Abstract

A newly developed Michelson interferometer for Fourier spectroscopy is described. It utilizes a nutating retroreflector (cube-corner mirror) to generate alterations in geometrical and optical paths. The nutation is achieved by the rotation of the retroreflector eccentrically, and it is tilted with reference to the optical axis of the interferometer. The forward–backward stop-and-go movement of a reflecting element of conventional Michelson interferometers is thus replaced by a continuous rotation. The design aims at a fast, simple, rugged and service-free, reliable spectrometer for field or airborne atmospheric monitoring. For Fourier spectroscopy, the instantaneous and the maximum difference between the two optical paths of the interferometer is of substantial importance. Performing the Fourier transform requires knowledge of the instantaneous difference; the maximum difference determines the spectral resolution of the device. The mathematical deduction of the path of a beam of radiation traversing the retroreflector is performed. First ray tracing is calculated for a fixed retroreflector, dependent on the angle of incidence. Then the path of a ray is deduced for the rotating retroreflector as a function of angle of incidence, eccentricity, and angle of rotation. It is shown that the path difference changes sinusoidally with the angle of rotation. The maximum path difference is expressed as a function of eccentricity and angle of incidence. Numerical results are presented.

© 1991 Optical Society of America

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References

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  1. M. V. R. K. Murty, “Modification of the Michelson interferometer using only one cube-corner prism,” J. Opt. Soc. Am. 50, 83–84 (1960).
    [CrossRef]
  2. P. Burkert, “Zweistrahl-Interferometer zur Fourierspek-troskopie,” European patent application0034325 (February10, 1981).
  3. V. Tank, “Method and arrangement of an interferometer,” U.S. Patent4,652,130 (March24, 1987).
  4. V. Tank, “Interferometer,” European patent0146768 (February1, 1989).
  5. V. Tank, “Pathlength alteration in an interferometer by rotation of a retroreflector,” Opt. Eng. 28, 188–190 (1989).
    [CrossRef]
  6. P. Haschberger, O. Mayer, V. Tank, H. Dietl, “Michelson interferometer with a rotating retroreflector: a laboratory model for environmental monitoring,” Appl. Opt. 29, 4216–4220 (1990).
    [CrossRef] [PubMed]
  7. O. Mayer, “Aufbau eines Infrarot-Michelsoninterferometers mit einem rotierenden Retroreflektor,” Diplomarbeit (Lehrstuhl für Elektrische Messtechnik, Technische Universität München, München, Germany, 1989).

1990 (1)

1989 (1)

V. Tank, “Pathlength alteration in an interferometer by rotation of a retroreflector,” Opt. Eng. 28, 188–190 (1989).
[CrossRef]

1960 (1)

Burkert, P.

P. Burkert, “Zweistrahl-Interferometer zur Fourierspek-troskopie,” European patent application0034325 (February10, 1981).

Dietl, H.

Haschberger, P.

Mayer, O.

P. Haschberger, O. Mayer, V. Tank, H. Dietl, “Michelson interferometer with a rotating retroreflector: a laboratory model for environmental monitoring,” Appl. Opt. 29, 4216–4220 (1990).
[CrossRef] [PubMed]

O. Mayer, “Aufbau eines Infrarot-Michelsoninterferometers mit einem rotierenden Retroreflektor,” Diplomarbeit (Lehrstuhl für Elektrische Messtechnik, Technische Universität München, München, Germany, 1989).

Murty, M. V. R. K.

Tank, V.

P. Haschberger, O. Mayer, V. Tank, H. Dietl, “Michelson interferometer with a rotating retroreflector: a laboratory model for environmental monitoring,” Appl. Opt. 29, 4216–4220 (1990).
[CrossRef] [PubMed]

V. Tank, “Pathlength alteration in an interferometer by rotation of a retroreflector,” Opt. Eng. 28, 188–190 (1989).
[CrossRef]

V. Tank, “Method and arrangement of an interferometer,” U.S. Patent4,652,130 (March24, 1987).

V. Tank, “Interferometer,” European patent0146768 (February1, 1989).

Appl. Opt. (1)

J. Opt. Soc. Am. (1)

Opt. Eng. (1)

V. Tank, “Pathlength alteration in an interferometer by rotation of a retroreflector,” Opt. Eng. 28, 188–190 (1989).
[CrossRef]

Other (4)

O. Mayer, “Aufbau eines Infrarot-Michelsoninterferometers mit einem rotierenden Retroreflektor,” Diplomarbeit (Lehrstuhl für Elektrische Messtechnik, Technische Universität München, München, Germany, 1989).

P. Burkert, “Zweistrahl-Interferometer zur Fourierspek-troskopie,” European patent application0034325 (February10, 1981).

