The authors are with the Department of Electrical Engineering, The Ohio State University, 205 Dreese Laboratory, 2015 Neil Avenue, Columbus, Ohio 43210.
Betty Lise Anderson and Rashmi Mital, "Polynomial-based optical true-time delay devices with microelectromechanical mirror arrays," Appl. Opt. 41, 5449-5461 (2002)
We previously reported optical true-time delay devices, based on the White cell, to support phased-array radars. In particular, we demonstrated a quadratic device, in which the number of delays obtainable was proportional to the square of the number of times the light beam bounced in the cell. Here we consider the possibilities when a microelectromechanical (MEM) tip/tilt mirror array with multiple stable states is used. We present and compare designs for quadratic, quartic, and octic cells using MEM mirror arrays with two, three, and five micro-mirror tilt angles. An octic cell with a three-state MEM can produce 6,339 different delays in just 17 bounces.
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With one transitional bounce (the 9th bounce, to mirror A), for m = 17 bounces and a maximum delay of 624Δ.
Table 2
Some Mirror Progressions for the Improved Quartic Cella
Delay
Mirror Progression
0Δ
AB AB AB AB AB AB AB AB
1Δ
AB AB AB AB AB AB AB AC
2Δ
AB AB AB AB AB AB AC AC
3Δ
AB AB AB AB AB AC AC AC
4Δ
AB AB AB AB AC AC AC AC
5Δ (first use of E)
AB AB AB AB AB AB AB EB
6Δ (5+ 1
AB AB AB AB AB AB AB EC
7Δ (5+ 2)
AB AB AB AB AB AB AC EC
8Δ (5+ 3)
AB AB AB AB AB AC AC EC
9Δ (5+ 4)
AB AB AB AB AC AC AC EC
10 (two visits to E)
AB AB AB AB AB AB EB EB
11Δ (10+ 1)
AB AB AB AB AB AB EB EC
⋮
24Δ (4 visits to E and to C)
AB AB AB AB EC EC EC EC
25Δ (first visit to F)
AF AB AB AB AB AB AB AB
⋮
124Δ (4 visits to C, E, and F) (note order reversal)
EC EC EC EC AF AF AF AF
125Δ (first use of D)
DB AB AB AB AB AB AB AB
126Δ (100 + 1)
DB AB AB AB AB AB AB AC
⋮
499Δ (3 visits to D plus 124Δ as above)
EC EC EC EC AF DF DF DF
500Δ
DB DB DB DB AB AB AB AB
⋮
523Δ (500 + 5 + 5 + 5 + 5 + 1 + 1 + 1)
DB DB DB DB EB EC EC EC
524Δ
DB DB DB DB EC EC EC EC
599Δ (3 times to F, 4 times to D, plus 24)
DF DF DF DB EC EC EC EC
600Δ (4 times to F and to D)
DF DF DF DF AB AB AB AB
604Δ (600 + 4)
DF DF DF DF AC AC AC AC
619Δ (3 visits to E, 4 visits to D, F, C)
DF DF DF DF AC EC EC EC
620Δ (4 times to E plus 600)
EB EB EB EB DF DF DF DF
621Δ (620 + 1)
EC EB EB EB DF DF DF DF
622Δ (620 + 2)
EC EC EB EB DF DF DF DF
623Δ (620 + 3) (max delay)
EC EC EC EB DF DF DF DF
(624Δ) (prohibited)
(DF DF DF D
E CE CE CE) or (CE CE CE C
F DF DF DF DF)b
With no transitional bounce, for m = 16. One delay fewer can be obtained compared with Table
1, but with one less bounce.
Boxes highlight prohibited progressions.
Table 3
Some Mirror Progressions in the Flower-Petal Octic Cella
Delay
Mirror Progression
0Δ
AB AB AB AB A BA BA BA BA
1Δ
AB AB AB AB A BA BA BA CA
2Δ
AB AB AB AB A BA BA CA CA
3Δ (first use of E = 3)
AB AB AB AB A BA BA BA BE
8Δ (two visits each to C, E)
AB AB AB AB A BA BA CE CE
9Δ (first use of D = 9)
AB AB AB AB A BA BD BA BA
26Δ (max number of times to C, D, E)
AB AB AB AB A CE CE BD BD
27Δ (first use of F = 27)
AB AB AB AB A BA FB AB AB
79Δ (max use of C, D, E, F)
AB AB AB AB A FD FD CE CE
80Δ (first use of G = 80)
AB AB AB AG A BA BA BA BA
239 (max use of C, D, E, F, G)
AB AB AG AG A FD FD BE CE
240Δ (first use of J = 240)
AB AB AB JB A BA BA BA BA
719Δ (max use of C, D, E, F, G, J), (79+ twice to G and J)
AB AB JG JG A FD FD BE CE
720Δ (first use of H = 720)
AB AB AB HB A BA BA BA BA
1000Δ
AB AB HB JB A FD AB CE AB
2159Δ (max use of C, D, E, F, G, J, H)
HB HB JG JG A FD BE CE
2160Δ (max use of C, D, E, F, G, J, H)
HB HB JG JG A FD FD BE CE
2160Δ (first use of K = 2160)
AK AB AB AB A BA BA BA BA
6399 (max delay)
JG JB HK HK A FD FD BE CE
With m = 17 and one transitional bounce.
