Next, we calculate the light-power ratio between two sources required for the emission of white light and the luminous efficiency of the photon-recycling semiconductor LED. We refer to X1 and X2 as the primary (short) and secondary (long) wavelength, respectively. For white light, X1 and X2 are pairs of complementary wavelengths. We define the color masses of the two light sources as

and

(21.6)

m = xx + yx + zx

m2 = x2 + y2 + z2

where xt, y1, z1, x2, y2, and z2 are color-matching functions at the two emission wavelengths X1 and X2, respectively (Judd, 1951; Vos, 1978; MacAdam, 1950, 1985). We define the power ratio of the two light sources as

(21.7)

R = P2 / P1

where P1 and P2 are the optical powers of the short-wavelength source (X1) and the long - wavelength source (X2), respectively. The chromaticity coordinates of the newly generated color are then given by

(21.8)

ynew

mx + Rm2

= P1 y1 + P2 y2 = y1 + Ry2

Px mx + P2 m2

and

(21.9)

new

xx + Rx2 mx + Rm2

For a white-light emitter, xnew and ynew can be chosen to coincide with the chromaticity coordinates of the Illuminant C standard (xc = 0.3101, yc = 0.3162; CIE, 1932; Judd, 1951), i. e. xnew = xc = 0.3101 and ynew = yc = 0.3162. Solving Eq. (21.9) for the power ratio R yields

R = Уі - Ус m. (21.10)

Ус m2 - y2

The power ratio as calculated from Eq. (21.10) is shown as a function of wavelength in Fig. 21.15.

Fig. 21.15. Calculated power ratio between the two optical output powers P, and P-, required to ob­tain white-light emission (after Guo et al., 1999).

380 400 420 440 460 480 500 Short wavelength Xj (nm)

21.8 Calculation of the luminous efficiency of PRS-LED

To produce the optical power P2 at the wavelength of X2 through the recycling of photons from the primary source with wavelength Х1з the optical power required from the primary source is given by

(21.11)

P2 x2 hc = P2 X2

П2 hc ^ П2 X1

where n2 is the optical-to-optical conversion efficiency of the photon-recycling light source. If P0 is the electrical input power, then the optical power emitted by the primary LED source is n^P0, where n1 is the electrical-to-optical power conversion efficiency of the primary LED. Thus, the optical power emitted by the primary LED is given by

P + - PLrL = П1 Po . (21.12)

П2 X1

Solving the equation for the electrical input power and using P2 = R P1 yields

1

(21.13)

+

П1П2 X

1J

П1

P0 = P

The total optical output power of the PRS-LED is given by

(21.14)

Pout = P1 + P2 = (1 + R) P1

so that the overall electrical-to-optical power efficiency of the photon-recycling dichromatic light source is given by

Po

1+R

out

(21.15)

П =

P0

R

1

1 П1

+

P1

П1П2 X1

П1

П1П2 X1J

P1(1 + R)

R

+

The luminous flux Olum of the device is given by

(21.16)

ФЬш = 683 W СУ1P1 + y2 P2) = 683 "W + y2 R)P1 .

W W

Then the luminous efficacy of radiation (measured in lumens per optical watt) of the photon - recycling semiconductor LED is given by

Ф

lum

(21.17)

Po

out

= 683 lm y1 + y2R

W 1 + R

Thus, the luminous efficiency of the source (measured in lumens per electrical watt) of the PRS - LED is given by

(21.18)

-й™ = 683 — n y1 + y2R

P0 ^1 + R

Using this formula, we calculate the luminous efficiency as a function of the primary wavelength. The result of the calculation is shown in Fig. 21.16 for ideal sources, i. e. for n1 = П2 = 100%.

The maximum luminous efficiency occurs if the primary source emits at the wavelength 440 nm. A theoretical luminous efficiency of 336 lm/W is obtained for this wavelength. Note that in the calculation we assume that both light sources emit monochromatic light. However, the spontaneous emission from semiconductors has a 1.8 kT spectral width. Taking into account a