V. Tank, “Method and arrangement of an interferometer,” U.S. Patent4,652,130 (March24, 1987).

V. Tank, “Interferometer,” European patent0146768 (February1, 1989).

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Figures (10)

Fig. 1
Fig. 1

Setup of a retroreflector.

Fig. 2
Fig. 2

Coordinate systems K and K*.

Fig. 3
Fig. 3

Segmenting the reference plane.

Fig. 4
Fig. 4

Geometry of the device.

Fig. 5
Fig. 5

Orientation of the different coordinate systems.

Fig. 6
Fig. 6

Orientation of the plane of radiation and the reference plane.

Fig. 7
Fig. 7

Example of the reference plane in front of plane of radiation.

Fig. 8
Fig. 8

Spectral resolution ΔStot−1 as a function of α and l.

Fig. 9
Fig. 9

Simplified mounting and orientation of the retroreflector.

Fig. 10
Fig. 10

Spectral resolution ΔStot−1 as a function of β and l0.

Equations (107)

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x + y + z = 2 d ,
x = ( x , y , z ) = ( d , d , 0 ) + u ( 1 , 0 , 1 ) + υ ( 0 , 1 , 1 ) .
P 1 = [ 0 x e A + y e x e D + z e ] ,
P 2 = [ x e + y e C 0 y e C D + z e ] ,
P 3 = [ x e + z e B y e + z e A B 0 ] ,
A = tan ( 45 ° + γ ) ,
B = tan ( 45 ° + ρ ) ,
C = tan ( 45 ° γ ) ,
D = tan ( 45 ° ρ ) .
x = P 3 + t R o ,
R o = ( 1 , A , D ) .
P a = P 3 + t o R o
P a = 1 1 + A + D [ 2 d x e ( A + D ) + y e + z e 2 d A + x e A y e ( 1 + D ) + z e A 2 d D + x e D y e D z e ( 1 + A ) ] .
s = | x 1 x e | | R i | + | x 2 x 1 | | R 2 | + | x 3 x 2 | | R 3 | + | x a x 3 | | R o | ,
R i = ( 1 , A , D ) , x 1 x e = x e ;
R 2 = ( 1 , A , D ) , x 2 x 1 = x e + y e C ;
R 3 = ( 1 , A , D ) , x 3 x 2 = y e C + z e B ;
R o = ( 1 , A , D ) , x a x 3 = 2 d + x e + y e z e B ( 1 + A ) 1 + A + D .
y e x e A ,
z e x e D ,
y e z e A B .
| R | = ( 1 + A 2 + D 2 ) 1 / 2 .
x e + y e + z e = 2 d .
s = ( 1 + A 2 + D 2 ) 1 / 2 1 + A + D 4 d .
x = T K * K x * ,
x * = T K K * x ,
x = ( x , y , z , 1 ) T ,
x * = ( x * , y * , z * , 1 ) T ,
ψ = arctan ( 2 d / 2 d ) = 54.736 ° .
T K * K = A 0 ( d ) A 1 ( β ) A 2 ( ψ ) ,
T K K * = A 2 ( ψ ) A 1 ( β ) A 0 ( d ) ,
T K * K = 1 6 [ 3 2 6 2 3 4 d 3 2 6 2 3 4 d 0 2 6 2 3 4 d 0 0 0 6 ] ,
T K K * = 1 6 [ 3 2 3 2 0 0 6 6 2 6 0 2 3 2 3 2 3 4 3 d 0 0 0 6 ] .
R i = T K * K R i * ,
R i * = ( x i * , y i * , z i * , 0 ) ,
tan γ = | n x y × R ixy | n x y · R ixy ,
tan ρ = | n z x × R izx | n z x · R izx ,
tan γ = 3 x i * y i * + 2 z i * ,
tan ρ = 3 x i * + 3 y i * 3 x i * y i * 2 2 z i * .
tan ( 45 ° + α ) = ( 1 + tan α ) ( 1 tan α ) ,
A = 3 x i * + y i * 2 z i * 3 x i * + y i * 2 z i * ,
B = 3 x i * + y i * 2 z i * 2 y i * 2 z i * ,
C = 3 x i * + y i * 2 z i * 3 x i * + y i * 2 z i * ,
