Abstract

The analytical expression of the phase profile of the optimum diffractive beam splitter with an arbitrary power ratio between the two output beams is derived. The phase function is obtained by an analytical optimization procedure such that the diffraction efficiency of the resulting optical element is the highest for an actual device. Comparisons are presented with the efficiency of a diffractive beam splitter specified by a sawtooth phase function and with the pertinent theoretical upper bound for this type of element.

© 2000 Optical Society of America

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References

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  1. See, for example, H. P. Herzig, ed., Micro-Optics: Elements, Systems, and Applications (Taylor & Francis, London, 1997).
  2. F. Wyrowski, “Upper bound of the diffraction efficiency of diffractive phase elements,” Opt. Lett. 16, 1915–1917 (1991).
    [CrossRef] [PubMed]
  3. F. Wyrowski, “Design theory of diffractive elements in the paraxial domain,” J. Opt. Soc. Am. A 10, 1553–1561 (1993).
    [CrossRef]
  4. U. Krackhardt, J. N. Mait, N. Streibl, “Upper bound on the diffraction efficiency of phase-only fanout element,” Appl. Opt. 31, 27–37 (1992).
    [CrossRef] [PubMed]
  5. See, for example, the contribution by J. Turunen in Ref. 1, p. 31 and references therein.
  6. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).
  7. M. Abramowitz, I. Stegun, eds., Handbook of Mathematical Functions (Dover, New York, 1972).
  8. F. Gori, “Diffractive optics: an introduction,” in Diffractive Optics and Optical Microsystems, S. Martellucci, A. N. Chester, eds. (Plenum, New York, 1997), pp. 3–22.
  9. See Ref. 1, p. 15.
  10. F. Gori, M. Santarsiero, S. Vicalvi, R. Borghi, G. Cincotti, E. Di Fabrizio, M. Gentili, “Analytical derivation of the optimum triplicator,” Opt. Commun. 157, 13–16 (1998).
    [CrossRef]
  11. A. P. Prudnikov, Yu. A. Brychkov, O. I. Marichev, Integrals and Series, V. 1 (Gordon & Breach, New York, 1992).

1998 (1)

F. Gori, M. Santarsiero, S. Vicalvi, R. Borghi, G. Cincotti, E. Di Fabrizio, M. Gentili, “Analytical derivation of the optimum triplicator,” Opt. Commun. 157, 13–16 (1998).
[CrossRef]

1993 (1)

1992 (1)

1991 (1)

Borghi, R.

F. Gori, M. Santarsiero, S. Vicalvi, R. Borghi, G. Cincotti, E. Di Fabrizio, M. Gentili, “Analytical derivation of the optimum triplicator,” Opt. Commun. 157, 13–16 (1998).
[CrossRef]

Brychkov, Yu. A.

A. P. Prudnikov, Yu. A. Brychkov, O. I. Marichev, Integrals and Series, V. 1 (Gordon & Breach, New York, 1992).

Cincotti, G.

F. Gori, M. Santarsiero, S. Vicalvi, R. Borghi, G. Cincotti, E. Di Fabrizio, M. Gentili, “Analytical derivation of the optimum triplicator,” Opt. Commun. 157, 13–16 (1998).
[CrossRef]

Di Fabrizio, E.

F. Gori, M. Santarsiero, S. Vicalvi, R. Borghi, G. Cincotti, E. Di Fabrizio, M. Gentili, “Analytical derivation of the optimum triplicator,” Opt. Commun. 157, 13–16 (1998).
[CrossRef]

Gentili, M.

F. Gori, M. Santarsiero, S. Vicalvi, R. Borghi, G. Cincotti, E. Di Fabrizio, M. Gentili, “Analytical derivation of the optimum triplicator,” Opt. Commun. 157, 13–16 (1998).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

Gori, F.

F. Gori, M. Santarsiero, S. Vicalvi, R. Borghi, G. Cincotti, E. Di Fabrizio, M. Gentili, “Analytical derivation of the optimum triplicator,” Opt. Commun. 157, 13–16 (1998).
[CrossRef]

F. Gori, “Diffractive optics: an introduction,” in Diffractive Optics and Optical Microsystems, S. Martellucci, A. N. Chester, eds. (Plenum, New York, 1997), pp. 3–22.

Krackhardt, U.

Mait, J. N.

Marichev, O. I.

A. P. Prudnikov, Yu. A. Brychkov, O. I. Marichev, Integrals and Series, V. 1 (Gordon & Breach, New York, 1992).

Prudnikov, A. P.

A. P. Prudnikov, Yu. A. Brychkov, O. I. Marichev, Integrals and Series, V. 1 (Gordon & Breach, New York, 1992).

Santarsiero, M.

