Abstract

We show how suitable combinations of cascaded diffractive optical elements (DOEs) can form a combined “moiré DOE” of adjustable refractive power and high diffraction efficiency. The optical power can be adjusted continuously by a mutual rotation of one DOE with respect to the other. Fresnel lenses and axicons of variable refractive power or spiral phase plates of adjustable helical charge can be realized this way.

© 2008 Optical Society of America

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References

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  1. B. Kress and P. Meyrueis, Digital Diffractive Optics (Wiley, 2000).
  2. L. B. Lesem, P. M. Hirsch, and J. A. Jordan Jr., “The kinoform: a new wavefront reconstruction device,” IBM J. Res. Develop. 13, 150-155 (1969).
    [Crossref]
  3. R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237-246 (1972).
  4. J. M. Burch and D. C. Williams, “Varifocal moiré zone plates for straightness measurement,” Appl. Opt. 16, 2445-2450(1977).
    [Crossref] [PubMed]
  5. S. Bará, Z. Jaroszewicz, A. Kolodziejczyk, and V. Moreno, “Determination of basic grids for subtractive moiré patterns,” Appl. Opt. 30, 1258-1262 (1991).
    [Crossref] [PubMed]
  6. Z. Jaroszewicz, A. Kolodziejczyk, A. Mira, R. Henao, and S. Bará, “Equilateral hyperbolic moiré zone plates with variable focus obtained by rotations,” Opt. Express 13, 918-925(2005).
    [Crossref] [PubMed]
  7. A. W. Lohmann, “A new class of varifocal lenses,” Appl. Opt. 9, 1669-1671 (1970).
    [Crossref] [PubMed]
  8. A. Kolodziejczyk and Z. Jaroszewicz, “Diffractive elements of variable optical power and high diffraction efficiency,” Appl. Opt. 32, 4317-4322 (1993).
    [Crossref] [PubMed]
  9. In this case Φ=ar2φ, i.e., the first condition, becomes 2arφ<π/p. Considering that |φ| can reach the maximal value of π, the condition finally becomes 2ar<1/p. On the other hand, the second condition becomes ar/π<1/p, which is less restrictive than the first one.
  10. S. N. Khonina, V. V. Kotlyar, M. V. Shinkaryev, V. A. Soifer, and G. V. Uspleniev, “The phase rotor filter,” J. Mod. Opt. 39, 1147-1154 (1992).
    [Crossref]
  11. N. R. Heckenberg, R. McDuff, C. P. Smith, and A. G. White, “Generation of optical phase singularities by computer-generated holograms,” Opt. Lett. 17, 221-223(1992).
    [Crossref] [PubMed]
  12. N. B. Simpson, D. McGloin, K. Dholakia, L. Allen, and M. J. Padgett, “Optical tweezers with increased axial trapping efficiency,” J. Mod. Opt. 45, 1943-1949 (1998).
    [Crossref]
  13. A. Jesacher, S. Fürhapter, S. Bernet, and M. Ritsch-Marte, “Size-selective trapping with optical cogwheel tweezers,” Opt. Express 12, 4129-4136 (2004).
    [Crossref] [PubMed]
  14. K. Ladavac and D. G. Grier, “Micro-optomechanical pumps assembled and driven by holographic optical vortex arrays,” Opt. Express 12, 1144-1149 (2004).
    [Crossref] [PubMed]
  15. A. Jesacher, S. Fürhapter, C. Maurer, S. Bernet, and M. Ritsch-Marte, “Holographic optical tweezers for object manipulations at an air-liquid interface,” Opt. Express 14, 6342-6352 (2006).
    [Crossref] [PubMed]
  16. L. Torner, J. P. Torres, and S. Carrasco, “Digital spiral imaging,” Opt. Express 13, 873-881 (2005).
    [Crossref] [PubMed]
  17. J. A. Davis, D. E. McNamara, D. M. Cottrell, and J. Campos, “Image processing with the radial Hilbert transform: theory and experiments,” Opt. Lett. 25, 99-101 (2000).
    [Crossref]
  18. C. Maurer, A. Jesacher, S. Fürhapter, S. Bernet, and M. Ritsch-Marte, “Upgrading a microscope with a spiral phase plate,” J. Microsc. 230, 134-142 (2008).
    [Crossref] [PubMed]
  19. S. Fürhapter, A. Jesacher, S. Bernet, and M. Ritsch-Marte, “Spiral interferometry,” Opt. Lett. 30, 1953-1955 (2005).
    [Crossref] [PubMed]
  20. P. Török and P. R. T. Munro, “The use of Gauss-Laguerre vector beams in STED microscopy,” Opt. Express 12, 3605 (2004).
    [Crossref] [PubMed]