Table 4
Some Mirror Progressions for the Alternatie Flower-Petal Designa
With one transitional bounce (the 9th bounce, to mirror A), for m = 17 bounces and a maximum delay of 624Δ.
Table 2
Some Mirror Progressions for the Improved Quartic Cella
Delay
Mirror Progression
0Δ
AB AB AB AB AB AB AB AB
1Δ
AB AB AB AB AB AB AB AC
2Δ
AB AB AB AB AB AB AC AC
3Δ
AB AB AB AB AB AC AC AC
4Δ
AB AB AB AB AC AC AC AC
5Δ (first use of E)
AB AB AB AB AB AB AB EB
6Δ (5+ 1
AB AB AB AB AB AB AB EC
7Δ (5+ 2)
AB AB AB AB AB AB AC EC
8Δ (5+ 3)
AB AB AB AB AB AC AC EC
9Δ (5+ 4)
AB AB AB AB AC AC AC EC
10 (two visits to E)
AB AB AB AB AB AB EB EB
11Δ (10+ 1)
AB AB AB AB AB AB EB EC
⋮
24Δ (4 visits to E and to C)
AB AB AB AB EC EC EC EC
25Δ (first visit to F)
AF AB AB AB AB AB AB AB
⋮
124Δ (4 visits to C, E, and F) (note order reversal)
EC EC EC EC AF AF AF AF
125Δ (first use of D)
DB AB AB AB AB AB AB AB
126Δ (100 + 1)
DB AB AB AB AB AB AB AC
⋮
499Δ (3 visits to D plus 124Δ as above)
EC EC EC EC AF DF DF DF
500Δ
DB DB DB DB AB AB AB AB
⋮
523Δ (500 + 5 + 5 + 5 + 5 + 1 + 1 + 1)
DB DB DB DB EB EC EC EC
524Δ
DB DB DB DB EC EC EC EC
599Δ (3 times to F, 4 times to D, plus 24)
DF DF DF DB EC EC EC EC
600Δ (4 times to F and to D)
DF DF DF DF AB AB AB AB
604Δ (600 + 4)
DF DF DF DF AC AC AC AC
619Δ (3 visits to E, 4 visits to D, F, C)
DF DF DF DF AC EC EC EC
620Δ (4 times to E plus 600)
EB EB EB EB DF DF DF DF
621Δ (620 + 1)
EC EB EB EB DF DF DF DF
622Δ (620 + 2)
EC EC EB EB DF DF DF DF
623Δ (620 + 3) (max delay)
EC EC EC EB DF DF DF DF
(624Δ) (prohibited)
(DF DF DF D
E CE CE CE) or (CE CE CE C
F DF DF DF DF)b
With no transitional bounce, for m = 16. One delay fewer can be obtained compared with Table
1, but with one less bounce.
Boxes highlight prohibited progressions.
Table 3
Some Mirror Progressions in the Flower-Petal Octic Cella
Delay
Mirror Progression
0Δ
AB AB AB AB A BA BA BA BA
1Δ
AB AB AB AB A BA BA BA CA
2Δ
AB AB AB AB A BA BA CA CA
3Δ (first use of E = 3)
AB AB AB AB A BA BA BA BE
8Δ (two visits each to C, E)
AB AB AB AB A BA BA CE CE
9Δ (first use of D = 9)
AB AB AB AB A BA BD BA BA
26Δ (max number of times to C, D, E)
AB AB AB AB A CE CE BD BD
27Δ (first use of F = 27)
AB AB AB AB A BA FB AB AB
79Δ (max use of C, D, E, F)
AB AB AB AB A FD FD CE CE
80Δ (first use of G = 80)
AB AB AB AG A BA BA BA BA
239 (max use of C, D, E, F, G)
AB AB AG AG A FD FD BE CE
240Δ (first use of J = 240)
AB AB AB JB A BA BA BA BA
719Δ (max use of C, D, E, F, G, J), (79+ twice to G and J)
AB AB JG JG A FD FD BE CE
720Δ (first use of H = 720)
AB AB AB HB A BA BA BA BA
1000Δ
AB AB HB JB A FD AB CE AB
2159Δ (max use of C, D, E, F, G, J, H)
HB HB JG JG A FD BE CE
2160Δ (max use of C, D, E, F, G, J, H)
HB HB JG JG A FD FD BE CE
2160Δ (first use of K = 2160)
AK AB AB AB A BA BA BA BA
6399 (max delay)
JG JB HK HK A FD FD BE CE
With m = 17 and one transitional bounce.
Table 4
Some Mirror Progressions for the Alternatie Flower-Petal Designa