平衡光偵測技術，是經由將來自相同光源的、功率相等的兩道光的光訊號相減，以取得其間的相對訊號值的一種技術。經由相減的動作，可以消除兩組光訊號所具有的共模雜訊，而能偵測到很小的差動訊號。但是使光訊號能維持平衡並不容易，因此常不容易達到預期的消除雜訊的效果。
自平衡光偵測電路是在1990年由Hobbs發明，用以改善上述的困難。Hobbs對其消除雜訊的能力作了許多探討，發現可以接近理論預期的散粒雜訊極限(shot noise limit)，而消除雜訊效果與兩道光的功率比值有關。
我們在這個論文研究中，應用自平衡光偵測技術，嘗試測量空氣的兩種磁光偏振效應法拉第效應，以及Cotton-Mouton效應，探討影響測量靈敏度的因素。
我們對螺線管加上驅動電壓，產生頻率10 kHz、振幅1.31 mT的交流軸向磁感應，以產生空氣法拉第效應。再應用自平衡光偵測器作測量，發現可以測量到訊號，而靈敏度達到2.99×10^-8 rad Hz^-1/2，約是二極體雷射測試光束(testing beam)的散粒雜訊極限的2.7倍。由此確認自平衡光偵測器消除雜訊的能力。
由於在測量時，除了考慮降低雜訊，也必須考慮訊號的大小。我們因此採用靈敏度作為最佳化分光比的依據：對於測量相同大小的空氣法拉第旋轉，不同的分光比設定，會得到不同的訊雜比，也就對應到不同的靈敏度。我們發現最高靈敏度的設定，是在兩道光功率相等，也就是在分光比為1。在這個實驗中，我們使用功率較小的氦氖雷射光源，但在分光比為1時，靈敏度可以達到3.02×10^-8 rad Hz^-1/2，約是測試光束散粒雜訊極限的1.3倍。
我們也探討訊號放大的效果。我們將產生法拉第效應的線圈置於共振腔內作測量實驗，發現旋轉角度放大符合理論模型，而靈敏度達到7.54×10^-10 rad Hz^-1/2，與目前文獻中，Jacob等人測量空氣法拉第旋轉達到的最高靈敏度2.93×10^-10 rad Hz^-1/2比較，只大了約3倍，但是在Jacob等人的實驗中，所用的共振腔的放大倍率是我們的45倍以上。
在嘗試測量空氣Cotton-Mouton效應方面，由於其較法拉第效應小很多，因此要產生相同程度的訊號，所需要的磁場必須更大。我們是採用永久磁鐵提供所需要的橫向磁場，其中橫向磁場達到0.8 Tesla以上的長度有3 cm，然後是以旋轉磁鐵的方式造成低頻的磁場調制。在這裡，嘗試了兩種方法作實驗：其一是使用電光調制器，以較高的頻率調制光束的偏振，再作測量，在這方面，主要是存在著電光調制器造成的雜訊的限制；其二是直接以自平衡光偵測器測量，取其對數輸出(log output)作分析，在這方面，主要是經由各種測試確認存在許多假性訊號(spurious signals)。由於這兩個因素，我們並沒有測量到空氣Cotton-Mouton訊號。

Balanced photodetection is commonly used to detect a relative power change between two beams with equal power and from the same source. The advantage of balanced photodetection is removing the common mode noise in the photonic signals by subtraction. Therefore, the small differential signal could be sensed. The performance of balanced photodetection for measuring small effects has been proven to be very sensitive. However to balance photocurrents is critical and the noise subtraction is not always perfect.
In 1990, Hobbs devised an auto-balanced circuitry to automatically balance the two photocurrents. Hobbs had investigated the noise cancellation capability, and found that the noise floor is close to the shot noise limit. This capability had also been found to depend on the photocurrent ratio of the two beams.
The dissertation reports our studies for the auto-balanced photodetection. We measured the air Faraday effect and air Cotton-Mouton effect with the technology, and studied the measurement sensitivity.
The air Faraday effect was induced by an axial magnetic induction with frequency of 10 kHz and an amplitude of 1.3 mT, which was produced by a solenoid. The effect was measured with an auto-balanced photoreceiver. The signal was found, and the angular sensitivity is 2.99×10^-8 rad Hz^-1/2, which is about 2.7 times the shot noise limit of the testing beam from diode laser.
