Abstract

The number of phase levels in a Talbot array illuminator (TAIL) is an important factor for estimation of practical fabrication complexity and cost. We show that the number of phase levels in a two-dimensional TAIL (2D-TAIL) has a simple relation to the prime number. When the output array is alternatively π phase modulated, there are similar simple relations. These simple relations should be highly interesting for practical use. An experiment with the 2D-TAIL based on the joint-Talbot effect is given as well.

© 2001 Optical Society of America

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References

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  1. A. W. Lohmann, “An array illuminator based on the Talbot effect,” Optik (Stuttgart) 79, 41–45 (1988).
  2. A. W. Lohmann, J. A. Thomas, “Making an array illuminator based on the Talbot effect,” Appl. Opt. 29, 4337–4340 (1990).
    [CrossRef] [PubMed]
  3. L. Liu, “Lau cavity and phase locking of laser arrays,” Opt. Lett. 14, 1312–1314 (1989).
    [CrossRef] [PubMed]
  4. S. Nowak, C. Kurtsiefer, T. Pfau, C. David, “High-order Talbot fringes for atomic matter waves,” Opt. Lett. 22, 1430–1432 (1997).
    [CrossRef]
  5. C. Zhou, S. Stankovic, C. Denz, T. Tschudi, “Phase codes of Talbot array illumination for encoding holographic multiplexing storage,” Opt. Commun. 161, 209–211 (1999).
    [CrossRef]
  6. J. R. Leger, G. J. Swanson, “Efficient array illuminator using binary-optics phase plates at fractional-Talbot planes,” Opt. Lett. 15, 288–290 (1990).
    [CrossRef] [PubMed]
  7. L. Liu, X. Liu, L. Ye, “Joint Talbot effect and logic operated moiré patterns,” J. Opt. Soc. Am. A 7, 970–976 (1990).
    [CrossRef]
  8. V. Arrizón, J. Ojeda-Castañeda, “Multilevel phase gratings for array illuminators,” Appl. Opt. 33, 5925–5931 (1994).
    [CrossRef] [PubMed]
  9. M. V. Berry, S. Klein, “Integer, fractional and fractal Talbot effects,” J. Mod. Opt. 43, 2139–2164 (1996).
    [CrossRef]
  10. C. Zhou, S. Stankovic, T. Tschudi, “Analytic phase-factor equations for Talbot array illuminators,” Appl. Opt. 38, 284–290 (1999).
    [CrossRef]
  11. M. Testorf, V. Arrizón, J. Ojeda-Castañeda, “Numerical optimization of phase-only elements based on the fractional Talbot effect,” J. Opt. Soc. Am. A 16, 97–105 (1999).
    [CrossRef]
  12. V. Arrizón, E. Löpez-Olazagasti, “Binary phase grating for array generation at 1/16 of Talbot length,” J. Opt. Soc. Am. A 12, 801–804 (1995).
    [CrossRef]
  13. W. Klaus, Y. Arimoto, K. Kodate, “High-performance Talbot array illuminators,” Appl. Opt. 37, 4357–4365 (1998).
    [CrossRef]
  14. C. Zhou, L. Liu, “Numerical study of Dammann array illuminators,” Appl. Opt. 34, 5961–5969 (1995).
    [CrossRef] [PubMed]
  15. C. Zhou, L. Liu, “Zernike array illuminator,” Optik (Stuttgart) 102, 75–78 (1996).

1999 (3)

1998 (1)

1997 (1)

1996 (2)

M. V. Berry, S. Klein, “Integer, fractional and fractal Talbot effects,” J. Mod. Opt. 43, 2139–2164 (1996).
[CrossRef]

C. Zhou, L. Liu, “Zernike array illuminator,” Optik (Stuttgart) 102, 75–78 (1996).

1995 (2)

1994 (1)

1990 (3)

1989 (1)

1988 (1)

A. W. Lohmann, “An array illuminator based on the Talbot effect,” Optik (Stuttgart) 79, 41–45 (1988).

Arimoto, Y.

