FRET is becoming increasingly important as a method to determine distances at the molecular level and to study dynamic processes like binding of antibody/antigen pairs.

If two dye molecules are located close to each other, their transition dipoles can interact, and energy can be transferred radiationless from one dye molecule (donor) to the other (acceptor).
The rate of energy transfer kET is according to Förster theory:

NA Avogadro constant
n refractive index of solvent
t0 radiative decay time of donor
r distance between donor and acceptor molecule
F(λ) fluorescence spectrum of donor, normalized according to

ε(λ) molar decadic extinction coefficient of acceptor
κ2 orientation factor: κ2 = (cosφDA – 3 cosφD cosφA)2
φDA angle between transition dipoles of donor and acceptor
φD angle between donor transition dipole and line connecting the dipoles
φA angle between acceptor transition dipole and line connecting the dipoles
As can be seen from the formula, the rate of energy transfer decreases with the 6th power of the distance between the dye molecules. FRET is very efficient only when donor and acceptor are in close proximity. With typical dye molecules it becomes negligibly small at distances above 100 Å (10 nm). Furthermore its rate is proportional to the extinction coefficient of the acceptor dye in the wavelength range of the donor fluorescence (overlap integral, J):

FRET is most efficient, if there is a good spectral overlap between fluorescence of donor and absorption of acceptor.

The click reaction of ATTO 565-DBCO and ATTO 633-Azide provides an example of the efficiency of FRET. A "double molecule" is formed in which the two chromophores are coupled to each other via a triazole ring:

If this reaction is followed by fluorescence spectroscopy, it can be seen at the beginning of the reaction that the two reaction partners are spatially separated and independent of each other: At an excitation wavelength of 560 nm, the fluorescence spectrum of ATTO 565-DBCO (- - -) is obtained, while at 628 nm the fluorescence spectrum of ATTO 633-Azide (- - -) is detected.
Once the reaction is complete, the two chromophores are connected by the linker and are in close proximity. Now the excitation energy is almost completely transferred by the FRET mechanism: When excited at 560 nm, the fluorescence spectrum of ATTO 633 is detected now; only when magnified a small proportion of the fluorescence of ATTO 565 can be seen at the base of the band (in addition to the excitation peak).

A practical measure of FRET efficiency is the distance at which the rate kET of energy transfer equals the rate of donor fluorescence.
This so-called Förster-radius R0 is given by:

