Abstract

In the scheme of land-groove recording in magneto-optic disk data-storage systems, it has been shown that an optimum groove depth exists at which the cross talk from adjacent tracks diminishes. Cross-talk cancellation, however, is very sensitive to various parameters of the system, and, in particular, the presence of substrate birefringence can have devastating effects on system performance. We analyze the origin of the observed effects by using scalar diffraction theory, and we show the reasons behind cross-talk cancellation. We also explain the relation between substrate birefringence and cross talk in simple analytical terms. Extensive computer simulations have been performed to verify and extend the theoretical results of this paper; the results of some of these simulations are also presented.

© 1996 Optical Society of America

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References

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  1. A. Fukumoto, S. Masuhara, K. Aratani, “Cross-talk analysis of land/groove magneto-optical recording,” in Optical Memory & Neural Networks ’94: Optical Memory, A. L. Mikaelian, ed., Proc. Soc. Photo-Opt. Instrum. Eng.2429, 41–42 (1994).
  2. H. Honma, T. Iwanaga, K. Kayanuma, M. Nakada, R. Katayama, K. Kobayashi, S. Itoi, H. Inada, “High density land/groove recording using PRML technology,” in Optical Data Storage, Vol. 10 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), paper WD1-1.
  3. H.H. Hopkins, “Diffraction theory of laser readout systems for optical video disks” J. Opt. Soc. Am. 69, 4–24 (1979).
  4. G. Bouwhuis, J. J. M. Braat, “Recording and reading of information on optical disks,” in Applied Optics and Optical Engineering, R. R. Shannon, J. C. Wyant, eds. (Academic, New York, 1983), Vol. 9.
  5. G. Bouwhuis, J. Braat, A. Huijser, J. Pasman, G. Van Rosmalen, K. Schouhamer Immink, Principles of Optical Disc Systems (Hilger, Bristol, 1985).
  6. P. Sheng, “Theoretical considerations of optical diffraction from RCA video disc signals,” RCA Rev. 39, 512–555 (1978).
  7. M. Mansuripur, The Physical Principles of Magneto-optical Recording (Cambridge U. Press, Cambridge, England, 1995), Chap. 6, pp. 187–193.
  8. H. Fu, S. Sugaya, J. K. Erwin, M. Mansuripur, “Measurement of birefringence for optical recording disk substrates,” Appl. Opt. 33, 1938–1949 (1994).
  9. The computer program DIFFRACT is commercially available from MM Research Company, Tucson, Ariz. 85718.The theoretical basis of this program is described in the following papers: M. Mansuripur, “Certain computational aspects of vector diffraction problems,” J. Opt. Soc. Am. A 6, 786–805 (1989);“Analysis of multilayer thin-film structures containing magneto-optic and anisotropic media at oblique incidence using 2 × 2 matrices,” J. Appl. Phys. 67, 6466–6475 (1990).

1994 (1)

1979 (1)

1978 (1)

P. Sheng, “Theoretical considerations of optical diffraction from RCA video disc signals,” RCA Rev. 39, 512–555 (1978).

Aratani, K.

A. Fukumoto, S. Masuhara, K. Aratani, “Cross-talk analysis of land/groove magneto-optical recording,” in Optical Memory & Neural Networks ’94: Optical Memory, A. L. Mikaelian, ed., Proc. Soc. Photo-Opt. Instrum. Eng.2429, 41–42 (1994).

Bouwhuis, G.

G. Bouwhuis, J. J. M. Braat, “Recording and reading of information on optical disks,” in Applied Optics and Optical Engineering, R. R. Shannon, J. C. Wyant, eds. (Academic, New York, 1983), Vol. 9.

G. Bouwhuis, J. Braat, A. Huijser, J. Pasman, G. Van Rosmalen, K. Schouhamer Immink, Principles of Optical Disc Systems (Hilger, Bristol, 1985).

Braat, J.

G. Bouwhuis, J. Braat, A. Huijser, J. Pasman, G. Van Rosmalen, K. Schouhamer Immink, Principles of Optical Disc Systems (Hilger, Bristol, 1985).

Braat, J. J. M.

