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Ruby Crystal for Demonstrating Time- and Frequency-Domain Methods of Fluorescence Lifetime Measurements

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We present experiments that are convenient and educational for measuring fluorescence lifetimes with both time- and frequency-domain methods. The sample is ruby crystal, which has a lifetime of about 3.5 milliseconds, and is easy to use as a class-room demonstration. The experiments and methods of data analysis are used in the lab section of a class on optical spectroscopy, where we go through the theory and applications of fluorescence. Because the fluorescence decay time of ruby is in the millisecond region, the instrumentation for this experiment can be constructed easily and inexpensively compared to the nanosecond-resolved instrumentation required for most fluorescent compounds, which have nanosecond fluorescence lifetimes. The methods are applicable to other luminescent compounds with decay constants from microseconds and longer, such as transition metal and lanthanide complexes and phosphorescent samples. The experiments, which clearly demonstrate the theory and methods of measuring temporally resolved fluorescence, are instructive and demonstrate what the students have learned in the lectures without the distraction of highly sophisticated instrumentation.

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  1. 1.

    Any repetitive function can be expanded in a Fourier series. The expression for this series expansion can be expressed either in real form \(y(t) = A_0 /2 + \sum\nolimits_{m = 1}^\infty {(A_m \,\cos \,m\omega _0 t + B_m \,\sin \,m\omega _0 t)} \), or complex form: \(y(t) = \sum\nolimits_{m = - \infty }^\infty {C_m \,\exp (im\omega _0 t)} \). ω0 is the fundamental frequency (the period is \(T = {{2\pi } \mathord{\left/ {\vphantom {{2\pi } {\omega _0 }}} \right. \kern-\nulldelimiterspace} {\omega _0 }}\)), \(m\) is an integer. \(1/2A_0 ,A_m \;{\rm and}\;B_m \) are the DC-offset and the mth sine and cosine component amplitudes, and \(C_m = (A_m^2 + B_m^2 )^{1/2} \).

  2. 2.

    The Fourier expansion of a pure square wave of period T and minimum and maximum amplitude 0 and A is \(y(t)_{{\rm Sq}\,{\rm Wave}} = A/2 + 2A/\pi \sum\limits_{\quad\scriptstyle m = {\rm odd} \hfill \atop \scriptstyle {\rm positive}\;{\rm intergers} \hfill} {\frac{1}{m}\sin \left( {\frac{{m2\pi t}}{T}} \right)} \)

  3. 3.

    The frequency dependence of each component is described in complex notation by \(A_s (i\omega ) = A_{_{s,0} } /(1 + i\tau _s \omega )\). If there are S different lifetimes, for every frequency ω1, the equations corresponding to Equations 5 and 6 are: \({\rm Modulation} = [ {( {\sum\limits_{s = 1}^{s = S} {\frac{{\alpha _s }}{{1 + (\omega _1 \tau _s )^2 }}} } )^2 + ( {\sum\limits_{s = 1}^{s = S} {\frac{{(\alpha _s \omega _1 \tau _s )}}{{1 + (\omega _1 \tau _s )^2 }}} } )^2 } ]^{1/2}\), and \(\Phi _{{\rm F},\omega _1 } = \tan ^{ - 1} ( {{{\sum\limits_{s = 1}^{s = S} {\frac{{(\alpha _s \omega _1 \tau _s )}}{{1 + (\omega _1 \tau _s )^2 }}} } \mathord{/ {\vphantom {{\sum\limits_{s = 1}^{s = S} {\frac{{(\alpha _s \omega _1 \tau _s )}}{{1 + (\omega _1 \tau _s )^2 }}} } {\sum\limits_{s = 1}^{s = S} {\frac{{\alpha _s }}{{1 + (\omega _1 \tau _s )^2 }}} }}} \kern-\nulldelimiterspace} {\sum\limits_{s = 1}^{s = S} {\frac{{\alpha _s }}{{1 + (\omega _1 \tau _s )^2 }}} }}} )\), where α i is the fractional amplitude of the ith frequency component at low frequencies, and \(\sum\nolimits_s {\alpha _s } = 1\).

  4. 4.

