Abstract

We have derived the signal-to-noise ratio in direct-detection Random-Modulation Continuous-Wave (RM-CW) lidar in the presence of colored additive noise. In contrast to a known formula derived for the photon shot-noise regime, which may adequately describe experimental conditions in the near-infrared, our result is applicable mainly at longer, mid-infrared wavelengths. Unlike the former formula, our result is explicitly dependent on the pseudorandom code (PRC) used for modulation. Three known modulation codes, the M-, A1-, and A2-sequence are compared and shown to have practically equivalent signal and noise properties (provided that clutter inherent in the A1- and A2-sequence is neglected), except that the M-sequence has a near-zero-frequency noise pickup that degrades its performance in real measurement systems. This difference provides an alternative explanation of a better performance of the A1-/A2-sequence in a previous experiment [3], carried out in the near-infrared. It suggests the presence of an additive noise component and thus some applicability of our result also in near-infrared lidar. A need for balanced sequences – particularly in the mid-infrared – is explained, although in a different way than previously suggested in near-infrared, photon shot noise-limited lidar. Additional, sinusoidal carrier modulation is considered and shown to have significant drawbacks. Our results allow comparison of given modulation sequences, and construction of improved ones. Interestingly, the improved sequences will possess less “random” characteristics, seemingly against the underlying concept of random modulation.

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References

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  1. N. Takeuchi, N. Sugimoto, H. Baba, and K. Sakurai, "Random modulation cw lidar," Appl. Opt. 22, 1382-1386 (1983).
    [CrossRef] [PubMed]
  2. N. Takeuchi, H. Baba, K. Sakurai, and T. Ueno, "Diode-laser random-modulation cw lidar," Appl. Opt. 25, 63-67 (1986).
    [CrossRef] [PubMed]
  3. Ch. Nagasawa, M. Abo, H. Yamamoto, and O. Uchino, "Random modulation cw lidar using new random sequence," Appl. Opt. 29, 1466-1470 (1990).
    [CrossRef] [PubMed]
  4. J. L. Machol, "Comparison of the pseudorandom noise code and pulsed direct-detection lidars for atmospheric probing," Appl. Opt. 36, 6021-6023 (1997).
    [CrossRef] [PubMed]
  5. Y. Emery and C. Flesia, "Use of the A1- and the A2-sequences to modulate continuous-wave pseudorandom noise lidar," Appl. Opt. 37, 2238-2241 (1998).
    [CrossRef]
  6. C. M. Gittins, E. T. Wetjen, C. Gmachl, F. Capasso, A. L. Hutchinson, D. L. Sivco, J. N. Baillargeon, and A. Y. Cho, "Quantitative gas sensing by backscatter-absorption measurements of a pseudorandom code modulated ~8-�m quantum cascade laser," Opt. Lett. 25, 1162-1164 (2000).
    [CrossRef]
  7. A. B. Carlson, Communication systems. An introduction to signals and noise in electrical engineering (McGraw-Hill, 1986).
  8. S. Haykin, Digital communications (John Wiley & Sons, 1988).

Other (8)

N. Takeuchi, N. Sugimoto, H. Baba, and K. Sakurai, "Random modulation cw lidar," Appl. Opt. 22, 1382-1386 (1983).
[CrossRef] [PubMed]

N. Takeuchi, H. Baba, K. Sakurai, and T. Ueno, "Diode-laser random-modulation cw lidar," Appl. Opt. 25, 63-67 (1986).
[CrossRef] [PubMed]

Ch. Nagasawa, M. Abo, H. Yamamoto, and O. Uchino, "Random modulation cw lidar using new random sequence," Appl. Opt. 29, 1466-1470 (1990).
[CrossRef] [PubMed]

J. L. Machol, "Comparison of the pseudorandom noise code and pulsed direct-detection lidars for atmospheric probing," Appl. Opt. 36, 6021-6023 (1997).
[CrossRef] [PubMed]

Y. Emery and C. Flesia, "Use of the A1- and the A2-sequences to modulate continuous-wave pseudorandom noise lidar," Appl. Opt. 37, 2238-2241 (1998).
[CrossRef]

C. M. Gittins, E. T. Wetjen, C. Gmachl, F. Capasso, A. L. Hutchinson, D. L. Sivco, J. N. Baillargeon, and A. Y. Cho, "Quantitative gas sensing by backscatter-absorption measurements of a pseudorandom code modulated ~8-�m quantum cascade laser," Opt. Lett. 25, 1162-1164 (2000).
[CrossRef]

A. B. Carlson, Communication systems. An introduction to signals and noise in electrical engineering (McGraw-Hill, 1986).

S. Haykin, Digital communications (John Wiley & Sons, 1988).

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Figures (2)

Fig. 1.
Fig. 1.

