Research question: how can newcomers reproducibly build, align and mode-lock a compact bow-tie (ring) Ti:sapphire femtosecond laser, a task the authors say is well covered in theory but poorly documented in practice? Contribution: an experimental guide that compiles the design calculations (ray-transfer-matrix stability, Kerr-lens sensitivity near the stability limit, astigmatism-compensation angle, dispersion compensation with dispersion-compensating mirrors, Peltier thermal control) and a step-by-step alignment and scanning procedure that moves mirror M2 and the crystal toward mirror M1 unti

A field guide to a one-gigahertz femtosecond laser

Flashes of light lasting a few hundredths of a trillionth of a second can image living cells, assist surgery and turn light into a ruler for measuring frequencies. This laser fires about a billion of them every second from a ring of light 30 centimetres around — and getting it to pulse at all means pushing its mirrors to a knife edge.

Based on Pinto et al., University of São Paulo, 2025 Language

d1 (M1 → crystal) stays fixed at 15.75 mm

Mode-locked: a single pulse laps the ring and one beam leaves the output coupler.

Schematic top view, not to scale. The pulse is drawn slowed roughly two and a half billion times; a real pulse is only about 18 µm long. Beam widths come from the cavity model (exaggerated); the power meter, spectrum and detector trace are illustrative, anchored to the paper's measurements.

1.03 GHz pulse rate

61 fs pulse duration

0.94 W average power

≈1 nJ per pulse

≈15 kW peak power

Try it: slide M2 a quarter of a millimetre closer. Then find your way back.

Every lap is a tick

A laser is a trap for light. Mirrors send the light round and round through a crystal that amplifies it on every pass. Here the four mirrors form a closed loop — a ring — so the light circulates like a runner on a track.

One of the four mirrors, the output coupler (OC), is deliberately imperfect: it reflects 98 % of the light and lets 2 % escape. When the light inside is gathered into a single pulse, every lap sends one flash out through the output coupler. The laser is a clock: one lap, one tick.

Light covers about 30 centimetres in a nanosecond (a billionth of a second). This ring is about 30 centimetres around, so the pulse comes back roughly a billion times a second: about 1 gigahertz (GHz). The authors measured 1.03 GHz .

Predict first: if you halve the length of the ring, do the flashes come…

Slowed about a billion times; ring size drawn on a compressed scale. The time strip shows the last 10 nanoseconds of flashes leaving the output coupler.

Switch on Compare with a linear laser above. A linear laser — two mirrors with the light bouncing back and forth — ticks half as fast as a ring of the same length. Its pulse moves just as fast, but it must come back the way it came, so each round trip covers that length twice (rate = c ÷ 2L instead of c ÷ L, where c is the speed of light). The authors point out that ring cavities reach twice the repetition rate of linear ones, which is how a compact design can pass 1 GHz.

Fast ticking has a consequence. The laser's average power is shared among all the flashes, so more flashes means less energy in each: 0.94 W spread over 1.03 billion flashes a second is just under one nanojoule (a billionth of a joule) per flash. The authors point out that such frequent, gentle pulses of a few nanojoules suit applications in imaging and nanomedicine.

Many colours, one flash

Why would light in a ring bunch up into one flash at all? Start with what the ring allows. A wave survives a lap only if it fits a whole number of times around the ring, so that each crest lands where an earlier crest was. These allowed waves are the ring's modes . Neighbouring modes differ in frequency by exactly the tick rate — about 1 GHz here.

Ti:sapphire — sapphire with a sprinkling of titanium atoms — can amplify an unusually wide band of colours, from deep red into the near-infrared. So many thousands of modes can lase at the same time.

What they do together depends on how their crests line up. With crests at random, they add up to a flickering but steady glow: continuous-wave (CW) light. Line every crest up at one instant and they add into a single bright flash once per lap and cancel everywhere else: mode-locking . The more modes in step, the shorter the flash.

The real laser is far beyond this toy. Its mode-locked spectrum is 42 nm wide — about 22 terahertz. Divided into 1.03 GHz steps, that is roughly 20,000 modes marching in step, which is how a flash only 61 femtoseconds long is possible.

So what makes the colours lock? Nothing in the ring is told to pulse…

A lens made of light

Glass and crystals bend light because light slows down inside them; how much it slows is the refractive index . In very intense light, sapphire's refractive index rises very slightly — most where the light is brightest. This is the optical Kerr effect .

A laser beam is brightest at its centre, so its centre is slowed most and the crystal acts as a weak converging lens: a Kerr lens , a lens that only bright light feels. The steady CW glow barely notices it. The intense peak of a pulse squeezes itself narrower.

Why would narrower help? The green pump laser energises only a small spot in the crystal — the gain region . A narrower red beam overlaps that spot better and collects more amplification on each pass. Bright peaks gain more; dim light gains less. Lap after lap, the peaks win and the light organises itself into a pulse. That is Kerr-lens mode-locking , and it needs no extra component: no slit, no special absorber — just the crystal and the right mirror spacing.

