Abstract

The intermodulation of signal frequencies, or moiré, is an effect generally encountered in situations in which periodic or quasi-periodic structures are superimposed. Most familiar are the striking low-frequency patterns achieved by the multiplicative superposition of two structures. In other situations, the intermodulation of signal frequencies may be caused by nonlinear distortions of an input signal. This can be a desired feature in measuring applications or a noise source in transmission or display. Effects of nonlinear transfer characteristics on input signals consisting of more than one frequency are shown. The dependence of resulting moiré frequencies on the nonlinearity is evaluated, and common nonlinearities are compared by simulating a moiré topography experiment.

© 1988 Optical Society of America

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References

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  1. Rayleigh, “On the manufacture and theory of diffraction gratings,” Philos. Mag. 47, 81–93 (1874).
  2. D. Meadows, W. Johnson, J. Allen, “Generation of surface contours by moiré patterns,” Appl. Opt. 9, 942–947 (1970).
    [CrossRef] [PubMed]
  3. H. Takasaki, “Moiré topography,” Appl. Opt. 9, 1467–1472 (1970).
    [CrossRef] [PubMed]
  4. P. S. Theocaris, Moiré fringes in Strain Analysis (Pergamon, Oxford, 1969).
  5. F. K. Ligtenberg, “The moiré method—a new experimental method for determination of moments in small slab models,” Proc. Soc. Exp. Stress Analysis 12, 83–98 (1955).
  6. O. Kafri, “Noncoherent method for mapping phase objects,” Opt. Lett 5, 555–557 (1980).
    [CrossRef] [PubMed]
  7. D. D. Weiner, J. F. Spina, Sinusoidal Analysis and Modeling of Weakly Nonlinear Circuits (Van Nostrand Reinhold, New York, 1980).
  8. M. Schetzen, The Volterra and Wiener Theories of Nonlinear Systems (Wiley, New York, 1980).
  9. M. Maqusi, “Analysis and modeling of intermodulation distortion in wide-band cable TV channels,” IEEE Trans. Commun. COM-35, 568–572 (1987).
    [CrossRef]
  10. O. Bryngdahl, “Characteristics of superposed patterns in optics,” J. Opt. Soc. Am. 66, 87–94 (1976).
    [CrossRef]
  11. O. Shimbo, “Effects of intermodulation, AM-PM conversion, and additive noise in multicarrier TWT systems,” Proc. IEEE 59, 230–238 (1971).
    [CrossRef]
  12. E. Imboldi, G. R. Stette, “AM to PM conversion and intermodulation in nonlinear devices,” Proc. IEEE 61, 796–797 (1973).
    [CrossRef]
  13. N. Nielsen, Handbuch der Theorie der Cylinderfunktionen (Teubner, Stuttgart, 1968); reprint of the 1904 edition.
  14. G. N. Watson, A Treatise on the Theory of Bessel Functions (Cambridge U. Press, Cambridge, 1962).
  15. J. B. Allen, D. M. Meadows, “Removal of unwanted patterns from moiré contour maps by grid translation techniques,” Appl. Opt. 10, 210–212 (1971).
    [CrossRef] [PubMed]
  16. M. Halioua, R. S. Krishnamurthy, H. Liu, F. P. Chiang, “Projection moiré with moving gratings for automated 3-D topography,” Appl. Opt. 22, 850–855 (1983).
    [CrossRef] [PubMed]
  17. O. Bryngdahl, “Beat-pattern selection—multi-color-grating moiré,” Opt. Commun. 39, 127–131 (1981); “Orthogonal-states-grating moiré,” Opt. Commun. 41, 249–254 (1982).
    [CrossRef]
  18. B. Braunecker, O. Bryngdahl, “Spatial carriers with orthogonal subcodes,” J. Opt. Soc. Am. 73, 823–830 (1983).
    [CrossRef]
  19. R. Eschbach, O. Bryngdahl, “Subcoded information carriers: hybrid moiré system,” J. Opt. Soc. Am. 73, 1123–1129 (1983).
    [CrossRef]
  20. J. Wasowski, “Moiré topographic maps,” Opt. Commun. 2, 321–323 (1970).
    [CrossRef]
  21. T. Yatagai, M. Idesawa, “Use of synthetic deformed gratings in moiré topography,” Opt. Commun. 20, 243–245 (1977).
    [CrossRef]

1987 (1)

M. Maqusi, “Analysis and modeling of intermodulation distortion in wide-band cable TV channels,” IEEE Trans. Commun. COM-35, 568–572 (1987).
[CrossRef]

1983 (3)

1981 (1)

O. Bryngdahl, “Beat-pattern selection—multi-color-grating moiré,” Opt. Commun. 39, 127–131 (1981); “Orthogonal-states-grating moiré,” Opt. Commun. 41, 249–254 (1982).
[CrossRef]