Arrizón, V.

Berry, M. V.

M. V. Berry, S. Klein, “Integer, fractional and fractal Talbot effects,” J. Mod. Opt. 43, 2139–2164 (1996).
[CrossRef]

David, C.

Denz, C.

C. Zhou, S. Stankovic, C. Denz, T. Tschudi, “Phase codes of Talbot array illumination for encoding holographic multiplexing storage,” Opt. Commun. 161, 209–211 (1999).
[CrossRef]

Klaus, W.

Klein, S.

M. V. Berry, S. Klein, “Integer, fractional and fractal Talbot effects,” J. Mod. Opt. 43, 2139–2164 (1996).
[CrossRef]

Kodate, K.

Kurtsiefer, C.

Leger, J. R.

Liu, L.

Liu, X.

Lohmann, A. W.

A. W. Lohmann, J. A. Thomas, “Making an array illuminator based on the Talbot effect,” Appl. Opt. 29, 4337–4340 (1990).
[CrossRef] [PubMed]

A. W. Lohmann, “An array illuminator based on the Talbot effect,” Optik (Stuttgart) 79, 41–45 (1988).

Löpez-Olazagasti, E.

Nowak, S.

Ojeda-Castañeda, J.

Pfau, T.

Stankovic, S.

C. Zhou, S. Stankovic, C. Denz, T. Tschudi, “Phase codes of Talbot array illumination for encoding holographic multiplexing storage,” Opt. Commun. 161, 209–211 (1999).
[CrossRef]

C. Zhou, S. Stankovic, T. Tschudi, “Analytic phase-factor equations for Talbot array illuminators,” Appl. Opt. 38, 284–290 (1999).
[CrossRef]

Swanson, G. J.

Testorf, M.

Thomas, J. A.

Tschudi, T.

C. Zhou, S. Stankovic, C. Denz, T. Tschudi, “Phase codes of Talbot array illumination for encoding holographic multiplexing storage,” Opt. Commun. 161, 209–211 (1999).
[CrossRef]

C. Zhou, S. Stankovic, T. Tschudi, “Analytic phase-factor equations for Talbot array illuminators,” Appl. Opt. 38, 284–290 (1999).
[CrossRef]

Ye, L.

Zhou, C.

C. Zhou, S. Stankovic, C. Denz, T. Tschudi, “Phase codes of Talbot array illumination for encoding holographic multiplexing storage,” Opt. Commun. 161, 209–211 (1999).
[CrossRef]

C. Zhou, S. Stankovic, T. Tschudi, “Analytic phase-factor equations for Talbot array illuminators,” Appl. Opt. 38, 284–290 (1999).
[CrossRef]

C. Zhou, L. Liu, “Zernike array illuminator,” Optik (Stuttgart) 102, 75–78 (1996).

C. Zhou, L. Liu, “Numerical study of Dammann array illuminators,” Appl. Opt. 34, 5961–5969 (1995).
[CrossRef] [PubMed]

Appl. Opt. (5)

J. Mod. Opt. (1)

M. V. Berry, S. Klein, “Integer, fractional and fractal Talbot effects,” J. Mod. Opt. 43, 2139–2164 (1996).
[CrossRef]

J. Opt. Soc. Am. A (3)

Opt. Commun. (1)

C. Zhou, S. Stankovic, C. Denz, T. Tschudi, “Phase codes of Talbot array illumination for encoding holographic multiplexing storage,” Opt. Commun. 161, 209–211 (1999).
[CrossRef]

Opt. Lett. (3)

Optik (Stuttgart) (2)

A. W. Lohmann, “An array illuminator based on the Talbot effect,” Optik (Stuttgart) 79, 41–45 (1988).

C. Zhou, L. Liu, “Zernike array illuminator,” Optik (Stuttgart) 102, 75–78 (1996).

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Figures (3)

Fig. 1
Fig. 1

Symmetric structure of phase factors in 2D-TAIL within one period for (a) odd Mx and My, (b) even Mx and My, (c) even Mx and odd My. Owing to the symmetries, the number of phase levels of (a), (b), and (c) are given by Eqs. (5), (6), and (7), respectively.

