GDD is simply a product of GVD with the length of the material. The dispersive properties of several optical materials are shown in Table 2.

TRUMPF lasers facilitate the very compact design of modern electronic power and control units. Using the laser precision tool, the internal contacts can be safely welded in a robust plastic housing with one-sided access in a final joining step. Alternative methods such as soldering, spot welding or connectors require a great deal more space for seam points. The laser also makes the most varied types of contactings possible - virtually spatter-free and with no adverse effect on adjacent sensitive components in very confined spaces.

where φMat(ω - ω0) is the spectral phase added by the material and R(ω) is an amplitude scaling factor which for a linear transparent medium can be approximated by, R(ω) ≈ 11.

Group velocity dispersionformula

In comparison to screw connections, laser welding external connecting contacts facilitates a much more compact installation of IGBT boards into their surroundings, for example in electric cars. However, it must be ensured when welding these contacting connections that the generated heat input which is sometimes fed into the component's interior, is kept to a minimum. This means that the sensitive electronics are not damaged. This is made possible by the high welding speeds (reduction of distance energy) in combination with the TRUMPF BrightLine Weld technology, which effectively avoids spatter, protecting the component from contamination.

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Dispersion in materials is defined by the group velocity dispersion. In order to estimate amount of GDD introduced by a material of length L, one has to calculate the wavelength dependent index of refraction, n(λ), typically in the form of a Sellmeier’s type equation, and then calculate second derivative at the wavelength of interest. GVD is related to the second derivative of refractive index with respect to wavelength by

(for the sake of brevity, negative frequency components are omitted). The electric field is now expressed as a function of frequency, Δω and Δt are related through the uncertainty relation1

Group velocity dispersionexample

TRUMPF image processing detects characteristics on components and ensures that welding is always implemented at the correct location. The operator no longer has to reprogram each weld seam individually with this solution.

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It is a common convention to express spectral phase as a Taylor expansion around the carrier frequency of the pulse as shown below,

This approach allows a more straightforward understanding of the effect of material dispersion on properties of the pulse. Taking into account that

DCB (Direct Copper Bondings) substrates are components for electronic circuits and systems which must be connected with each other in function via contact surfaces. Their specific design (ceramics coated with copper) makes the contact very sensitive as the coating may not delaminate due to thermal load nor may the ceramic itself be damaged. Using the green laser wavelength and the corresponding higher absorption rate in copper, long-lasting electric contacts can be created on the sensitive substrate. The DCB substrate remains completely intact.

hence phases in the frequency domain are simply additive. This result underscores the advantage of performing these types of calculations in the frequency domain.

Group velocity dispersionderivation

You are flexible as you can easily process different applications with the same tool – and you can also weld remotely with our programmable focusing optics of the PFO range. TRUMPF also has a unique selling point in the market with the green laser for copper welding.

Whether a green laser with green wavelength, BrightLine Weld or special sensor systems – TRUMPF offers countless innovative manufacturing solutions for power electronics in electric cars.

High-strength connections which are also entirely reproducible and have high conductivity can be achieved with non-contact tools.

Our newly developed beam source with visible green wavelength – whether pulsed or continuous wave – welds copper connections with minimum heat input, high reproducibility and with almost no spatter.

To arrive at the new pulse duration, it is necessary to transform the spectral envelope of equation (5) back into the time domain. Performing this Fourier transform, the pulse envelope is given by,

TRUMPF facilitates a multitude of applications within the production of control and regulation assemblies and power converters. For hardly any other tool is better than the laser for processing these intricate and sensitive components for electric cars. Our laser sources are under constant further development, contributing to the improvement of specific joining results, for example in pinpoint contacting in complex and thermally sensitive electronic components. Furthermore, industrially mature high-power lasers with visible wavelengths help ensure that copper contacts in electric cars can be processed more productively and with almost no spatter, also due to their high absorption characteristics.

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Figure 2 shows the width of a Gaussian pulse at 800nm before and after propagation through 20 mm of BK7 glass calculated using equation (8) and data from Table 2.

provides an expression for the pulse duration. Finally, by solving equation (8) for group delay dispersion while replacing the transform limited pulse duration with the spectral bandwidth of the pulse, GDD can be expressed completely in terms of observables (i.e. pulse width and spectrum),

You can weld contacts and copper connections with no contact and pin-point precision. You benefit from the defined energy input and can rely on reproducible process results. TRUMPF provides the laser source, sensor system and optics from a single source – offering a customized, turnkey complete package for every welding application in the area of power electronics.