1980 (1)

O. Kafri, “Noncoherent method for mapping phase objects,” Opt. Lett 5, 555–557 (1980).
[CrossRef] [PubMed]

1977 (1)

T. Yatagai, M. Idesawa, “Use of synthetic deformed gratings in moiré topography,” Opt. Commun. 20, 243–245 (1977).
[CrossRef]

1976 (1)

1973 (1)

E. Imboldi, G. R. Stette, “AM to PM conversion and intermodulation in nonlinear devices,” Proc. IEEE 61, 796–797 (1973).
[CrossRef]

1971 (2)

O. Shimbo, “Effects of intermodulation, AM-PM conversion, and additive noise in multicarrier TWT systems,” Proc. IEEE 59, 230–238 (1971).
[CrossRef]

J. B. Allen, D. M. Meadows, “Removal of unwanted patterns from moiré contour maps by grid translation techniques,” Appl. Opt. 10, 210–212 (1971).
[CrossRef] [PubMed]

1970 (3)

1955 (1)

F. K. Ligtenberg, “The moiré method—a new experimental method for determination of moments in small slab models,” Proc. Soc. Exp. Stress Analysis 12, 83–98 (1955).

1874 (1)

Rayleigh, “On the manufacture and theory of diffraction gratings,” Philos. Mag. 47, 81–93 (1874).

Allen, J.

Allen, J. B.

Braunecker, B.

Bryngdahl, O.

Chiang, F. P.

Eschbach, R.

Halioua, M.

Idesawa, M.

T. Yatagai, M. Idesawa, “Use of synthetic deformed gratings in moiré topography,” Opt. Commun. 20, 243–245 (1977).
[CrossRef]

Imboldi, E.

E. Imboldi, G. R. Stette, “AM to PM conversion and intermodulation in nonlinear devices,” Proc. IEEE 61, 796–797 (1973).
[CrossRef]

Johnson, W.

Kafri, O.

O. Kafri, “Noncoherent method for mapping phase objects,” Opt. Lett 5, 555–557 (1980).
[CrossRef] [PubMed]

Krishnamurthy, R. S.

Ligtenberg, F. K.

F. K. Ligtenberg, “The moiré method—a new experimental method for determination of moments in small slab models,” Proc. Soc. Exp. Stress Analysis 12, 83–98 (1955).

Liu, H.

Maqusi, M.

M. Maqusi, “Analysis and modeling of intermodulation distortion in wide-band cable TV channels,” IEEE Trans. Commun. COM-35, 568–572 (1987).
[CrossRef]

Meadows, D.

Meadows, D. M.

Nielsen, N.

N. Nielsen, Handbuch der Theorie der Cylinderfunktionen (Teubner, Stuttgart, 1968); reprint of the 1904 edition.

Rayleigh,

Rayleigh, “On the manufacture and theory of diffraction gratings,” Philos. Mag. 47, 81–93 (1874).

Schetzen, M.

M. Schetzen, The Volterra and Wiener Theories of Nonlinear Systems (Wiley, New York, 1980).

Shimbo, O.

O. Shimbo, “Effects of intermodulation, AM-PM conversion, and additive noise in multicarrier TWT systems,” Proc. IEEE 59, 230–238 (1971).
[CrossRef]

Spina, J. F.

D. D. Weiner, J. F. Spina, Sinusoidal Analysis and Modeling of Weakly Nonlinear Circuits (Van Nostrand Reinhold, New York, 1980).

Stette, G. R.

E. Imboldi, G. R. Stette, “AM to PM conversion and intermodulation in nonlinear devices,” Proc. IEEE 61, 796–797 (1973).
[CrossRef]

Takasaki, H.

Theocaris, P. S.

P. S. Theocaris, Moiré fringes in Strain Analysis (Pergamon, Oxford, 1969).

Wasowski, J.

J. Wasowski, “Moiré topographic maps,” Opt. Commun. 2, 321–323 (1970).
[CrossRef]

Watson, G. N.

G. N. Watson, A Treatise on the Theory of Bessel Functions (Cambridge U. Press, Cambridge, 1962).

Weiner, D. D.

D. D. Weiner, J. F. Spina, Sinusoidal Analysis and Modeling of Weakly Nonlinear Circuits (Van Nostrand Reinhold, New York, 1980).

Yatagai, T.