Predict: where should the Kerr lens matter most — with the mirrors in the comfortable middle of their range, or near the edge where the laser stops working?

Beam radius from the cavity model (a thin Kerr lens at the crystal centre, solved until the beam and the lens agree). The pumped-spot radius of 9 µm and the overlap measure are illustrative; widths are exaggerated in the drawing.

Near the edge, 100 kW of peak power squeezes the beam from 10.4 to 7.0 µm. In the middle of the range, the same power barely changes it. The Kerr lens itself is identical in both cases; what changes is how sensitive the whole ring is to that tiny extra lens. So where is the ring most sensitive?

The knife edge

A ring works as a laser only if the beam can repeat itself: after one lap it must come back with the same width and shape, ready to go round again. Whether it can depends on the two curved mirrors, M1 and M2, which focus the beam into the crystal, and on exactly how far each one sits from it.

Laser designers check this with ray-transfer (ABCD) matrices . Every stretch of empty space and every focusing mirror gets a small 2 × 2 table of numbers describing what it does to a light ray. Multiply the tables in order around the ring and you get one table for the whole lap. If half the sum of its diagonal entries, (A + D)/2, lies between −1 and +1, the beam repeats itself. Outside that range it spreads further on every lap, and the laser cannot start.

For this ring that leaves a narrow window: each curved mirror must sit between 15.65 and 17.55 mm from the crystal. Inside the window the paper computes the Kerr-lens sensitivity, δ : how strongly the beam in the crystal shrinks for each extra megawatt of power. In the middle of the window δ is close to zero. Only in the last few tenths of a millimetre before the inner edge does δ plunge strongly negative — right next to where the laser stops working.

Both panels are computed live in this page with ray-transfer matrices and a thin Kerr lens, following the paper's Section 3; they reproduce the trends of the paper's Figures 5 and 6. The regime colours (where mode-locking is possible, where the spectrum flickers) are illustrative thresholds on δ.

A ray is described by its height x and slope x′. Each optical element maps them with a matrix (Eq. 4): [xout, x′out] = [[A, B], [C, D]] · [xin, x′in]. A gap of length L is [[1, L], [0, 1]]; a curved mirror of radius R acts as a lens with focal length f = R/2: [[1, 0], [−1/f, 1]].

A Gaussian beam is described by the complex parameter q (Eq. 5): 1/q = 1/R − i λ/(π w²), where R is the wavefront curvature and w the beam radius. It passes through any element by the ABCD law (Eq. 6): q₁ = (A q₂ + B)/(C q₂ + D).

For the round-trip matrix, a beam that repeats itself has radius (Eq. 7) w² = (|B| λ/π) · 1/√(1 − ((A + D)/2)²), which only exists when −1 < (A + D)/2 < 1.

The Kerr-lens sensitivity (Eq. 8) is δ = (1/w)(dw/dP) at P = 0. In this page the Kerr effect is a thin lens at the crystal centre with strength 1/fK = 8 n₂ P l/(π w⁴) (n₂: sapphire's nonlinear index; l = 3 mm), solved together with the beam until they agree.

Three details that make or break it

Finding the window is not enough. Three practical details decide whether a pulse can live there: keeping its colours together, keeping the beam round, and keeping the crystal cool.

a Keep the colours together

A 61 fs pulse is made of many colours — the modes from step 02. In sapphire, redder light travels slightly faster than bluer light. On every lap the 3 mm crystal adds about +180 fs² of group-delay dispersion (GDD), the measure of how strongly a material pulls a pulse's colours apart in time. Left alone, the pulse would smear out: red ahead, blue behind.

The authors' answer is mirrors with the opposite habit. A dispersion-compensating mirror (DCM) is a stack of thin layers in which redder light penetrates deeper before it is reflected, so it is delayed more. Each one contributes about −70 fs² . Switch on the three in this laser — M1, M2 and P1:

Crystal: +180 fs² per lap (always there)

Simple (linear) dispersion only, applied for ten laps; in the running laser, gain and the Kerr effect also shape the pulse. Colours are exaggerated: the real light is near-infrared.

Real DCMs are not perfectly smooth: their dispersion wiggles with colour by about ±30 fs². The mirrors come in pairs whose wiggles are designed to be opposite, so together they give a smooth correction — without the prisms that many such lasers use, and without making the cavity bigger.

b Keep the beam round

M1 and M2 meet the beam at an angle, and a tilted curved mirror focuses horizontal and vertical directions differently — a defect called astigmatism that turns a round beam into an ellipse. The crystal's end faces are cut at Brewster's angle , so the laser light enters without reflection losses, and that slanted entry adds astigmatism of the opposite kind.

At one fold angle the two cancel. The paper's Equation 1 gives it from the crystal's length and refractive index and the mirrors' curvature: about 24°. Turn the fold angle and watch the bow tie open or close, the curved mirrors tilt, and the beam spot change shape.

Eq. 1 gives 24°

Left: the bow-tie seen from above, with the fold angle at M1 and M2 drawn to scale (distances are not to scale). Right: from the cavity model with both curved mirrors 16.2 mm from the crystal, the beam spot in the long arm and the range of mirror distances where each direction is stable. A round, stable beam needs both bars to cover the operating point.

