1. Introduction

Diode lasers play an increasingly important role in industrial applications and already substitute in some cases conventional solid state or gas lasers. They are efficient, small and easy to handle. However for the most important industrial applications like cutting and welding of metals their beam quality is not sufficient [1]. This is mainly due to the large number of laser diodes and corresponding optical paths used in diode lasers. One intensively studied approach to overcome this is coherent coupling, i.e. the locking of frequency and phase of all lasers. This leads in theory to an increase of possible focus intensity proportional to the number of locked lasers. Many investigators proved the feasibility of this but up to now it was not possible to achieve powers necessary for industrial applications [2]. A further promising method is wavelength multiplexing, where many laser beams with different wavelengths are multiplexed [1]. An entirely different approach is the use of time multiplexing, i.e. temporally separated laser pulses are produced by different laser diodes and the laser pulses are guided onto a common optical path [3, 4]. Reference [3] describes how this was realized for low power laser diodes with a conventional optical multiplexer taken from telecommunication technology. Reference [4] proposed for the first time the application of this principle to high power laser diodes and provides some basic work on that topic.

With time multiplexing a quasi-continuous laser beam with the beam quality of one laser and an average power scaling with the number of involved lasers is obtained. To achieve a nearly continuous power the time-gap between two pulses should be small against the pulse duration, which is in the order of 300ns. The latter value was proposed in Ref. [5] to achieve a significant pulse enhancement without damage. Hence, the switching speed of the optical multiplexer should be well below 300 ns. The switching frequency is therefore in the MHz-range.

The optical multiplexer is based on a digital scanner proposed in Ref. [6]. It uses a cascade of optical binary switches made of an electro-optic polarization switch (e.g. a Pockel’s cell) followed by a polarizing cube beam splitter. The Pockel’s cell is made of a z-cut LiNbO₃-crystal transversally driven by a harmonic electrical field in x-direction (coordinates referring to the crystallographic axes explained in Refs. [6–9]).

The high switching frequencies required for the desired purpose (above 1 MHz) forced us to go into the frequency regions of piezo-electric resonance. We observed that in resonance the necessary amplitude of the electric field drops by a factor ten and more due to the photo-elastic effect, which supports the electro-optics in a favorable manner. Hence, we found a simple polarization modulator well suited for the purpose of time multiplexing.

2. Time-multiplexing of four laser diodes

Figure 1 schematically shows our setup, which combines two branches that can in principle be driven independently from each other (which is not desirable of course).

In each branch, two laser diodes are alternately emitting laser pulses, which are polarization multiplexed and afterwards transformed to circular polarized light with a quarter wave plates. Each LiNbO₃-crystal mechanically oscillates on an eigenfrequency in a shear mode and influences the polarization due to the photo-elastic effect. In one of its two extreme positions during an oscillation period the effect of the quarter wave plate is doubled, leading to a 90° turn of polarization. In the other extreme position the effect of the quarter wave plate is annihilated, thus the polarization is changed back to its initial one before the quarter wave plate. When the emission of the laser pulses takes place at the extreme positions of the oscillation a kind of polarization rectification at the output of the crystal is obtained. This allows further polarization multiplexing (with the polarization filter PF3) of the output of the two branches. When the crystals, which are assumed to oscillate in phase, cancel the effect of the quarter wave plate the laser diodes LD1 and LD3 emit a pulse and when the effect is doubled the diodes LD2 and LD4 are addressed.

To make things clearer consider the first branch for the laser diodes LD1 and LD2. The two laser diodes alternately emit laser pulses. The light of laser diode LD1 is horizontally polarized and goes straight through the polarization filter PF1. Then it passes the quarter wave plate Q1 to become circularly polarized (it does not matter whether left or right). The timing must be now such that the crystal C1 is passed when it is in the extreme position, where it acts as a further quarter wave plate (the deformation amplitude and hence the amplitude of the exciting voltage must be well adjusted to achieve this) but with opposite sign as the first fixed quarter-wave-plate Q1 such that the net retardation is zero and the light leaves the crystal C1 with the initial horizontal polarization. The light of laser diode LD2 is horizontally polarized, too, but is turned to vertical polarization by the half-wave-plate H1 to enable reflection in the desired direction in the polarization filter PF1. The laser diode LD2 pulses now when the crystal C1 is in the other extreme position such that the net retardation of the crystal and the quarter-wave-plate Q1 is that of a half-wave-plate and the vertical polarization is turned back to horizontal polarization. Hence, the pulses of both laser diodes leave the branch with horizontal polarization. In the other branch, the same happens in an analog way but synchronized such that the output polarization is vertical, so that the output of both branches can be polarization multiplexed via the polarization filter PF3.

