Differential Spectral Responsivity Method 17

An alternative approach that overcomes both, daily spectral variations and imperfect replication of the solar spectrum, is represented by a spec-trally resolved method. As the short circuit current generated by a solar cell with spectral responsivity s(λ) under spectral irradiance E_λ(λ) is given by

ISC= Z

s(λ)Eλ(λ) dλ, (2.24)

theI_SC^STCis obtained from I_SC^STC=

Z

s(λ)ESTC(λ) dλ. (2.25) SinceESTC(λ) is a standardized solar spectrum, onlys(λ) contributes to the uncertainty of I_SC^STC, stressing the importance of providing accurate

s(λ) measurements for successful application of this spectrally resolved method.

The most widely spread and accepted approach for measurings(λ) at high accuracy is the differential spectral responsivity (DSR) method [14].⁷ In the DSR-method the solar cell under test is simultaneously illuminated by chopped monochromatic radiation (∆E(λ)) and steady broadband bias irradiation (E_bias).⁸ Measuring both the steady (dc) current response to the bias irradiation (j_b=j_SC(E_bias)) and the differential (ac) current response to the chopped monochromatic radiation (∆jSC(Ebias)) reveals the test device’s differential spectral responsivity

˜ at wavelength λ and dc current level jb. Assessing the bias irradiation level via jb is important, as the differential spectral responsivity evalu-ated by ∆jSC(Ebias) might vary with the current cell condition. In other words, the currently present carrier injection density induced byE_biasand evaluated byjb might affect the differential cell response. This injection dependence of differential measurands can be assessed by varying Ebias

in a DSR-setup and taking readings of ˜s(λ, j_b=j_SC(E_bias)) at each bias level.

In Fig. 2.2(a) an example for bias dependency is shown by plottingib

overEbias. For ideally linear solar cells, the slope of the curve is invariant ofE_bias. Thus, the differential response of the cell, ˜s(λ, j_b=j_SC(E_bias)), is invariant of Ebias (and, therefore, independent of changes of injection

7The subsequent discussion of the DSR-method follows most closely that one pre-sented byWinterin [24].

8The broadband bias irradiation injects an excess carrier density to the solar cell, thereby setting the test cell to a specific operating condition. For obtaining carrier generation profiles similar to natural conditions a spectral distribution similar to the AM1.5g is preferable. However, this spectral constraint is much less demanding than for solar simulators (see Section 2.2.2.1) and e.g. tungsten halogen lamps (equivalent to a Class C solar simulator according to IEC 60904-9, Ed. 2) are sufficiently similar in practically all cases.

Figure 2.2: Schematic representation of DSR-measurement outcomes for a linear (black line) and a nonlinear (red line) solar cell. (a) Bias dc currentjb plotted over bias irradiation levelEbias, (b) STC-weighted slopes of (a) according to Eq. (2.27). The reddish area in (b) representsI_SC^STCand illustrates the integration according to Eq. (2.28) withESTCas upper integration boundary.

carrier density orjb) and the trivial cases(λ) = ˜s(λ) is obtained. Conse-quently, the correspondingI_SC^STCcan be readily computed from Eq. (2.25) with ˜s(λ).

However, solar cells generally exhibit a more or less pronounced non-linearity that is e.g. being caused by injection-dependent carrier lifetime effects (see Section 2.1.3). In these cases the differential cell response varies with bias irradiation level (see red line in Fig. 2.2(a)), thereby making above’s trivial identity of differential and absolute responsivity invalid. Instead, a specific bias current level or cell operating condition has to be identified that corresponds to STC.

For identifying that bias level, the measured differential spectral re-sponsivities from Eq. (2.26) are weighted by the STC solar spectrum ac-cording to depen-dency for irradiation under STC (see Fig. 2.2(b)), a simple integration alongEbiaswould yield

I_SC^STC=

However, in practical applications the varying bias irradiation level Ebias is not known or rather difficult to assess. On the contrary, the dc current generated by the bias light, jb, is readily available from dc current measurements. Therefore, instead of integrating alongEbiasas in Eq. (2.28), it is rather integrated alongj_b with

Ex= The upper integration boundaryIx that yieldsEx =ESTC= 1000 W/m² is equivalent to theI_SC^STC.

Finally, the spectral responsivitys(λ) is obtained from

s(λ) =I_SC^STC

The DSR-method is implemented into the laser-based measurement fa-cility that has been developed in this work and is applied for measurements presented in Chapters 6 to 8.