    The digital transform of a repetitive function can be calculated easily from the data points in one period, if the points are equally spaced. Call the kth data point D k . Then form the following summations: \(F_{\sin } = \sum\nolimits_{k = 1}^K {D_k \sin (\theta _k )}\), \(F_{\cos} = \sum\nolimits_{k = 1}^K D_k\cos(\theta_k), \sum\nolimits_{k=1}^n X^2_i\), and \(F_0 = 1/K\sum\nolimits_{k = 1}^K {D_k} \); \(\theta _k = 2\pi k/K\), and K is the total number of points in one period. The modulation and phase of the fundamental frequency component is the easily calculated. Defining \(F_\omega = (F_{\sin }^2 + F_{\cos }^2 )^{1/2}\), we have \(M_{F,\omega } = F_\omega /F_0 \) and \(\Phi _\omega = \tan ^{ - 1} (F_{\sin } /F_{\cos } )\). This can be extended to higher harmonics.

  5. 5.

    The power spectrum of a repetitive signal is related to the squares of the Fourier coefficients. Let the Fourier decomposition of the time dependent signal be \(y(t) = A_0 /2 + \sum\nolimits_{m = 1}^\infty {(A_m \cos \,m\omega _0 t + B_m \sin \,m\omega _0 t)} \) ; see Footnote 1. The total power in the mth component is then defined as \(P_y (m\omega _0 ) = 1/2(A_m^2 + B_m^2 )\). The last expression plotted versus \(\log (m\omega _0 )\) constitutes the power spectrum.


  1. 1.

    Cubedda R, Comelli D, D’Andrea C, Taroni P, Valenti G (2002) Time resolved fluorescence imaging in biology and medicine. J Phy D: Appl Phy 35:R61--R76

  2. 2.

    Schneider P, Clegg R (1997) Rapid acquisition, analysis, and display of fluorescence lifetime-resolved images for real-time applications. Rev Sci Instrum 68(11):4107–4119

  3. 3.

    Redford GI, Clegg RM (2005) In: Periasamy A, Day RN (eds) Molecular Imaging: FRET Microscopy and Spectroscopy. Oxford University Press, New York, pp 193–226

  4. 4.

    Clegg RM, Schneider PC (1996) In: Slavik J (ed) Fluorescence Microscopy and Fluorescent Probes. Plenum Press, New York, pp 15–33

  5. 5.

    Clegg RM, Holub O, Gohlke C (2003) Fluorescence lifetime-resolved imaging: measuring lifetimes in an image. Methods Enzymol 360:509–542

  6. 6.

    Gratton E, Jameson DM, Hall R (1984) Multifrequency phase and modulation fluorometry. Ann Rev Biophys Bioeng 13:105–124

  7. 7.

    Goldman S (1948) Frequency analysis, modulation, and noise. Dover, New York

  8. 8.

    Willison JR (1985) Lock-in amplifiers and measurement techniques. Lasers and Applications March, 73–76

  9. 9.

    Hieftje GM (1972) Signal-to-noise enhancement through instrumental techniques. Part I. Signals, noise, and S/N enhancement in the frequency domain. Anal Chem 44(6):81A–88A

  10. 10.

    Hieftje GM (1972) Signal-to-noise enhancement through instrumental techniques. Part II. Signal averaging, boxcar integration, and correlation techniques. Anal Chem 44(7):69A–78A

  11. 11.

    Billington C (1960) Phosphorescence mechanisms. I. Approach and general analysis. Phys Rev 120(3):697–701

  12. 12.

    Billington C (1960) Phosphorescence mechanisms. II. Description of phosphorometer. Phys Rev 120(3):702

  13. 13.

    Billington C (1960) Phosphorescence mechanisms. III. Method of analysis. Phys Rev 120(3):708–709

  14. 14.

    Billington C (1960) Phosphorescence mechanisms. IV. Decay rate spectrum of ruby, uranium glass, and mn-activated zinc sulfide. Phys Rev 120(3):710-714

  15. 15.

    Aizawa H, Uchiyama H, Katsumata T, Komuro S, Morikawa T, Ishizawa H, Toba E (2004) Fibre-optic thermometer using sensor materials with long fluorescence lifetime. Meas Sci Technol 15:1484–1489

  16. 16.

    Temple PA (1975) An introduction to phase sensitive amplifiers: An inexpensive student instrument. Am J Phys 43(9):801–807

  17. 17.

    Scofield JH (1994) Frequency-domain description of a lock-in amplifier. Am J Physics 62(2):129–133

  18. 18.

    Alcala JR, Liao S-C, Zheng J (1996) Real time frequency domain fiberoptic temperature sensor using ruby crystals. Mod Eng Phys 18(1):51–56

  19. 19.

    Birks JB (1970) Photophysics of aromatic molecules. Wiley, London

  20. 20.

    Parker CA (1968) Photoluminescence of solutions. Elsevier, Amsterdam

  21. 21.

    Cherry RJ (1979) Rotational and lateral diffusion of membrane proteins. Biochim Biophys Acta 559:289–327

  22. 22.