In measurements, in additional to noise level, we should also consider the signal size. Therefore, we used the signal-to-noise ratio in the same air Faraday rotation measurement as the index to optimize the power split ratio to the best sensitivity. We found the best sensitivity appears at the photocurrent ratio of 1. Particularly, in this experiment, we used He-Ne laser and achieved the best sensitivity of 3.02×10^-8 rad Hz^-1/2, which is about 1.3 times the shot noise limit.
We also studied the performance of cavity enhancement. In this experiment, an optical cavity enclosing the solenoid was built to examine the enhancement effect. At resonance with this configuration, the Faraday rotation was amplified and the sensitivity also improved to 7.54 × 10^-10 rad Hz^-1/2, which agrees well with Jones matrix analysis. This value is only 3 times the best sensitivity of air Faraday rotation measurement, 2.93×10^-10 rad Hz^-1/2, obtained by Jacob et al. using a cavity with more than 45 times larger finesse.
The air Cotton-Mouton effect is smaller than Faraday effect, and was induced by a permanent magnet with transverse magnetic field in our experiment. This transverse magnetic field is larger than 0.8 Tesla within 3 cm. The magnetic field was modulated by rotating the magnet in low frequency. Then, two schemes are used in the experiment. First, we modulated the polarization of the testing beam with electro-optical modulator, and measured the effect with an auto-balanced photoreceiver. The important finding is that the EOM produced noise to limit the sensitivity. Second, we measured the effect directly with log output of the auto-balanced photoreceiver. We proceeded many tests and verified there were some spurious signals with the measurement. Owing to the two issues, we did not find the air Cotton-Mouton signal.

[1] P. S. Pershan, J. Appl. Phys., 38, 1482 (1967).
[2] D. Budker, W. Gawlik, and D. F. Kimball, A. Weis, Rev. Mod. Phys., 74, 1153 (2002).
[3] L. R. Ingersoll, and D. H. Liebenberg, J. Opt. Soc. Am., 44, 566 (1954).
[4] A. D. Buckingham, W. H. Prichard, and D. H. Whiffen, T. Faraday Soc., 63, 1057 (1967).
[5] C. Rizzo, A. Rizzo, and D. M. Bishop, Int. Rev. Phys. Chem., 16, 81 (1997).
[6] A. Cadène, D. Sordes, P. Berceau, M. Fouché, R. Battesti, and C. Rizzo, arXiv: 1307.6394v1 (2013).
[7] E. Hecht, Optics, 4th edition, Sec. 8.11.2 (Addison-Wesley, 2002).
[8] D. E. Gray, American Institute of Physics Handbook, 3rd edition, p. 6-230 ( McGraw-Hill, 1972).
[9] L. H. Siertsema, K. Ned. Akad. Van Wet. B, 1, 296 (1898).
[10] L. H. Siertsema, K. Ned. Akad. Van Wet. B, 2, 19 (1899).
[11] J. F. Sirks, K. Ned. Akad. Van Wet. B, 15, 773 (1912).
[12] L. R. Ingersoll, and D. H. Liebenberg, J. Opt. Soc. Am., 46, 538 (1956).
[13] L. R. Ingersoll, and D. H. Liebenberg, J. Opt. Soc. Am., 48, 339 (1958).
[14] R. Serber, Phys. Rev., 41, 489 (1932).
[15] A. Rizzo, and S. Coriani, Adv. Quantum Chem., 50, 143 (2005).
[16] 國立清華大學普通物理實驗室, 普通物理實驗課程講義, 實驗1 基本度量, http://www.phys.nthu.edu.tw/~gplab/exp001.html.
[17] L. R. Ingersoll, and W. L. James, Rev. Sci. Instrum., 24, 23 (1953).
[18] P. C. D. Hobbs, Proc. SPIE, 1370, 216 (1990).
[19] K. L. Haller, and P. C. D. Hobbs, Proc. SPIE, 1435, 298 (1991).
[20] H. Adams, D. Reinert, P. Kalkert, and W. Urban, Appl. Phys. B, 34, 179 (1984).
[21] C. B. Carlisle, and D. E. Cooper, Opt. Lett., 14, 1306 (1989).
[22] H. Wenz, W. Demtröder, and J. M. Flaud, J. Mol. Spectrosc., 209, 267 (2001).
[23] C. M. Lindsay, R. M. Rade Jr., and T. Oka, J. Mol. Spectrosc., 210, 51 (2001).
[24] M. G. Allen, K. L. Carleton, S. J. Davis, W. J. Kessler, C. E. Otis, D. A. Palombo, and D. M. Sonnenfroh, Appl. Opt., 34, 3240 (1995).