F. Gori, M. Santarsiero, S. Vicalvi, R. Borghi, G. Cincotti, E. Di Fabrizio, M. Gentili, “Analytical derivation of the optimum triplicator,” Opt. Commun. 157, 13–16 (1998).
[CrossRef]

Streibl, N.

Vicalvi, S.

F. Gori, M. Santarsiero, S. Vicalvi, R. Borghi, G. Cincotti, E. Di Fabrizio, M. Gentili, “Analytical derivation of the optimum triplicator,” Opt. Commun. 157, 13–16 (1998).
[CrossRef]

Wyrowski, F.

Appl. Opt. (1)

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

Opt. Commun. (1)

F. Gori, M. Santarsiero, S. Vicalvi, R. Borghi, G. Cincotti, E. Di Fabrizio, M. Gentili, “Analytical derivation of the optimum triplicator,” Opt. Commun. 157, 13–16 (1998).
[CrossRef]

Opt. Lett. (1)

Other (7)

A. P. Prudnikov, Yu. A. Brychkov, O. I. Marichev, Integrals and Series, V. 1 (Gordon & Breach, New York, 1992).

See, for example, H. P. Herzig, ed., Micro-Optics: Elements, Systems, and Applications (Taylor & Francis, London, 1997).

See, for example, the contribution by J. Turunen in Ref. 1, p. 31 and references therein.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

M. Abramowitz, I. Stegun, eds., Handbook of Mathematical Functions (Dover, New York, 1972).

F. Gori, “Diffractive optics: an introduction,” in Diffractive Optics and Optical Microsystems, S. Martellucci, A. N. Chester, eds. (Plenum, New York, 1997), pp. 3–22.

See Ref. 1, p. 15.

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Figures (8)

Fig. 1
Fig. 1

Wyrowski’s upper bound as a function of the ratio r.

Fig. 2
Fig. 2

Values of |τ0|2 and |τ1|2 for the blazed beam splitter for different values of the phase jump Φ0.

Fig. 3
Fig. 3

Phase profiles of the blazed diffractive beam splitter for several values of the power ratio r.

Fig. 4
Fig. 4

Efficiency versus r for a blazed beam splitter. Dotted curve, Wyrowski’s upper bound.

Fig. 5
Fig. 5

Behavior of the powers of the diffraction orders for the optimum beam splitter as functions of the a parameter. Solid curve, zero order; dotted curve, first order.

Fig. 6
Fig. 6

Relation between a and the power ratio r.

Fig. 7
Fig. 7

Phase profiles of the optimum beam splitter for different values of the parameter r.

Fig. 8
Fig. 8

Efficiency of the optimum beam splitter as a function of the balancing parameter r (solid curve), together with that of a blazed grating (dashed curve) and Wyrowski’s upper bound (dotted curve).

Equations (29)

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|τ1|2=r|τ0|2,
τn=-1/21/2τ(x)exp(-i2πnx)dx(n=0, 1).
η=|τ0|2+|τ1|2n=-|τn|2,
τ(x)=exp[iΦ(x)],
n=-|τn|2=1,
ηub=1π2(1+r)21+rE2π,4r(1+r)2,
ηub=8/π2.
Φ(x)=2πx(-12x<12).
Φ(x)=Φ0x,(-12x<12),
τn=sincn-Φ02π,
Φ0=2πr1+r.
η=(1+r)sinc2r1+r
τ0=201/2cos Φ(x)dx,
τ1=201/2cos[Φ(x)-2πx]dx.
I(Φ)=τ0+aτ1,
I(Φ)=201/2{cos[Φ(x)]+a cos[Φ(x)-2πx]}dx.
Φ(x)=Φ¯(x)+(x),
δI=I(Φ¯+)-I(Φ¯)201/2(x){-sin[Φ¯(x)]-a sin[Φ¯(x)-2πx]}dx,
a cos Φ¯(x)sin(2πx)-(1+a cos 2πx)sin Φ¯(x)=0,
Φ¯(x; a)=tan-1a sin(2πx)1+a cos 2πx,
τ0=1π0π1+a cos t(1+a2+2a cos t)1/2dt=1π(1+a)E4a(1+a)2+(1-a)K4a(1+a)2,
τ1=1π0πa+cos t(1+a2+2a cos t)1/2dt=1aπ(1+a)E4a(1+a)2-(1-a)K4a(1+a)2,
ηub=|F(x)|2|F(x)|2,
f(ν)=δ(ν)+r exp(iφ)δ(ν-1),
F(x)=1+r exp[i(2πx+φ)],
|F(x)|=[1+r+2r cos(2πx+φ)]1/2.
|F(x)|2=01[1+r+2r cos(2πx+φ)]dx=1+r,
|F(x)|=12π02π1+r+2r cost+φ2π1/2dt=1+rπEπ,4r(1+r)2,
ηub=1π2(1+r)21+rE2π,4r(1+r)2,

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