Block diagram of signal-to-noise analysis of RM-CW lidar in presence of colored additive noise.

Fig. 2.
Fig. 2.

Comparison of noise pickup distribution of M-sequence and A1-/A2-sequence; N=7.

Equations (44)

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g ( t ) = c 2 A r β r ( R ) T r 2 ( R ) Y ( R ) R 2 ,
T r ( R ) = exp [ 0 R α ( r ) d r ]
( α absorption coefficient ) ,
ψ a a ' ( j ) = 1 N i = 1 N a i a i + j ,
S j ( 0 ) N = P 0 l = 1 N ψ a a ( l j ) G j ,
N + 1 2 N P 0 G j 1 2 P 0 G j ( large N )
S j ( 0 ) 2 N P 0 N 2 N G j = 1 2 P 0 G j ,
S j ( 0 ) 4 N P 0 2 N 4 N G j = 1 2 P 0 G j ,
R d · S j ( 0 ) N 1 2 R d · P 0 G j ,
1 k N < k N > a i j n i 1 T T a ( τ ) n ( τ ) d τ ,
G a n ( f ) = G a ( f ) * η ( f ) 2 ,
G a ( f ) = { R a ( τ ) }
G a ( f ) = n = c n δ ( f n f 0 ) ,
c n = 1 T 0 T 0 R a ( τ ) exp ( j 2 π f 0 τ ) d τ
G a n ( f ) = η ( λ ) 2 n = c n δ ( f n f 0 λ ) d λ = n = c n η ( f n f 0 ) 2
N out 2 ( t ) = G a n ( f ) · H ( f ) 2 d f ,
h ( t ) = 1 T Π ( t T ) ,
H ( f ) 2 = { h ( t ) } 2 = sinc 2 f T
N out 2 ( t ) = n = c n η ( f n f 0 ) 2 sin c 2 ( f T ) d f
sin c 2 f T = 1 T T sin c 2 f T 1 T δ ( f ) for k ,
N out 2 ( t ) = 1 T n = c n η ( n f 0 ) 2
( S N ) j = 1 2 R d · P 0 · G j 1 T n = c n η ( n f 0 ) 2
D * ( f ) = R d A d η ( f ) 2 ,
( S N ) j = 1 2 P 0 · G j 1 T n = c n η ( n f 0 ) 2 R d 2 = 1 2 P 0 · G j A d 1 T n = c n 1 D * 2 ( n f 0 )
n = c n = R a ( τ = 0 ) = 1
( S N ) j = 1 2 R d · P 0 · G j η 2 T = 1 2 P 0 · D * · G j A d 1 T
( S N ) R M = k ξ · l P 0 G j N l P 0 G ¯ + b ¯ ,
( S N ) R M 1 2 ξ P 0 G ˜ j b ¯ ξ k N
R a ( M ) ( τ ) = ( 1 + 1 N ) k = Λ ( τ k N T c T c ) 1 N
c 0 ( M ) = 1 N 2 ; c n M = N + 1 N 2 sin c 2 n N 1 N sin c 2 n N , n 0
R a ( A 1 ) ( τ ) = 1 N +
+ k = { ( 1 + 1 N ) [ Λ ( τ k 2 N T c T c ) Λ ( τ ( 2 k + 1 ) N T c T c ) ] 2 N Λ ( τ k 2 T c T c ) }
c n ( A 1 ) = N + 1 N sin c 2 [ ( n 1 2 ) 1 N ] 1 N sin c 2 [ ( n 1 2 ) 1 N ]
c 0 = 1 T 0 T 0 R a ( τ ) d τ = T c T 0 i = 0 N 1 ψ a a ( i ) = 1 N i = 0 N 1 ψ a a ( i ) =
= 1 N i = 0 N 1 j = 1 N 1 N a j a j + i = ( i = 1 N a i N ) 2
a ( t ) a ( t ) [ 1 2 + 1 2 cos ( 2 π f m t + φ m ) ] ,
a ( t ) a ( t ) cos ( 2 π f m t + φ m )
R a , m ( τ ) = R a ( τ ) R m ( τ ) = R a ( τ ) 1 2 cos 2 π f m τ
{ R a m ( τ ) } = { R a ( τ ) } * { 1 2 cos 2 π f m τ } =
= { R a ( τ ) } * 1 2 δ ( f f m ) + δ ( f + f m ) 2 =
= 1 4 G a ( f f m ) + 1 4 G a ( f + f m )
R a m , a m ( τ ) = a ( t ) a ( t + τ ) cos ( 2 π f m + φ m ) 1 2 cos [ 2 π f m ( t + τ ) + φ m ] =
= 1 4 R a a ( τ ) cos ( 2 π f m τ + φ m φ m )
c n 1 4 c n n + 1 4 c n + n

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