2008 (1)

C. Maurer, A. Jesacher, S. Fürhapter, S. Bernet, and M. Ritsch-Marte, “Upgrading a microscope with a spiral phase plate,” J. Microsc. 230, 134-142 (2008).
[Crossref] [PubMed]

2006 (1)

2005 (3)

2004 (3)

2000 (1)

1998 (1)

N. B. Simpson, D. McGloin, K. Dholakia, L. Allen, and M. J. Padgett, “Optical tweezers with increased axial trapping efficiency,” J. Mod. Opt. 45, 1943-1949 (1998).
[Crossref]

1993 (1)

1992 (2)

S. N. Khonina, V. V. Kotlyar, M. V. Shinkaryev, V. A. Soifer, and G. V. Uspleniev, “The phase rotor filter,” J. Mod. Opt. 39, 1147-1154 (1992).
[Crossref]

N. R. Heckenberg, R. McDuff, C. P. Smith, and A. G. White, “Generation of optical phase singularities by computer-generated holograms,” Opt. Lett. 17, 221-223(1992).
[Crossref] [PubMed]

1991 (1)

1977 (1)

1972 (1)

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237-246 (1972).

1970 (1)

1969 (1)

L. B. Lesem, P. M. Hirsch, and J. A. Jordan Jr., “The kinoform: a new wavefront reconstruction device,” IBM J. Res. Develop. 13, 150-155 (1969).
[Crossref]

Allen, L.

N. B. Simpson, D. McGloin, K. Dholakia, L. Allen, and M. J. Padgett, “Optical tweezers with increased axial trapping efficiency,” J. Mod. Opt. 45, 1943-1949 (1998).
[Crossref]

Bará, S.

Bernet, S.

Burch, J. M.

Campos, J.

Carrasco, S.

Cottrell, D. M.

Davis, J. A.

Dholakia, K.

N. B. Simpson, D. McGloin, K. Dholakia, L. Allen, and M. J. Padgett, “Optical tweezers with increased axial trapping efficiency,” J. Mod. Opt. 45, 1943-1949 (1998).
[Crossref]

Fürhapter, S.

Gerchberg, R. W.

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237-246 (1972).

Grier, D. G.

Heckenberg, N. R.

Henao, R.

Hirsch, P. M.

L. B. Lesem, P. M. Hirsch, and J. A. Jordan Jr., “The kinoform: a new wavefront reconstruction device,” IBM J. Res. Develop. 13, 150-155 (1969).
[Crossref]

Jaroszewicz, Z.

Jesacher, A.

Jordan, J. A.

L. B. Lesem, P. M. Hirsch, and J. A. Jordan Jr., “The kinoform: a new wavefront reconstruction device,” IBM J. Res. Develop. 13, 150-155 (1969).
[Crossref]

Khonina, S. N.

S. N. Khonina, V. V. Kotlyar, M. V. Shinkaryev, V. A. Soifer, and G. V. Uspleniev, “The phase rotor filter,” J. Mod. Opt. 39, 1147-1154 (1992).
[Crossref]

Kolodziejczyk, A.

Kotlyar, V. V.