G. Bouwhuis, J. J. M. Braat, “Recording and reading of information on optical disks,” in Applied Optics and Optical Engineering, R. R. Shannon, J. C. Wyant, eds. (Academic, New York, 1983), Vol. 9.

Erwin, J. K.

Fu, H.

Fukumoto, A.

A. Fukumoto, S. Masuhara, K. Aratani, “Cross-talk analysis of land/groove magneto-optical recording,” in Optical Memory & Neural Networks ’94: Optical Memory, A. L. Mikaelian, ed., Proc. Soc. Photo-Opt. Instrum. Eng.2429, 41–42 (1994).

Honma, H.

H. Honma, T. Iwanaga, K. Kayanuma, M. Nakada, R. Katayama, K. Kobayashi, S. Itoi, H. Inada, “High density land/groove recording using PRML technology,” in Optical Data Storage, Vol. 10 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), paper WD1-1.

Hopkins, H.H.

Huijser, A.

G. Bouwhuis, J. Braat, A. Huijser, J. Pasman, G. Van Rosmalen, K. Schouhamer Immink, Principles of Optical Disc Systems (Hilger, Bristol, 1985).

Inada, H.

H. Honma, T. Iwanaga, K. Kayanuma, M. Nakada, R. Katayama, K. Kobayashi, S. Itoi, H. Inada, “High density land/groove recording using PRML technology,” in Optical Data Storage, Vol. 10 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), paper WD1-1.

Itoi, S.

H. Honma, T. Iwanaga, K. Kayanuma, M. Nakada, R. Katayama, K. Kobayashi, S. Itoi, H. Inada, “High density land/groove recording using PRML technology,” in Optical Data Storage, Vol. 10 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), paper WD1-1.

Iwanaga, T.

H. Honma, T. Iwanaga, K. Kayanuma, M. Nakada, R. Katayama, K. Kobayashi, S. Itoi, H. Inada, “High density land/groove recording using PRML technology,” in Optical Data Storage, Vol. 10 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), paper WD1-1.

Katayama, R.

H. Honma, T. Iwanaga, K. Kayanuma, M. Nakada, R. Katayama, K. Kobayashi, S. Itoi, H. Inada, “High density land/groove recording using PRML technology,” in Optical Data Storage, Vol. 10 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), paper WD1-1.

Kayanuma, K.

H. Honma, T. Iwanaga, K. Kayanuma, M. Nakada, R. Katayama, K. Kobayashi, S. Itoi, H. Inada, “High density land/groove recording using PRML technology,” in Optical Data Storage, Vol. 10 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), paper WD1-1.

Kobayashi, K.

H. Honma, T. Iwanaga, K. Kayanuma, M. Nakada, R. Katayama, K. Kobayashi, S. Itoi, H. Inada, “High density land/groove recording using PRML technology,” in Optical Data Storage, Vol. 10 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), paper WD1-1.

Mansuripur, M.

H. Fu, S. Sugaya, J. K. Erwin, M. Mansuripur, “Measurement of birefringence for optical recording disk substrates,” Appl. Opt. 33, 1938–1949 (1994).

M. Mansuripur, The Physical Principles of Magneto-optical Recording (Cambridge U. Press, Cambridge, England, 1995), Chap. 6, pp. 187–193.

Masuhara, S.

A. Fukumoto, S. Masuhara, K. Aratani, “Cross-talk analysis of land/groove magneto-optical recording,” in Optical Memory & Neural Networks ’94: Optical Memory, A. L. Mikaelian, ed., Proc. Soc. Photo-Opt. Instrum. Eng.2429, 41–42 (1994).

Nakada, M.

H. Honma, T. Iwanaga, K. Kayanuma, M. Nakada, R. Katayama, K. Kobayashi, S. Itoi, H. Inada, “High density land/groove recording using PRML technology,” in Optical Data Storage, Vol. 10 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), paper WD1-1.

Pasman, J.

G. Bouwhuis, J. Braat, A. Huijser, J. Pasman, G. Van Rosmalen, K. Schouhamer Immink, Principles of Optical Disc Systems (Hilger, Bristol, 1985).

Schouhamer Immink, K.

