Abstract

Wavefront coding system can realize defocus invariance of PSF/OTF with a phase mask inserting in the pupil plane. Ideally, the derivative of the PSF/OTF with respect to defocus error should be close to zero as much as possible over the extended depth of field/focus for the wavefront coding system. In this paper, we propose an analytical expression for the computation of the derivative of PSF. With this expression, the derivative of PSF based merit function can be used in the optimization of the wavefront coding system with any type of phase mask and aberrations. Computation of the derivative of PSF using the proposed expression and FFT respectively are compared and discussed. We also demonstrate the optimization of a generic polynomial phase mask in wavefront coding system as an example.

© 2016 Optical Society of America

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References

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    [Crossref]
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    [Crossref] [PubMed]
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    [Crossref]
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2014 (1)

2013 (1)

2010 (1)

2008 (4)

2007 (1)

2004 (1)

A. Janssen, J. Braat, and P. Dirksen, “On the computation of the Nijboer-Zernike aberration integrals at arbitrary defocus,” J. Mod. Opt. 51(5), 687–703 (2004).
[Crossref]

2003 (1)

S. Prasad, T. C. Torgersen, V. P. Pauca, R. J. Plemmons, and J. van der Gracht, “Engineering the Pupil Phase to Improve Image Quality,” Proc. SPIE 5108, 1–12 (2003).
[Crossref]

2002 (1)

2001 (1)

S. Sherif, E. Dowski, and W. Cathey., “Logarithmic phase filter to extend the depth of field of incoherent hybrid imaging systems,” Proc. SPIE 4417, 272–280 (2001).
[Crossref]

1995 (1)

Bagheri, S.

Barwick, S.

Braat, J.

A. Janssen, J. Braat, and P. Dirksen, “On the computation of the Nijboer-Zernike aberration integrals at arbitrary defocus,” J. Mod. Opt. 51(5), 687–703 (2004).
[Crossref]

J. Braat, P. Dirksen, and A. J. Janssen, “Assessment of an extended Nijboer-Zernike approach for the computation of optical point-spread functions,” J. Opt. Soc. Am. A 19(5), 858–870 (2002).
[Crossref] [PubMed]

Caron, N.

Cathey, W.

S. Sherif, E. Dowski, and W. Cathey., “Logarithmic phase filter to extend the depth of field of incoherent hybrid imaging systems,” Proc. SPIE 4417, 272–280 (2001).
[Crossref]

Cathey, W. T.

Christensen, M. P.

Cisotto, L.

de Farias, D. P.

Dirksen, P.

A. Janssen, J. Braat, and P. Dirksen, “On the computation of the Nijboer-Zernike aberration integrals at arbitrary defocus,” J. Mod. Opt. 51(5), 687–703 (2004).
[Crossref]

J. Braat, P. Dirksen, and A. J. Janssen, “Assessment of an extended Nijboer-Zernike approach for the computation of optical point-spread functions,” J. Opt. Soc. Am. A 19(5), 858–870 (2002).
[Crossref] [PubMed]

Dowski, E.

S. Sherif, E. Dowski, and W. Cathey., “Logarithmic phase filter to extend the depth of field of incoherent hybrid imaging systems,” Proc. SPIE 4417, 272–280 (2001).
[Crossref]

Dowski, E. R.

Filho, L. C.

Janssen, A.

A. Janssen, J. Braat, and P. Dirksen, “On the computation of the Nijboer-Zernike aberration integrals at arbitrary defocus,” J. Mod. Opt. 51(5), 687–703 (2004).
[Crossref]

Janssen, A. J.

Komatsu, S.

Konijnenberg, A. P.

Kumar, N.

Li, G.

Li, Y.

Mauger, T.

Narayanswamy, R.

Pauca, V. P.

S. Prasad, T. C. Torgersen, V. P. Pauca, R. J. Plemmons, and J. van der Gracht, “Engineering the Pupil Phase to Improve Image Quality,” Proc. SPIE 5108, 1–12 (2003).
[Crossref]

Pereira, S. F.

Plemmons, R. J.

S. Prasad, T. C. Torgersen, V. P. Pauca, R. J. Plemmons, and J. van der Gracht, “Engineering the Pupil Phase to Improve Image Quality,” Proc. SPIE 5108, 1–12 (2003).
[Crossref]

Prasad, S.

S. Prasad, T. C. Torgersen, V. P. Pauca, R. J. Plemmons, and J. van der Gracht, “Engineering the Pupil Phase to Improve Image Quality,” Proc. SPIE 5108, 1–12 (2003).
[Crossref]

Sheng, Y.

Sherif, S.

S. Sherif, E. Dowski, and W. Cathey., “Logarithmic phase filter to extend the depth of field of incoherent hybrid imaging systems,” Proc. SPIE 4417, 272–280 (2001).
[Crossref]

Silveira, P. E. X.

Somayaji, M.

Takahashi, Y.

Torgersen, T. C.

S. Prasad, T. C. Torgersen, V. P. Pauca, R. J. Plemmons, and J. van der Gracht, “Engineering the Pupil Phase to Improve Image Quality,” Proc. SPIE 5108, 1–12 (2003).
[Crossref]

Urbach, H. P.

van der Gracht, J.

S. Prasad, T. C. Torgersen, V. P. Pauca, R. J. Plemmons, and J. van der Gracht, “Engineering the Pupil Phase to Improve Image Quality,” Proc. SPIE 5108, 1–12 (2003).
[Crossref]

Wei, L.

Zhao, H.

Zhao, T.