With the TruDisk green continuous wave laser, you can produce copper weld seams of the highest quality, in a process that is practically spatter-free and has a high reproducibility rate. Find out more about the tried and tested TruDisk disk laser technology with green laser light, its advantages and possible applications.

it is easy to see that first term in (4) adds a constant to the phase. The second term, proportional to 1/νG, adds delay to the pulse. Neither of these terms affects the shape of the pulse. The third term, referred to as group delay dispersion (GDD), is proportional to

where φ2 is the sum of the group delay dispersion of the material and the group delay of the pulse. In order to get the new pulse duration, Δtout, it is necessary to obtain the intensity, Iout(t), by squaring the electric field in equation (6) and then relating Iout(t) to the general form for a Gaussian pulse,

As a result of their modular system, TRUMPF processing optics can be adjusted to suit different applications and spatial conditions.

Group velocityand phasevelocity

By measuring the spectrum and autocorrelation for a Gaussian pulse, equation (9) can be used to determine the amount of GDD. Figure 1 illustrates the results of a numerical simulation of the electric field for three pulses, all containing 100 nanometers of bandwidth, centered around 800 nanometers. The black curve corresponds to a pulse with the GDD set to zero, the red curve corresponds to a pulse with the GDD set to 5 fs2 and the blue curve corresponds to a pulse with the GDD set to -5 fs2. The pulse with the minimum time duration corresponds to the pulse having zero GDD. For the red pulse (positive chirp), the higher frequency components are lagging behind the lower ones and for the blue pulse (negative chirp), the lower frequency components are lagging behind the higher ones.

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TOD is the frequency dependence on the GVD. The dispersion properties are specified in units of fs3. TOD of several optical materials are shown in the Table below.

The description of the Gaussian pulse given by (1) is intuitive in the sense that it is fairly straightforward to conceptualize a pulse in the time domain. However, when dealing with pulses traveling through dispersive media, it can be problematic to work in the time domain. For example, in order to determine the duration of a pulse after traveling through some dispersive material, it is necessary to solve a convolution integral1 which in general must be done numerically. However, due to the fact that convolutions become products upon a Fourier transformation2, it is convenient to solve this type of problem in the frequency domain.

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Group velocity dispersionandgroupdelaydispersion

The BrightLine Weld option for TruDisk lasers offers the option of virtually spatter-free welding, great welding depth with a high feed rate as well as outstanding beam quality.

We see ourselves as your application consultant and enabler for new production opportunities. You benefit from our consolidated industrial expertise and our decades of experience as a laser manufacturer.

also known as group velocity dispersion (GVD). It introduces a frequency dependent delay of the different spectral components of the pulse, thus temporally changing it. The GDD and GVD are related through

where c.c. denotes the complex conjugate. In this expression, At is the amplitude of the pulse, ω0 determines the color of the pulse, Δt determines the minimum pulse duration and consequently the bandwidth of the pulse, and θ(t) determines the temporal relationship among the frequency components contained within the bandwidth of the pulse. θ(t) plays an important role in altering the pulse duration. It is the term that is responsible for pulse broadening in dispersive media and can be thought of as adding a complex width to the Gaussian envelope.

In the time domain, the electric field for a Gaussian pulse with a carrier frequency, ω0, pulse duration, Δt, and phase, θ(t), can be described by,

The fourth term, referred to as Third Order Dispersion (TOD) applies quadratic phase across the pulse. For the purpose of this tutorial, we will truncate the series at the third term, GDD, only making references to higher order terms when necessary. Truncating equation (4) at the third term allows us to rewrite equation (3) for a Gaussian pulse as,

The amount of introduced GDD in this case is about 1000 fs2, and is equivalent to propagating the beam through only a few optical components. It is clear that the effect is not significant for pulses longer than 100 fs. However, a 25 fs pulse broadens by a factor of 4.

where Δν = cΔλ/λ2. In general, cB is a function of the pulse profile as shown in Table 1. It should be noted that equation (9) is strictly for Gaussian pulses.

As well as the laser source, you will also receive the corresponding consultation services for innovative welding solutions from TRUMPF, for example when welding copper hairpins.

Using your actual component, our industry experts and application engineers will support you during application development and optimization in the Laser Applications Center – with our extensive portfolio of versatile laser processing systems. We would love to welcome you to our TRUMPF Laser Applications Center in Ditzingen - spanning an area of over 4,000 m², it is one of the largest laser application centers in the world.

When an input pulse, Ein(ω), passes through a dispersive medium, the phase added by the material is given simply by the product of the input field with the transfer function of the material. The emerging pulse Eout(ω), is given by,

Benefit from our optics for remote welding: using mirrors, the laser beam can be quickly positioned in any specified position within the processing field, or be guided along any path to create different seam geometries.

Time and frequency along with position and momentum represent a class of variables known as Fourier pairs2. Fourier pairs are quantities that can be interconnected through the Fourier transform. Performing a Fourier transform on equation (1) yields,

and the spectral phase, φ(ω), describes the relationship between the frequency components of the pulse. In equation (2), ω as well as Δω represent angular frequencies. Angular frequency can be converted to linear frequency, ν (i.e. the observable quantity), by dividing it by 2 π,

where k is the propagation constant, and L is the length of the medium, while also considering that the group velocity is defined as