Abstract

A spectral analysis of the electrical signal from the detector in degree of polarization (DOP) measurements that use a rotating polarizer shows base band frequencies that create a noise floor. The noise floor arises from phase and intensity modulation of the optical field owing to the varying thickness and transmission of the rotating optical polarizer in the DOP apparatus. A physical model is presented for the noise floor arising from the phase and intensity modulation, and a calibration procedure including configuration guidelines is provided to minimize the effects of the unwanted modulations.

© 2009 Optical Society of America

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References

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  1. P. D. Colbourne and D. T. Cassidy, “Imaging of stresses in GaAs diode lasers using polarization-resolved photoluminescence,” IEEE J. Quantum Electron. 29, 62-68 (1993).
    [CrossRef]
  2. D. T. Cassidy, S. K. K. Lam, B. Lakshmi, and D. M. Bruce, “Strain mapping by measurement of the degree of polarization of photoluminescence,” Appl. Opt. 43, 1811-1818 (2004).
    [CrossRef] [PubMed]
  3. J. H. Scofield, “Frequency-domain description of a lock-in amplifier,” Am. J. Phys. 62, 129-133 (1994).
    [CrossRef]
  4. P. D. Welch, “The use of fast-Fourier transform for the estimation of power spectra: a method based on time averaging over short, modified periodograms,” IEEE Trans. Audio Electroacoust. AU-15, 70-73 (1967).
    [CrossRef]
  5. L. J. Giacoletto, “Generalized theory of multitone amplitude and frequency modulation,” Proc. IRE 35, 680-693 (1947).
    [CrossRef]

2004

1994

J. H. Scofield, “Frequency-domain description of a lock-in amplifier,” Am. J. Phys. 62, 129-133 (1994).
[CrossRef]

1993

P. D. Colbourne and D. T. Cassidy, “Imaging of stresses in GaAs diode lasers using polarization-resolved photoluminescence,” IEEE J. Quantum Electron. 29, 62-68 (1993).
[CrossRef]

1967

P. D. Welch, “The use of fast-Fourier transform for the estimation of power spectra: a method based on time averaging over short, modified periodograms,” IEEE Trans. Audio Electroacoust. AU-15, 70-73 (1967).
[CrossRef]

1947

L. J. Giacoletto, “Generalized theory of multitone amplitude and frequency modulation,” Proc. IRE 35, 680-693 (1947).
[CrossRef]

Bruce, D. M.

Cassidy, D. T.

D. T. Cassidy, S. K. K. Lam, B. Lakshmi, and D. M. Bruce, “Strain mapping by measurement of the degree of polarization of photoluminescence,” Appl. Opt. 43, 1811-1818 (2004).
[CrossRef] [PubMed]

P. D. Colbourne and D. T. Cassidy, “Imaging of stresses in GaAs diode lasers using polarization-resolved photoluminescence,” IEEE J. Quantum Electron. 29, 62-68 (1993).
[CrossRef]

Colbourne, P. D.

P. D. Colbourne and D. T. Cassidy, “Imaging of stresses in GaAs diode lasers using polarization-resolved photoluminescence,” IEEE J. Quantum Electron. 29, 62-68 (1993).
[CrossRef]

Giacoletto, L. J.

L. J. Giacoletto, “Generalized theory of multitone amplitude and frequency modulation,” Proc. IRE 35, 680-693 (1947).
[CrossRef]

Lakshmi, B.

Lam, S. K. K.

Scofield, J. H.

J. H. Scofield, “Frequency-domain description of a lock-in amplifier,” Am. J. Phys. 62, 129-133 (1994).
[CrossRef]

Welch, P. D.

P. D. Welch, “The use of fast-Fourier transform for the estimation of power spectra: a method based on time averaging over short, modified periodograms,” IEEE Trans. Audio Electroacoust. AU-15, 70-73 (1967).
[CrossRef]

Am. J. Phys.

J. H. Scofield, “Frequency-domain description of a lock-in amplifier,” Am. J. Phys. 62, 129-133 (1994).
[CrossRef]

Appl. Opt.

IEEE J. Quantum Electron.

P. D. Colbourne and D. T. Cassidy, “Imaging of stresses in GaAs diode lasers using polarization-resolved photoluminescence,” IEEE J. Quantum Electron. 29, 62-68 (1993).
[CrossRef]

IEEE Trans. Audio Electroacoust.

P. D. Welch, “The use of fast-Fourier transform for the estimation of power spectra: a method based on time averaging over short, modified periodograms,” IEEE Trans. Audio Electroacoust. AU-15, 70-73 (1967).
[CrossRef]

Proc. IRE

L. J. Giacoletto, “Generalized theory of multitone amplitude and frequency modulation,” Proc. IRE 35, 680-693 (1947).
[CrossRef]

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Figures (9)

Fig. 1
Fig. 1

DOP experimental apparatus. The He–Ne laser light is modulated by the optical chopper before exciting the sample under study. The PL is passed through a rotating polarizer and on to a photodetector. The reflected He–Ne light is also collected for optical mapping of the facet. ND, neutral density.