In the lab, the authors found that angles slightly smaller than 24° worked best. The cavity model here agrees: the beam is roundest near 22°.

c Keep the crystal cool Measured

The pump pours watts of green light into a crystal only 3 mm long. A warm crystal becomes a weak lens of its own — a thermal lens — which quietly shifts the knife edge you just learned to find. The authors clamped the crystal to a copper plate cooled by a Peltier module (a small solid-state heat pump) with an air-cooled heatsink. That held it between 18 and 28 °C, with fluctuations under 2 °C.

Find the pulse yourself

Here is the authors' own procedure as a game. M1 is fixed on a rail; you can move M2 and you can move the crystal. You can't see the map of regimes — the authors couldn't either. They watched the spectrum and counted output beams.

Start from the best continuous alignment (where you are now).

Move M2 closer until the spectrum starts to flicker between sharp spikes and a broad band.

Move the crystal toward M1, keeping the broad band.

Repeat, creeping toward the edge, and tap P1 to jolt the light into a pulse.

Success: a smooth broad spectrum, no spikes, and only one output beam.

no repeating beam

continuous

on the edge

mode-locking possible

Spectrum flickers — you have reached the edge

Crystal moved toward M1 (at least 0.1 mm)

Pulse triggered with a tap

Pulse held for 3 seconds

Map and regimes from the cavity model; the power meter and spectrum are illustrative. In the lab the authors also fine-tuned the flat mirrors and the pump lens between steps, which this game leaves out.

What the bench delivered

Everything in this step is the real laser, in the authors' own measurements.

12.7 %

At 7.4 W of pump power the mode-locked laser delivered 0.94 W on average — 12.7 % of the pump, compared with about 8 % reported for other gigahertz Ti:sapphire ring lasers.

Flash calculator

Energy per flash = average power ÷ flash rate. Peak power ≈ energy ÷ flash duration (the paper's approximation).

Room for even shorter pulses

The width of a spectrum sets a limit on how short a pulse can be — the transform limit. A 42 nm band could support pulses of about 20 fs. The measured 61 fs is three times longer: the time–bandwidth product is 1.34, where a perfectly compressed bell-shaped pulse would give 0.44.

The authors note that better-matched dispersion-compensating mirrors or an added pulse compressor could close that gap; related work has reached pulses shorter than 10 fs.

Back to the knife edge

Remember the slider at the top: a quarter of a millimetre separated a pulsing laser from a dark one. The paper's contribution is a route to that narrow ledge that others can follow — a stability calculation to place the mirrors, one equation for the fold angle, dispersion-compensating mirrors instead of bulky prisms, an inexpensive Peltier cooler for the crystal, and a step-by-step scan to the edge.

The aim, in the authors' words, is to let newcomers build such lasers in significantly shorter time-frames. Where could that lead? The authors point to:

Imaging and nanomedicine Gigahertz trains of gentle, nanojoule pulses.

Surgery and spectroscopy Laser-assisted surgery and precise measurements of light and matter.

Frequency combs Rulers made of light for telecommunications, spectroscopy and metrology.

Cheaper, smaller pumps Green laser diodes in place of a large pump laser.

More power Ytterbium-doped crystals instead of Ti:sapphire.

Lasers on chips Ti:sapphire integrated into photonic circuits: portable and frugal.

Even shorter flashes Better dispersion control and pulse compression, toward pulses under 10 fs.

The smooth band is a ruler Illustrative

Zoom into the smooth mode-locked spectrum. Far enough in, it breaks into separate, evenly spaced lines — the ring's modes, 1.03 GHz apart. An evenly spaced set of lines like this is a frequency comb.

Window 120 nm wide (about 63,000 GHz): the 21,000 lines merge into a smooth band.

Source and credits

Vinícius Pereira Pinto, Giovana Trevisan Nogueira, Fátima Maria Mitsue Yasuoka and Jarbas Caiado de Castro Neto. Experimental Guide for Compact Bow-Tie Femtosecond Solid-State Laser Development. Preprints.org, posted 7 January 2025.

Published version: Photonics 2025, 12(6), 548 — doi:10.3390/photonics12060548

Preprint: doi:10.20944/preprints202501.0318.v1

São Carlos Institute of Physics, University of São Paulo; Federal University of Juiz de Fora; BR Labs Tecnologia Óptica e Fotônica, São Carlos, Brazil.

The paper is published under a Creative Commons Attribution 4.0 (CC BY 4.0) licence. Figures 8, 11, 12, 15, 16, 17, 18 and 19 are reproduced from it under that licence. They were cropped from the paper and resized; the two panels of Figure 19 are shown separately and the four panels of Figure 15 are arranged in a grid.

The interactive cavity model on this page uses ray-transfer matrices with a thin Kerr lens, following the paper's Section 3, and reproduces the trends of its Figures 5 and 6. Regime thresholds, the power meter, spectra and detector traces are illustrative. The measured values quoted come from the paper's Section 6.