G. Bouwhuis, J. Braat, A. Huijser, J. Pasman, G. Van Rosmalen, K. Schouhamer Immink, Principles of Optical Disc Systems (Hilger, Bristol, 1985).

Sheng, P.

P. Sheng, “Theoretical considerations of optical diffraction from RCA video disc signals,” RCA Rev. 39, 512–555 (1978).

Sugaya, S.

Van Rosmalen, G.

G. Bouwhuis, J. Braat, A. Huijser, J. Pasman, G. Van Rosmalen, K. Schouhamer Immink, Principles of Optical Disc Systems (Hilger, Bristol, 1985).

Appl. Opt. (1)

J. Opt. Soc. Am. (1)

RCA Rev. (1)

P. Sheng, “Theoretical considerations of optical diffraction from RCA video disc signals,” RCA Rev. 39, 512–555 (1978).

Other (6)

M. Mansuripur, The Physical Principles of Magneto-optical Recording (Cambridge U. Press, Cambridge, England, 1995), Chap. 6, pp. 187–193.

The computer program DIFFRACT is commercially available from MM Research Company, Tucson, Ariz. 85718.The theoretical basis of this program is described in the following papers: M. Mansuripur, “Certain computational aspects of vector diffraction problems,” J. Opt. Soc. Am. A 6, 786–805 (1989);“Analysis of multilayer thin-film structures containing magneto-optic and anisotropic media at oblique incidence using 2 × 2 matrices,” J. Appl. Phys. 67, 6466–6475 (1990).

G. Bouwhuis, J. J. M. Braat, “Recording and reading of information on optical disks,” in Applied Optics and Optical Engineering, R. R. Shannon, J. C. Wyant, eds. (Academic, New York, 1983), Vol. 9.

G. Bouwhuis, J. Braat, A. Huijser, J. Pasman, G. Van Rosmalen, K. Schouhamer Immink, Principles of Optical Disc Systems (Hilger, Bristol, 1985).

A. Fukumoto, S. Masuhara, K. Aratani, “Cross-talk analysis of land/groove magneto-optical recording,” in Optical Memory & Neural Networks ’94: Optical Memory, A. L. Mikaelian, ed., Proc. Soc. Photo-Opt. Instrum. Eng.2429, 41–42 (1994).

H. Honma, T. Iwanaga, K. Kayanuma, M. Nakada, R. Katayama, K. Kobayashi, S. Itoi, H. Inada, “High density land/groove recording using PRML technology,” in Optical Data Storage, Vol. 10 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), paper WD1-1.

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Figures (21)

Fig. 1.
Fig. 1.

MO readout system utilizing the differential detection scheme. The differential module consists of a polarizing beam splitter (or Wollaston prism), two identical photodetectors, and a differential amplifier. The output of this amplifier, ΔS, is the MO read signal. A quarter-wave plate (QWP) with its fast axis oriented at 45° to the direction of incident polarization precedes the differential module in some practical systems in use today. The QWP is used to eliminate the ellipticity of polarization of the beam, which may be present in the MO layer itself or may be induced by the substrate birefringence. When the QWP is present, the differential module is rotated around the optical axis until readout signal ΔS from a recorded pattern of domains is maximized; this would yield an optimal setting for the module. In the absence of the QWP, the optimum setting of the module is at 45° to the direction of incident polarization.

Fig. 2.
Fig. 2.

MO data marks of varying length and spacing written on (a) the central track (in this case, a groove) and (b) the two adjacent tracks (in this case, lands). These patterns of data are used to compute the MO read signal and the cross-talk signal, respectively, which are then used to compute the cross-talk ratio.

Fig. 3.
Fig. 3.

Differential signal output as the central track in Fig. 2 is scanned: (a) the MO read signal and (b) the cross-talk signal, computed with the data patterns in Figs. 2(a) and 2(b), respectively.

Fig. 4.
Fig. 4.