Fig. 2
Fig. 2

Power spectrum for an unstrained sample with the chopper and rotating polarizer. Several frequencies are labeled, including the chopper and its harmonics, the DOP/ROP signal at twice the polarizer frequency, and the AM sidebands between the chopper and the rotating polarizer. Many additional and regularly spaced harmonics are present.

Fig. 3
Fig. 3

Power spectrum for an unstrained sample with no chopper and no polarizer. Several regularly spaced components are present, but careful inspection shows they are all power line harmonics. Several frequencies are labeled.

Fig. 4
Fig. 4

Power spectrum of an unstrained sample with a chopper frequency of 1070 Hz and no polarizer. Strong signals are due to the fundamental frequency and harmonics of the chopper. Notice that there are several frequencies in the chopper signal very close to the fundamental between 20 and 45 dB (e.g., 1070 ± 25 Hz ) caused by the imperfect shape of the chopped wave.

Fig. 5
Fig. 5

Power spectrum of an unstrained sample with the polarizer rotating at 115 Hz and no chopper. The spacing of the harmonics is also 115 Hz . The magnitude of the components as frequency increases suggests it is related to Bessel functions (the inset shows a bar graph comparing normalized harmonic power for λ / 3 and λ / 8 phase modulation). This scan was performed on a second independent apparatus with a different optical flat, thus it has a slightly different profile than that of Fig. 2.

Fig. 6
Fig. 6

Expanded view of Fig. 2 in the vicinity of 230 Hz . The primary and secondary harmonics are clearly visible.

Fig. 7
Fig. 7

Surface profile of a PolarCor polarizer taken using a Zygo NewView 6000 interferometer. The area scanned is approximately 1.3 mm by 1.8 mm . (Left) Contour maps of the glass surface. (Right) Surface profile along the line indicated in the figure. The surface imperfections appear to be of the order of several hundred nanometers. The ringing around the edges is likely caused by diffraction effects and is unlikely representative of actual surface elevation.

Fig. 8
Fig. 8

Optimal configuration of the polarizer and chopper frequencies. The beat frequency is determined by the difference between the chopper and the nearest primary harmonic. In the case above, the nearest primary harmonic is 1035 Hz , creating a beat frequency of 38 Hz . While increasing the beat frequency will move the secondary harmonics further from the polarizer, it will also move the harmonic closer to the chopper.

Fig. 9
Fig. 9

Effect of secondary harmonics in a poor configuration. The device is a ridge waveguide laser. (Left) The PL is focused near (but not at) the center of the 230 Hz rotating polarizer, while the chopper is configured for 1040 Hz , resulting in a beat frequency of 5 Hz with significant noise. (Right) The chopper is configured for 1070 Hz , yielding a beat frequency of 35 Hz , but the PL is focused at approximately 2 / 3 of the optical flat’s radius from the center of rotation. The measured DOP offset in the unstrained region is 3.260% as opposed to an offset of 0.123% for alignment of the beam near the center of the polarizer.

Tables (1)

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Table 1 Harmonic Generation for a 115 Hz Polarizer and a 1070 Hz Chopper

Equations (8)

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DOP y = 0 [ L x ( E ) L z ( E ) ] R ( E ) d E 0 [ L x ( E ) + L z ( E ) ] R ( E ) d E ,
P = | Ω , k m ( r , θ ) cos [ ω t + k l ( r , θ ) + ϕ ( k , t ) ] d r r d θ d k | 2 .
P = | Ω m ( r , θ ) cos [ ω t + k l ( r , θ ) ] d r r d θ | 2 .
P = | Ω [ 1 ρ ( r , θ ) ] × m ( r , θ ) cos ( ω t + k l ) + ρ ( r , θ ) × m ( r , θ ) cos [ ω t + k l ( r , θ ) ] d r r d θ | 2 .
P = | ( m ( r , θ ) ¯ ρ ( r , θ ) × m ( r , θ ) ¯ ) cos ( ω t + k l ) + ρ ( r , θ ) × m ( r , θ ) cos [ ω t + k l ( r , θ ) ] ¯ | 2 = | ( m ( r , θ ) ¯ cos ( ω t + k l ) + ρ ( r , θ ) × m ( r , θ ) cos [ ω t + k l ( r , θ ) ] ¯ | 2 .
P = m ( r , θ ) ¯ 2 cos 2 ( ω t + k l ) + 2 m ( r , θ ) ¯ cos ( ω t + k l ) × ρ ( r , θ ) × m ( r , θ ) cos [ ω t + k l ( r , θ ) ] ¯ + ρ ( r , θ ) × m ( r , θ ) cos [ ω t + k l ( r , θ ) ] 2 ¯ ,
2 m ( r , θ ) ¯ cos [ ω t + k l ] × ρ ( r , θ ) × m ( r , θ ) cos [ ω t + k l ( r , θ ) ] ¯ m ( r , θ ) ¯ × ρ ( r , θ ) × m ( r , θ ) cos [ k l k l ( r , θ ) ] ¯ ρ ( r , θ ) × cos [ k l k l ( r , θ ) ] ¯ ,
E ( t ) = n 1 = n 2 = n k = J n 1 ( z 1 ) J n 2 ( z 2 ) J n k ( z k ) × cos [ { ω ( 1 + a o ) + i = 1 k n i i ω m } t + i = 1 k n i θ i ] ,

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