[25] G. Durry, I. Pouchet, N. Amarouche, T. Danguy, and G. Megie, Appl. Opt., 39, 5609 (2000).
[26] P. Vogel, and V. Ebert, Appl. Phys. B, 72, 127 (2001).
[27] R. Claps, F. V. Englich, D. P. Leleux, D. Richter, F. K. Tittel, and R. F. Curl, Appl. Opt., 40, 4387 (2001).
[28] W. A. Challener, R. R. Ollmann, and K. K. Kam, Sens. Actuators B, 56, 254 (1999).
[29] X. Wang, M. Jefferson, P. C. D. Hobbs, W. P. Risk, B. E. Feller, R. D. Miller, and A. Knoesen, Opt. Express, 19, 107 (2011).
[30] G. A. Smith, S. Chaudhury, and P. S. Jessen, J. Opt. B, 5, 323 (2003).
[31] P. Gangopadhyay, R. Voorakaranam, A. Lopez-Santiago, S. Foerier, J. Thomas, R. A. Norwood, A. Persoons, and N. Peyghambarian, J. Phys. Chem. C, 112, 8032 (2008).
[32] Nirvana Auto-Balanced Photoreceivers: Model 2007 & 2017 User’s Manual (New Focus, 2002).
[33] B. E. A. Saleh, and M. C. Teich, Fundamentals of Photonics, 1st edition, p. 203 (John Wily & Sons, Inc., 1991).
[34] B. E. A. Saleh, and M. C. Teich, Fundamentals of Photonics, 1st edition, Sec. 6.6, and Sec. 18.1 (John Wily & Sons, Inc., 1991).
[35] K. Iizuka, Elements of Photonics I: In Free Space and Special Media, Ch 6 (John Wily & Sons, Inc., 2002).
[36] B. E. A. Saleh, and M. C. Teich, Fundamentals of Photonics, 1st edition, pp. 196-197 (John Wily & Sons, Inc., 1991).
[37] B. E. A. Saleh, and M. C. Teich, Fundamentals of Photonics, 1st edition, pp. 223-224 (John Wily & Sons, Inc., 1991).
[38] M. Faraday, Philos. T. R. Soc. Lond., 136, 1 (1846).
[39] E. M. Verdet, C. R. Hebd Acad. Sci., 39, 548 (1854).
[40] O. Knudsen, Arch. Hist. Exact Sci., 15, 235 (1976).
[41] B. E. A. Saleh, and M. C. Teich, Fundamentals of Photonics, 1st edition, pp. 225-226 (John Wily & Sons, Inc., 1991).
[42] H. Becquerel, C. R. Hebd Acad. Sci., 125, 679 (1897).
[43] D. A. V. Baak, Am. J. Phys., 64, 724 (1996).
[44] J. T. Hougen, J. Chem. Phys., 32, 1122 (1960).
[45] A. D. Buckingham, and P. J. Stephens, Annu. Rev. Phys. Chem., 17, 399 (1966).
[46] D. M. Bishop, and S. M. Cybulski, J. Chem. Phys., 93, 590 (1990).
[47] W. A. Parkinson, S. P. A. Sauer, Jens Oddershede, and D. M. Bishop, J. Chem. Phys., 98, 487 (1993).
[48] M. Jaszunski, P. Jørgensen, A. Rizzo, K. Ruud, and T. Helgaker, Chem. Phys. Lett., 222, 263 (1994).
[49] S. Coriani, C. Hättig, P. Jørgensen, A. Halkier, and A. Rizzo, Chem. Phys. Lett., 281, 445 (1997).
[50] A. K. Bhatia, and R. J. Drachman, Phys. Rev. A, 58, 4470 (1998).

[51] M. Krykunov, A. Banerjee, T. Ziegler, and J. Autschbach, J. Chem. Phys., 122, 074105 (2005).
[52] U. Ekströma, P. Norman, and A. Rizzo, J. Chem. Phys., 122, 074321 (2005).
[53] P. C. D. Hobbs, Appl. Opt., 36, 903 (1997).
[54] D. Jacob, M. Vallet, F. Bretenaker, A. L. Floch, and R. L. Naour, Appl. Phys. Lett., 66, 3546 (1995).
[55] C. E. Lin, C. J. Yu, Y. C. Li, C. C. Tsai, and C. Chou, J. Appl. Phys., 104, 033101 (2008).