(a) In land-groove recording, the information is recorded on both lands and grooves. During read–write–erase operations, the objective lens, having a numerical aperture NA, focuses the laser beam on the land center (or on the groove center). Typically, the land and the groove have rectangular cross sections with equal width W. The groove depth is d, causing a phase shift of Φ = 4πn s d0 between the light reflected from the land and that reflected from the groove. (Here n s is the refractive index of the substrate and λ0 is the vacuum wavelength of the laser beam.) (b) A normally incident beam on the disk surface is diffracted into several orders on reflection. Only the zeroth order, the plus first order, and the minus first order are shown. The deviation angle (from the normal) of the first-order beam is θ = sin−10/2W). For W = λ0 we have θ = 30°. (c) The baseball pattern at the exit pupil of the objective lens consists of the superposition of the zeroth-order, plus first-order, and minus first-order diffracted beams. The overlap areas are indicated by A 0+1 and A 0−1, each covering a fraction equal to 2 3 3 / ( 2 π ) 0.39 of the aperture. The remaining area, A 0, which is solely occupied by the zero-order beam, has a fractional area equal to 3 / π 1 3 0.22.

Fig. 5.
Fig. 5.

Normally incident beam, linearly polarized along the X direction, is diffracted into multiple orders on reflection from the disk surface. The magnetization pattern is assumed to be uniform along Y but varying along X with a period of 4W. The MO generated component of the reflected light is polarized along Y, and its nth-order diffracted beam has angle θ n = sin−1(nλ0/4W) with the Z axis. For W0, the diffraction angles are θ±1 = 14.5°, θ±2 = 30°, and θ±3 = 48.6°. (b) Baseball pattern at the exit pupil of the objective lens for the MO generated component of polarization, showing the superposition of zeroth order and plus or minus first-, second-, and third-order diffracted beams.

Fig. 6.
Fig. 6.

Three cases of practical interest for the reflected amplitude a (x) at the disk surface of Fig. 5(a). The Fourier coefficients of each function are shown in the box below the graphical representation of the function itself. The central groove and its nearest neighbor grooves are always magnetized in the same direction (up). The two neighboring lands are either both magnetized up (a) or both magnetized down (b), or one is magnetized up while the other is magnetized down (c). The phase coefficient exp(iΦ) represents the difference in the height of the lands relative to the grooves. The (local) reflection coefficient is multiplied by −1 whenever the magnetization direction is reversed.

Fig. 7.
Fig. 7.

Reflected complex amplitudes a ∣∣ and a at the exit pupil of the objective lens (a) in the region denoted by A 0 in Fig. 4(c), where only the zeroth order beam resides; (b) in the regions denoted by A 0+1 and A 0−1 of Fig. 4(c), where the zeroth order beam interferes with the plus or minus first-order beams. In each region, a ∣∣ is a function of Φ and of r ∣∣, independent of the magnetization pattern. In contrast, a is a function of Φ, r , and the magnetization pattern. The three cases of a shown in each diagram, i.e., a ( I ) , a ( II ) , a ( III ) , correspond to the magnetization patterns depicted in Fig. 6.

Fig. 8.
Fig. 8.

Mask placed before the differential detection module in the system of example 2. The region blocked by the mask is the region A 0 of Fig. 4(c).

Fig. 9.
Fig. 9.

Differential readout signal obtained by scanning of (a) the six MO marks shown on the central track in Fig. 2(a), (b) the erased central track in Fig. 2(b), containing the cross-talk signal from the six MO domains on the two adjacent tracks. The mask shown in Fig. 8 has been placed before the detectors and the assumed groove depth is 0.143λ0. The peak-to-valley amplitude of the cross-talk signal is 0.036.

Fig. 10.
Fig. 10.

Phase shift of Δϕ between r and r ∣∣ will rotate the three complex amplitudes, a ( I ) , a ( II ) , and a ( III ) , away from a ∣∣. The effect of this rotation on cross-track cross talk is different for the case of readout from land (Φ > 0) shown in (a) and readout from groove (Φ < 0) shown in (b).

Fig. 11.
Fig. 11.