 

Fig. 1. Scheme for time-multiplexing of four laser diodes: LD1-4…laser diodes, PF1-3…polarization filters 1-3, Q1-2…quarter-wave-plates, C1-2…LiNbO₃-crystals with electrodes on the yz-surfaces, H1-2…half-wave-plates, xyz1-2…local crystal coordinate systems

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Synchronization is done with a multi-vibrator (Timer 555) giving at the desired resonance frequency a master pulse sequence, from which the control pulses for the laser diode drivers and the harmonic excitation for the crystals is derived. The phase between the latter and the control pulses is simply adjusted by monitoring and maximizing the output power on the desired output path.

3. Beam switching by harmonic modulation

For optical time multiplexing of periodically produced laser pulses in the manner described above a polarization switch is needed, which periodically switches between two orthogonal linear polarizations. The switching time should be at least below the pulse duration of 300ns to get a high duty cycle in the output beam (we realized a switching time of 200ns). The voltage applied to an electro-optic switch made of LiNbO₃ can usually not be switched on and off with high frequency since - due to the piezo-electricity of LiNbO₃ - ringing would disturb the desired effect. To avoid this we use a harmonic excitation.

We consider now the transmission Tr for the laser diode LD1 emitting horizontally polarized light through the elements PF1, Q1, C1, PF3 of the setup shown in Fig. 1. The crystal is resonantly driven by a harmonic voltage course on yz-electrodes, such that a shear oscillation mode is excited and both the electro- and the elasto-optic effect cause an artificial birefringence along the optical axis. Light traveling along this axis is therefore decomposed into two orthogonal linear polarized components with corresponding slow and fast axis refractive index n _s, n _f. In the case of z-cut LiNbO₃ with x-excitation, the difference of the refractive indices along the optical axis in a first order approximation is given by:

with n ₀…ordinary refractive index, E _x…electrical field in x-direction, $r_{22}^S$…electro-optic coefficient for constant strain, s ₁₂ and s ₃₁…shear components of the strain tensor, $p_{66}^E$ and $P_{14}^E$…photo-elastic coefficients at constant electric field, ∆B ₁₂…variation on positions 12 and 21 of the symmetric 3×3-matrix of index ellipsoid. The difference n _s - n _f is used in the definition of the retardance δ [10]:

where L is the crystal length and λ the wavelength of light. The transmission Tr depends now on the actual retardance δ imposed by the crystal C1 for which Stokes-calculus gives (the sign before π depends on the orientation of the quarter-wave-plate):

Figure 2(b) shows the graph of Tr (δ).

 

Fig. 2. Optical switching in Fig. 1(a) retardance course of crystal C1; (b) transmission function Tr(δ) for the laser diode LD1 emitting through the crystal C1, the quarter wave plate Q1 and the polarizers PF1 and PF3; (c) resulting temporal transmission Tr(t); (d) laser pulses from laser diodes LD1, LD3 (dashed) and LD2, LD4 (drawn through)

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For negative quarter-wave retardation (δ = -π/2) the transmission is zero, i.e., the polarization after the crystal is vertical and hence the light is blocked in the filter PF3. For positive quarter-wave retardation (δ = -π/2) the transmission is one, i.e., the polarization after the crystal is horizontal and hence the light passes the filter PF3. If the crystal is at rest or passes during the oscillation through its undeformed shape (the so-called zero crossing, where deformation velocity takes its maximum) the transmission is 0.5, i.e., the circular polarization (imposed by the quarter wave plate Q1) is not affected in the crystal and only half of the light intensity can pass the polarization filter PF3.

Now for a harmonic excitation the electric field, the deformations and hence the strains are all temporally harmonic and so is the retardation δ, due to the linearity of Eq. (1) and Eq. (2).