Fig. 2
Fig. 2

Symmetric structure of phase factors of π-phase-modulated 2D-TAIL within one period for (a) odd MIx and MIy, (b) even MIx and MIy, (c) even MIx and odd MIy. Owing to the symmetries, the number of phase levels of (a), (b), and (c) are given by Eqs. (13), (14), and (15) respectively.

Fig. 3
Fig. 3

Experimental result of a separable 2D-TAIL based on the joint-Talbot effect.

Tables (2)

Tables Icon

Table 1 Simple Relations between Number (L) of Phase Levels in 2D-TAIL and Opening Ratio (1/M) of the Generated Array a

Tables Icon

Table 2 Simple Relations between Number (L) of Phase Levels of Π-Phase-Modulated 2D-TAIL and Intensity-Opening Ratio (1/MI) of the Generated Array a

Equations (38)

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ϕ(k)=(rk2/M)πforevenM,
ϕ(k)=rk(k-1)M πforoddM,
pr=krM+1,
ϕ(i, j)=ϕ(i)+ϕ(j)=ryi2My π+rxj2Mx π.
ϕ(i1, j1)=ϕ(i0, j0)+2cπ,
L(O, O)=(Mx+1)(My+1)/4-δLforoddMx andoddMy,
L(E, E)=(Mx/2+1)(My/2+1)-δLforevenMx andevenMy,
L(E, O)=(Mx/2+1)(My+1)/2-δLforevenMx andoddMy,
φ(i1, j1)=i1(i1-1)My+j1(j1-1)Mx
=(i0+m)(i0+m-1)My+(j0+n)(j0+n-1)Mx
=φ(i0, j0)+2mi0+m2-mMy+2nj0+n2-nMx.
2mi0+m2-mMy=k,wherekisaninteger,
i0=kMy+m-m22m=kMy2m+1-m2.
i0=lMy-m+12,i1=i0+m=lMy+m+12.
L=Mx+12×My+12=(t1+1)(t2+1)4.
ϕ(k)=rk(k-1)MI πforevenMI,
ϕ(k)=rk2MI πforoddMI,
pr=krMI+1,
L(O, O)=(MIx+1)(MIy+1)-δLforoddMIx andMIy,
L(E, E)=MIxMIy-δLforevenMIx andMIy,
L(E, O)=MIx(MIy+1)-δLforevenMIx oddMIy.
φ(i1, j1)=i12MIy+j12MIx
=(i0+m)(i0+m)MIy+(j0+n)(j0+n)MIx
=φ(i0, j0)+2mi0+m2MIy+2nj0+n2MIx.
i0=kMIy-m22m=kMIy2m-m2.
i0=lMIy-m/2,i1=i0+m=lMIy+m/2.
i0=lMIy2-m2,i1=lMIy2+m2,
(a)l=1,m=1, 3 ,, (MIy-2);
(b)l=3,m=MIy.
L=(MIx+1)(MIy+1)-δL=(MIx+1)(MIy+1)/2.
forMx=My=2nModd,
L(2nModd)=(2n+1)L(Modd);
forMx=My=t1 t2 t3 . . . tnModd,
L(t1 t2 t3 . . . tnModd)=t1 t2 t3 . . . tnL(Modd),
forMIx=MIy=2nMIodd,
L(2nMIodd)=2n-1L(MIodd),
forMIx=MIy=t1 t2 t3 . . . tnMIodd,
L(t1 t2 t3 . . . tnMIodd)=t1 t2 t3 . . . tnL(MIodd),

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