Three slightly different methods proposed for the automatic correction of the beam's ellipticity prior to signal detection. In all cases a low-frequency wobble signal is needed to modulate the envelope of readout signal ΔS. This envelope modulation, when isolated and compared with the wobble itself, can produce a bipolar signal for feedback control and automatic adjustment of the phase difference between r and r ∣∣ . The same mechanism (or actuator) that produces the wobble can also be used to adjust the relative phase of r and r ∣∣. In (a) the differential detection module as a whole is placed in a voice-coil type of actuator and is wobbled around the optical axis. In (b) the differential detector is stationary and a half-wave plate is used to rotate the direction of polarization relative to the module. The system in (c) uses a voltage-controlled adjustable phase plate whose fast and slow axes are parallel and perpendicular to the direction of incident polarization. The voltage driving the variable phase plate has a single-frequency wobble component plus a slowly varying feedback signal that automatically adjusts the phase difference between r and r ∣∣.

Fig. 12.
Fig. 12.

Pairs of marks representing a 3T data pattern on a grooveed MO disk. These patterns are used in the computer simulations to analyze the cross-talk ratio. (a) Data on the central land for calculating the MO read signal; (b) data on both of the adjacent grooves for calculating the cross-talk signal.

Fig. 13.
Fig. 13.

Peak-to-valley (P–V) amplitude of the readout signals for a MO disk with a glass substrate. (a) Solid curve corresponds to the MO read signal arising from the data pattern shown in Fig. 12(a), and the dashed curve corresponds to the cross-talk signal arising from the pattern in Fig. 12(b). (b) Cross-talk ratio as a function of groove depth shows a minimum at d = 0.154λ0.

Fig. 14.
Fig. 14.

(a) Random pattern of marks on adjacent tracks of an erased central track. (b) cross-talk signal from the random pattern of marks shown in (a) for two different groove depths. The solid curve is obtained with d = 0.117λ0, corresponding approximately to a λ0/8 groove depth used in current MO disks. The dashed curve is computed with d = 0.15λ0, corresponding approximately to the optimum groove depth necessary for cross-talk cancellation. P–V, peak to valley.

Fig. 15.
Fig. 15.

(a) Peak-to-valley (P–V) amplitude of the readout and cross-talk signals versus defocus for a disk with a glass substrate and a groove depth of d = 0.154λ0. The data pattern used to compute the readout signal (solid curve) is that of Fig. 12(a), and the pattern used to compute the cross-talk signal (dashed curve) is shown in Fig. 12(b). (b) Cross-talk ratio versus defocus indicates that a defocus of ±0.25λ0 will not increase the cross talk above − 40 dB.

Fig. 16.
Fig. 16.

(a) Peak-to-valley (P–V) amplitude of the readout and cross-talk signals versus off-track error for a disk with a glass substrate and a groove depth of d = 0.154λ0. The data pattern used to compute the readout signal (solid curve) is that of Fig. 12(a), and the pattern used to compute the cross-talk signal (dashed curve) is shown in Fig. 12(b). (b) Cross-talk ratio versus off-track error indicates that a slight shift toward an adjacent track will increase the cross talk, but a tracking error of ±0.15λ0 should be acceptable.

Fig. 17.
Fig. 17.

(a) Peak-to-valley (P–V) amplitude of the readout and cross-talk signals versus comatic aberration of the objective lens for a disk with a glass substrate and a groove depth of d = 0.154λ0. The orientation of coma is chosen to place the tail of the focused spot in the cross-track direction. The data pattern used to compute the readout signal (solid curve) is that of Fig. 12(a), and the pattern used to compute the cross-talk signal (dashed curve) is shown in Fig. 12(b). (b) Plot of cross-talk ratio versus the magnitude of coma shows that over the range of comatic aberration considered, the cross-talk ratio remains below − 45 dB. (Note that ±0.28λ0 of coma corresponds to ±5 mrad of disk tilt.)

Fig. 18.
Fig. 18.

Cross-talk ratio versus groove depth, computed with the patterns shown in Fig. 12. The assumed disk substrate is polycarbonate, having a vertical birefringence of Δn = 6 × 10−4 and an in-plane birefringence of Δn ∣∣ = 2 × 10−5. The solid curve represents the case of landreading, the dotted curve corresponds to groove reading, the dashed curve shows cross-talk ratios for the case of land reading with only in-plane birefringence, and the dashed–dotted curve shows that with only vertical birefringence.

Fig. 19.
Fig. 19.