    Zidovetzki R, Yarden Y, Schlessinger J, Jovin TM (1981) Rotational diffusion of epidermal growth factor complexed to cell surface receptors reflects rapid microaggregation and endocytosis of occupied receptors. Proc Natl Acad Sci USA 6981–6985

  23. 23.

    Vanderkooi JM, Maniara G, Green TJ, Wilson DF (1987) An optical method for measurement of dioxygen concentration based upon quenching of phosphorescence. J Biol Chem 262(12):5476–5482

  24. 24.

    Mik EG, Donkersloot C, Raat NJH, Ince C (2002) Excitation pulse deconvolution in luminescence lifetime analysis for oxygen measurements in vivo. Photochem Photobiol 76(1):12–21

  25. 25.

    Holst G, Kohls O, Klimant I, Koenig B, Kuehl M, Richter T (1998) A modular luminescence lifetime imaging system for mapping oxygen distribution in biological samples. Sensors Actuators B 51:163–170

  26. 26.

    Bell JH, Schairer ET, Hand LA, Mehta RD (2001) Surface pressure measurements using luminescent coatings. Annu Rev Fluid Mech 33:155–206

  27. 27.

    Parker D, Senanayake PK, Williams JAG (1998) Luminescent sensors for pH, pO2, halide and hydroxide ions using phenanthridine as a photosensitiser in macrocyclic europium and terbium complexes. J Chem Soc Perkin Trans 2:2129–2139

  28. 28.

    Marriott G, Clegg R, Arndt-Jovin D, Jovin T (1991) Time resolved imaging microscopy. Biophys J 60:1374–1387

  29. 29.

    Selvin P, Hearst J (1994) Luminescence energy transfer using a terbium chelate: Improvements on fluorescence energy transfer. Proc Natl Acad Sci USA (91):10024–10028

  30. 30.

    Silfvast WT (1996) Laser fundamentals. Cambridge University Press, New York

  31. 31.

    Davis CC (1996) Lasers and electro-optics: Fundamentals and engineering. Cambridge University Press, Cambridge

  32. 32.

    Butz T (2006) Fourier transformation for pedestrians. Springer-Verlag, Berlin

  33. 33.

    Bracewell RN (1965) The Fourier transform and its applications, 2nd ed, McGraw-Hill, Kogakusha Ltd., Tokyo

  34. 34.

    Hamming RWH (1989) Digital filters, 3rd ed, Dover Publications, Inc., Mineola

  35. 35.

    Dern H, Walsh JB (1963) In: Nastuk WL (ed) Physical techniques in biological research; Electrophysiological methods Part B. Academic Press, New York, pp 99–217

  36. 36.

    Wolfson R (1991) The lock-in amplifier: A student experiment. Am J Phys 59(6):569–572

  37. 37.

    Brigham EO (1988) The fast Fourier transform and its applications. Prentice Hall, Englewood Cliffs

  38. 38.

    Lakowicz JR (1999) Principles of fluorescence spectroscopy, 2nd ed, Kluwer Academic, New York

  39. 39.

    Hamming RWH (1973) Numerical methods for scientists and engineers. Dover Publications, Inc., New York

  40. 40.

    Millman J, Halkias CC (1972) Integrated Electronics: Analog and digital circuits aqnd systems, 1st ed, McGraw-Hill Book Company, New Your

  41. 41.

    Delaney CFG (1969) Electronics for the physicist. Penguin Books Inc., Baltimore

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We thank the students of the Physics 590OS optical spectroscopy class for their enthusiastic participation, Ulai Noomnarm and Chittanon Buranachai for reading the manuscript and giving feedback, Jack Bopari, the director of the undergraduate teaching laboratories in the Physics department, for his generosity loaning equipment and the ruby sample, Eugene Colla for assistance assembling Setup 1 for partial fulfillment of the senior thesis project of DEC, and RMC thanks Gerard Marriott for his enjoyable participation years ago when the idea of listening to phosphorescence lifetimes was born.

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Correspondence to Robert M. Clegg.

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Chandler, D.E., Majumdar, Z.K., Heiss, G.J. et al. Ruby Crystal for Demonstrating Time- and Frequency-Domain Methods of Fluorescence Lifetime Measurements. J Fluoresc 16, 793–807 (2006). https://doi.org/10.1007/s10895-006-0123-7

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  • Fluorescence
  • Luminescence
  • Lifetimes
  • Time-domain
  • Frequency-domain
  • Phosphorescence
  • Ruby
  • Phase
  • Modulation
  • Lockin