Therefore its temporal course takes the form δ(t) = A sin(2π f_rt) (A…retardation amplitude, f_r …resonance frequency, here 1 MHz). The exciting voltage amplitude can be tuned such that the retardation amplitude is slightly larger than the quarter wave retardation π/2, namely the value A = 0.6π, which is shown in Fig. 2(a). From this a transmission course Tr(t) = Tr(δ(t)) results, which is shown in Fig. 2(c) (An oscilloscope shot of this curve is shown in Ref. 4). One observes now the following: For time gaps of 300ns, the transmission takes values above 95%. Laser diode LD1 is now synchronized such that its laser pulses are centered in these time gaps, such that most of the pulse energy goes into the desired direction. Approximately 5% is lost and takes the wrong direction. The latter value depends on the actual pulse shape and is obtained by integration over the product of the temporal, spatial and angular pulse intensity distribution with the temporal, spatial and angular dependent transmission. If now the crystal C2 oscillates in phase with the crystal C1 laser diode LD3, too, emits within this time gap. The simultaneous emission of these laser diodes is indicated by the dashed graph in Fig. 2(d), showing the temporal course of the pulses emitted by the laser diodes. After a transition time of 200 ns the transmission falls below 5% for a duration of again 300ns, indicating that most of the polarization is now turned. Within this period laser diodes, LD2 and LD4 emit a laser pulse. Again 95% of the emitted pulse energy takes the desired direction. The simultaneous emission of the laser diodes LD2 and LD4 is indicated by the continuous graph in Fig. 2(d).

It should be noted that when the crystal is excited in y-direction (referring to the crystallographic coordinate axes defined in Refs. [6–9]) thickness oscillation modes can be used in the same manner but with an analyzer oriented 45° to the x-axis. The advantage would be that for the same aperture the eigenfrequency is higher than in the case of shear oscillations and faster switching could be obtained. However, in our setup an additional half-wave plate after the crystals would be needed, orientated such that the polarization is turned by 45° into horizontal or vertical polarization. A 45° rotation of the crystals around the z-axis would not be possible since then the astigmatic beam cross-section of the laser diodes would not fit into the available aperture.

4. Analysis of off-axis light rays

Up to now all considerations hold exactly only for the central light path through the crystal. Figure 3 shows an experimental setup for visualizing the effect of divergent rays.

Without the quarter wave, plate the pattern in Fig. 3 would show the well-known Maltese cross produced by uniaxial crystals in this configuration, which is used for adjustment of Pockel’s cells. We published a shot of this pattern in Ref. [4]. The pattern shows that due to birefringence light rays diverging from the optical axis experience a change of polarization. When the crystal oscillates in an eigenmode, the pattern is temporally varying.

In the strict sense, the laser beam has a certain cross-section before it hits the diffuser. Each ray within this cross-section generates its own pattern on the screen. We concentrate now on the pattern produced by the central ray and make at the end some comments on off-center rays. We calculated how this pattern evolves during an oscillation period of the crystal using the Stokes-calculus for polarized light [10]. With the finite element solver ANSYS we simulated such an eigenmode to determine the temporally and spatially varying strain distribution. The crystal dimensions are 1.8×5×30mm (x-y-z-values) The first resonance was found at 1 MHz in accordance to experimental results. The assumed amplitude of the driving voltage was 5V. For this amplitude the desired retardation course shown in Fig. 2(a) is achieved.

Post processing was done with the program MATHEMATICA. We divided the oscillation period into twenty equidistant time points and calculated for every time step the corresponding pattern. To every pixel there is a corresponding ray emerging from the center of the diffuser. Now the course of the polarization (the Stokes vector) along this ray must be calculated. We parameterize the ray with a length variable s and have now to determine the s-dependent index ellipsoid given by N + ∆B with N…3×3 matrix of the index ellipsoid (or indicatrix) of the unperturbed material with the entries (1/$n_0^2$, 1/$n_0^2$, 1/$n_{\mathrm{e}}^2$) on the main diagonal and zeros elsewise, n _e …extraordinary refractive index, ∆B…variation of N due to electro- and elasto-optic effects. One entry of ∆B is given by Eq. (1), the calculation of all its components, the meaning of the index ellipsoid and a detailed description of the following considerations are found in Ref. [7].

 

Fig. 3. Setup to visualize the effect on polarization for different directions of propagation: xyz…crystallographic coordinate system, SC…screen, PF…polarization filter (analyzer), C…LiNbO₃-crystal with electrodes on its yz-surfaces, DF…diffuser, QWP…quarter wave plate, LD…laser diode (collimation not shown); α, β… angle coordinates. On the left the pattern, which can be seen on the screen SC in case of a resting crystal, is shown.