S. N. Khonina, V. V. Kotlyar, M. V. Shinkaryev, V. A. Soifer, and G. V. Uspleniev, “The phase rotor filter,” J. Mod. Opt. 39, 1147-1154 (1992).
[Crossref]

Kress, B.

B. Kress and P. Meyrueis, Digital Diffractive Optics (Wiley, 2000).

Ladavac, K.

Lesem, L. B.

L. B. Lesem, P. M. Hirsch, and J. A. Jordan Jr., “The kinoform: a new wavefront reconstruction device,” IBM J. Res. Develop. 13, 150-155 (1969).
[Crossref]

Lohmann, A. W.

Maurer, C.

C. Maurer, A. Jesacher, S. Fürhapter, S. Bernet, and M. Ritsch-Marte, “Upgrading a microscope with a spiral phase plate,” J. Microsc. 230, 134-142 (2008).
[Crossref] [PubMed]

A. Jesacher, S. Fürhapter, C. Maurer, S. Bernet, and M. Ritsch-Marte, “Holographic optical tweezers for object manipulations at an air-liquid interface,” Opt. Express 14, 6342-6352 (2006).
[Crossref] [PubMed]

McDuff, R.

McGloin, D.

N. B. Simpson, D. McGloin, K. Dholakia, L. Allen, and M. J. Padgett, “Optical tweezers with increased axial trapping efficiency,” J. Mod. Opt. 45, 1943-1949 (1998).
[Crossref]

McNamara, D. E.

Meyrueis, P.

B. Kress and P. Meyrueis, Digital Diffractive Optics (Wiley, 2000).

Mira, A.

Moreno, V.

Munro, P. R. T.

Padgett, M. J.

N. B. Simpson, D. McGloin, K. Dholakia, L. Allen, and M. J. Padgett, “Optical tweezers with increased axial trapping efficiency,” J. Mod. Opt. 45, 1943-1949 (1998).
[Crossref]

Ritsch-Marte, M.

Saxton, W. O.

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237-246 (1972).

Shinkaryev, M. V.

S. N. Khonina, V. V. Kotlyar, M. V. Shinkaryev, V. A. Soifer, and G. V. Uspleniev, “The phase rotor filter,” J. Mod. Opt. 39, 1147-1154 (1992).
[Crossref]

Simpson, N. B.

N. B. Simpson, D. McGloin, K. Dholakia, L. Allen, and M. J. Padgett, “Optical tweezers with increased axial trapping efficiency,” J. Mod. Opt. 45, 1943-1949 (1998).
[Crossref]

Smith, C. P.

Soifer, V. A.

S. N. Khonina, V. V. Kotlyar, M. V. Shinkaryev, V. A. Soifer, and G. V. Uspleniev, “The phase rotor filter,” J. Mod. Opt. 39, 1147-1154 (1992).
[Crossref]

Torner, L.

Török, P.

Torres, J. P.

Uspleniev, G. V.

S. N. Khonina, V. V. Kotlyar, M. V. Shinkaryev, V. A. Soifer, and G. V. Uspleniev, “The phase rotor filter,” J. Mod. Opt. 39, 1147-1154 (1992).
[Crossref]

White, A. G.

Williams, D. C.

Appl. Opt. (4)

IBM J. Res. Develop. (1)

L. B. Lesem, P. M. Hirsch, and J. A. Jordan Jr., “The kinoform: a new wavefront reconstruction device,” IBM J. Res. Develop. 13, 150-155 (1969).
[Crossref]

J. Microsc. (1)

C. Maurer, A. Jesacher, S. Fürhapter, S. Bernet, and M. Ritsch-Marte, “Upgrading a microscope with a spiral phase plate,” J. Microsc. 230, 134-142 (2008).
[Crossref] [PubMed]

J. Mod. Opt. (2)

S. N. Khonina, V. V. Kotlyar, M. V. Shinkaryev, V. A. Soifer, and G. V. Uspleniev, “The phase rotor filter,” J. Mod. Opt. 39, 1147-1154 (1992).
[Crossref]

N. B. Simpson, D. McGloin, K. Dholakia, L. Allen, and M. J. Padgett, “Optical tweezers with increased axial trapping efficiency,” J. Mod. Opt. 45, 1943-1949 (1998).
[Crossref]

Opt. Express (6)

Opt. Lett. (3)

Optik (1)

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237-246 (1972).

