Abstract

Four types of backward iterative quantization (BIQ) methods were proposed to design multilevel diffractive optical elements (DOEs). In these methods, the phase values first quantized in the early quantization steps are those distant from the quantization levels, instead of the neighboring ones that the conventional iterative method began with. Compared with the conventional forward iterative quantization (FIQ), the Type 4 BIQ achieved higher efficiencies and signal-to-noise ratios for 4-level unequal-phase DOEs. For equal-phase DOEs, the Type 4 BIQ performed better when the range increment of each quantization step was large (>15°), while the FIQ performed better when the range increment was small (<15°).

© 2005 Optical Society of America

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References

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Appl. Opt. (7)

IBM J. Res. Dev. (1)

L.B. Lesem, P.M. Hirsch, and J.R. Jordan, Jr., �??The kinoform: a new wavefront reconstruction device,�?? IBM J. Res. Dev. 13, 150-155 (1969).
[CrossRef]

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

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 (3)

J. Turunen and F. Wyrowski ed., Diffractive Optics for Industrial and Commercial Applications, (Akademie Verlag, Berlin, 1997).

M.B. Stern, �??Binary optics fabrication,�?? in Micro-optics: Elements, Systems and Applications, H.P. Herzig, ed. (Taylor & Francis, London, 1997), pp. 53-85.

J.W. Goodman, Introduction to Fourier Optics, 2nd Ed., (McGraw-Hill, New York, 1996), pp. 96-125.

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Figures (10)

Fig. 1.
Fig. 1.

Illustration of phase quantization in a intermediate step of designing a 4-level DOE using the (a) FIQ and (b) BIQ methods. (‘∙’: quantization level, ‘+’: pixel phase, bright area: free window)

Fig. 2.
Fig. 2.

Parameters and effect for the unequal-phase iterative quantization methods.

Fig. 3.
Fig. 3.

Phase projections of the iterative quantization methods between two adjacent levels.

Fig. 4.
Fig. 4.

Performance of 4-level equal-phase DOEs.

Fig. 5.
Fig. 5.

Performance of 4-level unequal-phase DOEs when ϕ(1)=0.

Fig. 6.
Fig. 6.

Performance of 4-level unequal-phase DOEs when ϕ(1)≠0.

Fig. 7.
Fig. 7.

The intensity distributions of (a) the target image and (b) the diffractive image of the 4-level DOE of the largest efficiency (η=0.8323, SNR=0.9816).

Fig. 8.
Fig. 8.

Instead of stagnation, a chaotic-like phenomenon was observed when the Type 3 BIQ method was used to design 4-level DOEs. An oscillation of several states resulted when the range increment of a quantization step was small. A chaotic instability resulted when the range increment was slightly larger. This phenomenon was generally observed when using the BIQ methods to design the 4-level DOEs.

Fig. 9.
Fig. 9.

The intensity distributions of the diffractive image of the 8-level DOE of the largest SNR (η=0.9077, SNR=4.308, ϕ(1)=0π, d1=1.027π, d2=0.489π, and d3=0.248π). The DOE was designed using the Type 2 BIQ method when Q=2.

Fig. 10.
Fig. 10.

The intensity distributions of the diffractive image of the binary DOE of the largest SNR (η=0.8605, SNR=1.028, ϕ(1)=0π and d1=1.020π). The DOE was designed using the Type 4 BIQ method when Q=60.

Tables (3)

Tables Icon

Table 1. Descriptive statistics of 220 DOEs with 4 equally-spaced phase levels

Tables Icon

Table 2. Descriptive statistics of 220 DOEs with 4 unequally-spaced phase levels when ϕ(1)=0

Tables Icon

Table 3. Descriptive statistics of 220 DOEs with 4 unequally-spaced phase levels when ϕ(1)≠0

Equations (21)

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G ( u , v ) = G ( u , v ) exp ( j Γ ( u , v ) ) = FT { g ( x , y ) } ,
G k ( u , v ) = F ( u , v ) exp ( j Γ k ( u , v ) ) , for ( u , v ) S .
g k + 1 ( x , y ) = 1 , or
g k + 1 ( x , y ) = g k ( x , y ) .
ϕ k + 1 ( x , y ) = FIQ { ϕ k ( x , y ) }
= { ϕ ( n , q ) , ϕ ( n , q ) Δ ( n , q ) ε ( q ) < ϕ k ϕ ( n , q ) + Δ ( n + 1 , q ) ε ( q ) ϕ k ( x , y ) , otherwise for n = 1 , , N
E ( q ) = n = 1 N ϕ [ C ( n , q ) , C ( n + 1 , q ) ) exp ( i ϕ ) exp ( i ϕ ( n , q ) ) 2 p aw ( ϕ ) ,
C ( n , q ) = ϕ ( n , q ) + ϕ ( n + 1 , q ) 2 for n = 1 , , N + 1 ,
C up ( n , q ) = C ( n + 1 , q ) Δ ( n + 1 , q ) ε ( q ) , and
C low ( n , q ) = C ( n , q ) + Δ ( n , q ) ε ( q ) .
ϕ k + 1 ( x , y ) = BIQ 1 ̲ q { ϕ k ( x , y ) }
= { C up ( n , q ) , C up ( n , q ) < ϕ k C ( n + 1 , q ) C low ( n , q ) , C ( n , q ) < ϕ k C low ( n , q ) ϕ k ( x , y ) , otherwise for n = 1 , , N .
ϕ k + 1 ( x , y ) = BIQ 2 ̲ q { ϕ k ( x , y ) }
= ( ϕ k ( x , y ) ϕ ( n , q ) ) β ( n , q ) + ϕ ( n , q ) for n = 1 , , N ,
ϕ k + 1 ( x , y ) = BIQ 3 ̲ q { ϕ k ( x , y ) }
= { ϕ k ( x , y ) , C low ( n , q ) < ϕ k C up ( n , q ) ϕ ( n , q ) , otherwise for n = 1 , , N .
ϕ k + 1 ( x , y ) = BIQ 4 ̲ q { ϕ k ( x , y ) }
= { ϕ k ( x , y ) Δ ( n + 1 , q ) ε ( q ) , Δ ( n + 1 , q ) ε ( q ) < ϕ k C ( n + 1 , q ) ϕ k ( x , y ) + Δ ( n , q ) ε ( q ) C ( n , q ) < ϕ k Δ ( n , q ) ε ( q ) ϕ ( n , q ) , otherwise for n = 1 , , N .
f k = w 1 η k + w 2 SNR k ,
η k = ( u , v ) S G k ( u , v ) 2 all G k ( u , v ) 2 , and
SNR k = min ( u , v ) S G k ( u , v ) 2 max ( u , v ) S G k ( u , v ) 2 .

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