Abstract

Building on the optimal-rotation-angle method, an algorithm for the design of inter-element coherent arrays of diffractive optical elements (DOEs) was developed. The algorithm is intended for fan-out DOE arrays where the individual elements fan-out to a sub-set of points chosen from a common set of points. By iteratively optimizing the array of elements as a whole the proposed algorithm ensures that the light from neighbouring elements is in-phase in all fan-out points that are common to neighbouring DOEs. This is important in applications where a laser beam scans the DOE array and the fan-out intensities constitute a read-out of information since the in-phase condition ensures a smooth transition in the read-out as the beam moves from one DOE to the next. Simulations show that the inter-element in-phase condition can be imposed at virtually no expense in terms of optical performance, as compared to independently designed DOEs.

© 2004 Optical Society of America

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References

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Appl. Opt. (5)

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

Opt. Eng. (1)

J. Turunen, A. Vasara, and J. Westerholm, "Kinoform phase relief synthesis: a stochastic method," Opt. Eng. 28, 1162-1167 (1989).

Optik (1)

R. W. Gerchberg and W. O. Saxton, "Practical algorithm for the determination of phase from image and diffraction plane pictures," Optik 35, 237-250 (1972).

Proc. SPIE (1)

J. Stigwall, S. Galt, and S. Hård, "Experimental evaluation of an ultra-fast free space optical analog-to-digital conversion scheme using a tunable semiconductor laser," presented at Microwave and Teraherz Photonics, Photonics Europe, Strasbourg, France. Proc. SPIE 5466 (2004).

Supplementary Material (2)

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Figures (6)

Fig. 1.
Fig. 1.

Schematic view of the optical analog-to-digital converter in [8, 9].

Fig. 2.
Fig. 2.

(a) DOE plane with pixel k as well as one of the target spot locations, m, indicated. (b) Complex-number plane showing the amplitude change, Δ|U| km , of the field in spot m from changing the phase modulation of pixel k by Δφk .

Fig. 3.
Fig. 3.

(a) Flowchart of the original ORA algorithm. (b) Flowchart of the extended ORA algorithm.

Fig. 4.
Fig. 4.

Groups of equal-phase spots for a five-bit optical ADC as in fig. 1. Black represents “one”, white “zero”. Zeros generate spots in the complementary word, resulting in 36 groups of spots that must be in-phase.

Fig. 5.
Fig. 5.

Movie clips showing the target spot plane as the beam is swept over a Gray-encoding ADC array consisting of 32 DOEs. (a) Array designed without inter-element coherence control (480 KB) (b) Array designed using the 36-group merger (467 KB).

Fig. 6.
Fig. 6.

Simulated intensities into all spot locations (different colours) while sweeping a beam over the 32-element Gray-encoding ADC DOE array. (a) Array designed without inter-element coherence control (b) Array designed using the 36-group merger.

Tables (1)

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Table 1. Comparison of the two merger methods with the original ORA algorithm. a

Equations (13)

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U km = A k inc exp [ j ( φ k inc + φ k ) ] A km exp ( j φ km )
A km exp ( j φ km ) = 1 4 π ( j k z L km r km c { j k 0 1 r km c } )
× 4 A k inc r km c exp ( j k 0 r km c ) sin ( k ˜ x a 2 ) sin ( k ˜ y b 2 ) k ˜ x k ˜ y
S 1 = Σ m A km cos Φ km
S 2 = Σ m A km sin Φ km
α k = arctan ( S 2 / S 1 )
Δ φ k opt = α k if S 1 > 0 Δ φ k opt = α k + π if S 1 < 0 Δ φ k opt = π / 2 if S 1 = 0 and S 2 > 0 Δ φ k opt = π / 2 if S 1 = 0 and S 2 < 0
S 1 = Σ m w m A km cos Φ km
S 2 = Σ m w m A km sin Φ km
w m new = w m old ( I m desired I m ) p
1 . w n BD : = w n BD + b c ( I n I ̅ ) / I ̅
2 . w n BD : = w n BD min ( w n BD )
unif . err . = ( I mn I mn desired ) max ( I mn I mn desired ) min ( I mn I mn desired ) max + ( I mn I mn desired ) min

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