Other (2)

In this case Φ=ar2φ, i.e., the first condition, becomes 2arφ<π/p. Considering that |φ| can reach the maximal value of π, the condition finally becomes 2ar<1/p. On the other hand, the second condition becomes ar/π<1/p, which is less restrictive than the first one.

B. Kress and P. Meyrueis, Digital Diffractive Optics (Wiley, 2000).

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Figures (11)

Fig. 1
Fig. 1 Generic setup (not to scale): two diffractive optical elements (DOEs) are placed directly behind each other. They can be mutually rotated around a central axis. This combined optical element manipulates the wavefront of an incident light wave in a predesigned way that changes with the mutual rotation angle.

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Fig. 2
Fig. 2 A MDOE that acts as a Fresnel lens with a refractive power that depends on their mutual rotation angle. Gray values in the figure actually correspond to phase-shift values between 0 and 2 π . Note that the two DOEs are identical if one of them is reversed (i.e., flipped upside down).

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Fig. 3
Fig. 3 Superposition of the two DOEs in Fig. 2 at different mutual rotation angles of (upper row) − 75 ° , − 30 ° , and − 15 ° ; and (lower row) + 15 ° , + 30 ° , and + 70 ° . The results are perfect blazed Fresnel lenses with different refractive powers; however, they include a sector of an angular range that corresponds to the mutual rotation angle that comprises a Fresnel lens of a different focal length.

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Fig. 4
Fig. 4 MDOEs forming an adjustable Fresnel lens. The MDOE is similar to the one in Fig. 2; however, it is calculated according to Eq. (12) such that there are no phase discontinuities (with the exception of 2 π phase jumps) along circular paths around the center.

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Fig. 5
Fig. 5 Result of the superposition of the two DOEs of Fig. 4 at different mutual rotation angles of (upper row:) − 75 ° , − 30 ° , and − 15 ° ; and (lower row:) + 15 ° , + 30 ° , and + 70 ° . A sector formation like that in Fig. 3 is now avoided.

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Fig. 6
Fig. 6 Diffraction efficiency and refractive power of the Fresnel lenses in Fig. 5 as a function of the mutual rotation angle between the two DOEs. The joint DOE corresponds to a bifocal Fresnel lens with two refractive powers (red and blue curves), the relative efficiencies of which depend on the rotation angle.

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Fig. 7
Fig. 7 Sensitivity of MDOEs to transverse misalignment: MDOEs generating a Fresnel lens (left two columns), and a Fresnel lens with avoided sector formation (right two columns), are plotted for two mutual rotation angles of 10 ° and 30 ° , and for three different decentering values of 1   pixel (upper row), 5   pixels (middle row), and 10   pixels (lowest row). The total size of each MDOE is 512   pixels × 512   pixels . The misalignment introduces lens deformations but it does not completely destroy the performance of the element.

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Fig. 8
Fig. 8 Two DOEs calculated according to Eq. (18) produce a combined DOE acting as a varifocal Fresnel lens with an offset refraction power; i.e., at a zero mutual rotation angle, the focal length is f offset .

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Fig. 9
Fig. 9 Calculated DOEs for creating an axicon with a refractive power that depends on the mutual rotation angle. All DOEs in (a), (b), and (c) have to be combined with a second DOE that is identical to the first one, but flipped upside down. The DOE in (a) is calculated according to Eq. (19) and produces a perfect blazed axicon with, however, an undesired sector [see example (d)] for a mutual rotation angle of 25 ° ). The DOE in (b) is calculated according to Eq. (20) and produces axicons without the undesired sector formation [see example (e)]. The DOE in (c) is calculated according to Eq. (21) and produces an axicon with a variable refractive power that is superposed by a Fresnel lens with a constant focal length.

