Quantum defect

The quantum defect of an optically pumped laser or amplifier is the energy difference between the pump photon and the emitted photon, expressed as a fraction of the pump photon energy. Each absorbed pump photon that produces one laser photon leaves this difference behind in the host as heat, through the fast non-radiative relaxations above the upper laser level and below the lower one. Ytterbium pumped at 976 nm and lasing at 1030 nm has a quantum defect of 5.2%; Nd:YAG pumped at 808 nm and lasing at 1064 nm, 24%; erbium pumped at 980 nm for 1550 nm amplification, 37%. It sets the minimum heat load of a conventionally operated laser.

Definition and heat load

With pump wavelength $\lambda_p$ and signal or laser wavelength $\lambda_s$,

$$q = 1 - \frac{h\nu_s}{h\nu_p} = 1 - \frac{\lambda_p}{\lambda_s}.$$

If every absorbed pump photon yields one laser photon and no anti-Stokes fluorescence carries heat away, the heat deposited per watt of output is $\lambda_s/\lambda_p - 1$, so

$$P_\text{heat} \ge P_\text{out}\left(\frac{\lambda_s}{\lambda_p} - 1\right).$$

A 1 kW ytterbium fiber laser at 1070 nm pumped at 976 nm ($q$ = 8.8%) therefore deposits at least 96 W in the fiber; a 10 W Nd:YAG laser at 1064 nm pumped at 808 nm deposits at least 3.2 W in the crystal. The same ratio, $\lambda_p/\lambda_s$, is the upper limit on the slope efficiency with respect to absorbed pump power: 0.76 for Nd:YAG at 808 → 1064 nm, before pump absorption and output-coupling fractions reduce it further.

Typical values

Thulium pumped near 793 nm is the exception to the simple rule. At concentrations of a few weight percent, cross-relaxation between neighboring ions can leave two ions in the upper laser level for each absorbed pump photon, which in the ideal case lowers the effective defect to $1 - 2\lambda_p/\lambda_s$, 18% at 1940 nm; real fibers fall between the two limits, as described under thulium fiber laser.

Measured heat versus quantum defect

The quantum defect is computed from the two wavelengths and is not itself measured. What is measured is the fractional heat load, the ratio of heat deposited to pump power absorbed, by calorimetry, by the temperature rise of the medium, or from the strength of its thermal lens. The measured value is usually larger than the quantum defect, because not every excited ion emits a laser photon: non-radiative decay from the upper level, concentration quenching, energy-transfer upconversion and excited-state absorption all add heat. It also depends on whether the laser is running. Below threshold the excited ions decay by fluorescence, and the heat fraction is $1 - \lambda_p/\bar\lambda_f$, with $\bar\lambda_f$ the mean fluorescence wavelength, plus any non-radiative losses; for Nd:YAG part of that fluorescence lies at wavelengths other than 1064 nm, so the heat load changes when lasing starts.

Where it matters

The small quantum defect is one of the main reasons ytterbium is used in most high-power fiber and thin-disk lasers: it means little heat per watt of output, which limits thermal lensing in crystals and, in fibers, the heat per meter that drives thermally induced mode instabilities and coating damage. Tandem pumping carries this further: pumping a 1070 nm ytterbium amplifier with ytterbium fiber lasers at 1018 nm reduces the defect to 4.9%.

A small defect has a cost. A pump close to the laser wavelength means the lower laser level is close to the ground state and is thermally populated, so the medium is quasi-three-level rather than a clean four-level laser: it reabsorbs at the laser wavelength, needs a higher pump intensity to reach transparency, and is sensitive to temperature. Narrow absorption lines, such as ytterbium's 976 nm peak, also call for wavelength-stabilized pump diodes. The 915 nm band of ytterbium is broader and more tolerant of pump drift, at the price of a defect of 14.5% at 1070 nm.

Pump-band choice is covered for ytterbium under ytterbium-doped fiber, and pump geometries under optical pumping.

Common questions

How is quantum defect different from quantum efficiency?

Quantum efficiency is the number of laser photons produced per absorbed pump photon; the quantum defect is the energy lost per photon even when that efficiency is 100%. The ideal optical-to-optical efficiency is their combination, quantum efficiency times $\lambda_p/\lambda_s$.

Can the quantum defect be negative?

The laser photon always has less energy than the pump photon, so q itself stays positive. The heat fraction of fluorescence, $1 - \lambda_p/\bar\lambda_f$, can be negative: when a medium is pumped at a wavelength longer than its mean fluorescence wavelength, each emitted photon on average carries away more energy than the pump photon brought in, and the host cools. This anti-Stokes fluorescence cooling has been demonstrated in ytterbium-doped glasses and crystals, and radiation-balanced lasers use it to offset the heat of lasing; the effect is small and requires very pure hosts.

References: W. Koechner, Solid-State Laser Engineering, 6th ed. (Springer, 2006); A. E. Siegman, Lasers (University Science Books, 1986); T. Y. Fan, "Heat generation in Nd:YAG and Yb:YAG," IEEE Journal of Quantum Electronics 29, 1457 (1993); D. J. Richardson, J. Nilsson and W. A. Clarkson, "High power fiber lasers: current status and future perspectives," Journal of the Optical Society of America B 27, B63 (2010).