Ultrashort Laser Pulses &

Nonlinear Optics

In this chapter the theoretical background on ultrashort pulses and nonlinear optics will be given that is relevant for the re-sults and discussions in the remainder of this work. Starting from a brief introduction on the basic principles of ultrashort pulses, the origin of nonlinear optical effects and a theoretical framework for describing them will be given. This will be used subsequently to outline some second and third order nonlin-ear optical effects, followed by a short introduction to phase-matching. Finally, the principle of supercontinuum generation from ultrashort laser pulses and photonic crystal fibers will be sketched concisely.

3.1 Basic Principles

In the course of this work a measurement system for solar cell spectral responsivities is developed that is being based on ultrashort laser pulses as illumination source. Owing to its quasi-monochromaticity this radia-tion source provides a significantly higher spectral power as available from typically used conventional white light sources (xenon or halogen lamps),

23

thereby enabling considerable reductions in measurement uncertainties (see Chapter 6). Naturally, the high spectral powers required for such a purpose are also available from continuous wave (cw) or Q-switched¹ lasers. However, the spectral range and availability of lasers is intrinsi-cally limited by the applied (gain) materials impeding their usage in the new measurement system that requires spectral tunability from the UV (approx. 300 nm) to the NIR (1200 nm or higher).

One way to overcome this bottleneck is to apply nonlinear optical ef-fects for generating radiation in the entire spectral range. Ultrashort pulses are especially advantageous in this context, as their extremely high pulse peak powers allow for efficient nonlinear optical processes (see Sec-tion 3.2) at moderate pulse energies and average powers (thus, in a quasi-cw² manner). For instance, a laser emitting ultrashort pulses of 100 fs full-width-at-half-maximum (FWHM) pulse duration at 80 MHz repeti-tion rate achieves a pulse peak power of more than 110 kW at an average output power of 1 W. In this section the basic principle of ultrashort laser pulse generation and typical properties of these pulses, that are important for the remainder of this work, will be briefly outlined. Readers interested in a detailed discussion of ultrashort pulses are referred to the textbook Ultrafast Optics byAndrew M. Weiner [25].

The underlying principle of ultrashort pulse generation is the superpo-sition of longitudinal resonator modes of different frequencies resulting in pulse durations that are essentially limited by the transform limit accord-ing to [26, p. 10]

τFWHM∆ω≥2πcTBP, (3.1)

with FWHM pulse durationτ_FWHM, FWHM frequency bandwidth ∆ω= 2π∆νand the minimum possible time-bandwidth-productcTBPthat takes different values depending on the temporal pulse shape. E.g. for a Gaus-sian pulse (cTBP = 0.441) withτFWHM = 100 fs a frequency bandwidth

1In Q-switching a repetitive variation of the cavity losses by means of mechanical or electro-optical shutters is used to achieve a very high population inversion in the laser gain medium (for closed shutters) that is depleted when the losses are turned off, thereby (typically) emitting ns-pulses.

2The importance of high repetition rates (quasi-cw operation) arises from the de-sired temporal continuity of the radiation used for the solar cell measurements and is addressed in Chapters 4 and 5.

of ∆ω≥2.77·10¹³ Hz (corresponding to ∆λ≥9.4 nm at 800 nm center wavelength) is required. In a laser cavity such ultrashort pulses are gen-erated bymode-locking longitudinal resonator modes so that they have a fixed phase relation to each other and constructively interfere in a periodic manner. Assuming a typical longitudinal mode spacing of 80 MHz (corre-sponding to approx. 1.9 m cavity length) the generation of an ultrashort pulse as given above requires more than 50.000 mode-locked longitudinal laser modes.

In practice, this mode-locking can be achieved by e.g. active or passive modulation of the laser resonator gain or loss. In active mode-locking, for example, an acousto-optic modulator (AOM) can be used to sinusoidally modulate the cavity losses at a frequency identical to the inverse of the cavity roundtrip time. Thereby longitudinal modes that are in phase with the modulation frequency are favored establishing mode-locking of all longitudinal modes. In analogy to the AOM, a saturable absorber can also be applied that bleaches with increasing intensity. As the cavity losses reduce with increased absorber transmission (thus, for a bleached absorber), the laser favors an operational mode where all longitudinal modes are in phase, thus mode-locked. However, as the losses are not actively controlled by the saturable absorber, this technique is referred to as passive mode-locking. Please note, that this is only a very short digest on mode-locking techniques. For a detailed discussion of mode-locking techniques and their theoretical treatment it is referred to [25, pp. 32-84].