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Fig. 10
Fig. 10 Example for a set of DOEs that generates a spiral phase element with a variable helical index that depends linearly on the mutual rotation angle between the two DOEs.

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Fig. 11
Fig. 11 Combined DOE elements of Fig. 10 at rotation angles of − 25 ° , − 5 ° , − 1.5 ° , + 1.5 ° , + 5 ° , and + 25 ° . The corresponding transmission functions correspond to spiral phase elements with helical indices of approximately − 13 , − 3 , − 1 , + 1 , + 3 , and + 13 , respectively.

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Equations (23)

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T joint ( x , y ) = T 1 ( x , y ) T 2 ( x , y ) = exp ⁡ { i [ Φ 1 ( x , y ) + Φ 2 ( x , y ) ] } .
T 1 ( r , φ ) = exp ⁡ [ i Φ ( r , φ ) ] , T 2 ( r , φ ) = exp ⁡ [ − i Φ ( r , φ ) ] .
T joint ( r , φ ; θ ) = exp ⁡ { i [ Φ ( r , φ ) − Φ ( r , φ − θ ) ] } .
T joint ( r , φ ; θ ) = exp ⁡ { i [ ∂ Φ ( r , φ ) ∂ φ θ − 1 2 ∂ 2 Φ ( r , φ ) ∂ φ 2 θ 2 + ... ] } .
T joint ( r , φ ; θ ) = exp ⁡ { i Φ r ( r ) [ d Φ φ ( φ ) d φ θ − 1 2 d 2 Φ φ ( φ ) d φ 2 θ 2 + ... ] } .
T joint ( r , φ ; θ ) = exp ⁡ { i Φ r ( r ) θ } .
Φ ( r , φ ) = − Φ ( r , − φ )
T 1 = exp ⁡ [ i a r 2 φ ] , T 2 = exp ⁡ [ − i a r 2 φ ] .
T joint = exp ⁡ [ i a r 2 φ ] exp ⁡ [ − i a r 2 ( φ − θ ) ] = exp ⁡ [ i a θ r 2 ] .
f − 1 = a θ λ / π .
f 1 − 1 = a | θ | λ / π , f 2 − 1 = − a ( 2 π − | θ | ) λ / π ,
T 1 = exp ⁡ [ i   round   { a r 2 } φ ] , T 2 = exp ⁡ [ − i   round   { a r 2 } φ ] ,
d Φ d r < π p and d Φ r d φ < π p ,
2 a r max φ max < π / p ,
a < 1 / 2 p r max , i . e . ,     a max = ( 1 / 2 p r max ) .
− f min − 1 < f − 1 < + f min − 1 ,
f min = 4 p r max / λ
T 1 = exp ⁡ [ i   round { a r 2 } φ ] exp ⁡ ( i π r 2 / 2 f offset λ ) , T 2 = exp ⁡ [ − i   round { a r 2 } φ ] exp ⁡ ( i π r 2 / 2 f offset λ ) .
T 1 = exp ⁡ [ i a r φ ] , T 2 = exp ⁡ [ − i a r φ ] .
T 1 = exp ⁡ [ i   round { a r } φ ] , T 2 = exp ⁡ [ − i   round { a r } φ ] .
T 1 = exp ⁡ [ i   round { a r } φ ] exp ⁡ ( i π r 2 / 2 f offset λ ) , T 2 = exp ⁡ [ − i   round { a r } φ ] exp ⁡ ( i π r 2 / 2 f offset λ ) ,
T 1 = exp ⁡ [ i a φ 2 ] , T 2 = exp ⁡ [ − i a φ 2 ] ,
T joint = exp ⁡ ( i a φ 2 ) exp ⁡ [ − i a ( φ − θ ) 2 ] = exp ⁡ ( i 2 a θ φ ) exp ⁡ ( − i a θ 2 ) .

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