T. Yatagai, M. Idesawa, “Use of synthetic deformed gratings in moiré topography,” Opt. Commun. 20, 243–245 (1977).
[CrossRef]

Appl. Opt. (4)

IEEE Trans. Commun. (1)

M. Maqusi, “Analysis and modeling of intermodulation distortion in wide-band cable TV channels,” IEEE Trans. Commun. COM-35, 568–572 (1987).
[CrossRef]

J. Opt. Soc. Am. (3)

Opt. Commun. (3)

O. Bryngdahl, “Beat-pattern selection—multi-color-grating moiré,” Opt. Commun. 39, 127–131 (1981); “Orthogonal-states-grating moiré,” Opt. Commun. 41, 249–254 (1982).
[CrossRef]

J. Wasowski, “Moiré topographic maps,” Opt. Commun. 2, 321–323 (1970).
[CrossRef]

T. Yatagai, M. Idesawa, “Use of synthetic deformed gratings in moiré topography,” Opt. Commun. 20, 243–245 (1977).
[CrossRef]

Opt. Lett (1)

O. Kafri, “Noncoherent method for mapping phase objects,” Opt. Lett 5, 555–557 (1980).
[CrossRef] [PubMed]

Philos. Mag. (1)

Rayleigh, “On the manufacture and theory of diffraction gratings,” Philos. Mag. 47, 81–93 (1874).

Proc. IEEE (2)

O. Shimbo, “Effects of intermodulation, AM-PM conversion, and additive noise in multicarrier TWT systems,” Proc. IEEE 59, 230–238 (1971).
[CrossRef]

E. Imboldi, G. R. Stette, “AM to PM conversion and intermodulation in nonlinear devices,” Proc. IEEE 61, 796–797 (1973).
[CrossRef]

Proc. Soc. Exp. Stress Analysis (1)

F. K. Ligtenberg, “The moiré method—a new experimental method for determination of moments in small slab models,” Proc. Soc. Exp. Stress Analysis 12, 83–98 (1955).

Other (5)

P. S. Theocaris, Moiré fringes in Strain Analysis (Pergamon, Oxford, 1969).

D. D. Weiner, J. F. Spina, Sinusoidal Analysis and Modeling of Weakly Nonlinear Circuits (Van Nostrand Reinhold, New York, 1980).

M. Schetzen, The Volterra and Wiener Theories of Nonlinear Systems (Wiley, New York, 1980).

N. Nielsen, Handbuch der Theorie der Cylinderfunktionen (Teubner, Stuttgart, 1968); reprint of the 1904 edition.

G. N. Watson, A Treatise on the Theory of Bessel Functions (Cambridge U. Press, Cambridge, 1962).

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Figures (8)

Fig. 1
Fig. 1

Generation of the moiré effect by superposition of two periodic structures: (a) the multiplicative superposition (I1I2) and (b) the rectified difference (|I1I2|) of the two input signals I1 and I2. Both plots show a strong low-frequency component.

Fig. 2
Fig. 2

Two nonlinearities, (a) squaring and (b) rectification, as functions of the input f.

Fig. 3
Fig. 3

Comparison of strengths of the moiré effect generated by squaring and rectification in the range of 0 ≤ S1 + S2π. (a), (b) A1,1 for the squaring and the rectification, respectively. (c) Difference between (b) and (a); with the rectification resulting in a larger A1,1 for S1, S2 ≲ 2. (d) A3,1 for the rectification; A3,1 = 0 for squaring. Note that the plots in (c) and (d) are magnified compared with those in (a) and (b), as indicated by the values of the A axes.

Fig. 4
Fig. 4

Moiré generated by (a) squaring and (b) rectification of two sinusoidel gratings. The gratings have periods of 45 pixels in the vertical direction and 90 pixels at an angle of 20 deg with the vertical, respectively. The sum (Σ) and difference (Δ) frequencies appear sharper for the case of rectification.

Fig. 5
Fig. 5

Sigmoidlike nonlinearity described in Eq. (11).

Fig. 6
Fig. 6

Change in the frequency spectrum dependent on the bias value S0. (a) Spectrum of the input signal with frequency ω1 denoted by (1, 0) and frequency ω2 denoted by (0, 1). (b) Spectrum obtained by applying the nonlinearity to the signal. Here the additional frequencies corresponding to (−1, 2) and (2, −1) appear, and no component appears at (−1, 1) or (1, 1), which represent the difference and sum frequencies, (c)–(e) Spectra resulting from the addition of a bias S0. Spectra in (c) and (d) show no symmetry with respect to the bias and consist of odd and even doublets (m, n). The bias in (e) was chosen to convert the sine of the sigmoid into a cosine, thus representing an even nonlinearity with respect to the bias. Only doublets with (m + n) even appear, and the spectra in (e) and (b) are mutually exclusive. Note that spectra in (a)–(e) are normalized independently to show the occurring frequencies.