Cross-talk ratio versus detector module orientation for a disk with a birefringent substrate, having a groove depth of 0.154λ0. The substrate contains the nominal amounts of both in-plane and vertical birefringence. A quarter-wave plate, set at 45° to the direction of the incident polarization, is assumed present in the read path.

Fig. 20.
Fig. 20.

Cross-talk ratio versus the magnitude of in-plane birefringence for land reading of a disk through its polycarbonate substrate. The vertical birefringence of the substrate is fixed at Δn = 6 × 10−4, the assumed groove depth is d = 0.154λ0, the read channel utilizes a quarter-wave plate set at 45° to the direction of incident polarization, and the detector module's orientation angle, θ, is chosen to correct the effects of birefringence. In the case of the solid curve, θ is fixed at 82°, correcting for the nominal value of Δn ∣∣ = 2 × 10−5. In the case of the dashed curve, the detector module is assumed to be controlled dynamically, its angle changing linearly from 88° to 76° as Δn ∣∣ increases from zero to 4 × 10−5.

Fig. 21.
Fig. 21.

Cross-talk ratio versus the orientation of the principal axes of in-plane birefringence for land reading of a disk through its polycarbonate substrate. The assumed groove depth is d = 0.154λ0, the read channel utilizes a quarter-wave plate set at 45° to the direction of incident polarization, and the detector module's orientation is set at 82° for the nominal value of Δn ∣∣ = 2 × 10−5. The curve indicates that the principal axes of in-plane birefringence can deviate from the nominal directions (i.e., radial and tangential to tracks) by as much as ± 20° without raising the cross-talk ratio above − 40 dB.

Tables (1)

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Table 1 Parameters Used to Model the MO Readout System a

Equations (8)

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cross-talk  ratio = 20   log 10 ( cross-talk  amplitude / data  signal  amplitude ) 16  dB .
= 1 2 [ 1 + exp ( i Φ ) ] r ,
= 1 π [ 1 + exp ( i Φ ) ] r .
a = [ ( 1 2 + 1 π ) + ( 1 2 1 π ) exp ( i Φ ) ] r [ 0.82 + 0.18 exp ( i Φ ) ] r in  A 0 + 1 , A 0 1 , a = [ 0.5 + 0.5 exp ( i Φ ) ] r in A 0 .
average  a   from  regions   A ^ 0 + 1 + 2 + 3 , A ^ 0 1 2 3     = [ a 0 + 1 2 ( a 1 + a 1 ) + 1 2 ( a 2 + a 2 ) + 1 2 ( a 3 + a 3 ) ] r , average  a   from  regions   A ^ 0 + 1 1 + 2 , A ^ 0 + 1 1 2     = [ a 0 + ( a 1 + a 1 ) + 1 2 ( a 2 + a 2 ) ] r , average  a   from  regions   A ^ 0 + 1 + 2 , A ^ 0 1 2     = [ a 0 + 1 2 ( a 1 + a 1 ) + 1 2 ( a 2 + a 2 ) ] r , average  a   from  regions   A ^ 0 + 1 , A ^ 0 1     = [ a 0 + 1 2 ( a 1 + a 1 ) ] r , a   within  regions   A ^ 0 + 1 1     = [ a 0 + ( a 1 + a 1 ) ] r .
average  a   from  regions   A ^ 0 + 1 + 2 + 3 , A ^ 0 1 2 3 , A ^ 0 + 1 1 + 2 ,            A ^ 0 1 + 1 2 , A ^ 0 + 1 + 2 , A ^ 0 1 2 = [ a 0 + 1 2 ( a 2 + a 2 ) ] r , average   a   from regions   A ^ 0 , A ^ 0 + 1 , A ^ 0 1 , A ^ 0 + 1 1                                                         = a 0 r .
a = [ 0.5 + 0.5 exp ( i Φ ) ] r , a = { [ 0.5 + 0.5 exp ( i Φ ) ] r , case I [ 0.5 0.5 exp ( i Φ ) ] r , case II 0.5 r , case III.
a = [ 0.82 + 0.18 exp ( i Φ ) ] r , a = { [ 0.82 + 0.18 exp ( i Φ ) ] r , case I [ 0.82 0.18 exp ( i Φ ) ] r , case II 0.82 r , case III.

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