavelength λUV = 385nm (FWHM=10nm). b) Negative 3D profile of the developed photo-sensitive polymer (AZ 5214 E, Microchemicals GmbH) after exposure to the illumination pattern (measured with SENSOFAR S neox Optical Profiler). c) Copper deposition on glass, revealing the over-exposed pattern on a photo-sensitive polymer. Variations in the intensity of the original beam lead to variations in the metallic spot sizes.

Equations (13)

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E G = z R E 0 z exp [ ( z R w 0 z ) 2 r 2 + i γ r 2 i tan 1 z z R ] ,
γ = π λ z .
E out = 1 2 z R E 0 z exp [ ( z R w 0 z ) 2 r 2 i tan 1 z z R ] × { exp [ i γ ( x S x 2 ) 2 + i γ ( y S y 2 ) 2 i α 2 ] + exp [ i γ ( x + S x 2 ) 2 + i γ ( y + S y 2 ) 2 + i α 2 ] } .
E out = E G exp [ i γ ( S x 2 4 + S y 2 4 ) ] cos ( γ S r + α 2 ) .
I out = I G cos 2 ( γ S r + α 2 ) .
E out 1 = E G exp ( i γ S x 2 4 ) cos ( γ S x x + α 1 2 ) .
E out 2 = E G exp [ i γ ( S x 2 4 + S y 2 4 ) ] cos ( γ S x x + α 1 2 ) cos ( γ S y y + α 2 2 )
I out 2 = I G cos 2 ( γ S x x + α 1 2 ) cos 2 ( γ S y y + α 2 2 ) .
E out 3 = E G 2 exp [ i γ ( S x 2 2 ) ] × { exp [ i γ S x x i α 2 2 ] cos ( γ S x ( x S x 2 ) + α 1 2 ) + exp [ i γ S x x + i α 2 2 ] cos ( γ S x ( x + S x 2 ) + α 2 ) } .
α 2 = α 1 + 2 μ π
γ = ν π S x 2
I out 3 = I G 2 cos 4 ( ν π x S x + α 1 2 + ν π 2 ) .
z ν = S x 2 ν λ