Appl. Opt. (4)

Biomed. Opt. Express (1)

J. Mod. Opt. (1)

A. Janssen, J. Braat, and P. Dirksen, “On the computation of the Nijboer-Zernike aberration integrals at arbitrary defocus,” J. Mod. Opt. 51(5), 687–703 (2004).
[Crossref]

J. Opt. Soc. Am. A (2)

Opt. Express (1)

Opt. Lett. (2)

Proc. SPIE (2)

S. Sherif, E. Dowski, and W. Cathey., “Logarithmic phase filter to extend the depth of field of incoherent hybrid imaging systems,” Proc. SPIE 4417, 272–280 (2001).
[Crossref]

S. Prasad, T. C. Torgersen, V. P. Pauca, R. J. Plemmons, and J. van der Gracht, “Engineering the Pupil Phase to Improve Image Quality,” Proc. SPIE 5108, 1–12 (2003).
[Crossref]

Other (1)

B. Frieden, Science from Fisher Information (Cambridge University, 2004).

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Figures (6)

Fig. 1
Fig. 1 The flow of the optimization of the phase mask.
Fig. 2
Fig. 2 Comparisons of the 2-D DPSF computed by FFT and the proposed expression, where (a) is computed by FFT, (b) is computed by proposed expression and (c) is the deviation between (a) and (b).
Fig. 3
Fig. 3 The computation of J under the diffraction limited condition (a), aberrated condition with 1/4λ spherical aberration, 1/12λ coma aberration and 1/12λ astigmatism aberration (b) and cubic phase mask coding condition with its parameter is 20 (c).
Fig. 4
Fig. 4 The layout of the system.
Fig. 5
Fig. 5 The MTF under different defocus errors without (a) and with (b) the phase mask.
Fig. 6
Fig. 6 The simulated blurred images (a) and the restored images (b) under different defocus errors.

Tables (1)

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Table 1 Coefficients of the Fringe Zernike Polynomials

Equations (19)

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U(x,y;ψ)= 1 π ν 2 + μ 2 1 exp[ iψ( ν 2 + μ 2 ) ]P(ν,μ)exp[2πi(νx+μy)]dνdμ = 1 π 0 1 0 2π exp( iψ ρ 2 )P(ρ,θ) exp[2πiρrcos(θϕ)]ρdρdθ
ψ= π D 2 4λ ( 1 f 1 d o 1 d i )
P(ρ,θ)={ exp{ i[ Φ(ρ,θ)+Θ(ρ,θ) ] }, | ρ |1 0, | ρ |>1
P(ρ,θ)= m,n α n m Z n m (ρ,θ) = m,n α n m R n m exp(imθ)
U(r,ϕ;ψ)= 1 π 0 1 0 2π n,m α n m R n m exp(imθ)exp(iψ ρ 2 )exp[2πiρrcos(θϕ)]ρdρdθ =2 n,m i m α n m exp(imϕ) 0 1 ρexp(iψ ρ 2 ) R n m (ρ) J m (2πρr) dρ =2 n,m i m α n m exp(imϕ) V n m (r,ψ)
H(r,ϕ;ψ)=U(r,ϕ;ψ) U * (r,ϕ;ψ) = | 2 n,m i m α n m exp(imϕ) V n m (r,ψ) | 2
H ψ = ( U U * ) ψ =U ( U ψ ) * + U * U ψ
U ψ =2 n,m i m α n m exp(imϕ) [ V n m (r,ψ) ] ψ =2 n,m i m α n m exp(imϕ)d V n m (r,ψ)
d V n m (r,ψ)=iexp(iψ) l=1 ( 2iψ ) l1 j=0 p u lj J m+l+2j (2πr) l (2πr) l 2iexp(iψ) l=2 ( l1 )( 2iψ ) l2 j=0 p u lj J m+l+2j (2πr) l (2πr) l
H ψ =4[ n,m i m α n m exp(imϕ) V n m (r,ψ) ] [ n,m i m α n m exp(imϕ)d V n m (r,ψ) ] * + 4 [ n,m i m α n m exp(imϕ) V n m (r,ψ) ] * [ n,m i m α n m exp(imϕ)d V n m (r,ψ) ] =8Re{ [ n,m i m α n m exp(imϕ) V n m (r,ψ) ] [ n,m i m α n m exp(imϕ)d V n m (r,ψ) ] * }
J( a;ψ )= ( H ψ 2 ) 2 = x y [ H ψ ] 2
J=64 r ϕ [ Re{ [ n,m i m α n m exp(imϕ) V n m (r,ψ) ] [ n,m i m α n m exp(imϕ)d V n m (r,ψ) ] * } ] 2
M= j J( a; ψ j ) = j x y [ H ψ ] 2 ψ= ψ j
PT= K[ 1Ψ(α) ] exp[ η(Ψ(α) Ψ ~ ) ]+1
MF= j x y [ H ψ ] 2 ψ= ψ j + K[ 1Ψ(α) ] exp[ η(Ψ(α) Ψ ~ ) ]+1
H ψ = 2 π 2 Re{ FT[ i ρ 3 exp( iψ ρ 2 )P(ρ,θ) ]F T * [ ρexp( iψ ρ 2 )P(ρ,θ) ] }
Φ= q=1 Q a q ( u 2q+1 + v 2q+1 )
V n m (r,ψ)= 0 1 ρexp[iψ ρ 2 ] R n m (ρ) J m (2πρr) dρ =exp(iψ) l=0 ( iψ πr ) l j=0 p u lj J m+l+2j+1 (2πr) 2πr
u lj = (1) p m+l+2j+1 q+l+j+1 ( m+j+l l )( j+l l )( l pj ) / ( q+j+l l )

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