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To determine now what happens with the polarization on a given light ray the local directions of the fast and slow axis and the corresponding refractive indices must be found. This is done by the determination of the section between the indicatrix and a plane containing the center point of the indicatrix and being perpendicular to the direction of the light ray. This yields an ellipse. Its main axes give the polarizations of the two light waves that can travel in the considered direction. The lengths of the two semi-axes give the corresponding refractive indices. Formally, one has to determine the eigenvalues and eigenvectors of the 2×2 matrix given by

with: (α, β)…direction of the considered light ray diverging from the central axis by α in the x-direction and by β in the y-direction (Fig. 3), A(α),B(β)∈ SO(3)…rotation matrices for rotations around the x- and y-axis by (α, β), P…2×3-projection matrix to extract from a 3×3 matrix the 2×2 matrix containing the positions 11, 12, 21, 22. The eigenvectors gives the slow and fast axis direction and the square root of the inverse of the eigenvalues yields the refractive indices n _s(s) and n _f(s) of the slow and fast axis.

The evolution of the Stokes vector along the considered ray is governed by a set of ordinary differential equations. The latter can be derived from the multiplication of the Müller-matrix for an infinitesimal thin retardation plate with the Stokes vector. The Müller transmittance matrix for an ideal retarder with optical interaction length L is [10]:

with c _φ = cos2φ, s _φ= sin2φ, φ… angle between the fast axis of the retarder and the horizontal, δ… retardance defined by Eq. (2). In Eq. (5), it is assumed that all variables are fixed along the light ray, which is not true in our case. One has therefore to consider what happens after a differential ray length ds. When the length L in the definition of δ is replaced by a differential ds and the corresponding infinitesimal retardation matrix is applied on the Stokes vector S(s) one derives:

with entries found by evaluation of Eq. (4). For every time t and pair (α, β) Eq. (6) has to be integrated from s = 0 to s=L (for α, β <<1) with an initial Stokes vector S(0) = (1,0,0,1)^T for circular polarized light and with functions n _f(s), n _s(s), cp φ(s) determined by Eq. (4) for which ∆B is determined by the simulation results of ANSYS with the most important contribution given by Eq. (1). To the solution vector S(L) the Müller matrix of the analyzer (polarization filter) M _pol has to be applied. The first component of M _pol S(L) is the intensity of the pixel of the screen in Fig. 3, where the light ray of interest intersects. To calculate the whole pattern the procedure must be carried out for a sufficient number of pixels. This was done for twenty time steps during an oscillation period. The movie in Fig. 4 shows this temporal evolution of the pattern.

 

Fig. 4. (0.375 MB) Movie of the patterns of the configuration of Fig. 3, when the crystal C oscillates in a shear mode; axis dimension…mrad. The red ellipse indicates the far field spot of the laser diode for removed diffuser and is only shown when a sufficient amount of linear polarization is achieved.

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During certain time windows indicated by a red ellipse showing the far field spot of the laser diode, which can be seen when the diffuser in Fig. 3 is removed, a certain numerical aperture of the crystal produces a high percentage (> 95%) of linear polarization. The pattern shows that only a small angular region of ~ 10×10 mrad can be used for efficient switching, which is due to the large natural birefringence of LiNbO₃ rapidly annihilating the effect of the artificial retardation when the light ray moves away from the optical axis [9]. A material with lower natural optical anisotropy would be preferable. One candidate with slightly lower birefringence and hence ~10% larger acceptance angle is KDP. Alas, its piezo-electric response is much smaller than that of LiNbO₃; hence, the amplitude of the driving voltage would have to be much higher.

Note that the patterns of Fig. 4 are only valid in the case of resonant operation and for light entrance in the middle of the crystal aperture. Since strain and hence the elasto-optic effect decrease towards the crystal boundary only a limited aperture can be used. The crystal cross-section is 1.8×5mm and a central section with the dimensions 0.9×2.5mm can be used for efficient polarization switching with 95% efficiency, which is defined as the ratio of intensity with the desired polarization to input intensity. This is true when the timing is as shown in Fig. 2 and the divergence is smaller as 10 mrad. Outside these pulse specifications, switching efficiencies rapidly fades away. The cross-section and the divergence of the output beam are therefore limited to 0.9×2.5 mm and 10 mrad.