D = 2 y i * 2 z i * 3 x i * + y i * 2 z i * .
s = ± ( x i * 2 + y i * 2 + z i * 2 ) 1 / 2 3 z i * 4 d .
P a = Hx = 1 1 + A + D × [ ( A + D ) 1 1 2 d A ( 1 + D ) A 2 d A D D ( 1 + A ) 2 d D 0 0 0 1 + A + D ] × [ x e y e z e 1 ] .
P a * = T K K * H T K * K x e *
P a * = [ x e * + ( 4 d x i * / 3 z i * ) y e * + ( 4 d y i * / 3 z i * ) 0 ] .
P 4 = 2 d 1 + A + D ( 1 , A , D ) .
x 12 = ( 0 , 0 , 2 d ) + t 1 [ 1 , A , ( 1 + A ) ] ,
x 23 = ( 2 d , 0 , 0 ) + t 2 [ ( A + D ) , A , D ] ,
x 31 = ( 0 , 2 d , 0 ) + t 3 [ 1 , ( 1 + D ) , D ] .
y 12 * = y i * 2 z i * x i * x * + 2 2 3 d ,
y 12 * = arbitrary , x * = 0 ,
y 23 * = 2 y i * + z i * 2 x i * + 3 z i * x * + 2 ( x i * + 3 y i * ) 3 ( 2 x i * + 3 z i * ) d ,
y 31 * = 2 y i * + z i * 2 x i * 3 z i * x * 2 ( x i * + 3 y i * ) 3 ( 2 x i * 3 z i * ) d .
P 4 * = T K K * P 4 = ( 2 d / 3 z i * ) ( x i * , y i * , 0 ) ,
y 12 > y 4 * = ( 2 d / 3 z i * ) y i * ,
x 23 < x 4 * = ( 2 d / 3 z i * ) x i * ,
x 31 > x 4 * = ( 2 d / 3 z i * ) x i * .
3 ( A 1 ) y e * 3 ( A + 1 ) x e * + 2 ( 2 A + 1 ) d ,
3 ( A 1 ) y e * 3 ( A + 1 ) x e * + 2 ( 5 A 2 ) d ,
3 ( 2 + D ) y e * 3 D x e * + 2 ( 2 D + 1 ) d ,
3 ( 2 + D ) y e * 3 D x e * + 2 ( 5 D 2 ) d ,
3 ( 2 + C D ) y e * 3 C D x e * + 2 ( 2 C D + 1 ) d ,
3 ( 2 + C D ) y e * 3 C D x e * + 2 ( 5 C D 2 ) d ,
3 ( 2 A B 1 ) y e * 3 x e * + 2 ( 2 A B + 1 ) d ,
3 ( 2 A B 1 ) y e * 3 x e * + 2 ( 5 A B 2 ) d .
cos x = c x , sin x = s x , T K * K = A 0 A 1 A 2 ( ω t ) A 3 A 4 , T K K * = A 0 A 1 A 2 ( ω t ) A 3 A 4 ,
T K * K = [ c 2 α ( c ω t 1 ) + 1 c α s ω t s 2 α ( c ω t 1 ) / 2 c α ( l + r s α ) ( c ω t 1 ) c α s ω t c ω t s α s ω t ( l + r s α ) s ω t s 2 α ( c ω t 1 ) / 2 s α s ω t s 2 α ( c ω t 1 ) + 1 s α ( l + r s α ) ( c ω t 1 ) 0 0 0 1 ] ,
T K K * = [ c 2 α ( c ω t 1 ) + 1 c α s ω t s 2 α ( c ω t 1 ) / 2 c α ( l + r s α ) ( c ω t 1 ) c α s ω t c ω t s α s ω t ( l + r s α ) s ω t s 2 α ( c ω t 1 ) / 2 s α s ω t s 2 α ( c ω t 1 ) + 1 s α ( l + r s α ) ( c ω t 1 ) 0 0 0 1 ] .
a x + b y + c z = w ,
a = x I , b = y I , c = z I .
w = x I L + y I 0 + z I 0 = x I L .
R I * = T K K * R I ,
P E * = T K K * P E ,
P E = ( L , 0 , 0 , 1 ) ,
P E * = [ L + L c 2 α ( c ω t 1 ) + c α ( l + r s α ) ( c ω t 1 ) L c α s ω t ( l + r s α ) s ω t L s 2 α ( c ω t 1 ) / 2 + s α ( l + r s α ) ( c ω t 1 ) ] ,
R I * = [ x I [ c 2 α ( c ω t 1 ) + 1 ] + y I c α s ω t + z I c α s α ( c ω t 1 ) x I s α c α ( c ω t 1 ) + y I s α s ω t + z I [ s 2 α ( c ω t 1 ) + 1 ] x I c α s ω t + y I c ω t z I s α s ω t x I s α c α ( c ω t 1 ) + y I s α s ω t + z I [ s 2 α ( c ω t 1 ) + 1 ] 1 ] ,