ηfl fluorescence quantum yield of donor, ηfl = τfl / τ0
τfl fluorescence decay time of donor
A table of Förster-radii R(0) for ATTO-dyes is presented in our „Support“ section.
Literature-reviews concerning FRET-Application in life sciences:
E. A. Jares-Erijman, T.M. Jovin, FRET imaging, Nature biotechnology 21, 1387 (2003).
S.-H.Cheng, N.-T. Chen, C.-Y. Wu, C.-Y. Chung, Y. Hwu, C.-Y. Mou et al., Recent Advances in Dynamic Monitoring of Drug Release of Nanoparticle Using Förster Resonance Energy Transfer and Fluorescence Lifetime Imaging, J. Chinese Chemical Soc. 58, 798 (2011).
A. Gust, A. Zander, A. Gietl, P. Holzmeister, S. Schulz, B. Lalkens et al., A Starting Point for Fluorescence-Based Single-Molecule Measurements in Biomolecular Research, Molecules 19, 15824 (2014).
Recommended donor-acceptor combinations of ATTO-labels according to the literature are:
ATTO 425 – ATTO 520
ATTO 488 – ATTO 550, ATTO 565, ATTO 647N, ATTO 655
ATTO 520 – ATTO 647N
ATTO 532 – ATTO 647N, ATTO 655
ATTO 550 – ATTO 590, ATTO 647N
ATTO 565 – ATTO 590, ATTO 647N
ATTO 590 – ATTO 620, ATTO 647N, ATTO 680
ATTO 620 – ATTO 680
Selected Literature using ATTO dyes:
B. Hellenkamp, S. Schmid, O. Doroshenko, O. Opanasyuk, R. Kühnemuth, J. Michaelis, C.A.M. Seidel, T.D. Craggs, T. Hugel et al., Precision and accuracy of single-molecule
FRET measurements - a multi-laboratory benchmark study. Nat Meth 15 (9), 669 (2018). → ATTO 550 – ATTO 647N
A. Auer, M. T. Strauss, T. Schlichthaerle, R. Jungmann: Fast, Background-Free DNAPAINT Imaging Using FRET-Based Probes, Nano letters 17 (10), 6428 (2017). → ATTO 488 – ATTO 647N
F. Castello, J. M. Paredes, M. J. Ruedas-Rama, M. Martin, M. Roldan, S. Casares, A. Orte, Two-Step Amyloid Aggregation. Sequential Lag Phase Intermediates, Scientific Reports 7, 40065 (2017). → ATTO 488 – ATTO 647N
U. S. Chio, S. Chung, S. Weiss, S. Shan, A protean clamp guides membrane targeting of tail-anchored proteins, PNAS 114 (41), E8585 (2017). → ATTO 550 – ATTO 647N
J. Funke, H. Dietz, Placing molecules with Bohr radius resolution using DNA origami, Nature Nanotech 11 (1), 47 (2016). → ATTO 550 – ATTO 647N
A. Andreoni, L. Nardo, R. Rigler, Time-resolved homo-FRET studies of biotin-streptavidin complexes. Journal of Photochemistry and Photobiology B: Biology 162, 656 (2016).
ATTO 550 (homo-FRET)
P. Ghenuche, J. de Torres, S.B. Moparthi, V. Grigoriev, J. Wenger, Nanophotonic Enhancement of the Förster Resonance Energy-Transfer Rate with Single Nanoapertures.
Nano letters 14 (8), 4707 (2014). → ATTO 550 – ATTO 647N
J. List, M. Weber, F. C. Simmel, Hydrophobic Actuation of a DNA Origami Bilayer Structure, Angew. Chem. Int. Ed. 53, 4236 (2014). → ATTO 532 – ATTO 647N
H. Höfig, M. Gabba, S. Poblete, D. Kempe, J. Fitter, Inter-Dye Distance Distributions Studied by a Combination of Single-Molecule FRET-Filtered Lifetime Measurements and
a Weighted Accessible Volume (wAV) Algorithm, Molecules 19, 19269 (2014). → ATTO 488 – ATTO 655
T. E. Tomov, R. Tsukanov, M. Liber, R. Masoud, N. Plavner, E. Nir, Rational Design of DNA Motors: Fuel Optimization through Single-Molecule Fluorescence, J. Am. Chem. Soc. 135, 11935 (2013). → ATTO 550 – ATTO 647N
E. Lerner, G. Hilzenrat, D. Amir, E. Tauber, Y. Garini, E. Haas, Preparation of homogeneous samples of double-labelled protein suitable for single-molecule FRET measurements, Anal. Bioanal. Chem. 405, 5983 (2013). → ATTO 488 – ATTO 647N
S. Winterfeld, S. Ernst, M. Börsch, U. Gerken, A. Kuhn, Real Time Observation of Single Membrane Protein Insertion Events by the Escherichia coli Insertase YidC, PLoS ONE 8, e59023 (2013). → ATTO 520 – ATTO 647N
S.L. Kuan, D.Y.W. Ng, Y. Wu, C. Förtsch, H. Barth, M. Doroshenko, K. Koynov, C. Meier, T. Weil, pH Responsive Janus-like Supramolecular Fusion Proteins for Functional Protein Delivery, J. Am. Chem. Soc. 135, 17254 (2013). → ATTO 425 – ATTO 520
R. Bienert, B. Zimmermann, V. Rombach-Riegraf, P. Gräber, Time-Dependent FRET with Single Enzymes: Domain Motions and Catalysis in H+-ATP Synthases, ChemPhysChem, 12, 510 (2013). → ATTO 532 – ATTO 655
K. Seyfert, T. Oosaka, H. Yaginuma, S. Ernst, H. Noji, R. Iino, M. Börsch, Subunit rotation in a single FoF1-ATP synthase in living bacterium monitored by FRET, Proc. SPIE 7905, 79050K-9 (2011). → ATTO 565 – ATTO 590 – ATTO 647N
L. Marcon, C. Spriet, T. D. Meehan, B. J. Battersby, G. A. Lawrie, L. Héliot, M. Trau, Synthesis and Application of FRET Nanoparticles in the Profiling of a Protease, Small 5, 2053 (2009). → ATTO 488 – ATTO 550 – ATTO 590