Pumping Rate and Pumping Efficiency:

We assume the diode laser electrical power to be P_p, this power is transformed into laser incident power P_pi,

P_pi=k_r k_t P_p

Where k_r is the diode radiative efficiency, k_t is the efficiency of pump transfer system. Also we assume the incident light is of Gaussian distribution, the light density is:

I_p(r,z) = I_p(0,0) exp[-(2r²/w_p²)] exp(-a z)

Where I_p(0,0) is the peak intensity at the entrance face of the active medium, z is the resonator axis, r is the transverse distance from the axis, w_p is the incident light spot size, a is the energy absorption coefficient.

For longitudinal diode laser pumping, clearly P_pi is related to I_p(r,0) by the following equation:

Integrate this equation then we get:

I(0,0)=2P_pi/p w_p²=2 k_r k_t P_p/p w_p²

Pumping rate is the number of atoms being raised from normal state to excited state per unit time per volume. So we have:

R_p(r,z)= a I_p(r,z)/h n_p

Where n_p is the pumping laser frequency, a is the absorption of pumping light.

Using the above relations we get:

R_p(r,z)= k_r k_t (P_p / hn _p)(2a / p w_p²) exp[-(2r²/w_p²)] exp(-a z)

It can be shown that, as far as the threshold condition is concerned, the effective pump rate of a mode is the average of R_p over the distribution field. We draw the conclusion:

<R_p>= k_r k_t (P_p / hn _p){2[1-exp(-a L)] / [p (w₀² + w_p²)L]}

Where L is the length of the laser rod, w₀ is the spot size at the beam waist. As a good approximation to the optimum condition, we can take w₀=w_p. The above equation gives the effective pumping rate for longitudinal laser pumping. If we define k_a=1-exp(-a L), and k_p= k_rk_tk_a, we can write <R_p> in a more convenient form:

<R_p>= k_p(P_p / hn _p){2 / [p (w₀² + w_p²)L]}

For transverse pumping, we have:

Where hn is the incident laser energy, R_p is the pumping rate, V is the active medium volume, k_a is the incident power absorption coefficient. Following similar computation as longitudinal pumping, using the simplified model that the laser active medium is within 0<r<a, we get:

<R_p>= k_p(P_p / hn _p)[1-exp(-2a²/ w₀²)] / (p a²L)

Where k_p= (k_r k_t k_a), L is the length of the active medium.

For comparison purposes, we list the average pump rate of normal lamp light pumping:

<R_p>= k_pl(P_p / hn _mp)[1-exp(-2a²/ w₀²)] / (p a²L)

Where k_pl= (k_r k_t k_a k_pq); k_pq is the energy quantum efficiency, which is the fraction of absorbed energy actually used to create population inversion between energy levels p and q, k_pq= (h n_mp)/(h n_p), n_mp is the minimum pump frequency for pumping energy to be absorbed, n_p is the actual pump frequency. The other parameters have the same meaning as diode laser pumping.

Longitudinal diode laser pumping threshold pumping power:

Transverse diode laser pumping threshold pumping power:

Normal lamp pumping threshold pumping power:

The meaning of the parameters involved in the above equations:

g is the single trip logarithmic loss of the laser resonator cavity;

s is the effective absorption cross section area of the active medium;

k_p= k_rk_tk_a, k_a=1-exp(-a L), k_r is the diode radiative efficiency, k_t is the efficiency of pump transfer system. a is the active medium absorption coefficient, L is the length of the active medium;

k_pl= k_rk_tk_ak_pq, k_pq is the energy quantum efficiency, which is the fraction of absorbed energy actually used to create population inversion between energy level p and q, k_pq= (hn _mp)/(hn _p), n _mp is the minimum pump frequency for pumping energy to be absorbed, n _p is the actual pump frequency, h is the Planck’s constant;

w₀ is the beam spot size at the waist, w_p is the pump beam spot size, a is the range within which active medium species were doped.

Comparison

Now we are ready to compare laser pumped lasers with lamp pumped lasers.

First, we compare their overall pump efficiency.

For lamp pumping, k_all= k_r k_t k_a k_pq. For diode laser pumping, after we add the k_pq which is the energy quantum efficiency, we also get k_all= k_rk_tk_ak_pq. The radiation efficiency k_r and energy transfer efficiency k_t are almost equal for lamp and laser pumping, but the absorption efficiency k_a for diode laser pumping is about 0.90~0.98, k_a for lamp pumping is about 0.17, so (k_a)_diode is about (5.3~5.76) times of (k_a)_lamp, and the energy quantum efficiency k_pq for diode laser pumping is about 1.4 times as large as lamp pumping, with typical value of 0.82 and 0.59 respectively. So the overall pumping efficiency of diode laser is about 7~8 times that of lamp pumping.

Next we compare their threshold pumping power.

We can divide the equation of P_th for diode laser pumping and lamp pumping to find out why there is such a big difference between them. Besides the threshold power reduction (about 10 times) for high pumping efficiency, the threshold power reduction is further reduced by a factor of (w₀² + w_p²) [1-exp(-2a²/ w_0l²)]/ 2a², w₀ is the laser spot size in the medium and w_p is the laser spot size at the surface of pumping interface, both are far less than the active species distribution range a; w_0l is the spot size for lamp pumping, which is about half of a. Let’s put into typical values, w₀=w_p=2 micron for a fiber coupled diode laser pumping, a=2mm, w_0l=1mm, the above factor is about 10^-6! How striking!

We can also find that the threshold pump power for longitudinal pumping is about half that of transverse pumping.

Do you see another big advantage as a consequence of the above results? For normal pumping processes, because of the low efficiency of pumping and the required high pumping power to maintain proper power output, a large fraction of the pumping power is wasted as harmful heat. This heat has to be properly removed, i.e., the laser has to be properly cooled to maintain proper working conditions. We see this when we study CO₂ lasers. While for diode pumped lasers, much of the absorbed power is used for final population inversion, the ratio of thermal generation from the absorbed radiation power for diode laser pumping is half that for lamp pumping. Remember the power required for diode pumping is far less than the lamp pumping, the absolute value of thermal burden of diode laser pumping is also strikingly small compared with lamp pumping. This makes it possible for more compact laser designs.

Example: The radiation efficiency, energy transfer efficiency, energy absorption efficiency and quantum efficiency for arc lamp pumping are 0.44, 0.82, 0.17 and 0.60 respectively, while the values for diode longitudinal pumping are 0.5, 0.8, 0.98, 0.82 respectively. Compute the overall pumping efficiency for lamp pumping and diode pumping.

Solve: Kall = Kr * Kt * Ka *Kpq

For lamp pumping, Kall = 0.44*0.82*0.17*0.6 = 3.68%

For diode longitudinal pumping, Kall = 0.5*0.8*0.98*0.82 = 32.144% =8.73 * 3.68 %

Diode laser pumping has other features: the power supply is of lower voltage and thus of small size; the maintenance cost is reduced because of the life time of diode lasers are becoming longer and longer; the life cycle costs are dropping rapidly with the development of diode laser techniques. Diode lasers can transfer laser light using fibers, the result laser beam can be diffraction limited. This technique makes laser diode pumping very flexible. More advantages can be added--the pumping powers are in rapid progressing, the diode lasers can be mass produced, etc.

That’s why diode pumped lasers are expanding their market rapidly. You can see the gallery for diode lasers and diode laser pumped laser systems.

G 2.11: A diode laser pumped lasers (courtesy of Thomson Components and Tubes Corp.)