5. The prototype

Figure 5 shows the compact realization of time multiplexing of four laser diodes from JDS Uniphase specified for 4W@960nm mounted on C-mounts. Their M² value after fast axis collimation (which typically increases the fast axis beam parameter product by a factor of two) is 30. At 1MHz, each diode emits laser pulses of 300ns duration and with peak power values up to 15 W. The average power is 3.6W corresponding to 90% of the specified cw-power to maintain lifetime. A long time test of 160 hours of one laser diode with pulsed operation at 4W average power showed no failure or degradation. We assume that due to the high pulse repetition rate the degradation is like in cw-operation.

 

Fig. 5. The prototype (dimensions 115×115 mm) for time–multiplexing of four laser diodes

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Cooling is done by three thermo-electric coolers (Melcor, PE-Series, nominal cooling power: 51.4 W) below the setup. These are fixed on a heat sink with cooling fins supported by a little blower.

The dimensions of the crystals are 1.8×5×30mm (x-y-z-values). They oscillate in phase with 1MHz in a shear mode. It is notable that a second usable resonance exists at 1.14MHz corresponding to a second shear oscillation mode with a motion direction perpendicular to the first one. The amplitude of the excitation voltage is 10V, twice as high as in the numerical simulation. This is due to additional damping caused by the friction of the crystal in its mount (the crystals are gently clamped between massive aluminum electrodes).

The experiment generates 10.5 W laser powers at the output. The total optical power emitted by all laser diodes is 4^*3.6 = 14.4 W, hence an overall optical efficiency of 10.5/14.4 = 73% is achieved. This rather low value is due to several trade-offs regarding adjustment, optical elements and most important the slightly different resonance behavior of the two crystals. The switching efficiency of the crystals can reach values of up to 95 % but a small detuning necessary for full operation of our experiment causes a decrease to 85%.

The beam quality of this diode laser is theoretically the same as that of one single laser diode. Two issues may affect this negatively in praxis. First, a pulsed laser diode may have a larger divergence and hence a lower beam quality than a continuous driven laser diode. We examined this at various power levels and found a decreasing beam quality at higher power levels but no difference in the beam quality of both operation modes. Second, the accuracy of adjustment of the laser diodes limits the overall beam quality since the axes of their beams must be perfectly aligned to the optical output path. Final alignment was done via tilting of the polarization filters. We achieved eventually collinear laser beams (at least within the accuracy of far field spots observed via a NIR-sensitive camera), i.e., an overall beam quality indeed being the same as that of one laser diode, namely M²=30.

6. Conclusions

A diode laser with an enhanced brilliance can be realized with the method of time multiplexing, where several laser diodes alternately emit a series of laser pulses, which are guided onto a common optical path with an optical multiplexer. The latter uses a scheme of polarization filters and periodically switched polarization switches. These consist of LiNbO₃-crystals excited with a harmonic low voltage course on a piezo-mechanical resonance frequency such that the desired modulation of polarization is achieved by the photo-elastic effect. We realized a compact setup for time multiplexing of four 4 W laser diodes driven in a pulsed manner with 3.6 W average power. It generates 10.5W optical power with the beam quality of one laser diode.

To judge the result it can be compared with polarization multiplexing of two 4 W-laser diodes. The efficiency of this is usually around 95%; hence, the output power would be 7.6 W with the beam quality of one laser diode. Compared to that our method yields an increase in brilliance of (10.5 - 7.6)/7.6 = 38%, which is rather low in contrast to the high complexity of the described system (We define brilliance as the ratio of power to the product of the beam parameter products in fast and slow axis direction. Since the beam parameter, products are the same in both cases only the increase of power matters). Hence, two main advantages of diode lasers, namely simplicity and compactness are lost. Besides the reliability of the system is much lower than that of continuously driven diode lasers. Hence, this type of diode laser will certainly not find an application in industry.

Of much higher value is the first (at least to our knowledge) usage of resonantly driven Pockel’s cells for the modulation of polarization and the low driving voltage to achieve a high retardation. This may find applications in ellipsometers, where a modulation of the polarization of a light beam is needed to determine Müller matrices of optical systems. Usually this is done by so called Kemp-Modulators described in Ref. [11]. There an isotropic optical glass (usually fused silica) is excited on a resonance frequency by a piezo-electric actuator. The polarization of light going through the glass is then modulated by strain-induced birefringence. This compound system, called “photo-elastic modulator” (PEM), may be in future realized by single crystal modulators as described in this work.