[56] C. Y. Chang, L. Wang, J. T. Shy, C. E. Lin, and C. Chou, Rev. Sci. Instrum., 82, 063112 (2011).
[57] W. B. Garn, R. S. Caird, C. M. Fowler, and D. B. Thomson, Rev. Sci. Instrum., 39, 1313 (1968).
[58] Magneto-optical glass MR3-2, Verdet constant: -96 rad T-1 m-1 at 632.8 nm, Xi’an Aofa Optoelectronics Technology Inc., http://www.xaot.com/.
[59] J. H. Moore, C. C. Davis, and M. A. Coplan, Building Scientific Apparatus, 3rd edition, p. 523, pp. 531-532, pp. 464-465, and p. 522 (Westview Press, 2002).
[60] Model SR830 DSP Lock-In Amplifier User’s Manual, p. 3-9, and p. 3-17 (Stanford Research Systems, Inc., 2006).
[61] Model SR830 DSP Lock-In Amplifier User’s Manual, p. 3-3 (Stanford Research Systems, Inc., 2006).
[62] J. G. Proakis, Digital Communications, 3rd edition, Sec. 2-2 (McGraw-Hill, Inc., 1995).
[63] D. B. Leeson, Proc. IEEE, 54, 329 (1966).
[64] P. Dutta, and P. M. Horn, Rev. Mod. Phys., 53, 497 (1981).
[65] 黃通隆, 穩頻雷射之研究, 國立中央大學博士論文 (1999).
[66] B. Brumfield, and G. Wysocki, Opt. Express, 20, 29727 (2012).
[67] B. E. A. Saleh, and M. C. Teich, Fundamentals of Photonics, 1st edition, Sec. 11.2 (John Wily & Sons, Inc., 1991).
[68] M. Vallet, F. Bretenaker, A. Le Floch, R. Le Naour, and M. Oger, Opt. Commun., 168, 423 (1999).
[69] R. Rosenberg, C. B. Rubinstein, and D. R. Herriott, Appl. Opt., 3, 1079 (1964).
[70] F. Bielsa, A. Dupays, M. Fouché, R. Battesti, C. Robilliard, and C. Rizzo, Appl. Phys. B, 97, 457 (2009).
[71] J. L. Hall, J. Ye, and L. S. Ma, Phys. Rev. A, 62, 013815 (2000).
[72] RP Photonics Encyclopedia, Finesse, http://www.rp-photonics.com/finesse.html.
[73] A. E. Siegman, Lasers, Sec. 11.2 (University Science Books, 1986).
[74] B. E. A. Saleh, and M. C. Teich, Fundamentals of Photonics, 1st edition, Sec. 9.2 (John Wily & Sons, Inc., 1991).
[75] J. W. Beams, Rev. Mod. Phys., 4, 133 (1932).
[76] R. W. Wood, Physical Optics, 3rd edition, p. 717 (The Macmillan Company, 1934).
[77] A. D. Buckingham, and J. A. Pople, Proc. Phys. Soc. B, 69, 1133 (1956).
[78] D. R. Lide, CRC Handbook of Chemistry and Physics, 88th edition, Sec. 14 (CRC Press, 2008).
[79] K. Cho, S. P. Bush, D. L. Mazzoni, and C. C. Davis, Phys. Rev. B, 43, 965 (1991).
[80] J. M. Liu, Photonic Devices, Ch7 (Cambridge University Press, 2005).
[81] A. Moritani, Y. Okuda, and J. Nakai, Appl. Opt., 22, 1329 (1983).
[82] P. Yeh, J. Opt. Soc. Am., 72, 507 (1982).
[83] C. Gu, and P. Yeh, J. Opt. Soc. Am. A, 10, 966 (1993).
[84] S. Hess, Phys. Lett. A, 30, 239 (1969).
[85] D. Chauvat, A. L. Floch, M. Vallet, and F. Bretenaker, Appl. Phys. Lett., 73, 1032 (1998).
[86] K. Muroo, M. Namikawa, and Y. Takubo, Meas. Sci. Technol., 11, 32 (2000).
[87] M. Bregant, G. Cantatore, S. Carusotto, R. Cimino, F. D. Valle, G. D. Domenico, U. Gastaldi, M. Karuza, E. Milotti, E. Polacco, G. Ruoso, E. Zavattini, and G. Zavattini, Chem. Phys. Lett., 392, 276 (2004).
[88] H. H. Mei, W. T. Ni, S. J. Chen, and S. S. Pan, Chem. Phys. Lett., 471, 216 (2009).