s 1 = s α ( c ω t 1 ) ( l + r s α + L c α ) [ ± ( x I 2 + y I 2 + z I 2 ) 1 / 2 ] x I s α c α ( c ω t 1 ) + y I s α s ω t + z I [ s 2 α ( c ω t 1 ) + 1 ] .
s = ± ( x I 2 + y I 2 + z I 2 ) 2 3 { x I s α c α ( c ω t 1 ) + y I s α s ω t + z I [ s 2 α ( c ω t 1 ) + 1 ] } 4 d .
P e * = [ z I L c ω t + y I L s α s ω t + ( z I c α x I s α ) ( l + r s α ) ( c ω t 1 ) x I s α c α ( c ω t 1 ) + y I s α s ω t + z I [ s 2 α ( c ω t 1 ) + 1 ] [ y I s α ( c ω t 1 ) z I s ω t ] ( l + r s α + L c α ) x I s α c α ( c ω t 1 ) + y I s α s ω t + z I [ s 2 α ( c ω t 1 ) + 1 ] 0 ] .
P 4 * = 2 d 3 [ x I [ c 2 α ( c ω t 1 ) + 1 ] + y I c α s ω t + z I c α s α ( c ω t 1 ) x I s α c α ( c ω t 1 ) + y I s α s ω t + z I [ s 2 α ( c ω t 1 ) + 1 ] x I c α s ω t + y I c ω t z I s α s ω t x I s α c α ( c ω t 1 ) + y I s α s ω t + z I [ s 2 α ( c ω t 1 ) + 1 ] 0 ] .
P a * = P e * + 4 d R i * 3 z i * P * ,
P * = ( 0 , 0 , 4 d / 3 ) ,
R i * = R I * = T K K * R I ,
P e * = P E * + t R I * .
P a * = T K K * P E + ( 4 d 3 z i * t ) T K K * R I P * .
P a = P E + ( 4 d 3 z i * t ) R I T K * K P * .
nx = w , x I x + y I y + z I z = x I L ,
x = a + sb , x = P a + s R I ,
x 0 = a + s 0 b ,
x 0 n = w .
( a + s 0 b ) n = w ,
s 0 = w an bn ,
P A = a + w an bn b ,
P A = P a + x I L P a · ( x I , y I , z I ) R I · ( x I , y I , z I ) R I ,
P A = g [ 3 [ L 2 c α ( l + r s α ) ( c ω t 1 ) ] x I 2 [ 4 d s α c α ( c ω t 1 ) + 3 L ] ( y I 2 + z I 2 ) + 2 x I { [ ( 2 d 3 r ) s α 3 l ] [ y I s ω t + z I s α ( c ω t 1 ) ] + 2 3 z I } 2 ( l + r s α ) s ω t y I 2 ( 4 d s α s ω t ) ( x I 2 + z I 2 ) + 2 y I { [ ( 2 d 3 r ) s α 3 l ] × ( x I c α + z I s α ) ( c ω t 1 ) + 2 d z I } 2 s α ( l + r s α ) ( c ω t 1 ) z I 2 4 d [ s 2 α ( c ω t 1 ) + 1 ] ( x I 2 + y I 2 ) + 2 z I { [ ( 2 d r ) s α l ] [ x I c α ( c ω t 1 ) + y I s ω t ] } ] ,
g = 1 3 ( x I 2 + y I 2 + z I 2 ) .
s 2 = 2 3 L x I + 4 d { x I s α c α ( c ω t 1 ) + y I s α s ω t + z I [ s 2 α ( c ω t 1 ) + 1 ] } 3 ( x I 2 + y I 2 + z I 2 ) 1 / 2 2 3 ( l + r s α ) [ ( x I c α + z I s α ) ( c ω t 1 ) + y I s ω t ] 3 ( x I 2 + y I 2 + z I 2 ) 1 / 2 ( x I 2 + y I 2 + z I 2 ) 1 / 2 [ 4 d + 3 s α ( l + r s α + L c α ) ( c ω t 1 ) ] 3 { x I s α c α ( c ω t 1 ) + y I s α s ω t + z I [ s 2 α ( c ω t 1 ) + 1 ] } .
s 0 = x I L ( x I , y I , z I ) P a R I ( x I , y I , z I ) ,
s 0 = 2 L x I x I 2 + y I 2 + z I 2 = 4 d + 3 sin α ( l + r sin α + L cos α ) ( cos ω t 1 ) 3 z I * + 4 d z I * 2 3 ( l + r sin α ) [ ( x I cos α + z I sin α ) ( cos ω t 1 ) + y I sin ω t ] 3 ( x I 2 + y I 2 + z I 2 ) .
S tot = sgn [ z * ( P E * ) ] s 1 + s + sgn ( z I s 0 ) s 2 .
y i * = 0 ;
α = 0 , l = l 0 ,
y i * = 0 .
x i * 0 .

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