Fig. 7
Fig. 7

Sigmoidal nonlinearity applied to the same two gratings as in Fig. 4. (a) Zero bias (S0 = 0), resulting in an odd symmetry and no sum or difference frequency. The low-frequency pattern corresponds to 2ω1ω2. (b) Effect of the same nonlinearity for a bias of 3π/4, resulting in an even symmetry and strong sum and difference frequencies.

Fig. 8
Fig. 8

Application of (a) squaring, (b) rectification, and (c) sigmoid in a simulated moiré topographic setup. The object has been chosen to yield four contour fringes as a difference frequency. As in Fig. 4, the fringes appear sharper for the case of the rectification. In (c) the sigmoid has been used to generate a fringe pattern with twice the number of contour fringes by selecting the second-order difference frequency [(m, n) = (2, −1)].

Equations (42)

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I 1 = S 1 0 + S 1 sin ( 2 π ω 1 x )
I 2 = S 2 0 + S 2 sin ( 2 π ω 2 x ) ,
I = I 1 I 2
I = | I 1 I 2 |
S 1 0 S 1 + S 2 0 + S 2 ,
M ( x ) = N [ S ( x ) ] ,
M ( x ) = k = c k exp [ i k S ( x ) ] ,
c k = 1 2 π π π N ( f ) exp [ i k f ] d f
S ( x ) = S 0 + S 1 sin ω 1 x + S 2 sin ω 2 x ,
| S 0 | + | S 1 | + | S 2 | π
M ( x ) = k = c k exp ( i k S 0 ) exp ( i k S 1 sin ω 1 x ) exp ( i k S 2 sin ω 2 x ) = k = c k exp ( i k S 0 ) m = J m ( k S 1 ) exp ( i m ω 1 x ) × n = J n ( k S 2 ) exp ( i n ω 2 x ) = m = n = exp [ i ( m ω 1 + n ω 2 ) x ] × k = c k exp ( i k S 0 ) J m ( k S 1 ) J n ( k S 2 ) = m = n = A m , n exp [ i ( m ω 1 + n ω 2 ) x ] .
A m , n = k = c k exp ( i k S 0 ) J m ( k S 1 ) J n ( k S 2 )
J μ ( α ) = ( 1 ) μ J μ ( α ) ,
A m , n = ( 1 ) m A m , n = ( 1 ) n A m , n = ( 1 ) m + n A m , n .
A m , n = 0 for all | m | + | n | > 1.
A m , n = 0 if k = c k J m + n ( k S ) = 0
N [ S sin ( ω x ) ] sin ( ω x ) .
A m , n = a 0 2 δ m , 0 δ n , 0 + k = 1 exp [ i π 2 ( m + n ) ] × { a k cos [ k S 0 + π 2 ( m + n ) ] + b k sin [ k S 0 + π 2 ( m + n ) ] } J m ( k S 1 ) J n ( k S 2 ) ,
A m , n = { a 0 2 δ m , 0 δ n , 0 + k = 1 a k J m ( k S 1 ) J n ( k S 2 )   m + n even i k = 1 b k J m ( k S 1 ) J n ( k S 2 ) m + n odd
a k squ . = 4 π 2 ( 1 ) k 1 k 2 , a 2 k + 1 rect . = 4 π 2 1 ( 2 k + 1 ) 2
k = 1 ( 1 ) k 1 k 2 J 2 m ( k ξ ) = 0 for m > 1.
N ( f ) = sin ( α f ) max [ sin ( α f ) ] , 1 < α < 1
b k = 2 π ( 1 ) k k α 2 k 2 .
I ( x , y ) 1 + cos { ω [ x + f ( x , y ) ] } ,
R ( x , y ) 1 + cos ( ω R x )
A m , n = 0 for all | m | + | n | > 1 ,
k = c k exp ( i k S 0 ) J m ( k S 1 ) J n ( k S 2 ) = 0 for all | m | + | n | > 1.
S 1 = S , S 2 = S S .
0 = k = c k 0 k S J m ( k S ) J n ( k S k S ) d ( k S ) k S
0 = k = c k J m + n ( k S ) .
A m , n = 2 i k = 1 ( 1 ) k 1 k J m ( k S 1 ) J n ( k S 2 ) ,
k = 1 ( 1 ) k 1 k J m ( k S 1 ) J n ( k S 2 ) = 0
| S 1 | + | S 2 | < π
| m | + | n | > 1 ,
| S 1 | + | S 2 | > π .
k = 1 j = 1 M ( 1 ) k 1 k J m j ( k S j ) = 0
j = 1 M | m j | > 1
j = 1 M | S j | < π .
k = 1 ( 1 ) k 1 k 2 p J 2 m ( k ξ ) = 0 ,
k = 1 ( 1 ) k 1 k 2 p + 1 J 2 m + 1 ( k ξ ) = 0 ,
π ξ π , m > p .
a k = ( 1 ) k 1 k 2 ,

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